JPRS ID: 10374 TRANSLATION MICROWAVE ANTENNAS AND EQUIPMENT: DESIGN OF PHASED ANTENNA ARRAYS ED. BY D.I. VOSKRESENSKIY
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8 March 1982
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MICRO'rNAVE ANTENNAS AND EQUIPMENT:
DESIGN OF PHASED ANTENNA ARRAYS
Ed. by
D.I. Voskresenskiy
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MICROWAVE AN?ENNAS AND EQUIPMENT;
;
DFSIGN OF �HASED ANTENNA ARRAYS
JPRS L/1Q374
8 March 1982
r:oscow ANTENNY I USTROYSTVA SVCH: PROYEKTIROVANIYE FAZIROVANNYKH
! ANTENNYKH RESHETOK in Russian 1981 (signed to press 13 Apr 81)
pp 1431
[Book edited by Professor D.I. Voskresenskiy: "Microwave Antennas
 and Equipment (Phased Antenna Array Design): Textbook for the Higher
Educational Institutes", approved by the USSR Ministry of Higher and
 Intermediate Special Education as a textbook for students in the
Radioengineering Specialties of the Higher Educational Institutes,
Izdatel'stvo "Radio i svyazl", 1981, 431. pages, 15,000 copies]
 CONTENTS
, Annotation����ee �.����s�s������.~��~~~�~~~~�����~~��~r�����~~~~~~��~���~~~�
Foreword 1
Section I. Antenna Arrays...1 4
Chapter 1. Microwave Antenna Design 4
;
1.1. Introduction 4
, 1.2. The Main Requirements Placed on Microwave Antenna Systeus and
 the Possibilities of Using Antenna Arrays 6
1.3.
Antennas
With Electrical Scanning
11
1.4.
Specific
Features of Phased Antenna Array Design
13
1.5.
Specific
Features of. Active Array Design:
15
Ghapter 2. Phased Antenna Arravs 19
2.1. Zhe Determination of the GeomeLric QZaracteristics of Phased
Antenna Arrays 19
2.2. Mutual Coupling Effects Among Radiators 25
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2.3.
1he Relationship Between the Directivity of a Radiator in an
Array and the Directional Qiaracteristics of a Fu1Zy Excited
Array
28
2.4.
Radiators of Phased Antenna Arraya
30
2.5.
Wide Angle Matching of Phaae Antenna Arr;tys..o
37
2.6.
Structural Configuration of Phased Antenna Arrays.................
41
2.7.
The Passband of a Phased Antenna Array.
47
2.8.
Switched Scanning ........o.........
53
2.9.
Switched Phase Shifters
54
2.10.
Discrete Phase Shifters and the Suppression of Switching
_
Lobes
56
2.11.
Beam 3umps in a Switched Array
59
2.12.
Design Procedure
60
Chapter
3. Frequency Scanning Antennas
61
3.1.
Fundamental Relationships for a Frequency Scanning Linear
Radiator Array ....................'e................
61
~ 3.2.
Channelizing Syatems of Frequency Scanning Antennas
69
, 3.3.
The Frequency Scanning Slotted Waveguide Array....................
72
3.4.
Tie Design Procedure for a FYequency ScJ :ning Linear Slotted
Waveg.:ide Array
76
Qlapter
4. Highly Directional Cylindrical and ARC Antenna Arraya..........
83
4.1.
General Information...............................................
83
; 4.2.
The Phase 'Distribution in Highly Directional Gylindrical
Arrays............................................................
86
4.3.
The Directional Patterns of Gylindrical Pencil Beam Arrays........
88
I
4.4.
Directional Patterns of Arc and Gylindrical Arrays
89
4.5.
The Di*ectional Gain of Cylindrical and Arc Arrays
94
4.6..
Bandwidth Properties of Arc Arrays
98
 4.7.
Some Structural and Circuit Design Varianta for Arc and
Gylindrical Arrays
100
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4.8. The Design Procedure for Gylindrical Arraya..o...................... 104
Chapter
5. Slotted Waveguide Arrays.........................................
107
5.1.
'Ilie Fimction 3nd Specific Features of Slotted Waveguide Arrays..... .
107
5.2.
The Major Parameters of a Slot in a Waveguide
107
5.3.
Dhe Typea of Slot ted Waveguide Arrays
112
5.4.
~
Methods of Designing Slotted Waveguir2e Axraya.......................
115
5.5.
Matching.a Slotte d Wave guide Array ta a Feed Waveguide
121
J
5.6.
1he I:.fluence of a(hange in Frequency on Antenna (haracteristics...
121
5.7.
The Directional Properi3es of Slotted Waveguide Arraya
122
5.8.
Possible Structural Configurations for Slo tted Wavegutde Arrays
and Struct;,.:al Deaign Examples
126
5.9.
A Sample Design Calculatioz Procedure for Slotted Waveguide
Arrays
129
_ Chapter
6. Accounting for Mutual Coupling Effecta in Slotted Waveguide
Arrays
131
6.1.
Basic Relationships
132
6.2.
Planar Slotted Waveguide Array
135
6.3.
An Analysis of Mutual Coupling Effects on the Mrectional
Pattern of an Array
136
6.4.
A Procedure for Synthesizing a Linear Slotted Waveguide Array
Taking Electrodynamic Mutual Coupling Effects Into'Accoim t..........
141
6.5.
~
A Procedure for Syntheaizing a Planar Slotted Waveguide Array
TakingMutual Coupling Into Account.................................
6.6. Design Calculation Recommendations ]SO
Chapte r 7. Phased Antenna Arrays With a Hemispherical Scan Space............ 7,51
7.1. General Governing Lawa 151
7.2. A Hybrid Phased Antenna Array With a Hemispherical Scan Space.
Operational Principle. Specific Structural Design Features of
a Phased Array With a Dome Shaped Lens 154
7.3. Conformal Phased Antenna Arrays 156
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7.4. Polyhedral Phased Antenna Arraya 263
(liapter 8. Beam Stee riug Systems for Phase d Antenna Arraya 168
8.1. Phased Antenna Array Control Problems............................... 168
8.2. Control Algoritiims ' for Phase Shiftera 171
8.3. Algprithms for Generating Directional Patterns of Special
Shapes 175
8.4. Switcher Control Algorithms 177
8.5. Adaptation Cont ml Algorithms....................................... 179.
8.6. Ihe Deaign of Beam Steering Systems for a Specified Precision
of the Directional Pattern Orientation in Space 184
Section II. Radiating Elements of an Antenna Array..........:............... 190
Qiapter 9. Prin ted* G`Lrcuit Antenna 190
9.1. 1he Ftiuiction and Specific Features of Printed Circuit Antennaa....... 190
9.2. The Majo rTypes of Printed Circuit An tennas and Their Operational
Principles ...190
9.3. 7he Ma3or Qzaracteristics and Design of Printed Circuit "
Resonator Antennas 193
9.4.
Antenna Arraya With Resonator Elements
1.97
9..5.
Printed Circuit Dipole Antennas
200
. 9.6.
Anfenna Arraya With Printed Circuit Dipole Elements
204
9.7..
Other Prtnted Circuit Radiating Systems
207
Gfiapter
10. Yagi Radiatora for Planar Phased Antenna Arrays.................
211
10.1.
Phased Arrays of Yagi Radiators
211
10.2.
Analysis of the Electmmagnetic Field of a Phased Antenna Array
of Yagi Radiators
211
10.3.
The (haracteristica of a Yagi Radiator in a Planar Phased
Antenna Array
213
10.4.
7he Optimization of a Yagi Radiator in an Array
216
10.5.
Designing the Input Circuit of a Yagi Radiator
218
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; 10.6. A Design Procedure for a Xagi Radiator for Phased Aatenna
! Arrays 220
Qiapter 11. Approximate Design Calculationa for Phaaed Waveguide
; Antenna Arraya Taking Mutual Coupling Into Account 221
11.1. Gennral Considerations ..............................................221
; 11.2. Deaign Grapha 221
; 11.3. Lieaign Recommendations....................o 225
~ Chapter 12. Wide Angle Matching of the Waveguide Radiators of Planar Phased Antenna Arrays .......................................4.. 228
12.1. Methods of Matching Waveguide Radiators in Planar Phased
Antenna Arrays 228
12.2. Matching With a Fixed Scanning Angle 233
Chagter 13. Slotted Resonator Radiators for Planar Antenna Arrays.......... 237.
13.1. Analysis of the Characteristics of a Slotted Resonator
Radiator............................................................ 237
13.2. The Characteristics of a Slotted Resonator Radiator as a
Independent Antenna 241
~ .
13.3. The Characteristics of a Slotted Resonator Radiator in a~ Planar Antenna Array......... 242
13.4. Zhe Optimization of the Characteristics of a Slotted Resonator .
Radiator in an Antenna Array 244
13.5. Examples of the Realization af Slotted Resonator Radiators.......... 246
13.6. The Design P m cedure 247
Chapter 14. Radiating Waveguide lrbdules With Reflective Phase
Shifters 249
14.1. Zhe Modular Design of a P'haaed Antenna Array 249
~ 14.2. Multiposition Phase Shifter for a Module 250
14.3. Microwave Bridge Devices for Feedthrough PhaBe Shifters............. 254
14,4. Zhe Design of a Radiating Module of an Antenna Array 255
14.5. Waveguide Directional Couplers 258
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_ 14.6. An Approximate Design Calculation Procedure for a Radiating
rb dule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Section III. Active Elements of Antenna Array Modules 264
(hapter 15. Mbdules of Tranamitting Phased Antenna Arraya Using
Semiconductor Devices
15.1. Zhe.Major Characteristics of the Active Elements of Modules........
~ 15.2. Major Structural Design Requir2metxts
15.3. Active Semiconductor Devices for Active Phased Antenna Array
Modules..........
15.4. The Radiation Power of Active Sendconductor Phased Antenna
Arrays
15.5. Active Phased Antenna Array Efficiency
15.6. Recommendations for the Selection of Module Circuits and
Parametera
QZapter 16. Externally Excited Oscillators and Amplifiera Using
Power Transistora
264
265
267
268
269
274
2 76
2 79
16.1. General Information 279
16.2. The Equivalent Circuit of a Microwave Transistor 279
16.3. A Time and Harmonic Analysis of Transistor Currents and
Voltages............................................................ 284
16.4. The Propertiea of Common Emitter and Common Base Generator
Configurations 289
16.5. The Procedure and Sequence for the Design Calculationa of the
Operating Mbde of an Oacillator/Amplifier 293
Qiapter 17. Externally Excited Microwave Circuits for Transistor
Oscillators and Amplifiers..................................... 300
17.1. General Information 300
 17.2. The Design of the Microwave Networks of Amplifiers and
Oscillators 304
17.3. Oacillator/Amplifier Power Supply Circuits ........................4 308
17.4. The Design of Micrawave Matching Transformer I"v'etworks Using
Lumped Elements.................................................... 310
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17.5. The Deaign of a Microwave Matching and Transforming Network
Uaing Elements With Distributed ~Parameters....................... 320
Chapter 18. Frequency Multipliera Uaing Nonlinear Capacitarice Diodes..... 328
18.1. General Information 328
18.2. Zhe Selection of the Multiplication Factor for the Frequency.
Multiplier of an Active Phased Antenna Array Module 329
~ 18.3. 7he Selection of Nonlinear Capacitance Diode sud Its Operating
Mode 332
18.4. The Pawer Deaign Calculation Procedure for the Operational
Mode of a Diode in a Parallel Type Multiplier 335
 18.5. 'IYze Deaign of the Micrawave Input and Output Networks of a
Multiplier 338
Chapter 19. . Micrnwave Amplifiers and Oacillators Uaing Avalanche
Transit Time Modea 343
19.1. Basic Characteristics 343
19.2.
The Parametera of IMPATT Diodes and Spectfic Featurea ~of
Iheir Appl.ications in the Modules of AcLive Phased Transmitting
Antenna .Arrays.:.................................................
345
19.3.
Microwavie Circuits of Oscillators Uaing IMPATT I3iodes............
353
 19.4.
Structural Design Principles..............................:......
357
19.5.
Principles of Design Calculationa of IMPATT Diode Microwave
Devices
360
Section
IV. Microwave Hardware
368
Chapter
20. Zhe Struc:tural Design of Microwave Hybrid Integrated
.
Circuit Components.......o ~
368
20.1.
General Information
368
, 20.2.
The Asymmetrical Transmission Stripline..........................
370
20.3.
Printed Circuit Inductance Coils..................................
375
20.4.
Capacitors
379
Chapter
21. Microwave Phasing Devices (Phase Shiftera)
383
21.1.
Semiconductor Phase Shifters.....................................
' 384
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21.2. Semiconductor Phase Shiftera Wi#h _a Qontinuoua Phase _Change....... 386
21.3. Discretely Switched Semiconductor Phase Shifters 391
(hapter 22. Microwave Filtera 404
22.1. The Qasaification of Microwave Filters 404
22.2. The Design of the Low Frequency F`ilter Prototype 405
22.3. 'Ihe Structural Execution of Micmwave Filtera 411
22.4. A Design Procedure for Micrawave FYltera 415
(hapter 23. Directional Couplers and Directional Filters Using
Coupled Striplines 417  23.1. 71ie Classification of Directional Couplers and Filters and
Their Operating Characteristics 418
23.2. 7he Main Design Equations for Single Section T lrbde Coupled
Line Directional Couplers.................... 420
23.3 Extended Bandwidths Directional Couplers Using Coupled Linea........ 426
23.4. :Ihe Characteriatic Impedancee of Coupled Lines in the Case .
ot inPhase and OutofPhaae Excitation 428
23.5. The Relationahip Between tYte Structural and Electrical
Cnaracteristics 432
23.6. 7he Ma.jor Design Relationahips for Single Loop I7ireCtional
. Filtera Using Striplines...............................:.......... 435
23.7. The Influence of Tblerances on the Parameters of Directional
Couplers.......................................................... 437
23.8. The Structural Design of 'Directional Couplers and Filters Using
Coupled Striplines 440
.23.9. 1he Design Procedure 443
~ Ghapter.24. Stripline Micx+owave Pawer Distribution Systems 447
24.1. Zhe Function and Major Character3stics of Micmwave Power
Mstribution Systems............................................... 447
24.2. The Comparative Performance of Varioua Types of Micmwave
= Pawer Distribution Systems 448
24.3. Ca.lculating the Electrical Parameters and Characteristics of
, 'Itao Channe l Pawe r Dis trib uto rs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 . . . . . . 450
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24.4. 'Ihe Calculation o� the Eletrical Paxameters and Chaxacteristics
of MuitiChanne'L Power Distribtuion Systems 455
24.5. An Approximate Design Procedure for Power Distribiition
Systems 457
Bibliography 459
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Annotation
[Text] Methods of calculating the characteristics of phased antenna arrays (PAA)
and active phased antenna arrays (APAA) as well as theii components are presented.
Arrays with various geometries, types of radiators and coritrol techniques are
treated as well as antennas with frequency scan,ning, switching, multislot, planar,
and cylindrical antennas, etc.
The book is intended for students in the radio engineering specialties of the
higher educational institutions in the performance of the diploma and course re
quired design, as well as for engineers engaged in the design of phased antenna
arrays and active phased antenna arrays.
D.I. Voskresenskiy, V.L. Gostyukhin, R.A. Granovskaya, K.I. Grineva, A.Yu. Grinev,
I.I. Guruva, N.S. Davydova, G.P., Zemtsov, M.V. Indenbom, G.I. Koptev, Yu.V. Kotov,
S.D. Kremenetskiy, S.M. Mikheyev, B.Ya. Myakishev, T.A. Panina, S.B. Petrov,.L.I.
Ponomarev, V.V. Popov, A.M. Razdolin, P.A.Solovtsov, V.I. Samoqlenko, V.S. Filippov,
V.V. Chebyshev, V.N. Shkalikov, V.Ye. Xamaykin. Reviewers: Department of Antenna Equipment and Radio I,Tave Propagation of Moscow
Power Engineering Institute (head of the departdent, doctor of the engineering
sciences, professor Ye.N. Vasil'yev) and the Department of Communications and Ratdio
Control of Ryazan' Radio Engineering Institute (head of the department,,doctor of
the engineering sciences, winner of the USSR State Prize, professor V.I. Popovkin)
Editorial Staff for Cybernetics and Computer Engineering Literature
Foreword
Material on the planning and desiga of phased antenna arrays (PAA), active antenna .
arrays (APAA) and their componeuts is collected and systematized in this book.
Engineering methods are given for electrically scanned antenna design to meet
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specific technical requirements, as well as the requisite information for design
work based on the parameters of existing equipment and a description of existing
designs. Special attention is devoted to flight systems.
The book "Microwave antennas and equipment. The planning and design of antenna
arrays and their radiating elements" which came out in 1972 under the editorship
of D.I. Voskresenskiy, to a known extent generalized the most widespread design
techiiiques. This book cont3ins materials which are an.extension of the indicated
work; the design methods.presented in it supplement and refine the methods
_ treated earlier, taking into.account the latest achievements in design automation
using computers. The range of problems considered has been significantly expand=
ed: questions of the design of new types of arrays are set forth, as well as
active and passive elements.
Considerable attention has been devoLZd to the cor_struction of active stripline
modules with semiconductor devices. Bringing the materials on the indicated
topics together in one book, based on the general requiremP:.c.s placed on array
elements, as well as the utilization of uniform criteria for a comparative
evaluation of these and other elements have significantly simplified the pro
blem of the goal directed design of an array, in particular, the selection of
an acceptable variant fo, the overall array configuration, as well as the type
of active and control elements.
The book consists of four sections. General questions of phased array design are
treated in the first. Here, questions of antenna design with phase, switching
and frequency scanning techniques are presented. Procedures are given for the
design of planar and cylindrical arrays, phased arrays with a hemispherical
scanning space as well as slotted waveguide arrays. Procedures are given in the second section for the engineering design of phased
array radiating elements, taking their interaction into aecount. Dipole, strip
line, slotted, director, waveguide and other phased array rp.:iiatnrs are treated.
The third section is devoted to the design of active phased :rrays and their
modules. Specific features of the construction and calculation of the character
istics of active phased arrays are presented; structural configurations are given
for active reflective and transmission type phased array modules as well as
methods of signal phasing in the modules. Modules of various types are compared
and possibilities of using various active elements are indicated. Procedures
are given for designing the modules of transmitting active phased arrays around
various semiconductor elements: oscillator stages using microwave transistors
and ItiPATT diodes, varactor multipliers, and hybrid IC microwave circuits.
Questions of the design of passive elements of phased arrays are treated in the
fourth section. Design procedures are given for directional couplers and coupled
line filters as well as multichannel stripline dividers, microwave phase shifters
and filters.
Widely known material existing in monographs and the periodical press is collected
and systematized in the book, and the published literature of the Problems
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Laboratory for Microwave Engineering of. Moscow Aviation Instltute are also used.
Topics from general microwave antenna and equipment theory are not treated; it
is assumed that the reader is already familiar with the ge:leral course given in
the radio engineering departments of the higher educational institutes.
It must be underscored that the design procedures incorporated in the book differ
substantially in terms of design precision and complexity. Along with simplified
calculations, which make no pretense of exhaustive completeness, some of the
latest techniques of computer assisted design are included in the book. Simpli
fied design methods are presented initially, which make it possible to design
phased antenna arrays or elements which meet the major technical requirements,
in the amount necessary for the course required or diploma design work, as well
as in the preliminary developmental work on antenna system designs. Further,
where it has proved possible, the authors provide more precise computational
methods which make it possible to optimize the device being designed with respect
to a particular criterion by means af the programs which have been developed.
A bibliography of the major literature is given at the end of the book, as well
as bibliographies for the chapters, which are recommended in the planning and
design of the given equipment.
The book is intended for students in the radio engineering specialties when doing
their diploma or course required design work, but can also be useful to engineers
engaged in the design of antenna arrays.
The book was wrirten by a collective of authors: D.I. Voskresenskiy (the Fore
word and Chapter 1); V.S. Fillipov (Chripter 2); R.A. Granovskaya (Chapter 3 and
17); L.I. Ponomarev (Chapter 4); V.L. Gostyukhin (Chapters 5 and 21); S.D.
Kremenetskiy (Chapter 6); V.Ye. Yamaykin (Chapter 7); V.I. Samoylenko (Chapter
8); V.V. Chebyshev (Chapter 9); M.V. Indenbom (Chapter 10); K.I. Grineva (Chap
ter 11); A.M. Razdolin (Chapter 12); A.Yu. Grinev and Yu.V. Kotov (Chapter 13);
V.V. Popov and S.M. Mikheyev (Chapter 14); G.P. Zemtsov (Chapter 15); G.I.
Koptev and T.A. Panina (Chapter 16); V.N. Shkalikov (Chapter 18); N.S. Davydova
(Chapter 19); S.B. Petrov (Chapter 20); B.Ya. Myakishev (Chapter 22); A.Yu.
Grinev (Chaprai 23); I.I. Gurova, B.Ya. Myakishev and P.A. Solovtsov (Chapter 24).
The overall editing of the book was done by D.I. Voskresenskiy.
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ANTEW?A ARRAYS SECTJJN I
l. Microwave Antenna Design
' 1.1. Introduction
The antenna and feedline, which provide for the radiation and reception of radio
waves, is an integral part of any radio engineering system. A number of technical
requirements are placed on the antenna, which follow from the f.unction of the
radio system in which it is used. The conditions for the placement and operation
of the antenna influence its characteristics. The feasibility of attaining the
requisite directional properties, frequency, power and other characteristics of
an antenna depend,in many respects on the working frequency band. The last two
decades have been marked by the wide scale i.ntroduction of radio equipment into
the economy and the use of microwave gear. Antennas in the microwave band pro
duce pencil beam radiation with a beam width measured in units and fractions of
degrees and have a gain reaching tens and hundreds of thousands. This makes it
possible to use the.antenna not just for radio wave transmission and reception,
but also for direction finding (in radar, navigation and radio astronomy), com
bating interference, providing for concealed operation of a radio system and for
other purpos :s.
Besides radars, microwave hardware is used in such sectors of electronics as
television, radio control, radio navigation, radio communications, telemetry, and
accelerators. The successful development of radio astronomy and the mastery of
space is related in many respect to the achievements of microwave engineering.
Pencilbeam scanning microwave antennas have become widesp�read at the present
time. The scanning makes it possible to scan the surrouitding space, track moving
objects and determine their angular coordinates. The replacement of poorly
directional or omnidirectibnal antennas (for example, coupled antennas) with
pencilbeam scanning antennas makes it possible to obtain not only a power gain
in the system because of the increase in the antenna gain, but also, in a number
of cases, to attenuate crosstalk between different radio engineering systems
operating at the same time, i.e., provide for electromagnetic compatibility of
these systems. In this case, the noise immunity, security and other character
istics of the system can also be improved. With mechanical scanning, which is
accomplished by means of rotating the entire antenna, the maximum rate of beam
travel in space is limited, and with the presently existing aircraft speeds,
proves to be insufficient. For this reason, it became necessary to develop new
types of antennas.
The application of phased antenna arrays (PAA's) to produce scanning pencilbeam
antennas makes it possible to realize a high space scanning rate and promotes an
improvement in the data obtained on the electromagnetic reflection or radiation
sources in the surrounding space. Modern microwave devices with vacuum tube or
semiconductor devices and electrically controllable media have made it possible
to not only create a controlled phase distribution in an antenna array (i.e.,
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effect electrical scanning), but to accomplish the initial processing of the in
coming information (summing of the fields, frequency conversion, amplification,
etc.) directly in the radio frequency channel of the antenna.
A further improvement in the characteristics of radio systems with phased arrays
is possible (resolution, speed, carrying capacity, detection range, interference
immunity, etc.) by refining the techniques of processing the signal transmitted
and received by the antenna (spacetime processing in the general case). The
antenna in this case is the primary processing unit and to a significant extent
governs the major characteristics of the system as a whole. Usually, far from
a.ll of the information contained in the wave impinging on a pencilbeam receiving
antenna is used, where the fields in the antenna from the individual radiators
are added together in a single radio frequency channel. The most complete infor
mation can be obtained by processing each received signal in the antenna array
separately, i.e., by processing a series of samples from the spatial distribution
of the incident wave. Antennas with different processing techniques are employed,
depending on the function of the system and the requirements placed on its
characteristics. One of the antenna variants with signal processing is the
adaptive array, which in a radio signal processing system can be treated as a
dynamic selftuning spacetime filter, in which the directional pattern, fre
quency properties and other parameters are changed automatically. Other signal
processing antennas are also known: selftuning, artificial aperture, with time
modulation of the parameters, digital processing, analog spacetime processing
using coherent optics methods, etc. '
Thus, the antennas being used in practice are very complex systems, having up to
tens of thousands and more radiators, active elements and phase shifters, which
are controlled by a special computer. The design of such aotennas is extremely
complex and basically determines the size and cost of the entire radio system.
 The characteristics of antennas presently predetermine the ma3or parameters of
an entire radio system, for example, in radars, the resolution and precision in
the determination of angular coordinates, the rate of beam travel in space and
the interference immunity.
The rapid development of microelectronics and its achievements have also found
their own place in antenna engineering. Integrated circuit stripline assemblies,
stripline and microstrip transmission lines and various microwave devices de
signed around them (phase shifters, switchers, rectifiers, amplifiers, etc.) have
come into widespread use in recent years. However, the potential possibilities
for reducing.the weight and volume of microelectronic radio equipment can be
_ realized with the appropriate design of the antennas, dispensing with traditional
types of them and making a transition to antenna arrays. The fact is that the
reflector antenna with,a fairing, the drive mechanism, waveguide channel and
microwave devices of the aircraft radars in operation has considerable size and
weight as compared to the other parts of the radar station. A radar in a micro
electronic design.using semiconductor microwave 3evices makes it possible to
achieve the greatest reduction in size and weight.  5 
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The increasing complexity of antennas during their development and their increas
ing role in radio systems have Ied to the expansion of the group of radio spec
ialists working directly in the field of antennas and feedlines. Not only the
specialists in these fields must be involved with the calculation of the major
characteristics of antennas and microwave devices, but also the designers of the.
entire radio system and its individual componentis, which are coupled to the
antenna. Their combined efforts duriag the preliminary design stage make it pos
sible to estimate the ultimately attainable characteristics of the entire radio
system, taking into account the feasibility of making the individual components.
The appearance of new types of antennas has led to a substantial expansion and
deepening of antenna theory as well as the development of new design techniques.
Considerable attention has been devoted to these questions in the literature:
a number of monographs have been published [01013] and a considerable number
of papers have published in journals. iiowever, the use of these materials by
radio engineers as well as students doing their diploma and course design work
also encounters considerable difficulties. Engineering methods of designing
prospective phased and active phased antenna arrays, as well as their elements,
are presented in this textbook. Considerable attention is devoted to the design
of aircraft and mobile antenna systems. The engineering design techniques are
supplemented with descriptions of designs of existing antennas and the requisite
reference material on the parameters of various microwave devices is given for
devices which can be used as the components of phased and active phased antenna
arrays.
The cited design techniques are sufficiently simple, based on approximate micro
wave antenna theory and suitable in the majority of cases for engineering prac
tice. These techniques make it possible in the initial design stages to approxi
mately determine the majai parameters and characteristics of the antennas, where
these parameters and characteristics can subsequently be made more precise where
 necessary by means of various more rigorous design methods. Also included in
the book are 3esign techniques developed on the basis of mathematical models of
antenna arrays and their components, close to the actual ones. The characteris
tics of director, waveguide, slotted resonator and slotted waveguide radiators
of a periodic array are studied and optimized by rigorous electrodynamic methods.
The calculated curves and the programs developed in the allpurpose algorithmic
languages of Algol60 Znd EortranIV for the BESM6 and M4030 computers are pre
sented. By basing the work on the general procedure for phased antenna array
design and using the materials of this book, one can design the radiating aperture
of a phased array in a rather well reasoned fashion.
The material presented here is intended for a reader already familiar with the
general course on antennas and microwave dev;.ces, which is studied in the radio
engineering departments of the higher educational inst3.tutes.
1.2. The Main Requirements Placed on Microwave Antenna Systems and the
Possibilities of Using Antenna Arrays
The major raquirements placed on an antenna are governed by the volume of infor
mation to be processed (or extracted) and are linked to the range, resolution,
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precision in the determination of coordinates, speed, reliability, interference
immunity and other characteristics of the radio engineering system. Establishing
the interrelationship between thg characteristics of various radio engineering
systems and the characteristics of the antenna and feedline is accomplished in
the relevant courses on radar, radio control, etc. Without going into the
details of the operation of these systems and establishing the interrelationship
cited above, one can state that in the final analysis, antennas and feedlines
should assure the appropriate: directionality, power, frequency and direction
f.inding characteristics, control characteristics and other general engineering,
operational and economic characteristics.
The requirements for antenna directianality predetermine the shape and width of
the spatial directional pattern (in the two main planes), thepermissible level
of sidelobes, the direction gain (KND) and the polarization characteristics of
the antenna. Antennas in the microwave band have needleshaped, cosecant, fan
shaped, funnelshaped and other directional pattern shapes. The polarization
characteristic determines the following: the polarization of the transmitted
and received waves, the permissible ijefficient of uniformity of the polariza
tion ellipse when using rotationally polarized waves and the permissible level
of crosspolarization in the case of linear polarization of the radiated field.
 When designing an antenna, the shape of the directional pattern, its width, the
sidelobe level, the directional gain and the polarization can be specified. It
must be noted that a relationship exists [01] between these characteristics
 which determine the directionality, and during the design work, frequently only
some of them are specified. Thus, in the electrical design, the starting data
can be the width of the directional pattern (beam width) or the directional gain.
It can be stipulated in this case that it is desirable to keep the sidelobe and
crosspolarized radiation levels to a minimum with the given relative antenna
dimensions.
The power characteristics of transmitting and receiving antennas make it possible
to determine: the signal power at the input to the receiver; the maximum permis
sible transmission power at which the electrical strength and permissible ther
mal mode are assured; the power needed to control the beam position in space;
the microwave power losses in the antenna and feedline channel as well as the
noise power in the receiving antenna. These powers are characterized by the
following parameters, as is we11 known [0.1, 0.2, 0.3, 0.6, 0.7]: the antenna
gain, the antenna efficiency and efficiency of the microwaye devices which are
used, the noise temperature, the input impedence (the matching in the feed chan
nel), the antenna Q[02] and the permissible electrical field intensity. In
contrast to ttechani.cally scanned antennas, in which the determination of the
power used to control the beam position in space is related to the electrical
drive design, in electrically scanned antennas, ttnis power is governed by the
 losses in the controllable microwave devices, and for this reason can have an
. impact on the thermal mode of the antenna. When designing microwave scanning
antennas, only individual values are egecified at times which characterize the
power indicators of the antenna. Thus, for example, the power (pulsed and
average) of a radio transmitter or the sensitivity of a radio receiver are speci
f ied .
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One of the tasks of design is to optimize the power characteristics of the
antenna being developed, taking into account the existing possibilities and
specific set requirements. Optimization reduces to bringing the feasibly attain
able characteristics close to the ultimately attainable theoretical characteris
tics, found for the specified optimality criteria. For example, such criteria
can be the maximwn gain or minimum noise temperature for the speci,fied relative
dimensions and losses in the microwave components being used.
The frequency properties of antennas are characterized by the greatest change in
the frequency of the transmitted (received) signal for which the major parameters
of the antenna do not go beyond the permissible limits. Depending on the require
ments placed on the radio system in which the antanna being designed will be used,
the frequency properties are detesrmined with respect to the change in the direc
tionality or the power characteristics. When calculating the frequency proper
ties of the antennas treated in this book, it is expedienr to draw a distinction
between the requirements placed on the working bandwidth of the antenna and the
bandwidth of the transmitted signals. The requisite passband is determined by
the condition of the simultaneous transmission or reception by the antenna of
a signal with a specified frequency spectrum. The range of frequencies is deter
mined by the condition of antenna operation sequentially in time at different
frequencies in the working band, i.e., permits a synchronous change in certain
antenna parameters with a change in the working frequency of the radio system.
For example, in an electrically scanned antenna array, the phase distribution
along the array is varied so as to preserve the direction of the beam in space
when the working irequency of the transmitter changes.
; In antennas and feedlines, a number of requirements are placed on the spatial
' scanning characteristics (such as tre scanning sector and time, etc.) as well as
requirements governing the change in the directional properties during the process
I of operating and switching the antenna from transmit to receive. These require
; ments also determine the requisite control characteristics for the antenna and
, feedline. The starting data with the choice of electromechanical or electrical
scanning for the performance of the design calculations for the selected type
of antenna are the spatial scanning sector of the beam, the scanning period (pace)
or time needed to set the beam to a specified point in space, method of spatial
scanning, precision of setting the beam to a specified point in space, etc. The
~ antenna switching time from transmit to receive is also to be included among the
control characteristics, as well as the requirements which arise in a number of
cases concerning the change during the operational process in the polarization of
 the transmitted field or the shape of the directional pattern. In mechanically
scanned antennas, the beam control characteristics are not related to the elec
trical design of the antenna and are determined during the design of the rotation
mechanisms.
The angular coordinates of ob3ects and the precision in the measurement of these
coordinates are determined by means of the direction finding characteristics
used in radar, radio direction finding, radio astronomy, etc. The requirements
placed on direction finding characteristics depend substantially on the direction
finding technique employed (monopulse, radio signal, amplitude, phase DF'ing,
etc.).
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Antenna usi.ng a monopulse direction finding method (monopulse antennas) have
become widespread in radar of late, the direction finding characteristics of
which are the slope and linearity of the characteristic, the depth of the "null"
in the difference pattern and the precision of its setting in a specified direc
Cion. The requirements placed on these characteristics, with the exception of
the latter, reduce to the creation of a special shape and symmetry in the direc
tional patterns, as well as to obtaining the maximum reception antenna gain. The
requisite precision in setting the "null" of the difference pattern in a specified
direction, within the bounds of a scanning sector, is governed by the scanning
technique and the characteristics of the devices which control the antenna beam
position. The realization of the requisite direction finding characteristics is
a most important and difficult task for many antennas. 
_ Overall engineering, operational and economic requirements are placed on an
antenna, just as on any radio engineering unit, such as: minimal size, w6ight
and cost, high reliability, adaptibilitq to specified conditions, as well as
control and repair convenience. Setting these requirements on an antenna being
developed is no less important than setting the electrical requirements, and
 meeting them is achieved not only through the appropriate structural design
solutions, f abrication technology and the use of the requisite materials, but
also through the selection of the appropriate scanning method, electrical circuit
configuration, operational mode for the system as well as the active elements
and microwave devices which are employed.
With the development of various radio engineering systems and the increased com
plexity of the design and engineeriag problems solved by them, the requirements
placed on the antenna characteristics are also increasing, and in a number of
cases, they become contradictory and altogether insoluble when attempting to
develop new antennas on analogy with those previously existing and presently in
operation. For example, the striving to increase the range and precision of the
determination of angular coordinates in radar leads to the requirement of increas
ing the antennas directionality, which causes an increase in their size and
weight. The increase in the f].ight velocity of aircraft leads to the necessity
of increasing the rate of beam motion in space. It is :iot possible to combine
the requirements of increasing the directionality and the rate of beam travel in
mechanically scanned antennas because of the inertia in their structure. Similar
contradictions also arise during attempts to simultaneously provide for high
directionality and the requisite frequency, power and direction finding characcer
istics. These circumstances force one to dispense with the traditional type of
antennas for the given class of radio systems and to change over to antenna
arrays.
The use of complex antennas in the form of arrays, consisting of systems of poorly
directional or directional radiators, significantly expands the possibilities
for realizing the requisite characteristics.
A system of radiators with an electrically controlled phase distribution  a
phased antenna array  accomplishes the electrical scanning of the beam in space
at a rate which can be several orders of magnitude greater than the speed of
mechanical scanning antennas. The setting time to a specified point in space for
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the.beam of a phased array is practically determined by the speed of the electric
al phase shifter or the frequency tuning or frequency scanning time, and is not
related to the weight or the dimensions of the antenna. With this "inertialless"
acanning, new, previously not used methods of spatial scanning and multitarget.
operation are possible (the simultaneous tracking of several targets in space),
Arrays made of narrow beam antennas.make it possible to increase the ultimately
realizable resolution, gain and maximum transmitted power. Arrays have been
designed and are being designed using large reflectors for the antennas of radio
telescopes for space communications, having a resolution of down to minutes of
an angle in the centimeter band [0.3]. The arrays make it possible to create
multiple function antennas, in which the shape and width of the directional pat
tern are changed by means of electrically controlled microwave devices, depending
on the functions being performed by the radio system.,
~ The realization of different kinds of amplitudephase distributions is signifi
cantly simpler in an antenna array than in reflector, horn, lens and other micro
wave antennas, since directional couplers, phase shifters, switchers and other
components can be inserted in the exciting radiators of the device (power dividers
of the antenna array), where these components provide for the requisite distri
bution or control. Various kinds of amplitudephase distributions make it possi
ble to realize socalled optimal directional patterns in practice (with minimal
sidelobe radiation), as well as directional patterns having deep troughs ("nulls")
in the direction of interference near a target outside the main lobe of the
antenna.
In terms of the structural design, the use of antenna arrays makes it possible
to reduce the longitudinal dimensions (in the direction of the normal to the
plane of the array) of pencilbeam antennas, and consequently, the volumes
occupied by them; and to use the exterior conducting surface of an object for
radiating. A highly directional antenna array made of horns or reflectors has
a smaller longitudinal dimension than one horn or reflector antenna with the same
directivity. An array of slotted radiators on the convex (conical, cylindrical,
spherical, etc.) exterior surface of an aircraft [OS], without increasing the
aerodynamic resistance, makes it possible to substantially reduce the occupied
volume as compared to the corresponding aperture antenna placed in a fairing.
Radio specialists have recently been devoting considerable attention to socalled
active phased antenna arrays, in which a selfexcited oscillator, amplifier,
converter, mixer, etc. are connected to each radiator or a group of them. This
new approach to the design of the entire radio system, where it is impossible to
single out such individual devices as a receiver, transmitter, etc., permits a
substantial expansion of system capabilities when processing the incoming infor
mation, as well as the construction of adaptive (selftuning) antennas and
achieving better interfacing of the �radio system to a computer.
From everything that has been presented here, the role of antenna arrays in
modern radio engineering systems, their possibilities in providing for the.
requisite antenna characteristics as well as for the entire radio system be
comes understandable. For this reason, the design principles and methods of
calculating the major parameters of prospective phased.and active pYiased antenna
~
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arrays with various radiators, geometries and control techniques are set forth
in this book.
1.3. Antennas with Electrical Scanning
We shall deal with the specific features of the construction and design of antenna
arrays with electrical scanning, which must be taken into account during the
planning. It should be noted that up to the present time, no final terminology
has been worked out in the field of antenna arrays with electrical scanning, and
conclusive engineering techniques for their design are also st311 lacking. We
shall employ borrowed terms and definitions, as well as the most widely dissemi
nated terminology, corresponding to the physics of the processes which take
place.
Electrically scanned antannas can be treated in the general case as arrays with
 a controlled phase or amplitudephase distribution. Various types of radiators
and channelizing systems are used in such antenna arrays, as well as diverse
ways of exciting the radiators and controlling the amplitudephase distribution
during scanning. Antenna arrays in this case have the most diverse structural
design. However, the directional properties of antenna arrays, when they are
correctly designed, can be determined just as for highly directional antennas
with a continuous radiating aperture, in which the directional properties depend
on the relative dimensions of the aperture (with respect to the wavelength) and
the field distribution in it. In linear and plaraar arrays, the equivalent radia
ting aperture changes during scanning, i.e., the pro3ection of the aperture onto
a plane normal to the direction of the beam, and consequently, the directional
properties also change. The changes in the beam width of the array during scan
 ning should be taken into account in the
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electrical design of the antenna. Graphs
which illustrate the.change in the 3irec
tional pattern width, 2Ap.5, are shown in
Figure 1.1 as a function of the relative
antenna size L/a and the direction of the
beam in space, A.
10 100 L/X 1,000
Figure 1.1. Beam width as a function
of array length and scanning angle for
the case of uniform excitation.
Linear, planar or axially symmetric
arrays (annular, conical, cylindrical,
spherical), as well as arrays with a more
complex shape (surface antenna arrays)
find application in practice.. Arrays
can be both equidistant (with a constant
spacing between the radiators) and non
�equidistant types. The directional pat
tern width of each radiator, the number
of them and their arrangement in the array
are governed by the requirements placed
on the directivity of the antennay the
spatial scanning sector and the conditions
for the placement and operation of the
antenna.
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Cylindrical antenna arrays and phased arrays with a hemispherical scan space are
also treated in the book. Such arrays can be constructed in the form of convex
polyhedra made of planar arrays and arrays which have been given the name
"conformal" in foreign literature.
Assuring the specified requirements for an array with electrical scanning during
the design work can be achieved with the use of different types of radiators,
different spacings between them, array configurations, etc. Cne of the main tasks
of the design work is to find the optimum array var.iant for the specified require
ments, taking into account the existing possibiiities for the excitation, place
mern.t, fabrication and operating conditions.
The radiators, the number of which in an antenna can reach tens of thousands can
be excited by means of waveguides, coaxial lines and striplines as well as other
types of channelizing systems using par2,S.?.el, series, branched and other feed
confi.gurations. A spatial excitation ter.hnique is also possible which is similar
to the method of exciting lenses and reflectors in which one (the primary) irrad
iator excites all of the radiators of the array simultaneously., The selection of
the excitation configuration durinl3 the design work is determined by the method
of scanning, the permissible losses in the antenna as well as the size and weight.
Beam scanning in a frequency scann3ng antenna* is ach3.eved by changing the oscil
lator frequency (in the transmitting antenna) and the receiver (in the receiving
antenna). The electrical spacing between the radiators, excited by the channel
izing traveling wave systems changes with the change in frequency, and'consequent
ly, the phase distribution in the array also changes. The determination of the
characteristics of these channelizing systems reduces, primarily, to the design
of frequency scanning antennas. Frequency scanning antenna arrays prove to be �
significantly simpler in their structure than other antenna arrays with electrical
scanning, since there are no other elements in them besides the channelizing and
radiating devices. The presence of a microwave receiver and generator with a
fast response, for example, with electrical frequency tuning is a necessary con
dition for the design of electrically scanning radio systems. However, the reali
zation of frequency scanning in th.e case of wide angle and twodimensional scan
ning encounter�s considerable difficulties. Moreover, the use of frequency scan
ning is not possible in all radio systems. In the case oi a constant working frequency for a radio system, the phase distri
bution in an antenna with electrical scanning can be controlled by means of phase
shifters. This technique has been given the name of phase beam scanning of an
antenna array. Ferrite, semiconductor, ferroelectric and other phase shifters
 have heen developed at the present time, in which the phase of the outgoing
electromagnetic wave changes either discretely or continuously from 0 to 360� as
a function of the control voltage or current. The incorporation of a system of
phasz shifters in the device exciting the antenna (the power divider) makes it
possible to realize electrical scanning, where the phase distribution control is
discrete in the ma3ority of cases. This occurs becauseof the discrete change in
*Questions of frequency scanning antenna theory and design were treated most
completely for the first time by L.N. Deryugin [010].
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the phase shift in a phase shifter or the control current (or voltage), which in
turn is due to the specific nature of the operation of the electronic device
controlling the beam position. Such an electrical scanning technique, which has
been termed switched scanning (or digitally switched scanning in previously pub
lished literature), is the most promising at the present time. With the
switching technique, as a result of the discrete change in the phase, the direc
tional properties of the pliased antenna array also change. These changes should
be taken into account when designing switched antennas.
The phase distribution of a scanning antenna array can also be controllod by
means of inechanical phase shifters, in which the phase change is accomplished by
means of inechanically moving or rotating special individual components or parts
of the channelizing system of the phase shifter [03]. With such a scanning tech
nique, which is termed electromechanical, the maximum rate of beam travel is
governed by the speefl of the phase shifter, and because of the low inertia of
the devices being moved, can be significantly greater than in mechanically.scanned
antennas. The calculation of the directional characteristics of antenna arrays
 with electromechanical scanning is the same as for electrically scanned arrays.
The choice of one scanning technique or the other during antenna design is deter
mined not only by the requisite characteristics, but also by the existing possi
bilities, the presence of the appropriate electronic devices, the characteristics
of the phase shifters and channelizing systems, power considerations, etc.
The transition from mechanical scanning to electrical led to increased complexity
in antenna structural design, which was due to the use, for example, of an array
of radiators with phase shifters instead of one dish antenna, as well as to a
sharp increase in the cost of the antenna unit. The use of phase shifters,
channelizing systems and other supplemental devices increases the phase errors
and thermal losses in an antenna and reduces the gain. For this reason, it is
expedient to change over to electrically scanned antenna arraqs only in those
cases where the mechanical approach does not assure the requisite beam control
characteristics and a certain degradation of the power characteristics and increase
in the cost are permissible.
1.4. Specific Features of Phased Antenna Array Design
The further development of. microwave antennas led to the woricing out of new
and increased complexity of known methods of computing the main characteristics.
The structural and computational design wbrk on the antennas became significantly
more complicated because of the increase in the number of parameters governing
antenna characteristics, as well as by the striving to optimize the characteris
tics or compute them more precisely. .
The design of scanning antennas with specified characteristics is accomplished
with the condition that these characteristics are assured for all antenna beam
positions. For this reason, the calculation of the directional, frequency and
other properties of arrays must be made for various beam positions in the spatial
scanning sector. In this case, the beam width, sidelobe level, directional gain
and other characteristics are determined not only by the array parameters, but
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also by the characteristics of the devices which control the phase distributiotz
(the discrete step for the phase change in a phase shifter, the deviation of the
dispersion characteristics of the channelizing systems from the requisite values,
etc.).
Complex interaction phenomena occur between the radiators in antenna arrays, which
are manifest in .a change in the directivity and input impedance of a radiator wheu
it is inserted in the array. As a result, the directional properties and power
characteristics of an antenna can change substantially in an array as compared to
A the characteristics found without taking the interaction into account.
Intense developmental work is under way at the present time on the theory of
accounting for interaction in microwave antenna arrays. Engineering methods of
calculating the interaction are known only for certain types of radiators and
a definite arrangement of them. Taking this interaction into account, which
changes when controlling the phase distribution, makes the design of phased micro
wave arrays considerably more difficult. ,
The interaction of the radiators in a phased antenna array depends on the type of
radiators used, their configuration and affects the antenna characteristics in
differant ways. Thus, the interaction of resonant poorly directional radiators
(resonant dipoles, resonant slot antennas) in an array leads to a substantial
change in the input impedance and the resonance properties, so that during scan
ning, the input impedance of eac:h radiator in the system and the matching of.the
driving channel depend on the beam direction in space. The change in the dis
tribution of the radiating current (field) and correspondingly, the directional
pattern of a radiator, is insignificant in this case.
The interaction of radiators in different types of antenna arrays (for example,
of the traveling wavedielectric rod type, helical antennas, yagi channels or
aperturewaveguide antennas, horns) is manifest in a change in the current dis
 tribution in the radiator and a corresponding change in the directional pattern
of an element. A change in the.directional pattern of a radiating element in
an array is manifest in a substantial change in its width and in the appearance
of deep nulls (indented pattern), something which leads to a significant drop in
antenna gain for certain,beam positions in space and to the corresponding mis
matching of the exciting channel. The mutual influence effect of radiatos can
be eliminated by means of the appropriate placement of the radiators, a choice of
their type an3 size as well as the use of dielectric coatings and other special
measures. For this reason, the design of the radiating elements of arrays is
treated in tnis book along with the general questions of phased array design.
Finding the optimal variant of a scar,ning antenna for given requirements, taking
into account the characteristics of the radiators, phase shifters, channelizing
systems and other microwave devices available to the designer, considerably
increases the volume of all of the calculations to be performed during the
design work.
Individual sections on the theory of microwave antenna arrays and electrical
scanning, which have been published in the literature, are intended primarily
 14 
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for specialists in antenna arrays. The study and utilization of this literature
in the designs of antennas for various functions require large expenditures of
time, which creates difficulties during engineering design work. For this reason,
some engineering techniques of designing scanning microwave arrays and their
elements are presented in the book, which make it easier for specialists familiar
only with the general theory and practice of the application of antenna devices
to determine the major characteristics. This has brought about the necessity of
intrcducing a number of approximations and simplifications, something which has
influenced the precision in calculating the characteristics and led to a limita
tion of their rangP of applicability.
Various methods exist for designing the antennas considered here, which differ in
the precision of the results obtained and the degree of complexity of the
calculations. The antenna characteristics found by means of the cited engineering
procedures can be made more precise by means of more rigorous computational tech
niques known from the literature (see the bibliographies for the relevant chap
ters). .
Along with the simplified design methods, where it has been possible, more rigor
ous computational methods are included using computers, which make it possiblP to
optimize the device being developed witYL respect to one criteria or another by
means of the programs which have been worked out.
T'he design of phased antenna arrays involves the solution of exterior and interior
electrodynamic pi�oblems from antenna theory. When using approximate analysis
' methods, the independent solution of the exterior and interior problems can be
, permitted. The solution of these problems, taking their mutual relationship
into account, makes it possible to calculate antenna characteristics and search
for the optimum variant of an antenna which best conforms to the  requirements.
Such an approach made it possible to create independent wethoL.. ,.ur che engineer
ing design of electrically scanned antenna arrays, arrays of radiators,and their
elements.
1.5. Specific Features of Active Array Design
The application of stripline and microstripline hardware makes it possible to
a significant extent to reduce the cost, improve the reliability and decrease
the size and weight of antenna equipment. Stripline and microstripline devices
can be used as channelizing systems, power dividers and directional couplers,
filters and circulators, isolators and phase shifters, etc. Such advantages of
printed circuit technology as repeatability of the parameGers during series pro
duction and the capability of integration have made it possible to also use these
devices in the structural design of microwave antenna, first in the decimeter and
meter bands, and then also in the centimeter band. The yagi (director antenna),
microstripline radiators, arrays of dipole radiators, compact resonator slot
antennas, etc. can be numbered among them. However, a substantial drawback to
stripline devices is the significant losses in the centimeter band and especially
in the short wave portion of this band. The insertion of an active element in
the microwave channel makes it possible to not only reduce the losses, but to
also increase the radiated power, simplify the microwave distribution system and
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ease the electrical requirements placed on it, as well as to miniaturize the
entire antenna system. The insertion of an active element (or device) in a
radiator or in its excitation channel transforms the antenna array from a passive
reciprocal [sic] device into an active antenna array, and a phased array into an
active phased antenna array, in which differsrtt active elements are used during
reception and transmission. In practice, antenna arrays are broken down into
receiving, transmitting and transceiving, depending on the function. The radia
tor, active elements, phnse shifter, lines connecting these microwave elements,
etc. are structurally combined into a single device, which has been given the
name of an active phased antenna array module.
The most diverse circuit configurations are known at the present time for receiving
= and transmitting modules. 'For example, in some ciYcuits, the active element is
coupled to each radiatior, while in others, it is coupled to a group of radiators.
There is a lack of unified terminology to an even greater extent for phased
antenna arrays for acti.ve arrays.
The designing.of the transceiving module of an active phased array with the
theoretic and component bases existing at the present time is actually broken
down into the solution of two independent problems: the development of the
transmitting module and the development of the receiving module. As is well
known, modern microelectronics has achieved significant successes; various inte
grated circuits have been created which are widely used in radio receivers. At
the same time, there is a lack of series produced high power microwave inte
gratF�d circuits for radio transmitting devices. This circiimstance has also led
to the necessity df a more detailed treatment of the questions of the design of
active tr.ansmitting modules in this book.
 When dPVeloping an active phased antenna array module, a solution which provides
4 for minimum antenna cost while assuring all of the requisite characteristics is
preferable. As stlidies show, the cost of the power generated in a circuit, where
each radiator is coupled to an individual active element, is higher, however,
this is compensated by the less expensive and lower power generators and phase
~ shifters, and the possibility of using more convenient power supplies as well
as facilitating the cooling of the elements of an array.
 When designing an active transmitting module, one can use eitller a selfexcited
oscillator or an externally excited generator (power amplifier), or a string of
series connected stages, among which there can be frequency multipliers. Because
of frequency multiplication, the distribution system operates at a frequency lower
than the output frequency, and as a rule, at a lower power level, which makes it
possible to substantially reduce the losses in the system.
The major Xequirements placed on the active elements of modules are assuring the
specified output microwave power, relatively high values of the efficienc; (no
less than 20 .*.0 40%) and power gain (more than 10 dB), operating mode stability,
comparatively wide passband (more than 5%), a small scatter in the parameters of
the individual models, operational stability in a wide range of temperature varia
tion, low levels of generator noise, filtering of spurious signals and those out
side the passband, as well as a number of structural design (small size and weight)
and economic requirements.
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Semiconductor microwave devices such as bipolar power microwave transistors, mul
tiplier diodes.(varactors and charge storage diodes) as well as microwave diodes
(IMPATT diodes and charge transfer diodes) have been finding increasingly wider
applications in active modules in recent years.
~ High power microwave transistors are the mnst sophisticated semiconductor devices
in the microwave band; they have working frequencies which as yet do not exceed
5 to 7 GHz. For this reason, when developing active phased array modules for a
working frequency in the 3 centimeter band using these transistors, it is neces
sary to provide a frequency multiplier, something which leads to the use of an
amplifier and multiplier chain in the module. Diodes with a nonlinear pn
junction capaci.tance are used as the nonlinear element in the multiplier, where
these diodes are distinguished by a high input to output signal power conversion
gain, small dimensions and weight and which practically require no power from
the power supply.
Microwave amplifiers designed around avalanche transit time diodes have a higher
output power (by an order of magnitude) and a greater efficiency (up to 5 to 15�6)
than charge transfer diodes.
Active modules can also be designed around selfexcited microwave device oscilla
tors (transistar or diodes) using a system of synchronization from a special
frequency source.
The design of the radiating system of an active phased antenna array is closely
tied to the development of active modules which assure the requisite characteris
tics of the antenna array. For this reason, when planning an active phased array,
it is necessary to select the circuit configuration.of the active modules, com
pute the operational modes of the generator stages and the microwave networks
matching them as well as execute the structural design of the generator circuit
components in the form of a hybrid integrated circuit. It should be noted that
calculations of semiconductor microwave generators are made at the present time
using approximate methods, since the devices themselves are complex nonlinear
_ microwave devices. However, these techniques make it possible to estimate the
major power and structural design characteristics of the stages with a precision
adequate for practice and to design the radiating system of an active phased
antenna array based on them.
The power engineering characteristics (output power, working frequency, efficiency,
gain, etc.) of microwave semiconductor devices are treated in the book for the
purpose of using them in the active modules of active phased arrays and attention
is drawn to the possibility of the appearance of thermal limitations with certain
structural design requirements related to the realization of beam scanning in
the array. Design procedures are given for the operating conditions of high
power microwave transistor oscillators and their matching networks, as well as
frequency multipliers using varactors and charge storage diodes, which make it
possible to design an active module using an amplifier and multiplier chain.
Special attention is devoted to the design of microwave oscillators and amplifiers
around avalanche transit time diodes, which meet many of the major requirements
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placed on the active elements of active phased array modules in the 3 cm band.
Reference materials are also given for the structural design of the components
of microwave networks.
,
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2. PHASED ANTENNA ARRAYS
2.1. The Determination of the Geometric Characteristics of Phased Antenna Arrays
Some of the most widespread types of phased antenna arrays are linear and plangr
arrays. The ma3ority of planar phased arrays consist of identical radiators,
positioned at the nodes of a plane coordinate grid with twofold periodicity.
The most useful grids are rectangular and triangular (or hexagonal) (Figure 2.1).
It is assumed in an elementary analysis that the directional pattern of a radiator
in an array does not differ from the directional pattern of an isolated radiatoro
Figure 2.1. Methods of radiator layout. Figure 2.2. Systems of coordi
_ nates
The excitation phase for the radiators in an array {n the case of narrow beam
radiation provides for the inphase addition of the fields in a specified direc
tion and depends on the position of the radiator in the array:
(j)nq 011in '1'r.a)   1~ (Xn g CUS 'Pivt " ] 1, nq 5111 (Prn) SI il nrn' (2.1)
where k= 2n/y 3.s the wave number; X.nq and Ynq are the coordinates of the radia
tors in the array; 9rn and Orn are the angles in a spherical system of coordinates
which determine the direction of the main lobe (beam) in space (Figure 2.2).
The directivity function of an array f(@, can be represented in the form of
the product of the directivity function of an isolated radiator F(9, times an
array factor FE(9, which can be treated as the directivity function of an
array consisting of isotropic radiators:
t (0, (p.) c (n, y) FE (0; (2.2)
where
Af, N 1 ~rt'nui ~'~mn~
~
rz (0. (1)) 2,
ug, it _ I In the cited expressions, Amn is the amplitude of the excitation for an array
element; (r;,,n =.k*(X,,,n co.,, ,I, I y,,,n 51n ,0 u '
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linetakes the form of a set of
.
switched coaxial cables of different
lengths. However, during wide angle
� scanning, the use of controlled delay
j lines in the channel of each radiator
~ does not prove to be possible because
of the impermissible increase in
N~ antenna cost and structural design in
7 conveniences related to the considera
ble overall length of the switched .
cables. For example, when scanning in
Figure 2.38. Configuration of a',phased an angular sector of emax  60% the
antenna array with controlled maximum length of the switched cables
. delay lines. differs only slightly from the aper
ture size. A widening of a phased
' . . array bandwidth can nonetheless be
attained through delay line control of not individual radiators, but rather groups
of radiators (Figure 2.38), while the phase control of individual radiators is
accomplished in a group by phase shifters. Even when the aperture is broken down
into two subarrays, the passband of a linear array is more than doubled.. Doubling
the passband of a planar aritenna array is accomplished when the array is broken
down into four parts.* In the general case, to increase the passband by a factor'
of N times, it is�necessary to break a linear ant,enna array down into N.subarrays,
and a planar array down into N2 subarrays, controlled by means of delay lines. By
employing the property of array symmetry, the number of delay.lines with long
cables can be substantially reduced [09].
The directional pattern of an antenna system, the~, subarrays of which are controlled
by means of changing the signal delay time, can li;e represented in the form:
f (0 (p) Fn (0+ T) Fzn (9, (P); (2.31)
where Fn(9, 0 is the directional pattern of the subarray; FEn is the multiplying
factor for the array, the elements of which arethe subarrays.
With a change in the frequency (Figure 2.39), the main lobe of the array factor
(1) remains in a constant position, since the phases of the subarray signals are
controlled by means of changing the clelay time, while the directional pattern of,
the subarray (2) is moved, 3ust as i:i the case of a radiating aperture, since the
radiators of the subarrays are cont;colled by phase shifters. For this reason,
the frequency proper.ties of the antennas considered here are determined by the �
frequency properties of the radiating aperture of the subarray. A significant
circumstance in the given case is the increase in the level of the sidelobes of
the directional pattern for the entire array at frequencies other than the center
frequency. This increase is due to the fact that when the main lobe of a subarray
directional pattern is shifted relative to the array factor, spurious maxim (3)
of the array factor (Figure 2.39) fall in the region of the maxima, where these
spurious maxima coincide at the center frequency in terms of direction with the
direction of the nulls of the subarray directional pattern. Therefore, the
level of sidelobe radiation in the direction of the spurious maxima of the array
factor increases. The sidelobe level for various signals is shown as a function
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of the argument u in Figures 2.37 and
2.38. It follows from the graphs that
if the drop in the gain does not exceed
1 dB, th3s level does not exceed 11 dB.
The graphs $hown in Figures 2.37 and
2.38 also characterizs the change in
the gain of an antenna array with sub
arrays controlled by delay lines, where
L in formulas (2.20) (2.29) is under
stood to be the corresponding dimension
of the subarray.
Figure 2.39.
The Frequency Properties of Power
The relative shift in the Divider Circuits. The Parallel Circuit
directional pattern of a Configuration. The various power
subarray with phase shifters divider circuits differ substantially
and the maxima of the array in their frequency properties. Parallel
factor with a change in the and double stepped circuit configura
frequency in an antenna tions with equal power division in each
system with delay line~. branch possess the best frequency pro
perties.' This is due to the fact that
the electrical length of the paths from
the antenna input to each radiator are the same and change identically with a
change in frequency. For this reason, the phase distribution remains constant
within the passband at the output of power dividers designed in these configura
tions. Other feed circuits introduce additional phase shifts, which lead to adisplacement of the beam of the array.
The Series Circuit Configuration. In the case of series pow2r division (Figure
2.40), where additional transmission line sections are absent (Figure 2.40a),
which equalize the signal path length from the antenna input to the radiators, a
frequency change which changes the phase relationships at the input to the radia
tors leads to a deflection of the beam of the antenna array. If a Tmode propa
gates in the main trunk faeder, then the shift in the beam caused by the linear
phase error occurring with the change in the frequency, is determined by the
expression [013]:
Aa^ 'T n f ~
k . I cosA~A ' (2.32)
where ao is the wavelength in the feeder trunk. The beam displacement which is
due to the properties ofthe series feed circuit for the radiators, eithex adds
to the displacement related to the frequency properties of the radiating aperture,
or cancels it: if the beam is deflected in the direction of the array input, then
the beam displacements add; if the beam is deflected in the direction of the load,
then they subtract. The passband, within which the arr,ay gain falls off no more
than 1 dB with a maximum beam deflection of 60�, is defined by the relationships
[013] :
A f20o.s
I 1 x(b/x
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(2.33)
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in the case of a long pulse with a changing frequency, and:
e _ f 400.6 (2.34)
.

_
in the case of a short pulse.
Bmom � .
~ � '
. , ~ ~ d ~D ~ ~ ~ ~ ~ �
. . ~ � . �
al d~
Figure 2.40. A series power division circuit configuration.
. When the main trunk feeder is fed in the
center (Figure 2.40b), the system can be
treated as two antennas with series power
division. If the beam is oriented with
;E ~~,~E respect to the normal to the line of posi.
tion of the radiators at the center fre
.~E quency, then when the frequency changes,
the beams of each of the halves of the
array will move in opposite directions
and the overall directional pattern will
Figure 2.41. An optical power division expand, without changing the direction.
circuit configuration. . As a result, the directivity of the
antenna array will be reduced. When the
array beam is deflected from the normal, the angular motion of the beam due to
the specific features of series power division, adds to the motion of the beam
due to the frequency properties of the radiating aperture. For the different
halves of the array, these motions are in opposition: for one half, the aperture
motion is cancelled by the displacementi due to the properties of the power divider,
and for the other, the motions add together with the same sign.
In the case of radiation along a normal, the series dividers are worse than paral
le1 dividers, but with a deflection from the normal through an angle of + 60�,
the degradation of the characteristics is approximately the same and does not
exceed 0.25 dB for the series circuit.
If the trunk feeder is a dispersion system, then:
0 f 200,G 1/44,
(2.35)
in the case of a long pulse with a changing frequency; and:
A f ,.s 400,6 (2.36)
in the case of a short pu.lse, i.e., the characteristics of the array are degraded.
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Optical Circuit Configurations. When using an optical power division configura
tion, the frequency properties of an antenna array depend on the relative focal
distance. If the focal distance is large (Figure 2.41), then the properties of
an optical power divider approach the properties of a parallel circuit with feeders
of equal length, though if the focal distance is small, the properties of the
optical power divider approach those of a series configuration with center
excitation. Since with a beam deflection through a maximum angle of + 60� from
a normal to the array aperture, the parallel circuit properties differ insigni
ficantly from the properties of a series, center fed configuration, the frequency
properties of optical power dividers in the case of wide angle scanning are prac
tically the same as the properties of a parallel feed circuit with feeders of
equal length.
TART.F. 9. 1 _
Type of Power Divider I Bandwidth nonxo 4.cror, %
� Tun aenxrenn wamxocrx
(2) � I 20p,b
1 ~7l/7ku
, 40o,n
nHraHNe c cepeAHtid
~
~ IIOCJICAo02TeJIbI1F.lA:
nxTauxe c KOHI(8
i
flapannenwio AeoxyNOsraxc
HWfI (1)
200, 6 1/14, I 400 .67l/.71,b
OnTHItecKNii 0PTICAL I 200,6 I 400,8
Key: 1. Parallel double stepped divider;
2. Series: end fed;
center fed.
The indicated results are summarized in Table 2.3. They correspond to a scan
sector of + 60� with a.permissible drop in the directivity of no more than 1 dB.
2.8. Switched Scanning
The beam position of a pencil beam antenna array is controlled by changing the
phase relationships between the currents in the radiating elements. A system
of phase shifters, inserted in the feeder system exciting the radiators can be
used for this purpose.
The major drawbacks to electrically controlled antennas with phase shifters, which
provide for a continuous phase change in the electromagnetic oscillations (ferrite,
semiconductor, ferroelectric phase shifters, etc.) are the instability (especially
the temperature instability), the complexity of the control circuits and the high
requirements placed on the stability of the phase shifter power supplies. These
deficiencies also exist in digital control systems, when individual operating
points on the characteristic of a continuous phase shifter are employed.
The indicated drawbacks are eliminated to a considerable extent with the switched
technique of directional pattern control proposed by L.N. Deryugin in 1960. The
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A11MNqN! MMIIYJIbCM ~ I KOPOTNNE NMf1yJIbCN
Long Pulses, I Short Pulses
2eo,g . 4eo,b
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essence o� the switching method consists in dispensing with phase shifters having
a continuous phase change and using switchers and switched phase shifters, at the
output of which the phase of the electromagnetic oscillations takes on definite
fixed values. The antenna beam control in this case reduces to the simplest opera
tions of switching radiators or feeder system branches on and off.
The stability of switched antennas is due to the fact that the phase control ele
ments (semiconductors, ferrites, ferroelectrics) operate in a mode in which only
the extreme portions of their characteristics are used. Moreover, switched anten
nas can have a simpler controller than a conventional antenna with a parallel
circuit configuration for the continuous phase shifters. The latter is related
to the fact that the position of the beam in space is not governed by the control
voltage, which is different for different antenna phase shifters, but only by its
presence at particular switchers.
However, switched antennas also have a number of deficiencies, the most important
of which is the presence of phase errors which arise because of the fact that the
radiator excitation phases change in steps and can assume only definite values.
This entails a reduction in antenna efficiency, an increase in the sidelobe radia
tion level and a jumplike motion of the beam. .
Among the various methods of constructing switched antennas, one can single out
the two most characteristic approaches. In the first, each radiator has a
definite set of phases, from which the selection of the requisite phase is made
by means of switching the switched phase shifter. With the second method, several
radiators are placed in each section of the antenna with a length of a/2, where
these radiai:ors are excited with different phases, and they a_re selectively turned
on. Some of the aspects of designing switched antennas based on the first approach
will be presented in � 2.10, since the realization of antennas with switched
radiators encounters serious difficulties related to the necessity of placing a
large number of radiating elements in a small portion of an antenna and consider
ably retarding the phase velocity of the electromagnetic waves in the feeder
 which excites the radiators.
2.9. Switched Phase Shiftera
Switched phase shifters are the major component of phased antenna arrays. They
can number up to several tens of thousands in highly directional scaLining arrays.
In this case, the spacing between the phase shifters usually falls in a range
,
of 0.5 a to X.
Switched phase shifters should have a high efficiency, sufficient electrical
strength, stability of the characteristics and consume the minimum power needed
for controlling their operation.
Moreover, the following requirements are placed on the structural design of phase
shifters: structural simplicity and suitability for production; small size and
_ weight; high reliability.
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Socalled digital phase shifters are used in the ma3ority of cases to control
the excitation phase of the radiators in a phased array. A feedthrough digital
phase shifter is broken down into p stages, each of which can be in one of two
states, characterized by the phase shift being introduced: 0 or n/2m1, where
m is the number of the stage. It is sufficient to employ p control signals which
take on values of 0 or 1 to select any of M= 2p possible states of�the phase
shifter. Then, for example, in a two place phase shifter, a signal of 00 corres
ponds to a ze:a phase shift, a phase shift of 90� has a control signal of 01,
etc. A reflective phase shifter for reflecting arrays can be derived from a
transmissive type by means of sharting the output. To preserve the phase shifts,
it is obviously necessary to cut the phase shift realized by each stage in half,
since the wave ina reflective phase shifter passes through each stage twice.
In ferrite phase shifters, the phase shift is due to the change in the magnetic
permeability of the ferrite with the action of an external magnetic field. The
switched elements of the majority of semiconductor phase shifters are PIN diodes.
Since the diodes usually operate in the ultimate modes, the tolerances for the
amplitude of the control signals are not stringent.
Merits of semiconductor phase shifters are the small size and weight, the fast
switching speed, the simplicity of the control devices, their reciprocity and
thermal stability. Semiconductor phase shifters are manufactured in stripline
and microstripline variants to reduce the size and weight, and improve the sta
 bility, which makes it possible to use printed circuit technology. Advantages
_ of ferrite phase shifterE are the relatively high microwave power level, since
a bulk ferrite medium.is used to control the phase; lower losses, since waveguides
are usually employed in making fersite phase shifters, the losses in which are
less than in lines using a Tmode, as well as a lower SWR. The switching speed
of diode and ferrite phase shifters amounts to 0.1 psec10 usec and 0.130 usec
respectively.
None of the indicated types of phase shifters has an absolute advantage over the
others and the use of 'a particular type depends on many factors: the power level,
range of.working temperatures, and requirements placed on switching speed and
stability. It must be noted that the high cost of phased antenna arrays, as a
consequency of the large number of microwave components used i.n them, limits the
wide scale application of phased array systems. Information of phase shifters
for phased antenna arrays is given in [2].
~ Semiconductor phase shifters have been developed at the present time which operate
at a transmitted power level in a CW mode on the order of hundreds of watts and
on the order of tens of kilowatts in a pulsed mode. In this. case, the losses in
a three place phase shifter in the ten cm band, for example, do not exceed 1 dB
[2]�
Ferrite phase shifters at wavelengths shorter than 5 cm have lower losses than
semiconductor types. The losses per place amount to about 0.3 dB in the 3*cm
band, while the pulsed and average transmitted powers are about 500 KW and 1,000
W respectively [2]. The advantage of ferrite phase shifters of some types is
an internal memory, which makes it possible to control the phase by means of
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feeding in short pulses. In the intervals between pulses, the phase shifter
remembers the phase shift, and no energq is expended to maintain it.
In contrast to ferrite phase shifters, semiconductor types using PIN diodes do
not have such a property, and this is a drawback to them. To preserve the reqUi
site phase shifts, it is necessary to expend considerable power: up to Reveral
kilowatts raith a large number of phase shifters. In fact, according to [013],
the control power for a diode phase shifter is 0.1 to S W, while the energy needed
to switch a ferrite phase shifter is 20 to 2,000 m.icrojoules.
Phase shifters using �ield effect diodes and resistive gates are being developed
at the present time [2], the utilization of which will make it possible to reduce
the control power for phase shifters from several kilowatts down to a few watts.
The voltage provided by standard logic gates is altogether sufficient for the
switching of these elements.
2.10. Discrete Phase Shifters and the Suppression of Switching Lobes
In the case of digital phasing, the phase distribution which can be realized ir: an array can be represented in.the following form:
'Dreal  1~initial + vA (Dpean  otla4+ vA'
(2.37)
where Oinitial is the initial phase distribution corresponding to the caEe where
all of the array phase shifters are in the same position, taken as the starting
position; v is the number of sequential switchings of a phase shifter with the
minimal discrete step of a phase change; A is the minimal phase jump (discrete
step) which can be realized by a phase shifter.
On the other hand, the feasible phase distribution differs from the requisite
distribution because of the discrete nature of the phase shifter by the amount
of the socalled switching phase errors:
(Dreal  Oreq + 80 ODCnn ~Tpc6'~ S~� (2.38)
In the majority of cases, the phasing is accomplished so that the phase errors
are minimal. With such phasing, the maximum value of the phase errors does not
exceed A/2.
In accordance with the indicated phasing principle:
v`E [(4)T1)CG(1)H84)/010151, (2.39)
where E[X] is the integer part of X.
In the case of digital phasing, the directional pattern of an antenna array having
N x Q radiators is : N, q 1(mTpeG+mny "f'"nq)
j(n, (p) F(0, (p) I.A,,4 e ~ 40)
n, Qal (2.
where Ang is the excitation amplftude of the nqth radiator; 0nq is the spatial
phase shift.
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We shall make e of the Poisson suu~ing formula and the Fourier expansion for
the factor eJ ~~s, treating it as a function of u=Oreq Oinitial, which as it is
easy to convince oneself, is periodic. If the phased antenna array takes the
form of a system of radiators positioned at the nodes of a coordinate grid of
a system of coordinates X, Y, then the directional pattern is [1]:
_
_    }.c, X(2) Y(2) )
sinA!2 ~ (1)le j r Ae m p th (2.41) j(0, (F) .'1/2 F(Q, Q~) ,`,r Mh} i Jdxdp dx d~,
p'!'ha ) y~l)
where
Xci> X(:) = Ndx/2;Y() =y(2) =QdY/2;
4)pth  (DTpe6 (1)� + Mh (0TpBO `~H69) F 2n p (X  Xj)/dx
` 2nt (YYi)I dy;
A(X,Y) is a continuous function which satisfies the condition A(XnYq) = Anq, where
Xn and Yq are the coordinates of the nqth radiator; M= 2w/A.
The sum of the terms of series (2.41) having a subscript of h= 0 defines the
direc:.ional pattern of an equivalent array: an array without switching phase
_ errors. The sum of the terms in (2.41) having a subscript of p= t= 0 represents
the directional pattern of a switched antenna array with a continuous distribution
of the radiators. The terms in the series having subscripts of p= t= h= 0
apply to the directional pattern of a continuously excited antenna without switch
ing phase errors. The terms in the series with subscripts of t# 0, p# 0 and
h= 0 correspond to the diffraction maxima of the directional pattern of an array
without switching phase errors. The terms of the series with subscripts of h# 0
and p= t= 0 define the additional lobes which arise in the directional pattern
of the array because of the presence of switching errors. We shall call these
lobes switching lobes in the following. Terms in the series having subscripts
of h# 0, p# 0 or t# 0 define the supplemental lobes in the directional pattern
of the array due to both the discrete nature of the operation of the phase snifters
as well as the discrete nature of the layout of the radiators. We shall call
I the indicated lobes combination lobes in the following. Because of the presence
~ of switching phase errors, the directional gain of a switched antenna array is
_j reduced: D =Do( sinA/2 l�
~ l e/2 / (2.42)
where Do is the directional gain of an equivalent antenna array without switching
phase errors (2.11). .
One of the drawbacks to phased arrays with discrete phase shifters is the presence
of switching and combination lobes, which in the case of discrete phase change
steps of 0=w/2  n/4 can be of a rather high level. For this reason, one of
the problems of practical interest is the suppression of the indicated lobes.
The concept of switching and combination lobe suppression consists in the follow
ing. The configuration of these lobes, in accordance with (2.41), depende on
'Dinitial, Where Oinitial, as can be seen from the given formulas, does not
influence the directional pattern of an array without switching phase errors,
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defined by the sum of the terms of series (2.41) having a subscript of h= 0.
Therefoi:e, it is necessary to choose oinitial so that the switching and combina
tion lobes have a minimal level. This is achieved with the uniform "erosi_on" of
the indicated lobes in space, i.e., the optimal shape for lobe suppression is
rectangular. In this case, the extent of the suppressed lobes should be such
that they do not overlap one another, since when the lobes are superimposed in
space, their total level is increased. It can be demonstrated that with such
deformation of the combination lobes, their extent is proportional to the suhscript
h. Therefore, such a value of Oinitial cannot be ciiosen so that the indir_atF1
condition is met simultaneously for all h. As a resiilt, optimal suppreosion can
be provided.only for lobes with a definite va1Le aF the subscript h. Est:l.mates
 of the level of additional sidelobe radiation, due to switched phase errors, show
that with optimal suppression of the switching lobes with h = + 1, the level of
 the overall sidelobe radiation due to the switched phase errors will be minimal.
 If it is required that the absolute value of thE ixitegrals of (2.41) not depend
on the angular coordinates, then in the case af a linear array, one of tne equa
tions for the determination of Oinitial assumes the form [1]:
d' O,ley/dx2 = 2n A'l j tDh dX Mh, (2.43)
where fPh is the level of t:he uniformly washedout lobe.
' As has been demonstrated, maximum suppression of switching and conbination lobes
occurs in the case where the washedout lobes are not superimposed on each other
in space. By employing this condition, ane can derive a second equation for the
determination of the minimum value of f'h and the optimal function 4~initial [11:

Mh tt mlyna (X (2~) _ d (Dnay (X(,)) ?n  (2.44)
( dX dX , dx
Equation (2.44) in conjuncCion with (2.43) completely detines the optymal initial
phase distribution which assures the maximum suppression of the switching and
combination lobes, as well as the level of the suppressed lobes.
Solving the system of equations (2.43) and (2.44),. we derive the following for
a uniform amplitude distribution (h 1) :
=1/(M t 1) (2.45)
where
y = ~,l(2dxNM);
(L.46)
qh=+1 is the level of the suppressed switching and combination lobes.
The overall sidelobe radiation level which is due to switching phase errors is:
qE = 21MYN . . (2.47)
In the case of a cosine amplitude distribution:
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2n X9 NdX s 2rcX
0naa  MNdz L 2 ~ 2n ) COS Ndx J~ (2.48)
9z n ~ ,
Y2 MYN
! The results obtained for a linear array are easily extrapolated to two dimensional
planar arrays. For example, for a planar reciangular array with the radiators
arranged at the nodes of an orthogonal coordinate grid, with a uniform amplitude
distribution:
oney=K(Vx X2 {1'y ' (2:49)
where YX = a/2dXNM; Yy =X/24yQM; N and Q are the number of rows and columns in
the planar array respectively.
The level of the suppressed combination and switching lobes of the directional
pattern of a planar array is:
qE = 21M ;INQ . . (2.50)
~
~
Quite substantial suppression of the lobes due to the discrete change in the phase
can be obtained in arrays with a large number of radiating elements. This makes
it possible in some cases to emp}.oy coarser, and consequently, simpler and less,
expensive phase shifters with lower losses. The optimal initial phase distribu
~ tion can be produced either by means of phase shifters with a fixed phase value,
inserted at the output of the power divider, or by means of phase shifters for
an array using a particular change in the phasing algorithm.
~
2.11. Beam Jumps in a Switched Array
i The main lobe of the direc*_ional pattern of an equivalent array without switching
phase errors is oriented precisely in a specified direction, Amain� When switch
ing lobes are present in the immediate vicinity of the main lobe, the maximum of
their sum, i.e., the maximum of the array directional pattern, is slightly
shifted relative to the direction emain� This shifti, which is due to the switch
ing phase errors,'determines the error in setting the array beam in a specified
direction. The error depends on the level of the switched beams, and consequently,
on the discrete phase change step, A. Moreover, the position of the initial phase
readout has an impact on the precision in setting the beam. Thus, if one of the
end radiators of a linear array is chosen as the initial coordinate point, the
beam steering precision proves to be four times higher than the precision with
phase readout from the center of the array.
Beam steering precision is directly related to its jumplike motion, which is due
to the discrete change in the phase. The average value of ajump change, when
the readout origin is positioned in the center of the array, is:
SO = 2Ao,s A12N.
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It is also necessary to note that with the same beam :travel speed, the switching
frequency of the end phase shifters will be different, depending on the position
of the phase readout origin. This must be taken into account when assessing the
operational speed of a phase shifter.
2.12. Design Procedure
The directional gain or directional pattern width, the scan sector, the sidelobe
level and the beam steering precision are usually specified.
The specified sidelobe level and the requisite beam steering precision govern the
discrete phase change step, i.e., the number of phase shifter positions and the
amplitude distribution in an array.
The antenna dimensions are determined from the specified values of the directional
gain or directional pattern width, the selected amplitude distribution, as wel.l
as the scan sector yising the formulas of Table 2.1, as well as formulas (2.8) and
(2.9). The spacing tetween the radiators and the number of phase shifters is
found based on the specified scaa sector by means of formulas (2.3) (2.6).
It is expedient when determining the number of positions of the discretely switched
phase shifters with respect to the maximum level of the sidelobes to represent
the specified sidelobe level in the form of the sum of two terms, one of which
is taken as the maximum switching lobe level, while the other is taken as the
antenna sidelobe level without switching phase errors. Then one can determine
A from the value of the first term of formulas (2.47) and (2.50) and the nature
of the amplitude distribution in the array in accordance with the data of Table
2.1, based on the value of the second term.
The maximum level of the switching lobes is chosen so that the number of requisite
positions of the phase shifter, 2n/A, is the least. This makes it possible to
use phase shifters of the simplest structural designs. On the other hand, one
cannot choose a second term which is too srall, i.e., the level of the sidelobes
of an ideal antenna, since this necessitates the use of amplitude distributions
which fall off sharply towards the edges, something which leads to the necessity
of increasing array dimensions to assure the specified directional pattern width
or specified directional gain. A compromise solution is found, depending on the
specific requirements based on the antenna array in each case.
Then the scheme is chosen for the energy distribution and
the phase shifters, the type of phase shifters, radiators,
and these assemblies are designed; the directional pattern
then the structural design is worked out.
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the configuration of
coupling elements, etc.
is calculated and
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3. FREQUENCY SCANNING ANTENNAS*
3.1. Fundamental Relationships for a Frequency Scanning Linear Radiator Array
[07, 010, 1, 2]
Frequency control of an antenna beam is one of the techniques of electrical
control and is based on changing the electrical spacing between radiators excited
 by a traveling wave with a change in the generator frequency. With this beam
steering technique,a generator which is electrically tuned in a wide range of
frequencies is needed to scan space in a rather large sector.
In microwave antennas with frequency'beam control, the radiators are, as a rule,
positioned directly in the exciting system. Linear arrays of radiatora formed
 by slits cut in one of the walls of a rectangular waveguide are shown in Figure 3.1.
A twodimensional array of radiators is needed to obtain a controlled narrow direc
tional pattern. Such an array can be created from linear arravs, arranged in a
definite manner on a specified surface. Some of the possible variants of such
antennas ar�e shown in Figure 3.2.
O
Op
Op ~
&,7,uoBV,11,W ecv*.4ide1rua Coanocy~orvcA Naaoys~v (2)
(a) 01 (b) JI
Figure 3.1. Slitted waveguide
radiator arrays.
Key: 1. Direction of
excitation;
2. Matching load.
In antennas which take the form of linear radiator arrays, the excitation is most
often accomplished using series or parallel configurations (Figure 3.3). The
direction of radiation of A linear array with an equally spaced arrangement of the
radiators is determined by the equation:
sin 0 1)1,11d  pX/d, 3.1
where 9 is the beam deflection angle from the normal to the axis of the array of
radiators; y Tc/v is the,phase velocity retardation in.thp ehanneliz�ing system
exciting Ehe radiators; c= 3 108 m/sec; a'is the generator wavelength; p= n+
+ 0/2n, n = 0, +1, +2, ; . , is the beam number; O =is the'tixed,phase ~'shift between
adjacent radiators, due to,the insertion of the supplemental phase shifters
~
Questior_s of frequency scanning antenna design and theory were most campletely
treated for the f irst time by L.N. Deryugin [010].
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(Figure 3. 3c); Zd is the geometric difference in the lengths of the channelizing
systems of two adjacent radiators; d is the apacing between the radiators; t is
the period of the retarding interaction system.
Je
~ Qd a~
~o~ a i ~ ~o Q
i~ .
BI s
~ (
a~(a) ' (b)
Ao~poBr,eHUe
BoadyerdeHUA ~
(2)
 ~
(c) aJ(d)
Figure 3.2. Antennas formed by linear twodimensional arrays of
radiators.
Key: a. Planar; b, Arranged on a cylindrical surface;
c. Planar "fanshaped"; d. Arranged on a conical
surface.
1. Beam direction;
2. Direction of excitation.
   o
rA"evav~.fv
m~`~am
d d a5. ~d
~F~~11~  ~
~ ~
Inpudt
~xod
Figure 3.3. The excitation of a linear radiator array.
Key: a. Using a parallel configuration;
b. Using a series configuration;
c. With a periodic retarding inCeraction system.
When.the generator frequency changes, because of the dependence of Y and a/d on
the frequency f, the.radiation angle changes and the antenna beam moves in space.
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Figure 3.4. Tkie dispersion
characteristic y(X) of
a periodic interaction
system.
The angular frequency sensitivity of
the antenna is the tem for the rate
of change in the antenna beam positidn
in space with a change in the frequency
(the wavelength):
a9 _ 0,673 !d
a ~
A =  y~p Slil A).
~,ia~ ~ose  d
(3.2)
where YrP =[ygr] = c/vgr is the retard
ation of the group velocity of the wave
propagating in the channelizing system;
the coefficient of 0.573 is introduced
when converting the angular frequency
aenaitivity from dimensionleas units to
degrees for the percentage change in
frequency.
If follows from expression (3.2) that the angular frequency sensitivity depends
on the beam direction, the dispersion properties of the system and the ratio Zd/d.
The greater A and (Zd/d)Ygr, the greater the angular frequency sensitivity.
The retarding of the group and phase velocities are related by the expression:
 Ygr = yrp = y  ,%dy/dA.. 0.3)
If the dispersion characteristic of the channelizing system is known, Y= Y(X).
(Figure 3.4), then ygr is determined graphically by the segment on the ordinate .
axis, intercepted by the tangent to the curve Y(a), run through the.point corres
ponding to the value of Y in the system. The slowdown in the group velocity Ygr is also relatEd to the power P flowing
through the system and the per unit, length electromagnetic energy W accumulsted
in the system: Ygr = cW/P
where P = vgrW.
(3.4)
To improve the angular frequency sensitivity of an antenna, it is necessary to
employ chan;lelizing systems with a large value of ygr, something which in turn can
be achieved by increasing the ratio W/P.
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The ultimate value of the power flowing along the channelizing system is:
p = W V = CW IY = Pnpclt � wnpeJ[ vrp  CWaAoWYrpr (3 5)
ult. ult. gr ult. gr
where Wult. is the ultimate value of the per unit length electromagnetic energy
of the system, which is limited by the effective crosssection of the syatem and
the electrical strength. Expression (3.5) makes it possible to establish the relationship of the power
Pult. to the angular frequency sensitivity A, since both of these quantities
depend on ygr, and to draw the conclusion that with an increase in. A, the ulti
mate power always falls off. For a specified value of A, the increase in the
ultimate power for any system can be achieved only by increasing Wult� However,
it must be stipulated that in a number of cases, the ultimate which can be passed
is limited by the electrical strength of the radiators.
The thermal losses in the walls of the channelizing system are due to the attenu
ation of the wave propagating in it. The attenuation coefficient is:
a = P loss /2P = a = PDO,l2P, (3.6)

where Ploss i$ the power of the losses per unit length of tt:e system.
~ The attenuation in the channelizing syatem at the distance of a wavelength taking
j expression (3.5) into account, is defined as:
I
Yrpn/Q, (3.7)
where Q= wW/.P is the 'quality factor of the channelizing system (w = 27rf). For retardive periodic structure type channelizing systems with a period of t,
tHe Q does not exceed QmaX = t/d 0 is the depth of field penetration into the
metal). In actual structures, Q x 0.3QmaX, which makes it possible to estimate
the anticipated losses in a system.
It is also not difficult to draw the conclusion from expreasions (3.2) and (3.7)
that an increase in the angular frequency sensitivity is always accompanied by a
rise in the system losses. The presence of losses in a channelizing system places
a limitation on the length of a radiator array, since with an increase in the
, length, its efficiency falls off, which in turn limits the generation of narrow
directional patterns by an array of radiators.
~ The directional pattern width and the efficiency also depend on.the distribution
of the power radiated along the array. The exponential distribution has become
widespread in practice (each element of the array radiates an identical fraction
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of the traveling wave power fed to it), as well as a uniform distribution (each
element radiates the same power) and other special kinds of distributions (for
example, symmetrical relative to the array center and falling off towards it edges).
In the case of uniform distribution, the efficiency of a radiator array is governed
by an expression which is justified when aJ1 � 1(which is usually observed in
practice):
ex 2aL pL 2a1,
~n~[ P~ Po Iexp(2aL) ' (3.8)
where Pp is the power 3t the antenna input; PL is the power at the end of the
antenna; L is the antenna length.
The half power level directional pattern width for the case of radiation close to
the normal to the axis of the array is determined from the formula:
290.5 [degrees ] = 50 . 7a /L 290,0 frPa1t1 = 50,771/L. (3.9)
Takingexpressions (3.8) and (3.9) into account, we derive the relationship
between 290,5, a and nA:
11. 7 �i
IIn = L exp r11,7 lpo s ) po ] 20o.s ( 3.10)
` iexp (II,7 aA.
1
~ 200.6
/
In the case of an exponential distribution:
 1 + 2aL
=1 p~
~1n 1 Po ) ( ln (PL /Pa) (3.11) '
~
The directional patttern width dependa on the relative power getting through to
the end of the antenna. When PL/Pp = 0.05 (the aperture utilization coefficient
is 0.83 in this case):
26o,5 [degrees] = 54.4a/L
20a,6 (r'pa,ql = 54,471/L.
(3.12)
Taking expressions (3.11) and (3.12) into account, we obtain the following when
PI,!Pp = 0.05: ~ 
ri A  0,95 (1  4,17aX/20o,6). (3.13)
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(3.13)
When determining directional pattern width using formulas (3.9) and (3.12), th2
quantities a and L have identical units of ineasurement.
?A
0,5
.
~
94
q0005 0,001 q005 O,O! 0,05 O,>~ ,dB/deg
Figure 3.5. The antenna efficiency as a.function of the
ratio of attenuation times wavelength to the
directional pattern width.
Curves for nAW/260,5), plotted using formulas (3.10) and (3.13), are shown in
Figure 3.5. Curves 1 and 2 were obtained for radiator arrays with a uniform
distributivn where PL/Pp = 0.05 and PL/Pp = 0 respectively. Curve 3 was plotted
for an exponential distribution where PL/Pp = 0.05. As follows from the graph,
an array with an exponential distribution has a higher efficiency: from 0.9 to 0.4.
Moreover, such an array permits switching the direction of excitation, something
which makes it possible to increase the beam ste ering sector with the,same fre
quency change and efficiency.
The working sector of space scanned by the beam of a radiator array can fall only
within the bounds of the transmittance ;Actor of the periodic structure used as
the channelizing system (Figure 3.3c). All periodic structures used in frequency
controlled antennas are ba.Ldpass filters, having frequency transmittance bands, to
which the angular transmittance sectors correspond. The width and orientation of
these sectors depend on the type of periodic structure, the specific features of
the radiators and the number of cells in the interaction structure between tha
radiators. As follows from expression (3.1), the beam direction for a radiator array in
space depends on the supplemental fixed phase shift 0 in the exciter unit between
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the adjacent radiators. Phase shifts of the same angle when making the transition
to each subsequent radiator are accomplished by fixed phase shifters. For example,
in the form of line sections of equal length running to the radiators (see Figure
3.3a). An additional phase shift tr can be realized ?ather simply. For example,
when a rectangular Hlp mode waveguide is used as the channelizing system, a phase
shift of n can be obtained by using radiating slots, which are coupled in an alter
nating phase fashion to the waveguide field.
The shape of the main lobe of the directional pattern changes when the beam moves
in space. As it approaches the axis of the array, the main lobe widens and becomes
asymmetrical with respect to the direction 6. The change in the width of the main
lobe will be small when scanning in an angular sector close to a normal to the
axis of the array and increases sharply as it approaches the axis of the array. It
is theoretically possible, but difficult in practice to preserve a constant width
of the main lobe during wide angle scanning.
The half power level width of the main lobe, taking its asymnetry into account, for
an array of length L� a, with a uniform distribution of the radiated power, can
be estimated from the expression;
200,6 = aresin (0,443%/L sin A) aresin (0,443%/L  sin 9).
(3.14)
In the case of axial radiation, the width of the main lobe proves to be 2.147__X
times greater than the width of the main lobe in the casP of radiation along a
normal.
The change in the width of the main lobe during its travel can be explained by.the
change in the effective length*, Leff, of an array of radiators and the amplitude
distribution along it. In a first approximation, for angles of 6< 7075 dv;grees
(depending the length of the array L), Leff can be determined as the projection
of the array length L onto a direction perpendicular to the main lobe of the
diredtional pattern:
Leff  L cos6
(3.15)
When L/a > 10, this assumption is quite well justified. Thus, the error in the
determination of Leff using formula (3.14) when L/71 = 10 and A= 70 degrees amounts
to about 1.5 percent with respect to the value of Leff determined from a mor e
rigorous formula (see [07, p. 354]). In some cases, the scan sector can be limited
by the permissible widening of the main lobe.
*
The effective length is understood to be the length of a uniform inphase linear
array which yields a directional pattern at the halfpower level of the same width
as the array under consideration.
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An integral part of a frequency scanning antennz is a frequencyi:tunable generator.
The precision in determining the beam position in space depends on the stability
and precision in setting the generator frequency. There are centimeter and deci
meter band generators at the present time, which can be electrically tuned in a
rather wide range of frequencies (from +10 percent up to an octave). The frequency
tuning range of a generator depends to a considerable extent on its power and work
ing frequency. Correspondingly, there are also wideband amplifiers which can be
used in the receiving equipment.
Zn a number of cases, one can use exciters to excite an antenna which are designed
in a complex circuit c onfiguration and contain a comparatively lowipawer generator
with a wide frequency tuning range and broadband power amplifiera. When the re
quisite range of working*frequencies is wider than the passband of a single ampli
fier, several amplifiers are employed; in this case, each of them operate in a
band of working frequencies set~,aaide for it. Such an approach can be used where
it is necessary to change the beam.direction in space while preserving its scanning
sector.
However, when designing a frequency scanning antenna, one must remember that the
use of a wide band of frequencies requires the use of radiators, transition and
decoupling elements, etc., having a wide passband and possessing a low attenuation
in this band. Otherwise, considerable changes may be observed in the power radi�
ated by the antenna and the shape of the directional pattern when the frequency
changes.
,
J
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3.2. Channelizing Systems of Frequency Scanning Antennas [010]
In the structural designs of centimeter band frequency acanning antennas, the
, radiators, as a rule, are placed directly in the exciting channelizing systems
(for example, a linear array of slotted radiators, with the slits cut in one of
the walls of a rectangular waveguide), which can be designed around waveguides,
coaxial lines, etc. The electrical properties of these channelizing systems are
evaluated by the alowdown in the phase velocity Y, the dispersion characteristic
Y= Y(X) and the attenuation factor a.
The major requirements placed on channelizing systems are as follows:
1. The retardation of the phase velocity y should not be large, since with an
increase in Y, the losses in the channelizing system increase, and greater
accuracy is required in the manufacture of the system. The latter is related to
the fact that minor relative changes can lead to the disruption of normal antenna
oFeration in a number of cases.
2. The attenuation factor a should be as low as possible, since the antenna effici
enGy depends on its value, as well as the possible directional pattern width (for
a specified efficiency).
3. The channelizing system should allow for the arrangement of radiators at a
spacing of d=X/2 in an axial direction to avoid a multiple lobed directional
pattern when the main lobe is deflected towards the axis .of the array.
4. In a twodimensional array, the transverse diraensions of the channelizing system
should be such that the spacing betwcen the radiators of adjacent linear arrays
does not exceed . Otherwise, the directional pattern will have multiple lobes.
5. The channelizing system should have as small a size and weight as possible.
This is especially important for aircraft antennas.
Waveguide Channelizing Systems (Figure 1.6).
Hlp Mode Rectangular Waveguide. The retardation y falls in a range of from 0 to 1.
In practice, Y= 0.360.86. The angular frequency sensitivity of the waveguide is
low and fluctuates on the average from tenths to units of degrees per percent
change in f.requency. The attenuation factor in the 3 cm band amounts to about 0.5
dB/m, which with an eff iciency of rA = 90 percent makes it possible to obtain a
directional pattern width of about 1�. .
Rectangular Waveguide Partially Filled with a Dielectric. The retardation y can be
regulatcd by changing the thickness of the dielectric and its dielectric permitti
vity e. The slowdown usually falls in a range of 0.7 to 1.5. The attenuation
factor is several times greater ttian f or a regular wavegu ide (a is about 1:2 dB/m
in the 3 cm band), and depends on the loss angle of the dielectric and its thick
ness h. A drawback to the system is the requirement that the dielectric proper
ties of the d ielectric employed be homogeneous.
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Waveguide with a Finned Structure. The retardation is y> 1, and can in practice
be made close to unity and even considerably higher. The system has considerable
dispersion and high angular frequency sensitivity. The attenuation factor in the
3 cm band for small values of Y(Y = 12) is about 2 dB/m. The system has a high
weight as campared to a regular wavegufde and requires a high fabrication precision.
Serpentine Waveguide. The retardation is y> 1 and can be regulated in a consider
able'range by changing the length (L + ALequiv), and in this case, the angular
frequenqy sensitivity is also ad3usted in a wide rarige. The attenuation factor in
this system in the 3 cm band is less than in systems with the same angular sensi
tivity, for example, in waveguide with an internal f inned structure) and amounts to
about 0.7 dB/m when y= 2.5. The considerable weight, great length (L + ALequiv)
and fabrication complexity must be numbered among the drawbacks to the system.
Helical Waveguide. The retardation is Y> 1 and is regulated by changing its geo
metric dimensions. The dispersion of the system is low. The attenuation factor
in the 3 cm band is about 2.5 dB/m when y= 4. A rectangular waveguide H plane
bend is most frequently used, since this makes it possible to reduce the spacing
between radiators.
Coaxial Channelizing Systems (Figure 3.7).
These are of interest,in those cases where systems are needed having a poor dis
persion and relatively simple control of the retardation. However, considerable
attenuation is inherent in coaxial systems. Only a coaxial line partially filled
with.a dielectric (Figure 3.7b) represents an exception. A cQaxial line with a
f inned structure on the inner canductor (Figure 3.7c) is distinguished from the
remaining systems by the presence of sharply pronounced dispersion properties. The
geometric dimensions of coaxial systems when they are used in the centimer band are
small, which substantially limits the power they can carry.
When using periodic structures as channelizing systems, for example, a waveguide
with a finned structure, a coaxial line with a diskonrod structure on the 3nner
conductor, and serpentine and helical waveguides, one can obtain a high angular
frequency sensitivity for an antenna. However, the considerable losses in such
systems do not make it possible to design an antenna with a high eff iciency and a
narrow directional pattern. Moreover, these systems., as a rule, have considerable
weight and are complex to manufacture, which limits the possibilities for their
applications in a number of cases, especially in aircraft antennas.
A rectangular H],p mode waveguide channelizing system ha's a number of valuable
qualities: low losses, relatively small size and weight, and a well mastered produc
tion technology. For this reason, linear arrays of radiators excite3 by this kind
of channelizing system have become widespread in antenna engineering. The maximum
theoretical scan sector of a waveguide antenna with radiators coupled to the wave
guide field in an alternating phase fashion, without taking into account the fre
quency properties of the radiators and the elements used to couple to them, runs
from 90� to +14� with a change in the retardation from 0.22 to 0.867 and a ratio
of a/2a from 0.975 to 0.5. An average angular frequency sensitivity of 1.61� per
percent and a change in the wa.velength by a factor of 1.95 times correspond to
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the indicated scan secror. Switching the direction of the traveling wave in such
an antenna makes it possible to cover a scan sector of 180�.
. d
(a)
Figure 3.6. Waveguide channelizing system
for frequency scanning
(b) antennas.
a
Key: a. Rectangular waveguide
(c) with slots, coupled to
the Alp mode of the
waveguide in an alter
r t nating phase fashion;
b. Rectangular waveguide,
(d) al ~ p a r t i a l l y f i l l e d w i t h
~ a dielectric;
6 ~ c ~,~sL+dL,~s c. Rectangular waveguide
6. ^ e with a finned structure
placed in it;
(e) d. Serpentine rectangular
. waveguide; .
e. Helical rectangular
t waveguide.
~ .
qNf Figure 3.7. Coaxial channelizing systems
 ~ for frequency scanning
antennas.
Key: a. Filled with a dielectric;
b. With dielectric disks;
7 77 c. With a finned structure
t, a, Et on the inner conductor;
t? d. Coaxial line with the
I~
inner conductor made
in the form of a spiral.
We shall give the major relationships and
~ design procedure for a frequency scanning
slotted waveguide array, in which a regular
rectangular HlO mode waveguide is used as the
channelizing system. When other channelizing
systems are employed, the design procedure will
be somewhat different, since the expressions
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which characterize the relationship of the dispersion properties of systems to
their geometric dimensions, as a rule, are rather complex. Moreover, the retarda
tion in these systems is greater than unity, which, it goes without saying, is
reflected in the recommendations for the choice of the antenna radiation zone.
3.3. The Frequency Scanning Slotted Waveguide Array [010, 2]
A slotted waveguide array (VShchR) is shown in Figure 3.1.. A regular rectangular
Hlp mode wa.veguide is used as the channelizing system for such an antenna. The
array radiators are slots cut in one of the waveguide walls. This.antenna is
excited from one end by a generator, and a matching load is connected to the other
end to provide for antenna operation in a traveling wave mode.
We shall give the majar characteristics of a regular waveguide with a Hlp mode
 (see Figure 3.6a) as well as those which determine their relationship.
1. The phase Mlocity retardation is:
Y = Y 1 (~,/2a)a,
(3.16)
where a is the generator wavelength in cm; a is the crosssectional dimension of
the waveguide in the H plane 3n cm. The dispersion characteristic y= y(a/2a) is
shown in Figure 3.8, plotted using formula (3.16).
2. The grouii veloc ity delay:
Ygr = 1 /Y
This follows from the well known relationship for a waveguide: vgrv = c2 or
ygry = l.
= 3. The ultimate transmitted power is:
(3.17)
~
p11pen [W TJ _ �bESv,u 2 1 ' (3.18)
ultimate
 where b is the crosssectional dimension of the waveguide in the E plane in
_ cm; EnPeA [Eult] is the ultimately permissible electrical field intensity in the
wavPguide for the specified temperature, pressure and humidity, in KV/cm; a and
a are chosen in csntimeters.
4. The attea.uation factor is:
[[~BB~j=793I 1~2 Q ( 2 b~al(a~'~ ' (3.19)
L l
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Here d is the conductance of the matPrial of the waveguide walls in mhos/m; a, b
 and a are chosen in centimeters.
5. The angular frequency sensitivity is:
d9 !,573 0,573 1 (3.20)
A a~,/~, cose ~Y''p+s~ne), cose C Y"~s~n8l.
/
In accordance with formula (3.16), the retardation of the phase velocity aan vary
fram 0 to 1, and it would seem that the angular frequency sensitivity can be made
as great as desired. However, the range of change in Y which can be realized is
~ considerably narrower. This is explained by the fact that when X + Xcr = 2a(y + 0),
the losses increase sharply and the power Pult falls off. The lower limit of Y can
br found, if one assumes that the Zosses approximately double as compared to a
conventional waveguide. In this case, X = 1.9a or a/2a = 0.95 and Ymin � 0.36.
The upper limit of a is related to the requirement for H20 mode suppression, where
this mnde occurs when a= a or J1/2a = 0.5. Under these conditions, ymaX = 0.867.
Thus, the retardatian of the phase velocity Y is limited to values of 0.867 > y>
> 0.36, while the retardation of the group velocity is limited to 2.77 > ygr > 1.15.
The direction of radiation of a linear radiator array excited by a wave traveling
along it is determined in accordance with equation (3.1) when ld = d using the
f ormula :
sin 0 = y  nXld
(3.21)
for radiators coupled in i hase to the waveguide field (0 = 0) and using the formula:
sin 0 = y  (i I 0,5) X/d (3.22)
for radiators with alternate phase coupling to the waveguide field (~D _7r).
The beam scanning with a change in frequency will occur by virtue of the change in
Y and X.
The curves for. X/d as a function of Y(the solid lines) are shown for convenience
in ana.lyzing and solving equations (3.21) and (3.22) in Figures 3.9 and 3.10 for
various values 'of the parameter 2a/d, plotted from the relationship derived from
the expression (3.16):
X/d = V1ya 2a/d.
(3.23)
The grid of lines for a/d is a function of Y is also shown in Figure 3.9 for various
values of the beam inclination angle 8 for n= 0 (the dashed lines). The values of
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a/d were calculated for Y= 0.5 and values of the parameter 2a/d, corresponding to
angles 9 from 0 to 90� in steps of 5� and the slope of these lines was determined
_ assuming that A= const., since this function is represented by a straight line
[see (3.22)], to construct the grid 6f Iines.
Tn Figures 3.9 and 3.10, the radiatiou caverage zone for the corresponding numbers
. of beams are bounded by lines w'ch different values of n. In Figure 3.10, a
radiation zone to the left of the line n= 0, running vertically, corresponds to
 the beam with the number n= 0. The radiation regions for n= 0(Figure 3.9) and
n= 1, 2(Figures 3.9 and 3.10) fall below the sloped lines corresponding to each n.
The choice of the spacing between adjacent radiators d, which should be such ttiat
during beam scanning in a specified sector, the possib3lity of the appearance of
major sidelobes is precluded, is of considerable importance in an antenna design.
This condition will be met if the spacing d satisfies the relationship:
d a). Alternating phase excita
tion of adjacent radiators is used to reduce the spacing between the radiators in
slotted wavegu ide arrays. In this case, d= ap/2. However, when all of the radia
tors are spaced at a distance of d= ag/2 from each other (so that the maitl lobe is
directed along the normal to the axis of the array), the waves reflected from all
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v
3,5
40
1,5
2A
f,5
1,0
0,5
D
~/d .
3 
? ns'(
2
1 n�2
~ n c
41 0,2 0,3 Q36'0,4 45 0,6 47 9,8 0,8670,9 y
FOR OFFICIAL USE ONLY
cr r
d a, Q, Q, a ~ .
.~c .>.ow.~ Q .
y~
s
E
i 
i
'
i
n
,
~
4
~
>.95

1,75
i
~
S
~

i
/
n=0
.
.90
70
60
4
5
40
35
30 Figure 3.9.
The radiation
25
coverage zones
'20
and scan angles
'15
>0
in the case of
radiators with
p
alternate phase
7
coupling to the
v
waveguide field.
~5
Figure 3.10. The radiation coverage zones ans scan
angles in the case of radiators which
are coupled in phase to the waveguide
f ield.
v '
dmux~.t
o,B ~oo 
0,6 Nf0 '
0.4
FigurE 3.11. Curves for amaX/X as a function of the
scan angle A. [sic] of the radiators add in phase at the antenna input, samething which sharply degrades
its matching (the socalled "normal" eff ect is observed). In the case of beam
deflection from the normal,.values of d other than aB/2 and when the waves re
f lected from the radiators are mutually cancelling to a greater extent, kvswr 1.
J
~
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To determine the minimum spacing between radiators, d, which differs from Xg/2
and for which the matching will be good throughout the entire working band of
frequencies, one can employ the expression: d 1, equat ion (3.22) makes sense only when n= 0, i. e. , when the
antenna operates with a null beam (n = 0), in which case, the beam was scanned
primarily in the region of negative angles A when the frequency changes (see
Figure 3.3c). '
3.4. The Design Procedurefor a Frequency Scanning Linear.Slotted Waveguide Array
It is asstuned in the design procedure cited here that the retardation of the phase
velocity in an excited waveguide slot is equal to the retardation in a regular
waveguide in which there are no radiators. In an actual slotted waveguide array,
because of the internal and external mutual coupling of the radiators, the retarda
tion in the waveguide will differ samewhat from Y. In this case, the deviation in
_ the delay from y depends on the number of radiator s, the spacing between them, and
 on the degree of their coupling to the waveguide field, etc.
 Accounting for the influence of radiator mutual coupling on the operation of a
slotted waveguide array is complicated and requires long and labor intensive calcu
lations (see Chapter 6). Becau se of this, it is exped ient in an approximate eng in
 eering calculation to neglect the mutual coupling of the radiators,,assuming that
the retardation is conszant and equal to y. In a number of practical problems,
one can be li,mited to just such a design. However, when designing a pencil beam
(200, 5< 1�) slotted waveguide array with high preeision in the determination of
its parameters and characteristics, following the preliminary approxima.*.e design of
' the antenna, a repeat design calculation is to be performed, using a m4re precisely
specified valtie of the delay in the exciting waveguide slot, taking the mutual
coupling of the radiators into account.
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The design of a frequency scanning slotted waveguide array consists in determining
the parameters of the waveguide which exCite the slot radiators, as well as the
spac ing between the rad iator s, d, taking the beam scanning in the spec if ied angu
lar sector into account and the design of the radiators and their coupling to tihe
wave,guide to assure the requisite distribution of the radiated power along the
array and then the calculation of the array directional pattern.
A specific feature of the determinatfon of the waveguide parameters and the spacing
d is the fact that the wavegui,de parameters y and d for a specified scan sector
AA and workin&wavelength a are related together by a single equation (3.21) or
(3.22). For this reason, to f ind one of the desired quantities, it is necessary
to specify beforehand the remaining quantities 3ncorporated in this equation. For
example, in order to determine Y, the values of 8 and d must be specified. By
changing the values of A and d, one can obtain several variants of the possible
waveguide excitation system, and then choose that one of them which makes it pos
sible to best satisfy the main requirements of the technical specif ications (for
example, minimal frequency variation during scanning, low attenuati,on factor in
the waveguide, high angular frequency sensitivity of the array).
We shall introduce .the following symbols: P is the power radiated by the antenna
in KW; amin, acp [aavg] and amax are the minimum, average and maximum wavelengths
of the generator respectively, in cm;
e% _ 2 )max%mtn ,100%
*%cp Tma1+2111n
is the relative change in the generator wavelength, in percent; Amin, 9cp [Aavgl
and Amax are the direction of the main lobe of the directional pattern for amins,
Xavg and amaX respectively, in degrees; 280.5 is the width of the main lobe of
the directional pattern at the half power level when a= aavg, in degrees.
Equation (3.22) at the limits of the scan sector, which is bounded by the angles
emax and Amin, has the form:
SitT9:nez  Ymin0,5~'ma:/d; ~3 . 26)
sinOro1. =Ymez0'5Xmtn/d� ' (3.27)
Different variants of the.problem can be encountered in design work. We shall
cite a few of them.
Var iant 1: P, aavg 9 AX/Xavg, 200.5 and 8avg are spec if ied . Determine the possible
scan sector: ;AA .
Variant 2: P? aavg, Aa/aavg, 290.5 and AA are specified. Determine the possible
beam dir.ection 6ayg.
Variant 3: P, aavg, AX/aavg and 29p,5 are specified. Determine the beam direction
6avg for which the scan sector A6 will be the greatest.
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Var iant 4: P, aavg, 280.5, eavg and DA are specif ied. Assure the specif ied scan
sector with as small as possible a relative change in the wavelength AX/aavg.
In doing the design calculations for any variant, it is recomanended that one use
the graphs of Figures 3.93.11 and the materials given in �3.13.3.
We shall consider an example of a procedure for the approximate design calculations
in the case where P, aavg AX/aavg, eavg and 290,5 are specified and it is neces
sary to determine the possible scan sector ee.
1. We choose the type of radiators and the number of the working beam. Taking into
account the considerations presented in �3.3, we choose slots with alternating
phase coupling to the waveguide field as the radiators of the antenna array, and a
beam number of n= 0.
2. By using the curves of Figures 3.9 and 3.12, we roughly calculate the possible
beam directions 9avg. Working frosn the specified values of aavg and A71/J1avg, we
f ind the wavelengths amax and amin. We start the calculat i,on with the choice of
the value of Yavg corresponding to Xavg. Consider3ng the fact that the angular
frequency sensitivity A(3.20) is larger for smaller values of y, it is desirable
to choose Yavg less than 0.5, however, it must be remembered in this case that with
a change in the frequency ymin cari prove to be less than 0.36 and the losses will
rise in the waveguide. For this reason, it is not expedient to chqose Ymin close
to 0.36. Using the graphs of Figure 3.12, we approximate yavg for ad > 1 to
obtain the requisite beam direction Aavg . Based on the curves of Figure 3.9, we
find the value of 2a/d for the known values of Yavg and 6avg; The value of 2a/d is
a structural design parameter for the antenna being planned, and consequently, will
stay constant during frequency scanning. We then determine ymax ana Ymin, Prelim
inarily determining the waveguide dimension a correspond ing to Yavg� To deter
g� mine a and the slowdown factors ymaX and
Ymin, one can use formula (3.16) or the graphs
C1771101 H I of Y= Y(a/2a) shown in Figure 3.8.
0,34J644 46 0,6 0,7 484867 y
Figur.e 3.12. The scan angle 8
as a function of Y
f or f ixed values
of a/d when the
ante.nna operates
with a ntYll beam.
To determine the angles 6maX and Amin, We find
the intersection points in the graphs of
Figure 3.9 of the ver.tical lines corresponding
to Ymin and Ymax with the line:;for a/d =X/d(Y)
when 2a/d = const. (the value of the parameter
2a/d has already been found). If the inter
section point lies above the line n= 0, then
such an operating mode is not feasible and the
calculat ion is to be repeated, speaifying
another value of Yavg� It is usually desir=
able to obtain the greatest scan sector A6
for the specified relative change in the wave
Iength da/aavg. Therefore, one may specify
two to three values of Y$vg in the calcula
tions and find the maximum possible sector.
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Considering the approximate nature of the performed calculations, related to the
error in the determination of the design values from the graphs, we shall further
 specify these quantities precisely (paragraphs 36).
3. We spec if y the spac ing between the rad iator s more pr ec isely f or the specif ied
value of Yavg based on equation (3.22) :
p).
d 0r5a'cp/(Vcp'"'"SI110,
Here, one must check to see that the condition d< dmax is met when a= amin
[see (3.24)] to aVoid the appearance of major sidelobes. 4. We determine the size of the wide wall of the wa*eguide more precisely from
the f ormula (3 .16) :
a _ %~p/2Y1y~t,.
5. We determine:
1~min = ~1(~'maz/2a)~~ Ymaz = ~'~1~~'m~n~~~=�
6. From equat ions (3 . 26) and (3.27), we f ind :
Omg: = aresin (1'rot110,5Xme:/d)
Omia = aresin (Ym$x0,5%m1p/d).
7. We determine the possible scanning sector:
~6 = Aniax ' Amin�
8. We find the angular frequency sensitivity at the average wavelength:
A 0,573 1
p( T'Ycp +sin 9cp1.
/
9. Using formula (3.22), we calculate the function e= e(a) in the working band
and plot the graph.
10. We select the waveguLde dimension.; b, being governed by considerations of
electrical strength, the essence of higher modes and the possibility of cutting
slots of widths lslot = aavg/2� .
11. We determine the ultimate transmission power Pu1t from formula (3.18).
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12. We choose the material for the waveguide walls and f ind the attenuation
factor a fram formula (3.19).
We select the distribution for the radiated power along the array of radiators,
working fram the requirements placed on the directional pattern and the gain of
the slotted waveguide array. We determine the length of the antenna array LA, its
efficiency nA, and the number of radiators in the array N. In the case where the
simplest distributions are selected for the radiated power (uniform or exponential),
the quantities LA, nA and N can be determined as indicated in paragraphs 1315.
13.. We select a uniform or exponential distribution for the radiated power along
the array, and working from the specified width of the main lobe, 290.5, we find
the approximate length of the antenna array from f ormula (3.15) : .
LA Leff./cosAavg
Lefg is determined form formula (3.9) or (3.12) assuming that L Leff When
a = Xavg. .
We shall determine Lp, more precisely, checking to see that the condition
2e0'.5  290.5 is met, where 290.5 is the width of the main lobe determined, from
formula (3.14). 14. We determine the efficiency of the slotted waveguide array using formula (3.8)
or (3.11) at the boundaries of the working frequency band.
15. We f ind the number of radiators in the antenna array:
N = LA/d + 1
16* We choose the dimensions of the slotted radiators and their arrangement in the
waveguide wa?1, taking into account the selected distribution for.the radia.ted .
power along the array of radiators.
17. We calculate the directional pattern when X = amin, aavg and amax. We deter
mine the coformity of the width of the main directional pattern lobe to the re
quisite width and the change in it during scanning.
18. We f ind the directional gain of the antenna array.
19. We draw the electrical schematic of the slotted waveguide array.
*
Points 16 through 18 are performed using the procedure set forth in Chapter 5.
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20. We design the feeder channel coupling the transmitter to the slotted waveguide
aray. '
. The structural design of the slotted waveguide array is accamplished taking ite
application into account.
 The procedure is basically retained when doing the design ealculations for variants
24; only paragraph 2 changes.
For variant 2, the rough calculation (paragraph 2) to determine the beam direction
Aavg, for which the requisite scan sector AA can be obtained, is carr ied out by
means of the graphs of Figure 3.9. Since the angular frequency sensitivity is
greater at small values of Y, then by specifying Ymin close to 0.36, we determine
Ymax by the method indicated in paragraph 2. Drawing two vertical lines correspond
ing to the values of Ymin and Ymax and a horizontal line for a/d = 1, we obtain a
region in the graph for the choice of 9avg in which the requisite scanning sector
can be obtained. The calculation reduces to the determinatj,on the spacing between
the radiators, d, which assures the requisite ee for the selected values of Ymax
and Ym in. By using the curves a/d(y) when 2a/a = const., we f ind a curve on the
graph in the resulting regions, which when we move along the curve from Ym8 to
ymin, we obtain the requisite value of AA. Then, having determined Yavg
= 1(X 2a) , we f ind Aavg �
For variant 3, the approximate calculations are performed in a manner similar to
the calculations for variant 2, with the difference that 9avg is determined for
which A9 will be a max imum.
For variant 4, the rough calculation reduces to obEaining the specified scan sector
A9 with a small a change as possible in the wavelength, i.e., it is desirable that
Aa/Xavg be small. For this purpose, we find the region of slowdown factors from
the graph of Figure 3:12 for which one can obtain tlie specified directtLon 6avg. We
select two to three values of Yavg corresponding to.8avg. Based on the specif ied
values of A6 and 9avg, we f ind the limits of the scan sector 9maX and 9min. For
each of the selected values of Yavg, We perf orm the following calculations. Based
on Yavg and Xavg, we f ind the waveguide dimension a and determine the parameter
2a/a. Then using the graphs of Figure 3.9, to determine the values of ymin and
Ymax corresponding to the intersectidn points of the straight lines 6~ 9max = const.
and 9= Amin = const., with the curve (X/d)(Y) for the found value of 2a/d. The
wavelengths Xmax and amin are determined fram the f ormulas: .
~'rnax = 2a j/ 1Xmin = 2aV1'pmear
while the range of change in the wavelengths 3s found using the.formula Aa = amax 
 amin. By repeating the same calculation f or other values of Yavg also, we will
f ind new values of Aa. As a result of the calculations, we determine the value of
Yavg corresponding to the least change in AX, which provides for the requisite
sector ee.
'
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In doir.g the design calculations for variant 4, it can turn out that a consi,derable
scan sector ee is required (for example, AA > 30�) . In this case, to reduce the
requisite value of Aa/aavg during scanning, a system of parallel waveguides can be
used which have different spacings between the radiators. Each waveguide, with the
same change in M/aavg will provide for scanning in the corresponding sector, while
the sum of these sectors should be pqual to the total sector. The structural
des3gn of such an antenna will be more camplex; it should consist of several wave
guides, switched when making the transition from one scan sector to another. The
design procedure for such an antenna is somewhat different than for a s3ngle slotted waveguide array, however, one can employ the procedure already considered
in the design calculations �or each waveguide. Breaking the total scan sector down
into component parts and determining the number of requisite waveguides can be
accomplished by using the graphs of Figure 3.9, as well as the book [010].
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4. HIGHLY DIRECTIONAL CYLINDRICAL AND ARC ANTENNA ARRAYS
4.1. General Information
Cylindrical antenna arrays take the form of a system of radiators arranged on a
cylindrical surface. A special case of cylindrical arrays is arc and ring antenna
arrays, the radiators in which are arranged along an arc or circle of a particular
radius.
Wire and slotted dipoles, open waveguide ends and horns, helical and dielectric
rod antennas as well as director radiators can be used as the radiators in cylin
drical arrays. The choice of the type of radiator depends on the working wave
length and the requisite passband, on the operational conditions and function,
as well as on the structural design requirements placed on the array as a whole.
In the centimeter band, the most convenient type of radiator for cylindrical
arrays is socalled diffraction type radiators, which take the form of openings
cut directly in the metal surface of a cylinder: a halfwave slot, an open wave
guide end or a small horn.'
A merit of such radiators is also the fact that they almost do not disrupt the
aerodynamic properties of the cylindrical surface, something which is especially
important when they are placed in aircraft. One of the important properties of
pencil beam cylindrical arrays is the capability of electrical cantrol of the beam
position in a wide sector of space without changing its width and shape. For
example, ring ~.!atenna arrays make it possible to have undistorted electrical
beam'scanning in the azimuthal plane. Cylindrical antenna arrays, as compared
to linear ones, possess yet a series of useful properties. Numbered among them
are a lower level of sidelobes (which are due to the discrete nature of the
radiator arrangement and switching phase errors in the case of switched beam
scanning), the possibility of expanding the working bandwidth, etc.
However, cylindrical antenna arrays also have a number of drawbacks as compared
to linear and planar arrays, the chief of which is the increased complexity of
the structural design of the antenna and its beam control system.
The Major'Requirements Placed on Cylindrical Scanning Arrays. The main parameters
of cylindrical antenna arrays are determined by working from their function,
installation site and operating conditions. For pencil beam cylindrical scanning
arrays, the main parameters specified during the design work are: the directional
pattern width, level of the sidelobes, directional gain, scan sector and beam
scanning rate, bandwidth properties, polarization of the radiated field, maximum
radiated power, efficiency, reliability, climatic operating conditions and cost.
The optimal configuration for the cylindrical array and type of radiator should
be selected during the planning process, the array dimensions should be determined
(radius, length, angular sector) as well as the amplitudephase distribution of
the current in the radiators and the law governing the current change during
scanning should be found, the directional pattern of the array calculated along
with its directional gain, overall gain, bandwidth properties; the method of
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scanning is determined and a device is chosen to realize the scanning, and the
structural design of the antenna array is worked out as a whole.
Strur_tural Configurations of Cylindrical Arrays. Cylindrical antenna arrays can
be broken down into three groups according to the method of microwave energy
distribution among the individual radiators: arrays with series and parallel
excitation, and arrays with a mixed feed circuit. Moreover, the array configura
tions in each of these groups can differ according to the method of energizing
the phase shifters.
We shall treat the main features of the indicated eircuit configurations using
the examgle of ring and arc arrays [04, 09, 013, 1, 2]. A ring array with a
parallel circuit for energy distribution between the radiators is shown in Figure
4.1a. A merit of this circuit is the fact that the antenna beam direction is
only a slight function of the frequency and there is the possibility of control
ling the amplitude distribution in the array by means of switching the inputs of
the feeder lines in a switcher (S). A drawback to the parallel excitation
configuration is the cumbersome feed system for energy distribution:
The variant of a ring array with spatial excitation (Figure 4.1b) is free of this
deficiency. The operational principle of such an array consists in the following.
The energy from the feed radiator is fed via a radial line to the receiving
radiators, and then to the phase shifters and is radiated by the ring array in
the requisite direction. The control of the antenna beam is accomplished within
small sectors by means of. phase shifters. In the case of wide angle scanning,
it is necessary to change the amplitude distribution over the ring array, for
example, by means of rotating the feed radiator or installing several feed
radiators and switching them in turn.
One of the circuits for a series excited ring array is shown in Figure 4.1c. A
merit of the circuit is the simplicity, as well as the fact that the volume
inside the array remains free and can be occupied by other devices, something
which is especially important when placing a ring array on the surface of an
aircraft. However, arc arrays, designed in a series excitation configuration,
also have a number of drawbacks, the main ones of which are the fact that the
array beam direction is a function of frequency and it is difficult to control
the amplitude distribution in the case of wide angle scanning.
A ring array formed from several arc arrays with a mixed circuit configuration for
power distribution among the radiators is free of this latter drawback (Figure
4.1d). The use of mixed excitation makes it possible, on one hand, to preserve
the advantages of parallel excited ring arrays, and on the other, to simplify the
energy distribution system, especially for arrays with a large electrical radius.
The most promising structural configurations for cylindrical arrays are the mixed
(Figure 4.2) and those with spatial excitation. The major properties of these
cylindrical array configurations are the same as for the corresponding ring arrays.
When selecting the circuit configuration for the phase shifters, it is expedient
to be governed by the following considerations. In the case of a series confi
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z 1 o 1
~V
aw
dl
, O~nyvm~ne~e
f 07
4J �1
Figure 4.1. Structural configurations of ring antenna arrays.
Key: 1. From the generator.
00
Figure 4.2. The structural configuration of a cylindrical antenna
array.
guration of the phase shifters, the maximum carrying capacity and the efficiency
are reduced, the dependence of the antenna directional pattern on the phaGe
setting errors is increased and the bandwidth properties of the antenna are
degraded. For this reason, the series configuratian of the phase shifters is
used rather rarely, primarily in linear antenna arrays, where such a circuit makes
it possible to simplify the controller for the phase shifters. In cylindrical
arrays, a series phase shifter circuit can be used in those portions of the feed
line which are arranged along the generatrix of the cylindrical surface, since
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this makes it possible to simplify the beam control unit in planes running
through the axis of the cylindrical array. In the remaining cases, parallel
phase shifter configurations are to be employed.
4.2. The Phase Distribution in Highly Directional Cylindrical Arrays
The amplitude distribution in cylindrical pencil beam arrays exerts a substantial
influence on the shape of the directional pattern and is chosen depending on
the requirements placed on the directional gain, the level of sidelobes and the
bandwidth properties of the array. This question is treated in more detail in
the following sections.
The phase distribution in
cylindrical arrays is chosen by working from the
_ requirement for beam formation in a specif ied direction. In this case, the phase
distribution in the radiators placed on the surface of a cylinder is usually
chosen so that the fields
radiated by each radiator add together in phase in the
direction 60, 1DO for highly directional arrays when generating a beam in the
direction 80, tD0�
We shall number the radiators of a cylin
z
drical array with a double subscript, mn.
In this case, the phase center* of the
A
00th radiator has cylindrical coordinates
of z= 0 and a= 0, while for the mnth
radiator, it has coordinates of z= zm
}(m40)
and a= an (Figure 4.3). The requisite
phase distribution in this case Omn(60,
 ,a
~0) of the current in the mnth radiator
y
(1
+OJ ~ ~
of the cylindrical array has the form:
_ .
l
,
(Dmn (0o, (po) _ [rcu sin 90 cos ((poan) f xzm cos 00 2T[k] (4.1)
k=0, f 1, :t: 2,...
X . .
.
Figure 4.3. The coordinate system and
scheme for the arrangement
of radiators in a cylindri
cal array.
In the special case of an arc array,
arranged in the z= 0 plane, the requi
site phase distribution is:
(Doo (0o, To) _
_.[rui sin Ao cos (To~) 2nk], (4.2)
k= 0, t 1, f 2....
The requisite law governing the phase control of the mnth phase Ghifter, 00a3 mn
Ppphase mnlt depends on the circuit configuration of the antenna feeder channel
and on the circuit configuration of the phase shifters themselves, and for a
cylindrical array, can be found from the relationship:
(Naa mn (DO m0) _[KQ 5111 Op COS i(p0 CCn)'+' lCZm COS ep
rca sin Oo cos ((poa�)xzm. cos Oo(Dfiy,m mn `I' (DonA m' n, I 21tk], (4.3)
k= Ot f 19 t 2,...t .
*The proposal of the presence of a phase center for the mnth radiator is justified
for radi.ators arranged on a cylindrical surface of considerable radius: a� X.
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where O$HA mn [Ofeed mn] designates the electrical length of the feedline from
the generator to the input terminals of the mnth radiator (without taking into
account the electrical length of the phase shifter 4~mn inserted in the feed channel
for the mnth radiator), while the subscripts m'n' designatea the phase shifter,
the phase of which is taken as zero.
Correspondingly, for an arc array:
0(bee on (0o, (Po) _  {xa sin Oo [cos (q)oan)cos (q)oan.J~.
. 'T" I~2nk 1 + k=0, f 1t f 29.... (4.4)
Q)~xll on ~~Blt On
We shall cite the expressions for ~Dphase mn for several specific configurations
of cylindrical and arc arrays. 1. A parallel excited arc array (Figure 4.1a). The electrical length of all of
 the feeder lines is the same, and the phase of the phase shifter of the radiator
with the coordinate a p= 0 is taken as the zero phase (n' = 0):
� 04as on (eo (Po) _ xa sin 90 [cos (tpoa ,)cos 4pol I ~_nk,
(4.5)
k= Ot f 19 t 29... 1
 Expression (4.5) is also justified for an arc array with spatial excitation (rigure
4.1b), if the phase center of the feed irradiator is placed in the geometric
center of the arc array and n' = 0. 2. A series excited arc array. The phase shifters are inserted in a parallel cir
cuit configuration (Figure 4.1c). The generator output.is connected to the 1`Tth
radiator, n' _ N: Od,88 0n (Do, (Po) = rca sin eo [cos (cpoa�)cos (To a_nr)1I
xay (a� a_N) 2nk, k=� 0p f 1p f 2l ...9 (4.6)
where Y is the retardation in the supply feedline. .
3. A
cuit
N:
series axcited arc array. The phase shifters are connected in a series cir
configuration. The generator output is connected.to the Nth radiator, n' _
04'es o. (eo, mo) � Kll Sitl Ap (COS ((p0OGn)cos (Toa_rv)1I
at
f Kay (a�  a_N)  0d,aa on 2nk, k�..!: 0. t 1, f 2,...
P~ N .
4. A cylindrical array with mixed excitation (Figure 4.2)_, n' = 0, m'
0Q)es mn \eo' ~0~ _ K(I Slil ep COS ((P0CGn~ _
KZm COS 0p + JUl SIII Ap COS (p0 K'pZm 2rck, h= 0, 1, t 2,...,
where y is the retardation of the wave in the feeders, arranged along
trix of the cylinder.
(4.7)
= 0:
(4.8)
the genera
The value of the.integer k in the expressions cited here depends on the type of
phase shifter. Thus, if the phase shifter can change the phase continuously in
a large range of phase values*, then k= 0 i.n expressions (4.3) (4.8). However,
socalled resetting phase shifters are used in electrically scanned antenna arrays,
. *The requisite range of continuous control of the phase of the wave in the phase
shifters depends on the size of the cylindrical array and the scan sector, and
 for pencil beam arrays can reach several tens and even hundreds of thousands of
degree:s.
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where the phase control range in such shifters is kept within a range of 0 to 27
radians. The advantage of such phase shifters consists in the smaller dimensions
and losses, as well as in the greater precision in setting the phase as compared
to phase shifters with largP phase control ranges. When using resetting phase
shifters, the value of k in the expressions cited here should be selected so thgt
the following inequality is observed:
0< myas Mn < 2t[. (4.9)
The choice of the number of controlled phase shifters depends on the requisite
scan sector, the directional pattern width and the amplitude distribution in
the array. The minimum possible number of controlled phase shifters tn the case
of wide angle scanning is chosen equal to the number of radiators.
4.3. The Directional Patterns of Cylindrical Pencil Beam Arrays
The normalized complex vector directional pattern, P(6, 0, of a cylindrical
 array when generating a beam in the direction 6o, ~0, can be written in the
_ form:   M. N, (m)
= F (e,(p)= A (1mn I Fmn (0+ (p) Gmn (  I rmn D9 x
m�_h(+n~IVj . (4.10)
x exP l(rca sin Ao cos (cpo (z�) Kz,� cos Oo 
xa sin A Cos ((p a�)  xtm Cos 01},
where IImni is the amplitude of the incident current (or voltage) waves in the
feeder of the mnth radiator; F' (A, _~mn(6, ~)Fmn(6, Fmn(6, emn
(6, 0) are the normalized amplitude and polarization patterns respectively of
the mnth radiator; G., is its gain; I'mn is the reflection factor from the input
of the mnth radiator; M1, M2, Nl(m) and NZ(m) are the numbers of the end
radiators of the cylindrical array; .
M, No (m)
~ ~ 1mn I Fmn (eot ~o) Gmn ~1 lTmn l~Z (4.11)
7 r m ~hl, li= N' (m)
is the normalizing factor.
In the following, we shall assume that the quantities Gmn and I'mn do not depend
 on the number of a radiator, i.e. Gmn = Gpp, Irmnl = Ir00l'
Expression (4.10) can be represented as the directYOnal pattern of an equivalent
linear radiator: "
F(0, = A I1mo I Fm (0+ (P) cxP jKZm (cos 80 cos 6)I,
m~_Ml � (4.12)
' V cl oo (i I roo 1,) N~ (m)
. rue r m(0, I/mo I I 1mn ( fi(e1 x
nmN, (m)
; X exp (jrca(sin8ocos (moa�)sinecos((pa�)]) (4.13)
is the complex vector directic,nal pattern of the mth arc array.
In a rather typical, although special case of a cylindrical array, formed by a
set of identical arc arrays, and where the current amplitude distribution divided
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along the coordinates a and z is 1ImnI= 1Im1I IOn 1l the directional pattern of the
cylindrical~array is determined by the product of the directional pattern of the
arc array, Fo (9, lying in the plane z= 0, times the factor for the linear
 system of radiators, fM(A): .
F(o, ~)Fo(e. ~)tM(e), (4.14)
M.
I 1mo I exp jeczm (cos 6acos 6)] (4.15)
rRe fM.(e) ~ 111 s~ Mt M
. ~
Lj 1 Imo~
rn e MI
Fo (0, (p) �
N,
I ionl Fo, (a, ip) exp {iKa [sin Ao cos (q+oa�)sin 6 cos (Ta�)J}
R�NI (4.16)
N
fon I Fon (eo, To) .
n N, �
Thus, the study of the directional pattern of a cylindrical antenna reduces
basically to the study of the directional pattern of the corresponding arc array.
Moreaver, the directionalpattern of the arc array is poorly directional in the
plane passing through the direction of the beam and the z axis. For this reason,
 when generating a pencil beam, the shape of the directional'pattern of a cylin
drical array in the indicated plane in the region of the msin lobe and the first
sidelobes is governed primarily by the factoz for the linear system of radiators,
' fM(9). However, the directional pattern completely matches the directional
 pattern of the arc array (4.16) in the orthogonal plane.
4.4. Directional Patterns of Arc and Cylindrical Arrays
When calculating the directional pattern of an arc array using expression (4.16),
~ it is first of all necessary to determine the directional pattern of an individual
_ radiator in the array, which is a rather complex and independent problem. The
complexity of the problem consists in the necessity of taking into accouni both
diffraction phenomena at the surface of the antenna and effects of radiator
interaction in the arc and cylindrical arrays. The techniques for solving this
AM problem can be partially found in the literature [1]. However,.in the initial
design stage, it is expedient to determine the directional pat:,.rn of an indivi
I dual radiator by means of simpler approximation�methods, iiithaut taking inter
_ action into account and with an approximate accounting for di,ffraction phenomena.
The essence of the approximation consists in the fact that the amplitude direct
ional pattern of a radiator, FOn(6, 0), in an array in a range of angles of an 
n/2 X/2,
Dmazd1 when
2,86rca sin p npx d L, dz> 7. .
dl d� . 2 . 2 .
Dma:= 4n when � ~ I (4.34)
, S.H. npH di < 2I da < 29
where Sequiv is the equivalent aperture area; dl and d2 are the spacings between
adjacent slots in the plane passing through the axis of the cylinder and in the
plane perpendicular to the axis of the cylinder respectively.
For the mth arc array, the maximum directional gain can be approximated using
the expression:
N~ 1
Droax" ~ Dmn Finn(2
ToJ. (4.35)
rt~Nt where Dnm is the directional gain of the mnth radiator irL the direction of the
radiation maximum axis.
The summing in cxpression (4.35) can be approximately replaced by integration.
In this case, fcir a system of identical radiators with Doo, we have:
aN. rmn (n12, To)
Dm ron: Doo J dl
a dan (4.36)
= Doo 2 d/k f r'nn (n'/2, ipo) dan.
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Expressions are given for Dm maX in Table 4.2 for a few directional patterns,
given the condition that a _N1  a NZ 2 0'
TABLE 4.2_
'n cos (~an) A }cos (~pan)
F
~ " C 2 ' (p) A +1
� � sin ~ sin2~
Dmma: D�� d~I +si2p 2~1 Doo d I~2As}4A A 20
J +s),
1
The maximum directional gain of a cylindrical antenna formed from radiators of
any type can be determined by summing the maximum directional gains of the
corresponding arc arrays from which the cylindrical antenna is formed.
When the amplitude distribution differs from the optimal the array directional
gain is reduced by a factor of v times (v is the surface utilization factor).
~ For a cylindrical array with a shared current distributior. of 1Imn1 _ IIml II0n1'
v _ V1V2 (4.37)
where vl is the antenna surface utilization factor for the z coordinate; v2 is
the arc array surface utilization factor.
For an arc array with a spacing between the radiators of about a/2, the coeffi
cient v2 can be determined from Table 4.1 for the appropriate distribution in
the equivalent linear radiator. If the spacing between the radiators is approximately a or more, then the coeffi
cient v2 must be determined from the more complex formula:
v21/0 1 Np), (4.38)
/y, L
 N' jOn opt v IOp lOp opt N,
_ p, 1 N, j,
Y~P I In  N, ~On
~ n/yj ` 1OP opt no_N,
I P N, .
N, (4.39)
, n
!on For ( 2 ~ ~o
t P ~Ni � n
1 on _ N, Fon ~ 2 ~~o) ~
n 2 ~
� _ ~ I por ( 2 ~ ~'o) I '
Pa ~N~
IImI IIOn opti is the current amplitude in the mnth radiator, having the maximum
directional gain; 1Ym1 IIDnj is the actual current amplitude in the mnth raiiia
tor.
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The coefficient vl depends on the form of the amplitude distribution II.1 with
respect to the coordinate z and can be found as in the case of linear antennas.
Figure 4.4. The surface utilization
coefficient v2 as a func
tion of the direction of
the beam of a ring array.
We will note that in the case of electrical
scannino in azimuth, it is necessary in the
general case to control not just the phase,
but also the amplitude distribution. The
most effective method of amplitude distri
bution control is that of "shifting" it in
azimuth through the scan angle wi*_hout
changing the shape. This technique is
feasible in cylindrical and arc arrays
with spatial excitation with eleetrically
or mechanically driven motion of the array
feed irradiator directional pattern.
However, it is frequently undesirable to control the amplitude distribution in
cylindrical arrays during scanning for a number of reasons, in garticular,
becauae of the increased complexity of the circuitry and structursl design of
the antenna. With beam scanning of a cqlindrlcal array in the azimuthal plane
solely through the control of the phase distribution, the directional gain of
the array changes. When the spacing between the radiators is about a/2, tlie
reduction in the directional gain during scanning can be determined from
expression (4.38), where:
, f I~eKn ~x~ ~ f 1aHe (x) dXl1aKe 12 dx
Q~P'^ iks 1 aK. (x) dx g (4 . 40)
AItH .
Iequiv(x) is the amplitude distribution in the equivalent radiator, perpendicular
to the direction of the beam. This expYaosion is to be used if the actual ampli
tude distribution in the equivalent radiator cannot successfully approximated by
one of the functions in Table 4.1. Otherwise, it is simpler to choose the
coefficient v2 from Table 4.1. �
If the spacing between the radiators is approximately equal to a, then the
change in the directional gain during scanning is to be computed from formulas
(4.38) and (4.39), taking into account the fact that:
Jon opt = Fojn/2, fpo), 1on =70*,jn12, 0). .
In two cases (radiators which are omnidirectional in azimuth and placed at a
spacing of approximately a or more, and.radiators with a directional pattern of
Fmn(7/2, = c0s4  an), positioned at a spacing of about a/2), the change
in the directional gain during scanning and with a constant amplitude distribution
is determined simply and shown graphically in Figure 4.4. .
In both cases, the optimal amplitude distribution is uniform. So that the direc
tional gain does not change during scanning, the beam direction 00 should not
exceed an angle of n/2  S. .
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The efficiency, n, of cylindrical (ring) antennas depends on the losses in the
feeder channel and phase shifters, as well as the antenna circuit configuration.
For this reason, the gain of cylindrical arrays should be computed in each
epecific case following the choice of the antenna circuit, the type of feedline
and phase shifter. During the preliminary calculation of the gain o� cylindrical
scanning antennas, the efficiency of these antennas can be taken as 50 to 60%.
4.6. Bandwidth Properties of Arc Ari�ays
 Arc antennas make it possible under certain conditions to obtain a poor depen
dence of the major directivity characteristics of the antennas on frequency
~ (beam direction and width, sidelobe level, directional gain) in a wide band
of frequencies. The bandwidth properties of arc arrays depend substantially on
, their circuit configuration, the type of radiators and control element.
Thus, in an arc array with spatial (or parallel) excitation (Figure 4.5), where
broadband phase sttifters are used [4], with a deviation of the frequency f from
the cEnter frequency f.0, with which the phasing is accomplished, a symmetrical
phase error occurs in the aperture CD:
~
e(D(y)=Ko f a1 i~t(a 2~zoa' ii a
(a.4i
)
when npe y 2
10
8
Figure 5.2. The resonant length of a Figure 5.3. The Q of a slot as a
longitudinal slot as a function of its rela
function of its displace tive width dl/X.
ment xl.
ficantly from a generator halfwavelength. Inclined slots in the narrow wall
have a resonant length equal to approximately half of the wavelength in free
~ space [01] (its precise value is usually chosen experimentally).
In design calculations for slotted waveguide arrays, it is important to know
the slot passband, which is characterized by the quality factor Q. The Q of
a longitudinal slot is shown in Figure 5.3 as a function of its relative width
dl/), for a waveguide with a phase velocity retardation of Y= 0.67 when the
center of the slot is shifted relative to the center line of the wide wall of
the waveguide by xl/a  0.185. It follows from the figure that with a slot
width of dl/a = 0.05  0.1, its Q changes insignificantly and does not exceed
10, which with a high carrier frequency in the microwave band corresponds to
; a considerable bandwidth (2Af/f = 10%).
The graph for the Q of a longitudinal slot as a function of its relative width
can also be used for a transverse slot in a roughly estimating its bandwidth.
The slot width in a slotted waveguide array is chosen by working from the
conditions for assuring the requisite electrical strength and the necessary
passband. When a slotted antenna operates only in a receive mode, the major
factor in the selection of the slot width is the bandwidth of the signals
being received.
When selecting the slot width dl, a safety margin of two or three times with
respect to the breakdown field intensity for the center of the slot should be
provided, where the field intensity, Eslot, is a maximum (21 = a/2). This
safety margin is chosen by working from the structural design requirements and
the operational conditions of the slotted antenna:
E =U'"< I E
Eslot �j dl ~(,2 3 �Q' (5.2)
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where U is the voltage amplitude at the antinode; Enp [Eult] is the ultimate
value of the field intensity at which electrical breakdown begins (for air under
normal atmospheric conditions, Eult  30 KV/cm).
_ In the case of a uniform amplitude distribution over the antenna aperture, when
the power radiated by the antenna is divided equally among the slots:
r 2P t
U"' v N GE ' (5.3)
where P is the power delivered to the antenna; G. is the radiation conductance
of the slot; N is the number of slots.
If the amplitude distribution over the aperture differs from a uniform distri
bution, the slot which radiates the greatest power is to be determined for the
specified amplitude distribution. Knowing the distribution of the radiated
power over the antenna slots and the delivered power, it is not difficult to
calculate what fraction of the total power goes for a given slot. Substituting
the value found in formula (5.3) in place of P/N, one can find Um.
Finally, the slot width is determined from (5.2):
dl::;~: (2�3) Um/E,,n"n� ult. (5.4)
If the slot is filled with a dielectric or covered with a dielectric plate, its
electrical strength is increased [9].
5.3. The Types of Slotted Waveguide Arrays
Distinctions are drawn between resanant antennas, nonresonant ones and antennas
with matched slots.
~
ii
i~
o (a)
_ 4
_ ~0. _ I~_ �
~  . _ ...........l_
!e/7
dl (b)
Figure 5.4. A resonant antenna with transverse (a) and longitudinal
(b) slots.
In resonant antennas, the spacing between adjacent slots is equal to XB (Figure
5.4a: the slots are coupled inphase to the waveguide field), or ag/2 (Figure
5.4b: the slots are coupled in an alternating phase fashion to the waveguide
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field). Thus, resonant antennas are inphase antennas, and consequently, the
direction of maximum radiation coincides with a normal to the longitudinal
axis of the antenna. Inphase excitation of longitudinal slots placed on
different sides of the center line at a spacing of aB/2 is assured by virtue
of an additional phase shift of 180�, due to transverse currents in opposite
directions on both sides of the cznter line of the wide wall of the waveguide.
In the case of inclined slots in the side wall, the additional 180� shift is
obtained by virtue of changing the direction of slot inclination (+d). Conse
quently, the resulting phase shift for ad3acent radiators in both cases proves
to be 360� or 0�, regardless of the type of load at the end of the antenna.
A resonant antenna can be quite well matched to the feedline in an extremely
narrow band of frequencies. In fact, since each slot is not individually
 matched to the waveguide, nll of the waves reflected from the slots are added
together inphase at the antenna input and the reflection factor of the system
becomes large.. It is obvious that this mismatching can be compensated at the
antenna input by means of any tuning element, but since the matching is dis
rupted with even small changes in the frequency, the antenna remains a very
narrow band type. For this reason, in the majority of cases one dispenses with
_ inphase excitation of individual slots and the spacing between them is chosen
as d ag/2.
ti characterli'stic feature of the nonresonant antenna obtained in this fashion
is the greater bandwidth within which there isgood matching, since individual
reflections are almost completely cancelled with the large number of radiators.
4' 2�
.
zl (d)
Figure 5.5. Configurations of nonresonant slotted waveguide antennas
with longitudinal (a, b), and transverse (c) slots in
. the wide wall of a waveguide, as well as with oblique (d)
slots in the narrow waTl of a waveguide.
However, when the spacing between the slots differs from aB/2, this leads to
outofphase excitation of the slots by the incident wave and the direction
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of the main radiation lobe is deflected from the i.ormal to the antenna axis.
This deflection is most often small (with the exception of special cases) and
changes in the shape of the main lobe and the level of the sidelobes caused by
the deflection pf the beam are still not noticeable. For this reason, the
directional properties of such an antenna can also be determined as in the
case of inphase excitation, with subsequent accounting for the beam inclination
angle.
A terminal absorbing load is usually installed to eliminate reflections from the
end of a waveguide. Circuits of nonresonant antennas with inphase coupling
of the slots are shown in Figure 5.5 (Figure 5.5a, c) as well as with alternate
phase coupling (Figure 5.5b, d) to the waveguide field, where the slots are cut
in both the wide and in the narrow walls of the waveguide. In all cases, the
phase distribution in the antenna can be considered linear if the mutual coupling
of the radiators via both the internal and external space is not taken into
 account.
v~ (a)
d=2 e/Z
.
 �Er ' Es
d~ (b )
Figure 5.6. Inclined slots in the narrow wall of a waveguide.
d_,~8
I _ _ ~
~ I
. ~
~ ~ I
While the slotted waveguide arrays
shown in Figure S.Sac have a radiation
field with oniy the dominant polariza
tion, antennas with oblique slots in
the narrow wall (Figure 5.5d) also have
a field with parasitic polarization.
The direction of the transverse currenCs
Figure 5.7. A slotted antenna with in the narrow wall of a waveguide and
obliquely displaced matched the field intensity vectors for the
electrical field excited in two oppo
slots. sitely inclined slots (+d) where the
spacing between them is aB/2 is shown
in Figure 5.6a with the arrows. The radiation of such slots is determined by
the horizontal components of the field intensity vector of the slots (Figure
5.6b). The vertical components produce a parasitically polarized field. To
reduce the parasitic component of the radiation field, the inclination angles
of the slots must be made d< 15�, for which the power lost to parasitic
polarization amounts to less than 1%. However, this limits the possibility of
obtaining the requisite normalized conductances of the slots, g. For this
reason, special steps are taken in practice [01] to suppress the parasitic
polarization field.
In antennas with matched slots, each slot (longitudinal, transverse or obliquely
displaced) is matched to the waveguide by means of a reactive dipole or a stop
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and does not produce any reflections. Consequently, a traveling wave mode is
established in such antennas with a terminal absorbing load. A schematic of
an antenna with obliquely displaced matched slots is shown as an example in
Figure 5.7. In such antennas, good matching to the feed waveguide is obtained
in a wide passband (5 to 10%). In the case of obliquely displaced slots in
the wide wall of a waveguide, through the choice of the inclination angle d
and the displacement xl, th.e normalized conductance of the.waveguide in the
crosssection of the slot is made equal to unity and the susceptance existing
in this crosssection is cancelled out by means of a reactive stub. Since the
stub is installed in the waveguide section, passing through the center of the
slot, with a change in the frequency there is a simultaneous change in the
susceptances of the stub and the slot and their mutual compensation takes place
in a certain range of frequencies. With a substantial change in frequency,
the antenna likewise remains matched to the feed waveguide, since it becomes
a nonresonant one.
The spacing between matched radiators in an array with alternate phase coupling
of the slots is usually taken equal to ag/2 at the nominal frequency. The
direction of the maximum radiation in this case is perpendicular to the axis of
the waveguide.
I
/
5.4. Methods of Designing Slotted Waveguide Arrays
There are several methods of designing slotted waveguide arrays. Strict design
techniques entail considerable mathematical difficulties, and for this reason
they are not used in engineering calculations and in synthesis problems.
Approximate methods are usually employed in engineering calculations.
Approximate design calculations can be performed for slotted waveguide arrays
by means of the energy technique of [07], which does not take into account the
mutual coupling of the slots via the internal and external spaces. It is
assumed that the phase shift between adjacent radiators through the feed wave
guide is equal to the electrical spacing between them of 27rd/aB, while the phase
distribution in the antenna aperture is linear. However, because of the external
and internal mutual coupling of the slots in the waveguide, there is a substan
tial deviation of the amplituciephase distribution from the requisite distribu
tion, while the attainable directional pattern deviates from the specified one,
which is primarily due to the cross coupling of the slots via the dominant mode
[5].
The method of recurrent relationships of [6] takes mutual coupling of the slots
via the dominant mode in the feed waveguide into account and provides for a
better approximation of the specified distribution in the antenna aperture by
the feasible distribution as compared to the energy technique.
The most precise design calculations for slotted waveguide arrays can be
performed using the method of successive approximations of [07], which takes
into account both external and internal iLteracCion of the slots in the wave
guide (via the dominant and higher modes). However, the design calculations
are more complicated in this case.
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We shall consider the method of recurrent relat;onships and the energy technique
for design calculations of slotted waveguide arrays.
owo.~
e, ~2
9 .
(a) al
1 2
~
,
~Z~l AdZ A~2~ ~ a1,v� ~d~ a~ 1 dr~7
.~f I ,~2 I ~M1 I 9M I ~M
2
4f
a2~A2 �+~d � ANea
?
. 6~ (b )
Figure 5.8. The equivalent circuits of a resonant slot, arbitrarily
cut in the wall of a waveguide (a), and a slotted
waveguida array (b). .
The Method of Recurrent Relationships [6]. The equivalent circuit of a slotted
waveguide array with arbitrary resonant slots in the form of a two wire line
with shunting conductances is shown in Figure 5.8b. The spacing between adja
cent conductances is composed of the distance between the slots and the two
wire line sections incorporated in the equivalent circuit of the slots. We
designate the complex amplitudes of the incident and reflected wave voltages at
the input as un_1 and un_1, and use the symbols un and un for the complex
amplitudes of the incident and reflected waves at the output of the nth four
pole network, into which the equivalent circuit of the antenna is broken down:
iln I nni + J Brt1 , un =An "'F' ) Bn+
t Cnt I J DnI, �n=L'nF] Dn�' (5.5)
By using fourpole network theory, one can establish the fact that the real
components An_1 and Cn_1 and the imaginary component Bn_1 and Dn_1 of the
complex amplitudes of the incident and reflected voltages at the input of the
nth fourpole network are expressed as follows in terms of the real An, Cn
and imaginary Bn and Dn components of the complex amplitudes of the incident
and reflected voltages atthe output of the same fourpole network:
AnI  r 1} 2n) (nn COS An Bn SI17 nn ) 2n (Cn COS An Dn Slil An)~
~
Bn_I _ (1 ~2n) (An 5111 nn 1 Bn COS /.~n) 1 � 2n ~Cn Slfl An 4 D. cos en),'l
~ (5.6)
Cn_l_~I nl lCn COS An Dn SICI L~n~ � (f~nCOS An Bn SI11 On~,~
2 2
~ (5.6)
Dn_I _ (1  (Un COS A~  L'n $I11 nn ) 2n (An 51110n  Bn COS On
\
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Here, gn is the normalized conductance of the nth slot; pn = pdn + 4n1)+ ~in)
is the electrical spacing between the (n1)th and the nth conductances in the
equivalent circuit; Adn is the electrical spacing between the slots along the
waveguide; Ain) and A2~nl) are the electrical lengths which are due to the
equivalent circuit of the nth and (n1)th slots.
Taking into account the symbols which have been introduced, the radiation power
and phase of the field radiated by the nth slot are as follows respectively:
=I�nunl2g'n=l(nn=FCJ2"(Bn+ DJ21 gn; (5.7)
pn
011 = arg (u~�{  arctg BIIFDn f /zn, (5.8)
where k= 0, l, 2, A"+C"
Using formulas (5.6) (5.8), one can perform the design calculations for a
slotted waveguide array taking into account the mutual coupling of the slots
via the dominant mode and without taking their interaction into account via
the external space or via higher order modes.
The distributions of the radiated powers Pn or the amplitudes F(zn) (zn :ts the
coordinate of the nth radiator) as well as the phases On of the fields radiated
by each slot are usually specified in the design of slotted waveguide arrays.
The distribution of, the radiated powers should be normalized so that:
N
~ Pn =1x, (5.9)
nal
where the power at the input to the antenna is taken equal to unity (Po = 1);
K= PL/PO is the ratio of the power absorbed in the load PL to the power at
the antenna input Po.
Since the amplitude distribution f(zn) is related to the distribution of the
powers Pn by a certain normalizing factor v:
pn =aP (Zn),
(5.10)
then by substituting the value of Pn from (5.10) in formula (5.9) instead of
Pn, we obtain:
~ /g(Zn(5.11)
Q=~1!C) InN1
N
After determining n~~,' j' (z�) from the specified distribution and the known
relative value of the power absorbed in the load (usually, K= 0.05  0.1 to
obtain the maximum antenna gain), the normalizing factor Q is found, and conse
quently also the power radiated by any slot Pn [formula (5.10)], given the
condition that the power at the antenna input is taken equal to unity.
The design of an antenna in the case of a specified amplitude distribution
(antenna synthesis) is managed using an equivalent circuit (Figure 5.8) from
the antenna end, i.e., from the last Nth fourpole network. The electrical
spacing between the slots is considered to Ue specified and constant in this
case.
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If there is a matched load (gH [gloadl  11 uN = 0) following the last Nth
slot in a nonresonant antenna, then in expressions (5.6) BN = CN = DN = 0 and
AN =vfK. Then we obtain for the pormalized conductarace of the last Nth slot
from formula (5.7):
9N   I 'N N.
The phase of the field radiated by the last slot is taken equal to zero (see
equation (5.8)). The quantities PN and K included in formula (5.12) are known:
the power PN is determined by expression (5.10) while K= 0.05  0.1 in the
usually employed antennas of the type considered here.
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(5.12)
Then, by using expressions (5.5) (5.7), the real and imaginary components of
the complex amplitudes of the incident and reflected waves are calculated: AN_19
BNT1, CN1 and DN_1 at the input to the Nth fourpole network, and consequently
also the conductance of the (N1)th slot:
b`N1
rN._
(5.13)
(/IN1 I ('JIYI'(l1N1 FON_ A) 3 .
By sequentially applying formulas (5.6) and (5.13) with the preliminary substi
tution of the current subscript n in the last formula for the subscript N1,
we determine the parameters of the equivalent circuit of the antenna.
The quantity pn = ~~n1)+ ~dn + Ain) takes on a simpler form, An = Adn, i
long'tudinal slots are used in the wide wall of the waveguide for which Dfn~ _
_Dlni = 0( igure 5.8a) [4] or transverse slots in the wide wall, for which bfn)
7r/2 and Ain~ _w/2. In the case of more comPlex slots (for exa le, obltayel
displaced slots in the wide wall of a waveguide), the quantities A ~pn) and A2
are determined by the expressions given in [4].
The deviation of the phase distribution in the antenna aperture from a linear
distribution, which is caused by the mutual coupling of the slots via the
dominant mode in the waveguide, is calculated from the formulas:
8fi==?n d(Nri)~bn
(5.14)
in the case of slots coupled in phase to the waveguide field, and
S(t)( 2nd..1 nl(Nit)d)� (5.14a)
~ Xn ~
in the case of alternate phase coupled slots, where 0 n is the phase of the field
radiated by the nth slot [formula (5.8)].
In calculations using formulas (5.14) and (5.14a), the number k in expression
(5.8) is chosen so that the difference between the quantities appearing on the
right sides of formulas (5.14) and (5.14a) will be the least.
One can correct the phase distribution in the aperture by changing the spacing
between the radiators d or by using more complex slots, but there no need for
this, since in the given design method, the external mutual coupling of the
slots and mutual coupling via higher modes have not been taken into account.
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The method of designing slotted waveguide antennas using the recurrent relation
ships (5.6) is applicable for any number of radiators in nonresonant antennas
and for any amplitude distribution over the aperture.
However, with a large number of radiators in an antenna, i.e., in a long* antett
na, its design is simplified. In fact, with a large number of slots, their
coupling to the waveguide proves to be rather weak and the reflections from the
slots are neglectably small. Moreover, since in a nonresonant antenna, the
adjacent radiators are excited with�a slight phase shift, then at the antenna
input, practically all of the waves reflected from the slots cancel each other
out and the input impedance of the antenna remains close to the characteristic
impedance of the feed waveguide in which a mode is established which is close
to the traveling wave mode.
In this case, one can use the energy technique to calculate the parameters of
the antenna. We shall indicate the approximate limit of applicability of this
technique for nonresonant antennas.
The design calculations for a slotted waveguide array where N= 12 for a speci
fied amplitude distribution [6] using the energy technique and the method of
recurrent relationships have shown that in the case of shart antennas (N = 12),
the energy technique yields too rough an estimate: the error in the feasible
distribution of the powers relative to the specified value in som,e radiators,
reaches +30%. Moreover, the amplitude distribution proves to be asymmetrical.
For this reason, in an approximate design of an antenna for a spe:cified ampli
tudephase distribution using the energy technique, one should relughly take
the number of radiators as N> 15, if the power absorbed in the IIiatched load
is K= PL/PO = 0.05  0.1. In the case of a greater power dissipated in the
load, the number of radiators N is correspondingly reduced.
The Energy Method for Design Calculations.
Nonresonanr n^.te^.r.as. Formula (5.10) determines the relative rad:Lation power
of any nth slot (i.e., the radiation power PN referenced to the power delivered
to the antenna no, which is taken as unity):
Pn''=6f2(ZnNx f2\Zn~�
~ /Z (ztt)
, n=1
The factor 1K in the numerator of this expression, without taking into account
the losses in the walls of the waveguide, is the antenna efficiency nA; there
fore: _ +ln fg(Zn)�
N
IZ (zn) (5.15)
Considering the relationship [07] between the relative radiation power Pn, the
slot coupling factor to the waveguide, a n, and the slot conductance gn:
*We will conditionally understand a long antenna to be one in which the per
unit length radiation power is low.
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'n
~i '  ~ ~i, a) /1' (5.16)
, ( ~ I ~ 1= . �
kn L~~ M. /(I an )r (5.17)
nne can initially determine the relative radiation powers Pn of all of the d1otg
based on the specified amplitude distribution as well as the antenna efficiency
by means of iterative conversion calculations from the last Nth slot to the
first, and then the coupling factors an, and finally, the equivalent normalized
slot conductances, gn (5.17). Based on the known slot conductances,'the coupling
elements are determined, i.e., the displacements of the slots relative to the
. waveguide axis, xl, or their inclination angle, d[see � 5.2, Table 5.11.
In the case of identical slot radiators (an exponential distribution of the
field amplitudes over the antenna), when the equivalent conductances (or
resistances) of all of the slots are equal, formula (5.17) can be used to deter
mine them from thespecified nA, where:
N
,
a ~A �(5.18)
Resonant Antennas. A resonant antenna with arbitrary resonant slots and a
spacing of d= aB/2 between them (or d= J1B) is designed by the energy technique,
 which consists in the follow�ing. If the amplitude distribution is designated,
as f(zn), Just as before, and one takes into account the fact that all of the
slots are resonant, then the equivalent normalized conductance of the nth slot
is [05] :
gn  gnxf a (Zn ) ' ~ fa (Zn nnt (5.19)
The antenna conductance gBX [gin] incorporated in the formula is chosen so as
to assure good matching of the antenna to the feed waveguide. Thus., the value
gin can be chosen equal to unity.
Antennas with Matched Slots. As was indicated in � 5.3, obliquely, displaced
slots in the wide wall of a waveguide are used along with'simple slots, where
the former slats are characterized by two geometric parameters: the displace
ment xl and the rotation angle d, by means of which one can independantly
adjust the amplitude and phase of the field radiated by the slot. Matched
obliquely displaced slots for which there is no mutual coupling of the radiators
via the dominant mode are of the greatest practical interest, since there are
no reflections from the radiators and a traveling wave mode is established in
the antenna,the designing of the antenna for a specified dietribution is '
accoinplished by the energy technique using the formulas for nonresonant antennas.
The methods set forth for designing slotted waveguide arrays with slots equiva
lent to parallel conductances, gn, inserted in a line equivalent to the wave
guide, also remain valid for slots equivalent to resistances rn, which are
 inser.ted in series in the line. For this reason, the design calculations for
an antenna are performed in a similar manner, with the condition that the
normalized resistances rn are substituted for the normalized conductances gn
in the appropriate expressions.
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5.5. Matching a Slotted Waveguide Array to a Feed Waveguide
The matching of a slotted waveguide array to the waveguide feeder is usually
3udged based on the value of the reflection factor from the antenna input.
Yn the case of a nonresonant antenna with a terminal matched load, the reflec
tion factor from the antenn3 input is [OS]:
2(gr HLn) cxp ~j lnd) .
" (5.20)
N ~
1 + ~ 2 (Kn bn)
. n=I
where gn +Jbn is the total equivalent normalized admittance of the nth.slot.
In the case of identical slot radiators, where the admittances of all of the
slots are identical, this expression assumes the form:
2(g j!~) exp C j~~ (N} I) dl sin RH ~ Nd1 . (5.21)
1~ 2 N(g+j b) N stn (~n d~ �
~
It follows from formula (5.21) that the reflection factor takes on a value of
zero (Kst =[SWR] = 1) when 21rNd/aB =w(N + 1). The spacing between the slo*_s,
d, is determined from this so that throughout the entire range of change in
a, there is no resonant excitation of the antenna and higher order major lobes
do not appear in the directional pattern:
d95 777d Deg.
V,epnd
the waveguide feeder for
as a function of the
the slots.
5.7. The Directional Properties of Slotted Waveguide Arrays
The same methods are used to calculate the directional patterns of slotted wave
guide arrays as for the calculation of the directional patterns of multiple
dipole antennas. In this case, the shape of the directional pattern is governed
by the amplitudephase distribution in the antenna aperture.
The following kinds of amplitude distributions are the ones most frequently used
in practice: uniform, symmetrically decaying relative to the antenna center and
exponential. The phase distribution is most often linear.
The normalized directional pattern of a linear array of radiators can be written
in the form:
F (0, rl (0, (p) rn (0, (p)i
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where fl(8, is the directional pattern of a single radiator; Fn(61 0) is the
antenna array factor, which depends on the number of slots in the antenna.
We $hal.l give expressions for the antenna factor for various amplitude distribu
tions in the antenna. In the case of a uniform amplitude and linear phase
distribution over the length of the array:
sin (Nip12) ~ (5.24)
N si n (ip/2)
where * = ko d sin 9~1 is the phase shift between the fields produced at the
observation paint by adjacent radiators; ko = 2ff/a is the phase constant of
free space; A is the angle read out from the normal to the line of position of
the slots (Figure 5.10); *1 is the phase difference between adjacent radiators
along the feed system; N is the number of slots. In an inphase antenna, *1 =
0; in a nonresonant antenna with inphase coupling of the slots to the waveguide,
* 1= 21Td/a,U and in the case of alternate phase coupling, *1= 2nd/aB  ff.
If the field distribution in the aperture of a discrete linear array of radiators
is exponential, then:
sh(E/N) sin~u�~�sl12~
rn  slit V,dti2(u/N)+SJ12(~/N) ' c5 . 25> *
Figure 5.10. The readout of the angles
in calculating the direc
tional pattern of slotted
waveguide arrays.
where a L/2 is a quantity which
characterizes the xonuniformity of the
amplitude distribution in the aperture;
a= a E+ a CT is the attenuation constant,
due to radiation losses as well as losses
in the waveguide walls, Np/m; in a wave
guide with low losses, a ST � aE and
a= aE; L= Nd is the length of the
antenna array; u= 0.5 kpi.(sin 9 sin
emain) is the generalized coordinar_e;
emain is the direction of the main lobe
of the antenna directional pattern.
The deflection of the main lobe of the
directional pattern from the normal to the line of position of the radiators
is determined from the formula:
sin 01.,, y  A,ld, (5.26)
main
where Y= a/aB is the retardation of the phase velocity in the waveguide; p= 0
applies to slots coupled to the waveguide field in phase, and p= 0.5 is for
alternate phase coupled slots.
The following obvious relationship can be employed to determine the attenuation
constant a E : *The formula was derived by G.A. Yevstropov and G.K. Fridman.
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~
~y 2Nd lil P .
i.
In the case of antennas with a symmetrical amplitude diatribution relative to
" the center which falls off towards the edges (for example, a cosine distribution),
 the calculation of the directional pattern in the case of a large number of
radiators involves labor intensive computations. In this case, one can use the
factor for an antenna with a continuous distribution of omnidirectional radiators,
 FL(9) [7], since the directional pattern of a discrete array and a continuous
one practically coincide when N> 6(d = a/2):
, 1"n (o) rL(0)'= I rAo sin u I_ Ai [sin (un/2)
A1012AI/ic f u 2 uW2 (5.27)
1
( sin (i1 1 n/2) If
~n/l '
u
~ where Ao is the amplitude of the field at the edges of the antenna.
When the amplitude distribution over the antenna is referenced to unity: A1 =
1 Ao. The directional pattern of a single slot F1(6) in the YOZ plane, which
passes through the line of position of the radiators (Figure 5.10), can be
determined from the formulas for the directional pattern of a slot in an infinite
shield in the case of engineering calculations: for a longitudinal slot, F1(8)
_[cos ((n/2) ' sin A)]/cos A, and for a transverse slot, Fi(9) = 1, since the
antenna length is usually great (several wavelengths), and moreover, the direc
tional properties of an antenna in this plane are determined primarily by the
array factor Fn(9). When determining the directional pattern in the transverse plane (YOX in Figure
5.10) for an antenna with longitudinal slots in the wide wall of a waveguide,
one must consider the fact that the finite dimensions of the shield (the trans
verse dimensions of the waveguide) have a substantial impact on the shape of
the directional pattern [07]: the limited nature of the shield imparts a .
direcCional nature to the radiation: the field in the direction of the shield
is reduced to approximately 40 to 50% relative to the value of the field in the
direction of the directional pattern maximum.
In order to simplify the determination of the directional pattern of a slot in
the plane normal to its longitudinal axis (the YOX plane), it is convenient to
replace the waveguide with a flat strip of the same width [06]. It then turns
out that for a waveguide width of a=(0.7  0.8)J1, the directional pattern will
be close to any of the patterns depicted in Figure 5.11.
In the case of transverse slots in the wide wall of a waveguide or slots which
are inclined in the narrow wall, the directional pattern in the YOX plane can
be approximated from the formulas for the directional pattern of a slot in an
infinite shield, since the shield dimensions in the direction of the slot axis
have little influence on the directional pattern in either the Eplane or the
Hplane of the slot [07].
Formulas are given in Table 5.2 for the determination of the width of the
directional pattern of inphase slotted waveguide arrays and the levels of the
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.tiG 60
I:4
1G Iip
� 1.40
?f.9
X'
~
270 300 740 ?70 .100
z ?l .t ?H=.t 7L .t 7.//=O,S.t
Figure 5.11. Calculated directional patterns of a halfwave slot
in the Eplane for various dimensions of the rectangular
.shield.
first sidelobes are indicated for various amplitude distributions in the antenna.
One can also use the indicated formulas in the case of nonresonant antennas,
since the spacing between the radiatore in such antennas (5.22) differs insigni
ficantly from the spacing in inphase arrays and the angle of beam deflection
from the normal to the array is small.
TABLE 5.2.
Yponenb
0 neptroro
; AwnnN'rYAnoe pacnpeJlenenNe 2 e0,6 nenecTKe,"/l13(1,
AmPlitude Distribution
PanuoMepnoe Uniform I 511INd I 713,5
3KC11011fD1tN;lAb110C (x = 1'L/Pa 0.05) I 54,4 1~/Nd I 12, l
F.~rnnnan t i a l
KoceirycotI,qanbuoe; aMnn+TyAa nona ua KpaAx aH
Teiunr: ~ 5(' X/Nd
Ao 17,8
Aap O,~ ~ (ql (A1 _  1) 0,.,) (2) 68 X/Nd
Key: 1. Level of the first sidelobe, dB;
2. Cosine; the amplitude of the field at the antenna edges.
In those special cases where1t is necessary to deflect the beam considerably
from the normal to the array, the effective length of the aperture Leff  Ndcos
emain is to be substituted in the formulas for the directional pattern width,
290.5, in place of the antenna length L= Nd.
i , The directional gain of an antenna with alternate phase slots in the wide or
narrow wal.ls of a waveguide when y= J1/aB < 1 and d= aB/2 =(0.6  0.9)a is
determined by the approximate formula:
, Do go (3 4 vN/X), . (5.28)
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where v= 2.for longitudinal slots in the wide wall and v= 4 for oblique slots
in the narrow wall of a waveguide (when S< 15�).
The aperture utilization coefricient, g0, incorporated in formula (5.28) depends
on the amplitude distribution in the antenna: in the case of a uniform distri
bution, g0 = 1; with an exponential distribution go = 0.85 and 0.92 respectively
for K PL/Po = 5% and 10%; with a cosine distribution, g0 = 0.81 and 0.965 for
Ao = 0 and Ao = 0.5 respectively.
The directional gain of an antenna can be estimated using formula (5.28) during
scanning, if the beam deflection angle 9main < 40%, d/X < 0.6 and the antenna
length is L= Nd � a, since a change in the antenna directional gain during
scanning in the indicated range, because of the change in the effective length
of the aperture, is compensated in that a linear antenna becomes directional
in two planes when 6main ; 90�, while for 8main = 0, the antenna is directional
in one plane [03].
In contrast to a linear array, a p2anar array of radiators is directional in both
main planes, and for this reason, its directional gain during scanning begins
to fall off immediately because of the reduction in the effective aperture of
the array. The efficiency of a nonresonant slotted waveguide radiator, nA, can be computed
from formulas (3.8) or (3.11). Since a shortcircuiting piston is usually installed in a resonant antenna instead
of an absorbing load, its eff iciency is higher than the efficiency of a nonreso
nant antenna of the same dimensions. With known values for the efficiency and
directional gain of an antenna, the overall gain is G= DOnA�
5.8. Possible Structural Conf igurations for Slotted Waveguide Arrays and Struc
tural Design Examples
Figure 5.12. Inclined slots in the
narrow wall of a waveguide
with isolat.ing metal pro
jections between the
radiators.
Depending on the function of an antenna,
it can be constructed in the form of a
linear or planar slotted waveguide array
or consisf of a set of linear slotted
arrays, arranged along the generatrices
of the surface of an aircraft (Figures
5.12  5.16). A schematic depiction of
a portion of a linear antenna with
oblique slots in the narrow wall of the
waveguide, which is used in marine radars,
is shown in Figure 5.12. To attenuate
the parasitic component of the radiation
field of such an antenna, which is polar
ized perpendicular to the axis of the
waveguide, metal isolation projections
[01] are installed between adjacent slots.
By utilizing the basic concepts of wave
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Figure 5.13. A nonresonant slotted waveguide array with slots ia
the side wall of the waveguide.
Shortcircuiting
piston 
/~'o,oom~oeorsbi~unoivuu
B61C0NOV0,C/JJ0/111161II
,00'38EM RF eonnector.
~ From the transmitter
~ Om 7e,oeda1,7y11�r4'
Figure 5.14. A nonresonant slotted waveguide array with longitudinal
slots in the wide wall of the waveguide.
~
attenuation in an overmoded waveguide in the case of propagation between para
llel metal plates [8] and knowing the spacing between the slots, one can deter
mine the spacing between the projections do (Figure 5.12), their length 11 and
thickness t.
Examples of the structural design of nonresonant slotted waveguide antennas with
' oblique slots in the narrow wall of the waveguide when they are excited from
' a rectangular waveguide (Figure 5.13) as well as with lonitudinal slots in the
wide wall when fed by a coaxial cable (Figure 5.14) are shown in Figures 5.13
' and 5.14.
An example of the structural design of a slotted waveguide array with electro
~ mechanical scanning (with a removable upper slotted wall) is shown in Figure 5.15.
' One of the variants of a two dimensional slotted waveguide array [9] consisting
af eight parallel waveguides, in each of which ten dumbbell slots are cut is
shown in Figure 5.16a. As compared to conventional rectangulars slots, dumbbell
' slots have a greater bandwidth [07]...,.A specific feature of the antenna is the
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1
Om aeHPn~ ,aa From the generator
6,,
�y Om
3neHm,oa
dBuaame~A
From the
electric
motor
Figure 5.15. An electromechanically scanned slotted waveguide array.
Key: 1. Housing;
2. Upper wall with the slots;
3. Moving metal projection  "knife";
4. Absorbing load;
5. Cover for the beam steering mechanism;
6. Cam; �
7. Push rod;
8. Return mechanism link; .
9. Return spring housing;
10. "Knife" guide bearing.
{
H 00 H 40 0~ 00 9e 0* ~ ; , ~
~O H H H ~4 f~
O~ H ~0 1~ H F~ ~
4 H 6.5 H �0
C
r~
al (a) 0 (b)
Figure 5.16. The antenna of an aircraft navigation system (a) and
its directional patterns (dashed lines) (b).
fact that the even and odd waveguides are fed from different sides by means of
power dividers and the entire aperture is used to generate four beams (Figure
5.16b). Such antennas are used, for example, in aircraft independent Doppler
navigation radars, intended for determining the speed and drift angle of an
aircraft.
For protection against atmospheric precipitation and dust, the aperture of the
slotted waveguidearray is covered with a dielectric plate or the entire radia
ting system should be housed in a radiotransparent fairing.
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5.9. A Sample Design Calculation Pxocedure for Slotted Waveguide Arrays
When developing or planning slotted waveguide arrays, the starting data, for
 example, can be the width of the directional pattern in the two main planes
or in one of them (290.5) and the level of the sidelobes; and the directional
gain Do.
We shall deal with the design procedure for the following variant: the direc
tional pattern width is specified in one or two main planes as well as the
sidelobe radiation level.
' The type of slotted waveguide antenna is chosen at the outset. If the angular
position of the main lobe of the directional pattern, 9main, is specified and
the antenna should provide for operation in a band of frequencies, a nonresonant
antenna is chosen. However, if according to the design specifications, the
antenna is a narrow band one, but should have a high efficiency, a resonant
antenna is preferable. Then the spacing between the radiators is found in the
waveguide selected for the construction of the antenna and the specified band
of frequencies. In a resonant antenna with alternating phase slots, d= aB/2..
In a nonresonant antenna, the quantity d can be chosen in two ways. If the
position of the main lobe of the directional pattern in space, emain is specified,
then the requisite value of d is found from formula (5.26). If the angle emain
is not specified thou�h, then the spacing between the radiators is chosen from
the condition that d< aB/2, as well as to assure that there is no resunant
excitation of the antenna at the edge frequencies of the specified band (5.22).
Then, the amplitude distribution for the antenna is selected which assures a
directional pattern with a specified sidelobe level. Based on the now known
ampli*_ude distribution, the length of the antenna is found (and correspondingly,
the number of radiators), which assures the requisite halfpower level directional
pattern width (see the formulas in Table 5.2).
Then the design calculations are carried out in the following order:
1. Based on the overall equivalent circuit for the antenna (Figure 5.8b), the
equivalent normalized conductances, gn (or resistances rn), are computed for
~ all N slots of the antenna (see � 5.4);
2. Knowing gn or rn,'the displacement of the center of the slots.relative to tt:e
center of the wide wall of the waveguide, xl, or their inclination angle in the
side wall, d, is determined frocri the formulas of Table 5.1.
3. Having calculated the radiation conductance of the slot in the waveguide, GE
(i.e., the external conductance), the voltage at the antinode Um (5.3), and
consequently also the slot widtli dl (5.4) are determined from the known value
of the power at the input (in the case of a transmitting antenna).
4. With a known position o� the slots in the waveguide wall and their width,
the resonant length of the slots in the waveguide is found from the data of � 5.2.
5. The directional pattern (see � 5.7), the directional gain and overall gain of
the antenna are calculated.
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Besides the electrical design of the antenna itself, design calculations are also
performed for the feedline and exciter; when called for by the design specifica
tions, the requisite type of rotating joint is selected and its main characteris
tics are determined.
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6. ACCOUNTING FOR MUTUAL COUPLING EFFECTS IN SLOTTED WAVEGUIDE ARRAYS
As has already been noted in Chapter 5, the energy and recurrent techniques of
design calculations for slotted waveguide arrays (VShchR) do not assure the
practical veasibility of antennas with the anticipated parameters in many cases.
This applies primarily to arrays with a low sidelobe level in the d irectional
pattern and is explained by the fact that in the indicated techniques, many electro
dynamic factors are not taken into account which occur in the actual antenna
structure.; It has been determined as a result of specif ic calculations and experi
ments that the limit of applicability of these methods for arrays with a compara
tively small number: of radiators (up to 30) in each waveguide [1, 2] can be con
sidered a directional pattern level of roughly* 15 dB.
To illustrate this fact, experimentally measured directional patterns and direc
tional patterns calculated by the energy and recurrent techniques for typical
nonresonant alternating phase slotted waveguide arrays with different numbers N1
of longitudinal slots in the wide wall of the waveguide are shown in Figure 6.1.
In these antennas, the identical amplitudephase distribution of the field in the
aperture of the arrays, Vn, was specif ied as:
Vn=^
0,95cos 2n nN1_1 111exIi(�jrcd(n1)sin0,��1, 1
X cos a~ qKS 1r< 1.5ag, the contribution of higher modes to the mutual ad
mittance YA) is less than 0.1 percent, and for this reason, one can assume that
under these conditions, mutual coupling is due on].y to the dominant HlQ mode. �
c. Finite Thickness of Waveguide Walls. In order to take into account the depend
ence of the equivalent slot width di on the thickness of the waveguide wall t, ,
it is suff icient to introduce in place of the actual width d1 of the slot, its
equivalent width d* in formula (6.13) for the calculation of the inherent internal
admittance YM. lIt follows from,Figure 6.4 that with an increase in t, the
equivalent 29th of the slot decreases, by virtue of which the value of the factor
d
CIL ~1 increases. This leads to the fact that to obtain a specif ied
~rc
21t) qdl 1 2
precision in the determination of YnnM, it is necessary to take into accounta
larger.number of modes than in parsgraphs a and b. Canputer calculations have
shown that with lthe same relative error of one percent as in paragraph g f or the
calculation of Ynn), it is sufficient to use the first 14 higher modes.
The directional patterns of an arraI calculated for the cases of the actual slot
width dl and the equivalent width dl are shown in Figure,6.3. A comparison of the
dashedanddotted and the dashed curvess attests to the necessity of taking the
f inite thickness of waveguide walls into account.
d. External Mutual Coupling of Radiators. The necessity of taking this factor
into account in planar slotted waveguide arrays is obvious, since at least the
nearest slots of adjacent waveguides have a substantial influence on each radiator.
The conductance and susceptance components of the external mutual admittances '
Y~n) can amount to up to 40 percent of the inherent external admittances Y~~) of
the slot. In order to take the external mutual coupling intn account, it ir
sufficient to substitute the values of Ymn), camputed from formula (6.8), in the
matrix for the mutual admittances of the system of equati,ons (6.2).
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; In those cases where the spacing between the slots, dmn, is no less than 0.5a, one
can use the following formula to calculate the external mutual admittances [01]:
1,fi4~, cos 1~ cos 0cas l 2 cos 0n 1_j ~ dmn (6.14)
Y(e) 7 ~ 1 \ l e
nm ~
7UInui SII] Um SIII Un
where 0m and On are the angles between the radius vector from one slot to another
and the longitudinal axes of the mth and nth slots respectively. In this case,
 since the slots are longintudinal, en = em = o, and therefore Y~n) = 0.
 Similar results are also obtained for other amplitudephase distributions of the
f ield in the aperture of the array.
Thus, when synthesizing planar slotted waveguide arrays, it is necessary to take
into account all four electrodynamic factors, and when calculating the internal
adm ittances, to consider only the f irst 14 higher modes. For this reason, the
superscripts in the twin summation signs of (6.13) are to be set equal to p= q=
= 3:
8: Isin 3 / np
~L n~ii q=i~ n1~KS ((n/2Kl)s (in,i] L ~~qdl J2a . � I cos=1 n �i~ ~l
 Y(l)
mn"
X ~sKrBnq II 5p,~ n~ rI n z~ (6.1.5$)
~ I~,~~~ (Jt/1K~)d}�P~9 ~ 2ki ZK1 ) ] ' MOIt,
L ~
(G.15a)
3 s titFa ~
1/ j FV V 6n9 qn L\ n4 ).I (_jp / J cos (~Q ~,in~ l X .
'1 .~i.ol Aiwl ' 1Q()KPP? Kh Kll \ R ~
P'=0 4==0.
( nQ tT,a Cxlnt~P9_~ pKfma1~9
X cos I yir i )
It ~ 4K2 (n 6 (JL/2KIIn)2I(in~ )C
~
('.rcl1I(1Jq 1 CnlnRP? K I ain l )
~6.I..Sb
~ _ it/lrcll,)" f P= ~dN!!l~li. (fi.1 ~ifi)
(
n~
6.4. A Procedure for Synthesizing a Linear Slotted Wavegu ide Array Taking
Electrodynamic Mutual Coupling Effects Into Account
The General Scheme. In order to design a slotted waveguide array, taking mutual
I coupling effects into account, it is necessar.y, as follows fron equations (6.2),
to f ind the'internal and external admittances of the slots for a specif ied number
and type of array radiators, as well as the geometry for their arrangement, which
realize the requisite distribution of the complex voltage amplitudes Vn. Since
this is a rather complicated nonlinear mathematical problem, it is expedient to
approach its solution from physical considerations.
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We shall f irst treat the case of a linear slotted waveguide array. It is obvious
tha.t the magnetomotive force in an arbitrary mth slot can be represented as:
f'm =v1j ~~Mrn~'Yr~irfn).
(6,16)
where V~1) is the voltage in the slot without taking the mutual coupling effect
into account.
Then we rewrite system (6:2) in the following manner:
y Vn (}'mn~'Yinn) v(nl) ~Yr~iitm+Y~mr~ii), 1 :m~Ni� (6.17)
It can be solved by an iteration technique, by employing certain physical consid
erations. It has been shown in paper [3] that f or this, it is suff ic ient to .
employ the following procedures: .
 1� Based on the lrnown values of Vn (the requisite distribution) using one of the
known methods, which do not take into acoount all of the mutual cou ling factors,
the array geometry is d eterm ined (the displacement of the slots xnO~ and the
spacing between them).
2. Using fornrulas (6.8) and (6.7) (or (6.14)) the matrix of the internal Ymn~ and
external y(e) admittances is calculated, where these admittances are due to
factors notntaken into account in paragraph 1. 3. The complex voltage amplitudes YM are determined from the lmown values of
Vn, Y~In~ and Ymn) from expression (T.17).
4. Based on the computed values of Vml~, just as in paragraph'l, the next approx
 imation of the array geometry is determined, and then paragraph 2, 3, etc. are
carried out.
_ It is clear from physical considerations that this iterative process converges.
As a result of the calculation, that array geometry will be found which assures
the realization of the requisite distribution in the antenna apertiire, taking all
of the electrodynamic mutual coupling factors into account.
The Initial Approximation. As can be seen from the synthesis scheme proposed here,
the most complex procedure in a computational sense is procedure 1, all the more
since the known computational methods cannot be used (the energy or recurrent
methods), since the camplex distribution of Vi1) in the general case has a non
 . linear phase characteristic. To carry out the procedure, a special method has
been proposed in a number of papers [2, 9111, which is based on the representa
tion of the slotted waveguide radiator in the form of a long line of fourpole
networks, which are characterized by the equivalent circuits of the slots.
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The equivalent circuits of various slots of length 1,n = 0.5X, cut in the wide wall
of a waveguide and their equivalent normalized admittances (or impedances) are
presented in Table 6.1, in which the same symbols are used as in Chapter 5. All
of the par meters of these circuits are uniquely related to the external y(e) 9nd
internal YM admittances and are found from the balance condition for thencomplex
powers in the waveguide slot crosssection [12].
Thus, a daminant mode slotted waveguide array can be represented in the form of
series connected fourpole networks.
It can shown by employing fourpole network theory that the normalized incident and
ref lected voltage waves, Un and Un, at the input and output of the nth f ourpole
network are interrelated by the following expressions:
.
Una1e n_.,aze6n
(U1_1: ~ . (illeA" 1b2eAn
173 cAn n4 c ',IL U"i
h1 c A,' G& eI,I (U. ,
(6.18)
where: ~1{.t~2f2titxl; aa=b R~a f2ti ~nl;
as='4(1ItsMita?~ttz1; Q4=9t., [I Ei~z f 2tiW;
!'j= 9~a [1 f ~t+~2+2ti W; ba q (1 W 1~tta�12~
Nbl. tt2�21, t2j; Ga=(I t,)l1 ~i~1+ 241+1b1I:
A� _ : (2n/l ) rl q t.  E, , Z, z,,; = f~, ~ for the equivalent, c ircuit of
Figure 6.7a; q=1, ~1 = z~n), C2  Yn, E3 = zin1) for the equivalent circuit of
Figure 6.7b; dn is the spacing between the nth and (n1)th radiators.
All of the equivalent conductances n= Re C2, bn = Im E2 and dn can be found based
 on the specified am litude fn = lVni and phase ~n = arg Vn distributions, with the
initial values of d~0) and the fractions of the power absorbed in. the load Pg
specified from physical considerations.
For this, the auxiliary normalized coefficient is determined initially:
N,
s'v' � I~Pi[ I V" 12.
(6.19)
If the incident volta$e wave to the slotted waveguide array load is normalized to
unity, i. e. , we set iJN = i, then the ref lected wave amplitude will be UNl = qT,
where I' is the reflection factor, while q=+1 for the equivalent circuits of
Figures 6.7a and 6.7b respectively. Then, as shown in the literature [10, 11], the
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TABLE 6.1 Ih~JlnNtrI111C ~t 3RI~uP;i I;nyT~1CII~IAA u~x~nnnnF~ocT~, utcno
~1~ ,irnrnos~ cxeMn u~c.ni~ I~2~ Y(~)G(r)_~.~~;(j)
ol
~~c
Z=Ra;jX
a~
a�
o~
Zo y 76
Y(t).~ x
r
3 ~Ki ~.i ~ ~ I Y
A"'1 ab
=u
/2npl\
s cus
l I 1
1 ~ a
~ l ~ _ (2~~~,t1
~bl
~
Sill2 ( n 11n
n \ 1) ~P9
_cxP ( ~
p nq
(G 7)
(1 p n n~ c1i a u n P. I3aui~nnn~~ U, n Da oupr:tcnema n(12).
Note: The quantities Dl and D2 are determined in [121.
.,.I
;Acnnnancnrmar np,.
u11nn~,ncTti ium rnu.
11(rTIIIMCIIIte
(3)
2G
qG(t)
7 y~..~ 1g(r)
s
Y ~ y
D'
ji;(i) '
D~
7a �j n, ;
7b 1 1
/.)z
Key: 1. Position and equivalent circuit of the slot�
2. Intiernal admittance of the slot, Y~i) = G(ij + jB(i);
3. Equivalent admittance or impedance.
 144 
i, Y
3 ~
x "0 � X
14A
nt v
?t
y X
tc ~d Z
X X
X 1Kdj . ig ad,, 1
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conductance (or resistance) of the slots i;s camputed from the formula:
from expression [13]:
b'n IVn I2/sly'jUri 4Un ja. . (6.20)
Re ~Y(/)
.
�
~rn2RC ~~n .0o)
,U) (6.21~j
~~~tineI jirn ~ 1 nn ~.~'n ) I
n
We determine the displacement x0) of the nth slot relative to the center line of
the wide, wall of the waveguide Oby any gradient technique, in particular in the
following manner:
(6.22)
s(0)x(0)  d si n
~ y g~$n ~ny) ~
rrv ny_ I
where .
1
rC' Re [Y(~)nn(�C(0nV
n
~ ~ _ 2Re YRn j Im [y~R> (Xn~))j
1
av_ (Sv_j; v1 xnx sign(g�gn�~) =s1gn(Zngny_q).
~8y 1/2, 5ignr(gngnv i) f' St6n (gngny_ Z).
It is natural to take the following as the initial value of xn0,'
a'' Ir a r,
n  :1rc5iI1  gn ,
~t a 7'n 2,09 cosa 0 l ( 6 . 23 )
2 A'D /
which realizes the requisite value of gn given the condition of slot resonance.
The initial step dp is chosen in a range of a/20 = a/10.
0) :
We determine the equivalent susceptance of the slot from the value found for x(n
2 1 m Re [ Ynfn (xnD)(6.24)
Y~`~JIm [Y~~~n n
~x~�~~] nn n
i �
~ The amplitude and phase of the field radiated by the nth slot are:
 I'R 1~~,'~~ ~ U,~ gU, 1; (6. 25)
~ 60�, the requisite
number of radiators practically does not change). Since an increase in 00 leads
to a decrease in do/A (when o0 = 60 to 90�1 d/a = 0.54  0.50), something which
is frequently undesirable because of structuril design considerations, values
of �0 = 50  60� are expedient.
The overall number of controlled radiators, in the case of dimensions of the
 radiating region of H 2ao0 and a rectangular grid is:
lll, , Nz Nvt (7.18)
and in the case of a triangular grid:
N, t  (1/,3/2) Nn; (7.19)
Here, the requisite number of radiators along the generatrix is:
NZ :1I ldz Cu, 5/200, L7, (1 I'. SIII O,,c), (7.20)
200.5Z  C0.5X/H is the width of the main lobe in the plane of the generatrix;
the requisite number of elements about the periphery of the cylinder is:
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(7.21)
200 .5 is the width of the main lobe in a plane normal to the axis of the cylinder
(the Airectional pattern of an element of the phased array is taken as cosinusoi
dal).
Circular conformal scanning is achieved in a cylindrical phased array while at
the same time using a comparatively small fraction of the overall nuiLiber of
radiators (30 to 40%).
a
Figure 7.6. A conical phased antenna
array.
Conical phased arrays (Figure 7.6) provide
for a hemispherical scan (with fluctuations
in the gain of several decibels). They
are used in cases where it is necessary to
place a hemispherical scanning phased
array in a conical (or ogive) housing of
an aircraft, as well as when the maximum
gain should be achieved in an axial direc
tion or a direction inclined to the axis
of the cone.
Conformal scanning (movement of the raL'i_a
ting region) is realized iri conical phased
arrays in the plane of the base, as well
as conventional sector scanning in the
plane of the generatrix. The radiating
region occupies a sector, the size of
which depends on the direction of the main
lobe, Omain'
In the case of hqmispherical scanning, some portions of the conical surface
radiate at large angles, and for this reason, it is expedient to take the step
of the array close to 0.5a (d/a = 0.50  0.55), although in a number of cases,
it must be increased up to 0.6a to 0.75a because of structural design considera
tions.
It is expedient to use radiators, the main directiona:l pattern lobe of which is
not directed along a normal to the generatrix of the cone, but rather in the
direction of the requisite maximum gain. The influence of mutual coupling
between the radiato.rs of a conical phased array on the directional pattern of
a radiator in the plane of the base is manifest, just as in cylindrical phased
arrays, in an indented central portion of the directional pattern of a radiator.
However, in contrast to cylindrical phased arrays, the deep nulls inthe direction
al pattern related to the appearance of additional interference maxima, are near
ly absent because of the conical shape of the surface (Figure 7.7). The width
of the main lobe, 200.5, and the gain, GmaX, of a conical phased array are
determined by the dimensions of the equivalent plane aperture of the radiating
region. ~ .
N 2~tn/d 4~ o.e I ~sinq~o
~v w' 200 5cp sin `Lyo+'l(pu '
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f�'rF~
s,olSp
~ joN

>,2
a~ +O
fMflV. ncp1,0
o,,s,t
8rl62� 08 '
0,4
~ 17,4
0, Z 0,1
170  80 40 0 40 84 50�
Figure 7.7. The directional pattern of
a conical phased antenna
array radiator.
~
0 15 3/7 4.'% 6n 7f 6,.,,
Figure 7.8. Sepa/SD as a function of
0main'
Key: 1. Planar phase antenna
array.
The graphs of Figure 7.8 make it possible to estimate the change in the equiva
lenfi plane aperture area of a conical phased array, Sepa/So (SQ = na2 is the
area of the base) with a change in the direction of focusing of Omain and the
vertex angle of the cone Ok (the angle between the axis and the generatrix).. As
a rule, radiators are not placed at the vertex of the cone, however, this area
is small and was not taken into account in the given case. An analysis of the
graphs shows that to reduce variations in the gain during scanning, it is
expedient to employ conical surfaces having Ok = 18 to 20�.
The impossibility of placing controlled radiators at the vertex of the cone leads
to the appearance of a nonradiating region in the equivalent plane aperture.
In the case of axial radiation,.it has the s::ape of a circle with a radius of
ap � a. The dimensions of the nonradiating rEgion, when ao/a ='0.1  0.3, have
little impact on the gain, but can substantially increase the level of the
aperture sidelobes q. The level of the sidelobes q as a function of the ratio
ao/a with a axial radiation for uniform (A (r) = 1) and optimal Aopt(r) amplitude
distributions are shown in Figure 7.9 [4].
The following phase distribution should be produced to focus the radiation in
the direction Omain, ~main on the radiating portion of a conical phased antenna
array: ~U~i~ '  K7v ~~R ~I~a SIt10,.~~ Cl)s (~P~,n Cc1S f)r,~~~ (7.22)
where the coordinates of the elements Z. < 0. The requisite number of radiators
is:
N = = Ss/Snr,
where:
Sit vn= (1 (a�l(7)il cosec q'�
(7.23)
(7.24)
is the area of the conical surface; and:
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zn
Aunr(r)
,tr
40
0 0,7.
P~
20~ ~ .
I ._~._i
0,4. 176 0,8 a,7117
x
~
I
Figure 7.9. The level of sidelobes Figure 7.10. A spherical phased antenn.a
where a nonradiating region array.
is present.
r{z in the case of a square grid,
S�~� 2d2/ I'i (7.25)
in the case of a triangular grid.
is the geometric area of a cell; d is the step of the array.
To obtain narrow directional patterns, the overall number of controll4d radiators
in conical phased arrays shauld be of the same order of magnitude (10 ) as for
cylindrical arrays. A hemispherical scan with gain variations of no more than
1 to 2 dB can be provided using a conical phased array.
Spherical phased arrays (Figure 7.10) provide for a hemispherical scan with
minimal changes in the directional pattern and gain variations (0.1 to 1.0 dB).
This is accomplished by arranging the radiators with a nearly uniform density
over the surface of the sphere and employing conformal scanning, i.e, maintaining
the shape and dimensions of the radiating region during scanning. The center
of the radiating region in usually located in the direction of the main lobe (o�r
the equal signal direction).. The radiating region is moved by switching the
feed for the radiators, while the phasing (identical within the limits of the
radiating region for any position of the region) serves to compensate for
phase errors (focusing). Disconnecting some of the radiators and.controlling
the shape of the radiating region make it possible to obtain directional patterns
with different parameters.
The most uniform filling of a spherical surface with radiators is obtained with
a triangular grid for the radiator layout.
To preclude large losses in the gain, the step of the array d/a when generating
an axially symmetric main lobe is to be chosen based on the coiidition (in a
manner similar to a cylindrical phased array):
dlX < (1} sin 0o) 1,
(7.26)
where 26 0 is the central angle of the radiating segment (Figure 7.10). The type
and dimensions of a radiating element are chosen based on the same considerations
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as Lur ottier convex phased antenna arrays. The influence of mutual coupling o�
the elements of a spherical phased array on the directional pattern of a radiator
is less pronounced than for other types. In the case of spheres with large radii
(a/A > 10), the directional pattern of a radiator is close to the directional
pattern of an element of a planar array having the same period. When condition
(7.26) is met, the directional pattern of a radiator of a spherical phased
antenna array is :
F" (0. ~p) cos (o  0 1') cos ((p
(7.27)
where 0~ and ~uv are the spherica~l coordinates of the uvth radiator.
The radiators of a spherical phased array should have either circular or con
trolled polarization.
The central angle of the radiating region, 60, can be optimized based on the
criterion of a minimum overall number of radiators to obtain the specified
width of the main lobe of the directional pattern or a specified gain [5].
Calculations show that the minimum number of controlled radiators is obtained
when 6= 50� to 90�, but because of the poor radiation efficiency of elements
having a directional pattern of the type of (7.27) at large angles, it is
expedient in practice to use phased arrays where 60 = 45� to 60�.
d/.t N' 0 d6 Cos A surface covered with radiators should
I ~ encompass the portion of the sphere bounded
by the angles:
0, 8
.f0
(76
40
94
3,7
02
2U
�
o
Figure 7.11. The parameters of a
spherical phased array
as a function of 6o.
uV ti(I (C(c 0,.,, CUS Qv ~ SI11 nJ.A SI fl n~ COS ~~~'ivi ~~~I~v~~�
(7.29)
The overall dimensions and requisite number of radiators in.a spherical ( or
spherocylindrical) phased array are calculated using the following procedure.
1. The radius of the circular equivalent planar aperture of the radiating seg
ment, aepa, is determined:
2(I:,Iipl% Cp,6I20O,bl
162
0 ~ 0' a/2100. (7.28)
where the spherical surface in the region
0' >7r/2 can be replaced by a cylindrical
one because of structural design considera
tions (a apherocylindrical phased array).
The total radiating area of the surface
does not change in this case. The follow
ing phase distribution [04] is produced
to focus the radiation in the directi.on
emains ~main on the radiating segment of
the spherical phased array:
>6  66
..20 62
22 60
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where in the case of a cosine directional pattern of a radiator (see (7.27)) and
vo = 45  60�, the coefficient CO.S = 62  64�, while the level of the aperture
sidelobes is q=20 to 21 dB (see Figure 7.11).
' 2. The radius of the spherical surface is determined:
a = aepa/sinAO (7.31)
 3. The step of the array is determined (see (7.26)); when vo = 45  60�, the
step is d/a = 0.59  0.54 (see Figure 7.11).
4. The requisite number of radiators N in the case of a triangular layout grid,
taking formulas (7.25) and (7.28) into account, is found from the expression:
N N'420o,n)z,. (7.32)
where
N'_=. "Co,, (l+sin0o)3
~/3 sii% ;lo (7.33)
(see Figure 7.11). The overall number of radiators in a pherical phased array
in the case of narrow directional patterns is N=104  10~.
7.4. Polyhedral Phased Antenna Arrays.
Classification. Excitation Techniques. Polyhedral phased antenna arrays take
the form of a system of planar phased arrays (subarrays), arranged on the
faces of a convex polyhedron.
Where the number of subarrays is Nsub 10, they are usually arranged on the
faces of a regular or truncated pyramid. Such polyhedral phased arrays are
called pyramidal. In the case of a large numbe�r of subarrays, they are placed
on the faces of a regular polyhedron (for example, an ieosahedron) on inscribed
in the sphere of a polyhedron derived from an icosahedron by the sequential
frrtion3zation of its faces. The overall number of subarrays can reach Nsub 
10  103, while the overall number of radiators runs to N = 104  105.
In terms of the directional pattern parameters, pyramidal phased arrays are
close to planar arrays, while polyhedral phased arrays where Nsub 102 are
close to conformal phased arrays (such phased arrays can be called quasicon
formal). A spatial excitation technique (in a transmissive variant) can also
be used to excite the subarrays, in addition to feeder and active excitation
methods.
, Polyhedral phased arrays have the following advantages over planar and hybrid
phased antenna arrays: the possibility of realizing hemispherical scanning with
smaller fluctuations in the gain and better utilization of the surface taken.up
by the radiators. As compared to conformal phased arrays, they are better suited
_ for production, something which is related to the use of not just modules of
the same type, but also subarrays of the same type, as well as the utilization
of a spatial feed system for the suba,:rays and thP simplification of the phase
distribution control system.
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In pyramidal phased arrays, the planar subarrays are placed on the faces of
regular or truncated pyramids where the number of side faces is M= 3 6
(Figure 7.12). lfao phasing modes are possible for the subarrays: independent
and combined. In the first mode, each subarray scans independently of the
others in a definite spatial sector; in the second, several subarrays scan ae
a single system. It is expedient to employ both modes in combination.
,
~
~
r
79,117
Figure 7.12. Pyramidal phased antenna arrays.
In the case of independent phasing, the design procedure for the structural and
radio engineering parameters of planar subarrays does not differ from the
procedure used for planar phased arrays.
The optimaZ slope angle for the side face of a pyramid, &oPt, can be determined
by working from one of the following requirements:
1) Maximum gain in a def inite direction, Omax, and in this case:
Eopt = Omax
411T  � 01o,1;
(7.34)
2) Miaimum variations in
the gain,.in_t.hQ_
hemisphere, in this case:
arcig (col;
~ 1
.
(7 . 35)
I
For a truncated pyramid
(using the upper
face, where a subarray can
be placed):
~,pt arccos
V coscc4
(Msec2 ~
cosec2 M 
36)
(7
1
/
Al
.
The minimum gain Gmin is obtained with maximum beam deflection from the normal
to the subarray:
arccos (sin ~ cos n/M).
(7.37)
Ttie drop in the gain at the boundary of the scan sectors of the subarrays in
the case of a cosine directional pattern of a radiator is:
AG [1tG1 10 lg (cosz Oc�)�
(7.38)
The main lobe uniqueness condition requires limiting the step of the subarrays:
dA < (I sin 0,011 (7.39)
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and therefore, the overall number of radiators in a pyramidal phased array with
a circular configuration for the subarrays N= AN' arranged on the faces of the
pyramid, where:  C , sc o.6 �
q( 160,r, in the case of a square grid,
n ~ n v3 Cn.ti '
g(l~~n) in the case of a triangular grid.
CO.S = 6267 when q=20 25 dB and with a quasioptimal amplitude distri
bution (or the equivalent thinning of the subarrays);
N, M(1sin Q,�)z fur a pyramid,
I(M J 1) (1 +sin 0,�)2 for a truncated pyramid.
Comparative data are given in Table 7.1 for pyramidal phased arrays where M=
36. An analysis of the data of Table 7.1 shows that pyramidal systems with
four sides have the best combination of parameters. In this case, a truncated
pyramid (five subarrays) has th,, least fluctuations in the gain, while a four
sided pyramid without a top face has the fewest number of radiators.
TABLE 7.1.
nAPAM,.Tn I iiNpaktNna I YceltenxeA mipnMSInn
M I
:3 I
4 (
5 'I
G
I:3
I 4
I 5
I fi
63,5
I54,7 I
51,0 I
49,1 I
82,5
I 74,5
I 69,6
( 66,8
O,! K
( 63,5
I .54,7
( 51,0
( 49,1
I 60,3
I 47,1
I 40,7
I 37,3
A G. ltli I
7,0 I
9.8 I
4,0 I
3,7 I
611 I
3,3 (
2,4 I
2,0
N'
I I0,s I
13,2 I
I>,A l
I8,5 (
14,0 I
15,0 (
I6,4 I
1811
The joint phasing of several adjacent subarrays of pyramidal phased arrays is
expedient to reduce the drop in the gain in the region of the boundaries of
adjacent sectors. Thus, for a pyramidal phased array where M= 4, the joint
phasing of two subarrays in'a range of + 20 to + 25� from the boundaries of
adjacent sectors leads to a drop in the gain from 4.8 down to 1.0 to 1.5 dB. In
this case, the considerable spacing between the centers of the subarrays leads
to the appearance of additional diffraction lobes, havtng a level of 10 to 13
dB.
Pyramidal phased arrays are the simplest in terms of structural design among
phased arrays with hemispherical scanning, but have considerable fluctuations
in gain within the hemisphere.
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Figure 7.13. A strip structure of a
array. Figure 7.14. An icosahedral structure
polyhedral phased antenna
for a polyhedral phased
antenna array.
It is expedient to use quasiconformal phased arrays in the form of polyhedra
inscribed in a sphere, where the polyhedra have a large number of almost iden
tical faces (20 to 400), on which identical planar subarrays are arranged, to
reduce fluctuations in the gain within the scan sector. The subarrays can be
realized in the form of strip (Figure 7.13) or icosahedral structures (Figure
7.14). The number of radiators in one subarray (10 to 100) is governed by the
requisite overall number of radiators in the phased array, N, the minimum
permissible number of subarrays, Nsub, and the operational convenience of the
system.
When Nsub = 102, estimates of the main structural design and radio engineering
parameters of a quasiconformal polyhedral phased array can be derived 3ust as
for a spherical phased array (see � 3.4). However, breaking the radiating
system down into subarrays and the polyhedral shape of the radiating surface
cause a certa3n change in the parameters, primarily in the gain and the direc
tional pattern.
We shall use the number of subarrays No arranged about the periphery of a great
circle.of the sphere as an independent parameter. When No > 10, the overall
number of subarrays Nsub, Placed on the faces of a polyhedron inscribed in a
spherical segment, bounded by the angles 0< 0' 10, the faces cannot be the same, while it is expedient to
standardize the form and dimensions of the subarrays, the entire area of the
 faces cannot be utiliaed. The losses in area lead to gain losses of:
AGS [dB] = 10 log (NCSC/Ssph) nGs[Jl6]=10Ig(Ne S,;1S,,11), (7.42)
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where Sc is the area of one subarray; Ssph is the truncated area of the sphere,
circumscribed around the polyhedron.
NRP
5102
2�101
101
50
20
10
>0
The gain losses AGloss for strip and
icosahedral structures of a polyhedral
phased array fall off with an increase
in no; when no = 10102, AGS =1.5 to
0.5 dB.
Figure 7.15. Nc as a function ot no
(the solid curves are for
an icosahedral structure;
the dashed curves are for
a strip structure).
The control of the phase distribution and
the motion of the radiating region in
quasiconformal polyhedral phased arrays
have some special features.
1. The steering of the radiating region
is accomplished by turning not individual
radiators on and off, but rather entire
subarrays; since 10 to 100 subarrays are
incorporated in the radiating region
(sometimes less than 10), this causes
jumps in the gain, which fall off with
an increase in no; when no = 10102
and 80 = 4560�, the jumps in the gain
when switching the subarrays amount to
from 2 to 0.1 dB; when no > 30, they do
not exceed 0.5 dB.
2. Multistage phase control is usually employed in polyhedral phased arrays:
a special computer specifies the phase distribution relative to the centers of
the subarrays included in the radiating region (see (7.29)); each planar sub
array is focused by conventional means in the common direction (Omaing Omain)�
The calculation of the directional pattern, the gain and the overall structural
design parameters'can be carried out just as in the case of spherical phased
arrays, while the design calculations for each subarray (structure, excitation,
matching, phasing) can be performed in accordance with the procedure used for
planar phased arrays.
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8. BEAM STEERING SYSTEMS FOR PHASED ANTENNA ARRAYS
8.1. Phased Antenna Array Control Problems.
Various electronically cantrolled radiofrequency devices are used for control
of the phase distribution of phased antenna array: phase shifters, switchers,
splitters and attenuators. Phase shifters are used in modern phased arrays in
which the phase shift can be varied discretelq: in quantum steps, where the
number of quanta is equal to 2�, v= 3, 4, 5, while the quantum step is A _
360� � 2�, i.e., 45�, 22�30', 11�15' respectively.
The phase shifting'elements in phase shifters can be ferrites and semiconductor
devices, in particular, PIN diodes. The structural design of the phase shifters
also differs correspondingly. The operation of phase shifting elements is
assured hy actuating amplifiers, which are an integral part of the phase Fhif
ters. The control signaT is fed to the input of the actuating amplifiers from
the phase shifter control unit.
Phase shifters also have such a configuration of the phase shifting elements
that the number of its inputs is equal to the digit capacity (v) of the phase
shifter. A binary quantized signal is fed to each input. Consequently, a
vplace parallel binary code is fed to the phase shifter. High frequency switchers provide for the connection of the input RF channel to
one of the two output channels in accordance with the binary control signal
which is fed in. Consequently, the signal controlling the switcher is a single
place binary code.
Attenuators are RF devices, the gain or the attenuation of which depends on the
control signal level. This control signal can be either continuous (analog) or
quantized. The problems of beam steering in phased antenna arrays can be treated the most
completely using the example of a circular scan receiving phased array, for
_ which a generalized structural configuration of the control system is shown
in Figure 8.1. To provide electronic scanning in a wide scan sector (for exam
ple, in the hemisphere above a ground surface), it is necessary to employ
phased arrays with a convex structure, for example, a spherical configuration.
The receiving elements and phase shifters are combined in subarrays, most fr~
quently planar arrays. The number of phased array elements can run up to 10
and more and the number of subarrays can reach several hundreds. The combining
of the components in subarrays is necessary to reduWe tre number of input
channels of the switcher and simplify its design. Moreover, the packing of the
spherical surface with planar subarrays makes it possible to use the very
simplest row and column algorithms and contr.ol devices.
Only the working region of a phased array surface participates in generating a
directional pattern in a specified direction. The smaller the number of sub
arrays incorporated in the working region, the fewer the possibilities for
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adaptation: spatial filtering of interference. The greater the number of sub
arrays incorporated in the working region, the more complicated the switcher is.
Taking these contradictory requirements into account, the number of subarrays in
the working region can fluctuate in a wide range (5 to 50).
The number of possible working zones is limited to several tens (depending on the
structural design of the phased array). Each working zone provides for scanning
in a definite sector of space, usually not exceeding + 15 to + 30� from the
center direction.
The switcher serves to connect the subarrays participating in the formation of
the working zone to the subsequent processing channels. The number of RF inputs
to the switcher is equal to the total number of subarrays on the surface of the
phased antenna array, while the number of outputs is equal to the number of sub
arrays included in the working zone. The switcher consists of a set of RF
_ switches, connected in accordance with a truth table and controlled by signals
generated in the switcher control unit (BUK).
Tyi I~_ _AymeHya i
Antenna l
(2) I Ay0
(3)~6AH~
I
I
i
~ 4~ ~ fi.4K T ~/r'c,0f07a1Vn,0 (8)
...I
(5) Adi;7;.~n~;c,oAdapter71
T.J~ I
~
(6) ( 9`
~ ~ ~ .
(7) ~ l~iK
~4rom computer To receiver
Figure 8.1. Structural configuration of
The phasing direction code (direction
cosines of the main lobe of the direc
tional pattern) is fed to the input of
the phase shifter controlle:r (BUF) from
the central computer. The codes which
are fed to the phase shifters are gene
rated in accordance with this code. For
this reason, it is most expedient to
design the phase shifter controllers in
the form of a special digital computer.
The phase shifter controller is the most
complex and most important assembJyo uf
the beam steering system (SU) and will
be treated in more detail in subsequent
sections.
a
receiving phased antenna
Modern circular scan pha.sed arrays take
ar
ray.
the form of extremely large and complex
Key: 1.
Beam steering system;
structures. These structures undergo
2.
Phase shifter controller;
deformations, both elastic and irrever
3.
Automatic tuning unit;
sible when e::posed to ext(arnal factors
4.
Switcher control unit;
(temperature changes, wind, rain, snow,
5.
adaptation control unit;
etc.). As a resuit, the position of the
6.
Monopulse signal pro
direction of the main lobe of the direc
cessing controller;
tional pattern in space changes. If the
7.
Automatic monitor;
errors caused by these deformations ex
8.
Switcher;
ceed the snecified aiming precision, it
9.
Monopulse unit.
becomes necessary to have tuning and
automatic fine tuning. Information on
the phased array geometry, external factors, true coordinates of the targets or
aiming points,
based on which the tuning
is accomplished, is fed to the input of
the automatic
tuning unit (BAN). The correction signals generated by the auto
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matic tuning unit are fed to the phase shifter controller, eliminating or reducing
the phase distribution errors.
,  . _
~
~ 7
~ F_.  ~
L ~
~ 11 12 13 l~i 21 22 73 24
, n
n
1~
ni r,1 n; n/i
137.3 n,f f~i24 174
4~~~ m rN
Figure 8.2. The principle of monopulse signal processing.
Modern radar is unthinkable without interference filtering (natural and artific
ial). Spatial filtration is also used along with frequency and time filtration.
4ny directional antenna realizes spatial filtration. However, even with spatial
filtering of interference from different directions, falling on the sidelobes of
the directional pattern, reception is disrupted if the interference power is
much greater than the power of the useful received signal. Adaptation systems have been developed to attenuate tae impact of interference in
 modern phased arrays. The amplitudephase distribution on the surface of a phased
array is changed in such a manner as a consequence of adaptation that reception is
substantially attenuated from interference source directions, retaining inthis
case a sufficient level of the useful received signal in the direction of the
main lobe of the directional pattern. As a result of adaptation, "nulls" or
"dips" are formed in the directional pattern in the directions of the interference.
Adaptation is all the more effective, the more the para�,:~2ters of the amplitude
phase distribution change and the greater the number of channels in the adapta
tion system. Usually, the gains and phase shift (amplitudephase adaptation) or
only the phase shift (phase adaptation) change in each channel during adaptation.
In the case of amplitudephase adaptation, each channel is split into two
quadrature channels (with a phase shift of n/2), and a controlled attenuator with
 a gain which varies from 1 to +1 is inserted in each subchannel. Thus, the
amplitude can be varied from 0 to v12__ and the phase varied from 0 to 27. In the
case of phase adaptation, a controlled phase shifter is used in each channel.
The adaptation controller (BUA) generates control signals for the attenuators and
phase shifters. The algorithms in accordance with which these control signals
� are generated depend on the presence and completeness of the data on the inter
ference sources, as well as on the choice of the adaptation quality indicator.
In the case of a multichatlne'L antenna and monopulse signal processing, it becomes
possible to simultaneously generate four directional patterns, which are shifted
in pairs by approximately half of the 0.5 power level directional pattern width.
For this, four splitting phase shifters each are installed in each channel, as
shown in Figure 8.2. The phase shifts in these phase shifters must be changed
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when the direction of phasing changes within one working zone and when changing
the working zones. Consequently, the splitting phase shifters are controllable,
and a monopulse signal processing control unit (BUM) is provided to control them.
An automatic monitor unit (BAK) for both the operability of the control system
itself and the antenna complex as a whole is included in the phased array control
system in addition to the units enumerated above (phase shifter controller, auto
matic tuning unit, switcher control unit, adaptation controller and monopulse
signal processing controller). It should be noted that the monitoring of the
operational status of the phased array requires a clear cut definition of entire
antenna failure. Experiments show that the failure of up to 10% of all of the
elements does not lead to the failure of the entire antenna. In this case, the
shape of the directional pattern changes somewhat, but the antenna remains
operable.
The structural configuration of a phased array described here, of course, is not
the only one. The control system is somewhat simpler in transmitting phase
arrays: there is no adaptation control unit, no monopulse processing, there
can be many fewer modules and the switcher can be altogether absent or substan
tially simplified. The control system for a phased array with a small scan
sector is simplified a great deal, when one can use an antenna with a planar
configuration.
8.2. Control Algorithms for Phase Shifters.
The major task of controlling phase shifters is the generation of a plane phase
front, perpendicular to the specified phasing direction.
The phasing direction is a unit vector rm, having coordinates of &xm, E YM and
EZm, which are direction cosines. The phasing direction can also be specified
by two angles 0 and however, the algorithms for computing the control codes
prove to be considerably more complicated in this case.
We shall consider the ith radiating element with phase center coordinates of
Ri(xi, yi, zi). In order to produce a plune wave front, perpendicular to rm and
passing through the origin, it is necessary to compensate for the spatial phase
change at a distance of di with the phase shifter (Figure 8.3). This distance is
determined by the scalar product:
di='(Rirm) =X(~xmIJI~UnI{ zt~rm�
(8.1)
For a specified wavelength a, at this distance the phase change (in radians) is
defined as follows:
2n ?rc e
~r~2:i x d( _ x (Xi Sxm'{'Yi ~1/in I 'Zi bzm~� (8.2)
The phase change can contain a definite number of whole periods and an additional
phase shift Oi:
'n~r2~~K ~R,.
where k is an integer.
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Since a phase shift of 2nk does not have any influence when combining harmonics,
the phase shifter should compensate for the phase shift of 4~i, determined from
(8.2) :
01= `l1[f ~t Exm~' ~ ~yrn'~' ~ tzmIAP . (8.~+)
here {�}AP is the operation of isolating the fractional part of the number.
w
Figure 8.3. On the determination of the
spatial phase change.
0i kA
. 
3A  ~ ~
/ I
_ ' I
211
i
6'(a) al 2n' trii
611 (b) Dl ?.n' Oyi
Figure 8.4. Rounding off to the least (a) and to the nearest (b)
value.
abscissa. The quantization operation is essentially one of roundingoff: in
Figure 8.4a, in the direction of the least, and in Figure 8.4b, in the direction
of the nearest permitted (quantized) value. A definite binary vdigit code,
ni, corresponds to each permitted value of the phase shift. It is convenient,
and so it is usually done, for this code to numerically determine the number of
quanta included in (pi qu or to determine a quantity proportional to this number.
To determine the control code when roundingoff in the direction of the least
value of ni, the phase shift 01 is not to be expressed in radians, but rather in
quanta by working from the fact that 27 radians correspond to 2� quanta, and
after this, the lower order digits following the decimal point are rejected, i.e.,
the integer part of the following number is singled out:
+
, v v J X~ Jt Zt ~ (8.5)
n`{2tc ~ }~2 l ~zm~nr)A,
where {�}u is the isolation of the integer portion of the number.
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As has already been noted, phase shifters
with a quantum step of A + 2w � 2� rad
are used at the present time for control
ling the beam of phased arrays. The
value of the quantized phase shift is
found by quantizing ~Di, i.e., by per
forming a nonlinear transformation,
defined by the graphs shown in Figure
8.4 for v= 2. In these graphs, the
phase shifts are plotted on the ordi
nate, where these shifts can be produced
by the phase shifter, while the conti
nuous value of (Di is plotted al.ong the
Oi 116
qu
,
2A
A ' (
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If all of the numbers included in (8.5) are expressed in binary notation, then
multiplying by 2� is simply shifting the number over by v places. This operation
~ can be performed by a separation method, not feeding the integer part of 2�@i/27
to the control inputs of the phase shifter, but rather the v high order digits
of the number 4)i/27r. Then the algorithm for computing the code for the ith
phase shifter can be written inthe following manner:
(8.6)
nl=siISfJiJ st (v+l.
where:
Si ==xi Exm{yl tum+Zi tzm; (8.7)
Xi  Xi/X; yi  yi/X; and zi = zi/X are the coordinates expressed.in wavelengths;
are the digits to the left of the decimal point, beginning with the first;
]�[v+1 are the digits to the right of the decimal point, starting with v+ 1.
Thus, to calculate the control code for a vplace capacity phase shiftPr, it is
necessary to perform three multiplication and two addition (8.7) operations,
after which the v digits to the right of the decimal point are isolated. Element
coordinates are stored in a read only memory; the codes for the direction cosines
are fed to the input of the beam control system.
Y
/Ay
4 A
4
0
Figure 8.5. A planar orthogonal phased Figure 8.6. On the beam steering of
antenna arra;~. a modular phased array.
 When rounding off to the nearest side, as can be seen from Figure 8.4b, prior
_ to rounding off it is necessary to add 2(1+1) to the number si, which corresponds
to adding half of a quantum step to 0i. Adding 2(v+1) corresponds to feeding
a carry to the (v + 1)th digit following the decimal point after finishing the
computation of the unit place.
The algorithm (8.6), (8.7) provides for calculating the code for any phase
shifter and is therefore applicable to any phased array design, however, its
realization requires either that an individual computer (and then the speed of
the control system will be a maximum) be installed at each phase shifter, or
that the phase codes be coniputed sequentially for a group of phase shifters.
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Zn the latter case, it is necessary to prcvide a switcher which connects the
computer sequentially to the requisite elements. The control circuitry can
be simplified in the special of a planar phased array with an equally spaced
configuration of the elements.
As can be seen from Figure 8.5, each element has its own column and row number,
akl. The coordinates of an element are xkl = kAx, ykl = lAy and zkl = 0.
Consequently, algorithm (8.7) reduces to the form:
 ';lal G�[~x~.rnt I lAy~'~'.~~�i (r.y.
(8.8)
This is the socalled row and column control algorithm. It is necessary to
compute the phase changes for the array step along the x and y axes to realize
this algorithm: eX = AxExm and ey= AyE , after which, by using expression
(8.8), the total phase shift skl 3s compVted, and finally, the quantization is
performed in accordance with (8.6).
The calculation of keX and ley can be simplified if one considers that k and 1
are integers. Thus, if k, 1= 2, 4, 8, 16, etc., then multiplication is accom
plished by simply shifting by one, two and three places respectively. Multipli
cation by 3, 5, 6, reduces to the addition of the two numbers already
obtained for the corresponding cells (2 + 1= 3, 4+ 1= 5, 4+ 2= 6, etc.).
In a similar manner, multiplication by other integers can also be realized, for
example, 7= 6+1=4+2+1, 11= 8+2+1.
If the working zone of an antenna consists of several planar modules, then the
phasing is accomplished in a single system of coordinates. In this case, the
reference phase s00 is calculated for each module, which is determined by the
spatial phase change of the ith module (di) relative to the phase at the origin
of the central system of coordinates XYZ, as shown in Figure 8.6. The phase
change in accordance with an expression similar to (8.8), but which takes into
account the position and orientation of the module, is added to this reference
phase.
Thus:
sh t  coo I h,h.r. I (F,". (8.9)
The further generation of the control code is carried out in accordance with
(8.6).
To calculate the phase changes for the array step eX, and E+, it is necessary
to determine the spatial phase changes in a single system ol coordinates for
two points on the X' axis ar_d two points on the Y' axis (for example, for the
points 0, 1 and 2 in Figure 8.6), and then to determine the difference in the
phase changes : slo ~ xlntZ,�m I l
r~iim 'I zio .~:n~'(�t"on ~,r~~ I uoo k!nII I
A.e' Exnt '1 AIl ' ~um I AZ' ~zm FA.l'" ~xnt I'AI/, 4m I nz bnnA.C' XIO.CoM. Ax" .roi xnn, (8.10)
Aj/'
rhn
(/nn,
nll'
!/ui !/uu,
AZ'
Zip
Z00
^zn
�'ul... ZUn;
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Xoo' y00 and zoo are the coordinates of the point 0'; X10, y10 and zlo are the
coordinates of point 1; XO1' y01 and zol are the coordinates.of point 2 in
Figure 8.6:
tuo Xou~ym'I'znntzM�
Row and column algorithms are applicable in the case of equally spaced modules
in a planar phased array. However, to reduce the directional pattern sidelobes,
it is expedient to introduced random components into the configuration of the
modules. In this case, the application of row and column algorithms proves to
be impossible in the general case. A quasirandom arrangement of the modules
at the nodes of a rather thick equally spaced array is sometimes used, the
majority of the nodes of which remain unfilled. Individual control with the
calculation of the code inuependently for each cell remains a universal approach.
8.3. Algorithms for Generating Directional Patterns of Special Shapes.
When generating a directional pattern of a special stiape (cosiecant, beavertail
pattern, etc.), the wave front in the direction of the radiation differs from
a plane front. A small deviation fr an a plane front is achieved by changing the
geometry of conventional parabolic antennas, for example, using special cosecant
hood s.
The requisite deviations from a plane front can be realized in phased arrays.by
feeding the appropriate control codes to the ptiase shifters. This makes it
possible to change the shape of the directional pattern during operation by
electronic means.
The shape of the directional is specified relative to the direction of radiation
by a functional relationship in the vertical and horizontal planes. For this
reason, the deviation in the phase distribution is specif ied relative to a plane
front, perpendicular to the phasing direction in the x', y' coordinates. The
OY' and OZ' coordinate axes and the phasing direction vector, rm, fall in one
plane9 and the OX' axis os perpendicular to OY' (Figure 8.7).
It can be shown that the unit vectors of the new system of system of coordinates
(i, j, k) are expressed in terms of the unit vectors of the main system of coordi
nates and the phasing vector rm:
k'rm ="i.~xnI Aym'I k~zun, �
i j k
IO U q 
~ kk' ~~�m ~,~~m ~zrn _ i~qm I J~Xm
~ I Ikk'l I I lkk'l 1 Fnn~ ,
(8.12)
....i~.rrn~zmAllm~:m�Ik~~xmf'tj
1' lkk,j ~m) ,
~t xn1. 1  ~ym
where [A B] is the vector product of the vectors A and B.
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i  �xEl/m+JExIn 
x'(R )
5xm 'i  E yrn
 y'(RJ')
, 4smazm'J~zm1 Z(t xrnUm~
V U m I U. .
(8.13)
The additional phase shift, which changes the direction of the directional pattern
beam, is given in general form by the function:
~ (x~ ~ ~l'ia~n''~'n~m(�~~~.4~1�
add $ add ' y ~
v
(8.14)
Consequently, the phase shift calcula
tion algorithm for the ith cell 'consists
of the following operation: 1) The cal
culation of the phase shift to produce
a plane front in accordance,with (8.7);
2) The determination of the new coordi
nates of the ce11 in accordance with
(8.13); 3) The calculation of the addi
tional phase shift for the ith cell in
accordance with (8.14); 4) The determina
tion of the overall phase shift for the
ith cell:
Figure 8.7. Rotation of the system of
coordinates.
sl'  S!�{mtAon: C8.15)
and 5) The determination of the code generated for the control of the ith pYiase
shifter, in accordance with (8.6).
The algorith for generating directional patterns of a special shape includes
extremely complex expressions for coordinate transfonaation (8.13), functional
transformation (8.14), supplemental addition (8.15), and moreover, there remain all
of the procedures for calculating the plane front (8.6) and (8.7).
Different ways of simplifying the computational algorithm are possible by means of
approximating the requisite phase distribution. Thus, with a modular configuration
of a phased array, one can calculate the supplemental phase angles only for the
ref erence phase of the modules, def ined by relationship (8.11). In this case, the
phase distribution in the antenna aperture is of a piecewise planeparallel nature:
each module produces a plane front, perpenddcular to the direction of radiatio.r},
while there are phase jumps between these plane sections, which also produce the
total approximation of the phase distribution.
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 The phase distributions can be more precisely approximated by the plane sections
, if these sections are not made mutually parallel, but tangential to the requisite
phase distribution. In this casQ, it is necessary to calculate the column for each
module using the complex algorithm of (8.13).
8.4. Switcher Control Algorithms
A switcher serves to generate the working area from the set of subarrays. The
output channels of the switcher are connected to the subarrays of the working zone
in a strictly def ined order, for example, the f irst channel to the center subarray,
the second to the next ane up, etc. A possible variant for the generation of the
working area from seven subarrays is shown in Figure 8.8. When the phasing direc
tion is changed and the working area is changed, the same subarray can appear in a
different place in the working zone. Thus, when producing the working area, which
is shown with the dashed line, the subarray which was designated with the number 1,
is now designated number 7. This same subarray can appear in any other location
in the working zone. Consequently, the switcher should provide for the capability
of connecting each antenna subarray to any output of the switcher.
If the number of input channels (number of antenna
O ~ subarrays) is N, while the number of subarrays
including the working zone M, then the number of
switches is:
O O Sn = M log2N (8.16)
00J A single digit control code (0 or 1) is fed to each
switch. Consequently, the control unit for the
" Figure 8.8. Shifting the switcher has Sn outpuCs: The d irection code (of
working zone. the direction cosines of the vector rm) is fed to
the input of this device. The operational algorithm
for the control device for the switcher is broken
down into two parts: the determination of the number of the working area which will
generate the d3.rectional pattern in the requisite directicn, and the generation of
the control signals for the switches of the switcher. The first task is handled by
a decoder and the second by an encoder.
Each working zone provides for scanning space in a rather narrow sector of +15 to
+20� relative to the cent:al direction of the given working zone. When the target
leaves this sector, the next working zone is switched on. For this reason, the
task of selecting the working zone consists in determining whether the specif ied
direction belongs to the scan sector of a working zone. This task is simplest to
solve by determing the angle between the central direction of the working zone and
the specified.:phasing direction.
That working zone is selected for which this angle is the least. To curtail the
calculations, one can determine the cosine of this angle and select that working
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'4a Yes
~
No Hem
. d) (b )
Q) (a)
Figure 8.9. Priority coupling configuration.
zone for which this cosine ts the greatest. The calculation of the cosine of an
angle is the determination of the scalar product of two unit vectors:
cos v =(rm ru) = gxm 4xqI" Eym tvq+$:m t:q. (8.17)
where v is the angle betwea:n the phasing direction and the central direction of
the working zone; r(~X ,~y ,tZ ) is the unit vector of the center direction of
the working zone.
The coordinates of the working zone center in spherical antennas are proportional
to the direction cosines of the center direction of the working zone:
Rn(xxt. yup zu)�RrRtExa, EL4. E:n)� (8.18)
For this reason, (8.17) may not be computed, but the quantites si determined by
(8.7) compared in calculating the reference phase of a module.
The algortHims for selecting the greatest from a set of values can differ. The
simplest is a sequential elimination of the least values, however, it requires the
performance of hundreds (based on the number of working areas) of sequential compar
isons, and for this reason, it is the worst in terms of operational speed. Since
the function (8.17) is monatonic with respect to v, there is no need to compare
the value of this function for a given working zone with the values for a11 other
zones; it is sufficient to compare it with the nearest zones on the surface of the
phased array. That working zone is selected for which the value of cos v is the
greatest ks compare3 to the nearest one, but not with all of them.
It can be seen from Figure 8.8 that the nearest working zones can number no more
than six. Consequently, the choice of the working zone is made by an AND gate with
six inputs. When the va lue of cos(v) is equal for two or even three working zones,
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it is necessary to give priority to a particular circuit. A schematic showing
priority based on a given direction (axial priority) is shown as an example in
Figure 8.9. Each arrow in this figure is a priority comparison gate, the opera
tional principle of which is shown in Figure 8.9b. When cos(v) is equal for the
right and left working zones, a"1" will be fed to AND gate of the right side
working zone; this zone enjoys the priority in this case.
Thus, the switcher control unit consists of two units: a decoder and an encoder.
The phasing direction code is 'fed to the decoder input. A"1" signal appears at
one of the NW,Z, outputs of the decoder, which corresponds to the selected working
zone, while a"zero" appears at the remaining ones (NW,Z, is the number of working
zones, which can be less than the number of subarrays, since not every subarray
can be the center of a working zone.
The encoder has NW,Z, inputs and Sn outputs (Sn is the number of binary switches
in the RF switcher). When a"1" is fed to one of the encoder inputs, a Sdigit
code is produced at its output, which provides for the connection of the requisite
subarrays into a working zone in accordance with the truth table.
For convex phased arrays, the truth table is drawn up manually. This is explained
by the fact that convex surfaces (sphere, ellipsoid of rotati,on, etc.) cannot be
packed regularly with a rather large number of points. It is known fram geametry
that no more than 12 points (an icosahedron) can be regularly arranged [equi
distantly spaced] on a complete sphere. For this reason, when packing a sphere with
a large number of subarrays, it is impossible to provide for an identical configur
ation of the working zones and to define a single phasing algorittmn. This leads to
the necessity of camposing the truth table for the switcher manually.
The circuit configuration for encoders are determined by the component base and
are described in the considerable literature on digital device design.
8.5. Adaptation Control Algorithms
Phased array adaptation is the generation of such a directional pattern that ar
improvement is assured in the.quality indicators for antenna functioning. Optimal
adaptation provides for the extremum of the generalized quality in3icator.
Adaptation of receiving phased arrays is accomplished for the spatial f iltering
of interference, i. e. , to suppress the gain in the directions of incoming inter
ference while retaining adequate gain in the requisite direction. In this case,
the quality indicator is either the signal power to noise power ratio at the
output of the phased array, or the mean square error between the requ isite ancl
actual output signals. Adaptation is realized by changing the complex frequency
response in the channels of the modules included in the working zone. Each channel
is split into two: a phase shift of w/2 is realized in one of the subchannels; an
electronically controlled amplif ier or attenuator is inserted in each subchannel.
The signals in all the channels are then added. Thus, the output signal:
x'. w.  WT X,
r
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where X=[xi] is the column vector of the si.gnals; W=[wi] is the column vectcr
of the weighting coeff icients; WT and XT are transposed vectors.
If the requisite signal is yp, then the mean square error is defined by the
expression:
= 1111n.I !l~ WTXXT2W7XtJ.�I !lii+ (H .19)
where is averaging with respect to time. The product XXT is the square
matrix for the interchannel crosscorrelation coefficients RXX _[xixk], while
Xy0 = XXyo is the corrOlation vector between the input signals and the requisite
output signal:: Taking this into account, expression (8.19) can be written as
follows:
HrTRZrHrlWTS,N.'*jyu� (8.20)
' The opt imal value of the vector f or the coef f icients W* correspond to the minimal
mean square error ~in. Any variation uU
W:::W*I ILU (8.21)
for an arbitrary vector U and a small coefficient of variation increases the
mean square error. By substituting (8.21) in (8.20), we obtain the function *
o2(11). This function is minimal when u= 0. Consequently, the following equation
is observed (the extremum condition) :
r)S2 00 (8.22)
$r1)"11U7~\ ~/2 a) .
r~
(b)
c) B)
Figure 8.10. Multichannel adaptation.
By substituting (8.21) in (8.20), (8.22) can be written in the following form:
orts (it) IIIA. 1UT ( R s.cHr* 0.
~)li (8.23)
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Since U is an arbitrary vector, then (8.23) can be satisf ied only when the
expression in parentheses is equal to zero, i.e., when RxxW* = SXyo. The vector
of the optimal coefficient is also determined: from this equation:
1
W* R  I S= Rsxl X/n. (8.24)
xx xp�
This is the equation for optimal Wiener multichannel filtration.
We determine the requisite output signal by the transform Wp on the useful output
signal Xo, so that:
# Xn Wu Wo X~~� (8.25)
By virtue of this, expression (8.24) is reduced to the form:
W`:: R.eci X(xn Wo) 'Rxa1 Kxx. Wa� (8.26)
In particular, if the input signal contains the expected signal Xp and interference
Xn, then:
 (RXo X. 1 kX. xif i Rx liX.) Rx 11 XJ11RX. X. I ' klie Xj W0' (8.27)
*
When the signal Xp and the interference Xn are independent:
W: (KX.x.+Rxn Xn)1RXo xo Wo. (8.28)
It is completely obvious in the absence of interference;
W* Rx; X� R,V.x,,wo= Wn (8.29)
and those coefficients Wp are established in system for irhich y= yo and there is
no error.
One of the most widespread adaptation control circuit conf igurations is shown in
Figure 8.10, where the control circuitry for a module and for the zrgument of the
complex frequency response tv2 is shown (Figure 8.10a), as well as its schematic
representation (Figure 8.10b). In the case of a linear control characteristic f.or
the weighting coefficients:
ta2 = au i
(8.30)
The behavior of the system depicted in Figure 8.10c is described by the expres
sions:
dul .
dl �kel. et=eXt; e:~ y_ ilo; f ~iflmXrn.
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Eliminating all of tlie intermediate variables froie these expressions and taking
into account the averaging narrow band response of the control system, we obtain
the system of equations:
dtv
(
dt akY xtxmu~tyozml, m= 1.2.....M. (8.32)
, ~t 1
We write this equation as follows in vector form:
dw
dt  _ ak (Rxx W Sxd.)=' ak (RXX WXyo). (8.33)
It follows from (8.33) that in the steadystate mode, the weighting coefficients
are:
W = WycT== RXX XJo. (8.34)
S. S.
In this case, as ean be seen from a cdmparison with (8.24), the mean square error
ey(t) ig minimal and the entir.e system provides for optimal Wiener filtration.
The adaptation circuit described here was proposed for the first time by Widrow.
The main drawback to this conf iguration is the necessity of apriori knowledge of
the form of the useful (requisite) signal yp. However, if it is knonw, then
adaptation is not necessar}r.
In the case of signals which are weak relative to the noise, the useful signal
power can be disregarded (yp = 0). In this case, adaptation is realized, however,
in accordance with (8.34) all of the channels are blocked, W} 0 and the antenna
ceases to respond not only to the interfence, but also to any other signals.
Various methods have been proposed for eliminating this deficiency. One of the
techniques is dual mode adaptation. In the f irst mode, it is assumed that the
useful signal is absent and that the interference is suppressed (along with the
useful signal, if it is present!). In the second mode, the inputs of all of the
channels are switched to a useful signal simulator for Xp. In this case, the
known output signal yp for the input signal being simulated ts fed to the control
c ircu it .
If prior to the start of the f irst mode, the coeff icient vector is designated as
W[n], then by the end of the first mode of duration T1, we obtain the increment of
the coefficients in a first approximation determined fram (8.33) when yp = 0:
AW [nJt tl dW  al:rl RXXW [nJ, n = 1, 2.
di W [n, (8.35)
g,=u
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Similarly, when Rx{ = RgO{O we determine from (8.33) the increment of the coef
f icient vector during the second mode which is of duration T2:
AW(n =t1dW _
1 ~ akis (Rae X. V1 [n]  Xo yo)�
RX. x. (8.36)
Thus, by the start of the next f irst mode, the coeff icient vector W(n+1J is
determined froin (8.35) and (8.36) by the recurrent equation:
w In + I I = W[n]  ak {(TL RXx {TS Rx. x.) W [n) Ta Xo 9n} _
=W fn1ak t(ti Rxxs, Ra�. xo) W [n]  ts Rx. X. Wo} (8.36a)
.
After completing the transition modes W[n+l] = W[n] = W[] and using (8.36a), we
f ind :
W["O1= (tt RXX Ta RX. x.)' T: Rx. X. �'o � (8.37)
It follows from (8.37) that with an increase in the signal level, adaptation
approaches optimal Wiener filtration when the interference and the useful signal
are independent, and in the absence of interference (X., = 0)., the adaptation
system provides for the requisite directional pattern.
Various circuits are possible which assure a compromise pattern between that needed
without interference and the optimal one in the presence of interference. Some of
them are shown in Figure 8.11.
s,
o) (a) (b) 0)
Figure 8.11. Variants of adaptation
circuits.
Key; 1. FNCh = low pass filter.
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The circuit configuration described hers with mode switching is shown in Figure
8.11a; open loop and closed loop single mode feed circuits for the desired distri
butions, tup, in the form of low frequency control signals are shown in Figures
8,1lb and 8.11c.
8.6. The Design of Beam Steering Systems for a Specified Precision of the
` Directional Pattern Orientation in Space
For the correct choice of the parameters for the beam steering system of a
phased array it is necessary to analyze the errors produced by the control unit
and their impact on the directional pattern. Some three types of control errors
can be singled out according to the point of occurrence: input errors, computer
~ error s and output quant ization errors. In this case, the phase error el enent of a
phased array can be represented by the expression:
Ft 1 =�im I lLtr{ FtIA, (8.38)
where uim is the phase distortion caused by the input errors; uir is the phase
distortion caused by computational errors; u i0 are the phase shifter quantization
error s.
If the input data in expression (8.1) are specified with errors of dRi = RiSri,
drm, then the resulting error in the computations will be:
6d; (R, 8ri,,) I(gll; r~~~) l~in, I Rtr� (8.39)
The quantity
ltin, Ri (r; 8ripl)
(8.40)
1
is the function for each element of the error in the representation of the vector
rm by a digital code, where the argument of this funetion, arm, which is common
 to all elements is determined only the direction of rm.
The second component
~ IZ1 (8.41)
! ir
is determined by the errors in the digital code repr.esentation of the vector Ri.
The quantity Sri di:ffers for each element in the general case.
The quantization error eip is distributed over the set of elenients uniformly in
a range of +A/2, where A is the phase shifter quantum step.
Thus, the phase distortion eim is functionally distributed over the elements of
the phased array. The quantities uir and up in the case of a large number of
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phased array elements not mutually correlated and are randomly distributed over
the elements of the phased array. The mathematical expectations of the errors eir
and eip are equal to zero by virtue of the symmetry of the distribution of these
quantities.
The dispersion of these errors is:
fi
[Pt�r I l~~ t..1 [I~ ~.A I Rlz qr112 I'Aa/11 .'Y)r I;//ns (8.42)
� ~ /r
where qr = 2Pr; pr is the number of digiL olaces in the camputer.
In analyzing the impact of control errrors an the shifting of the directional
pattern maximum with dev iations in thephase direction rm from the axis of symmetry
of a convex phased array through an angle of A< 1520�, the vector for the dis
placement in the directional pattern maximum can be represented as the vector sum
of two orthogonal components. .
It has been demonstrated that in a f irst approximation the position of a directional
pattern maximum in space is defined by the normal to the plane of regression of the
phase distribution. Taking this displacement of the directional pattern maxY.mum
into account, hX and hy are defined by the following ratios:
N N .
v'.~'!(lLir i ltiA) v~JI (E1 tr 1I~iA) (8.43)
t i
hY N , hy
a�i: ZJ12
N N z
I/fxl  xz ('~,..1 .~A)/ (Ari .e;Cl, (.9),. I~A); (8.44)
/
M lhx1na jh,Il o� (8.45)
In a rectangular planar array of m x m cells, the displacements of thL,' direc
tional pattern maxima in the XOZ and YOZ planes are independent and are determined
by the errors in the calculation of the phase change over an array step, xp, Y0+
and by the quantization errors of the cells, ukl:
ni m ur (8.46)
Ir� ` l,'L~tlm.~~~~ ~j h~ I Ion/ao;
!r ~ 1!== I h, I
rn
(h.1'I_. .,y)O/,Pn
(8.47)
where up is the error in calculating the phase shift for one step of the planar
array; ukl is the quantization error of the klth cell; 1:1 are the numbers of the
cells with respect to rows and columns.
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~ The shift in the directional pattern maximum because of the functional error is
def ined similarly:
N I 'V N N
!t ~Xi flIrn (8.48)
i r ; I � j �
Sin.:e the following condition is met Lar an axially symmetric phased array:
N N
v~ xiJi�.. v .cFZt=U,
1=I 1=1
Then expression (8.48) can be represented in the form:
(8.49)
6rm N ~5frn
J, N ~ xt Rt xt xt 1 xt i~
J a�; r= ' ~ x; ;:.N : i ~N `i (8. 50)
~ i ~
' .N
I k ~ Xt z` ,
1
1~= 1
6rin fixm, (8.51)
where qm = 2'Pm; pm is the number of digits which specify the components of a unit
vector.
An analysis of'the random quantities hX and hy shows that they have a two
dimensional gaussian distribution while the correlation coefficient between the
quantities hX and hy is equal to zero. In this case, the scalar value of h=
= hX~ has a Rayleigh distribution.
h ( h~ 1 (8.52)
cp(h) _ ~2 eXp l 20' I. '
. n rr /
where rt2 rt c:. . nN IIr,.] .h Ihi I�
The distribution of the system of randam quantities fX and fy is uniform in a
range of :
_'9md2 .l fx < 9m/2. qm12 fII < 4rn/2:
9m V Z ~l ' ; jX 1 f N ~~irt/4_jqiR~'1. 4m V 2 /2. (8.53)
.
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Thus, the input errors determine the discrete positions of a phased array beam in
space. Its displacement, which is causdd by these errors, depends only on the
number of digits in the code which specifies the direction cosines of the vector rm.
Computational errors and output quantization errors lead to a random shifting of
the beam from its discrete positions. The shift depends on the phased array
geometry, the number of elements in the array, the word length of the computer and
the word length of the phase shifters which are used.
We shall consider the procedure for calculating the parameters of the control
system computer by a phased array. Let the d iscrete step f or beam steering, fmax
(radians) and the ultimate pPrmissible steering error, hmax, be specified, where
f max rknax �
We det.ermine the number of digits pm in the input code from expression (8.53):
9mvl /'l G fmnx. (8.54)
Considering the fact that qm = 2pm, we obtain:
(8
Prn > (~or;sfm:ix0,5). .55)
We determine thw word step pr by means of relationships (8.42) or (8.44) given
the c ond it ion :
hioax 0,9:~ (h) _JC~~~~i~'Ir ~0s). (8.56)
For this, we express the permissible dispersion Dr:
hiiiax19CI, OA. (8.57)
I
For a known value of the quantum step of the phase shifter and a known phased
array geometry, we f ind:
2 N N
(8.58)
t a c,t
Taking (8.42) into account, we obtain:
1 12 p ~
9r ~ R ( 9CR A) (8.59)
' whence: Pr > _1092qr (8,60)
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The eamputation sequence established here makes it possible to determine tY3e word
length of the input data code and the word length of the computer based on the
requisite discrete step and precision of phased array beam steering, givert the
coiidition that the coordinates of the phased array elements and the phase shifter
quantum step are specif ied.
Sample Calculation. 1. The planar array m x m, m= 11, xp = yp = 2ff and 0=rt/4
is specif ied. The requisite discrete step for the beam is 1� and the beam steering
precision is 10'.
We express the angular quantities in radians:
f max = 1.75 � 102 rad; h max = 0.292 � 102 rad.
Substituting the value fmax in (8.47), we obtain:
pIII > (logL�1.75 � 102 + 0.5) = 55 [sic].

We choose pm = 6 digits.
We determine the square of the permissible beam deflection:
h2 (0.292 � 102)2 = 8.5 � 106 rad2.
max
The second term in equation (8.47) is:
On 4 2 I 12 1,63 � IQ1
e ~ / ) ~ ~ 4,~7.1U~ =p,341Uapa119.
ntzo : P2 11 (2n)2 2 (122 3a _142 .152)
~
nt
The first term in equation (8.47) is:
r !r' 8 5� 10e
ninx _ U~~~'1�~~e ~ 9 0~34�10'e=~,~�I~i'~ peJ(a.
We determine the computer word:
0r=xo9,1112� ~0r/xo=pr/12==0,ii�10a.
whence:
q,27,2� 10a, 9r=2Pr _2,G8109,
From the last expression, we have the following:
prlog2 _ 3 + 0.43 = 2.57; pr > 2.57/0.3 = 8.6
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We choose p2 = 9 digits.
2. A hemispherical array has a radius pl = 25 � 2w, A _w/4 and C1 = 0.2 � 104.
The digit capacity of the input code pm has already been determined. We substitute
the initial data in equation (8.58);
8'~'.1pa  (n/9)2 =4,72�10aI,G3�102=3,09�101 paA2.
~ ~ 9,02� 1()4 12 ,
 We obtain the following from formula (8.59):  
yr G 1 1/12�3,09� 102 . 6,09� 101 _3.87e 109.
25�2n 1,57�102
From this we obtain pr = 8 digits.
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RADIATING ELEMENTS OF AN ANTENNA ARRAY SECTION II
9. PRINTED CIRCUIT ANTENNA
9.1. The Function and Specif ic Features of Printed Circuit Antennas
A Printed circuit antennas differ in their structural design from other types of
microwave antennas. Not only radiators, but also transmission lines, r:,tching
~ elements, etc. can be made using the techniques af printed circuit technclogy.
I
More than other antennas, these meet Lhe requirement of miniaturiz3tion, one of
 the major requirements for aircraft equipment. This explains the increasingly
widespread use of printed circuit antennas.
We shall note tte major advantages of printed c3xcuit antennas:
Structural simplicity, small volume, weight and cost;
Convenience in combining antennas with printed circuit feed lines and devices;
High fabrication precision, because of which good repraducibility of antenna
_ characteristics is achieved;
The ability to design antenna structures for aircraft which protrude little or
not at all, in particular, structural designs which do not change their strength
characteristics.
_ The drawbacks to printed circuit antennas include poor� electrical strength, the
difficulty of designing tunable devices and measuring the parameters of printed
circuit components. .
Printed circuit antennas are used in a frequency range of from 100 MHz Y.o 30 GHz
at low and moderate power levels. At very low frequencies, the size and weight
of antennas which are comparable to the wavelength become quite considerable. At
higher frequencies, these antennas have no advantages as compared to others.
Printed circuit antennas are poorly directional and for this reason, they are
used primarily as constituents af antenna arrays.
9.2. The Major Types of Printed Circuit Antennas and Their Operational
 Pr inc ipl e s The major elements which form an antenna are the radiator (the antenna itself)
and the excitation device. Pr.inted c3rcuit antennas correspondingly differ in
the operational principle of the radiator and the manner of its excitation, as
well as in the type of transmission line. Moreover, the radiation characteristics
of the antennas and their structural parameters can also be distinctive
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attributes. Primarily the first group of attributes is treated in this chapter.
The radiation characteristics of the antennas are of the togic treated in this
chapter.
Striplines are most frequently used as the transmission lines. As a rul.e, the
type of striplines governs the structural design of the other antenna elements.
In the low frequency portion of the band, the excitation is accamplished by means
of coaxial linES. It is also possible to use a waveguide transmission line.
Resonator type printed circuit radiators designed around asyaQnetrical striplines
(see Chapter 2) also find widescale application. Another more traditional type
' of printed circuit antenna is dipoles of various configurations and slots cut in
' the metal wall of a symmetrical type transmission striplir.e. Developmental modi
fications of these antennas are stripline spirals and curvilinear radiators.
~ An example of a resonator printed circuit radiator is shown in Figure 9.1. This
radiator is used most often [1, 2]. It consists of a rectangular strip conductor
[1], placed on a thin dielectric layer (2) with a conducting substrate (3). The
radiator is excited by a strip transmission line. This system is a flat lossy
resonator filled with a dielectric for the transmission line, where the losses are
due to radiation. The edges of the resonator form two radiating slots A and B,
which are spaced Z apart, approximately equal to ad/2, where ad is the wave
length in the dielectric. At the edges of the resonator, the camponents of the
f ield which are normal to the canducting substrate are aut of phase. The f ield
components parallel to the conducting substrate add together in phase and form a
linearly polarized radiation field having a direction of maximuu: radiation along
to the normal to the plane of the substrate. The dimension b of the radiator
can differ.
To obtain a rotating polarization field, two pairs of radiating slots are needed
which are arranged perpendioular to each other and are excited with a phase shift
of 90� each. For this, a square radiator is chosen which is excited at two points
in the ce�tcr of adjacent sides of a strip conductor. The excitation is realized
most readily by a rectangular radiator with a single feed point, which is shown
in Figure 9.2. One side of the strip conductor of the radiator is greater than
ad/2 by the amount A, whl.le the other is smaller by the same amount, something
which provides for the 90� phase shift for each. The quantity 0 is chosen experi
menta.lly. The radiator is excited by astripline. A possible excitation variant
for this r.adiator is a coaxial line perpendicular to the conducting substrate.
The center conductor of the coax line is connected to the str.ip canductor of the
radiator.
Other types are discrete radiators in the form of printed circuit dipoles and
slots. The current in the strip conductor of the radiator serves as the radiation
source in this case. Slot antennas, excited by a stripline, are a direct analog
of slotted waveguide antennas. They are widely used as the radiating elements of
scanning antennas arrays. With the appropriate excitation, one can.use such
radiators to design antenna systems which realize extremely arbitrary directional
characteristics.  191 
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~
~
r b.  ~ ~
I
3 ~ ~ .
/lonocti�o/1on
2 ~.t plin~'"ar~
Figure 9.1. A printed circuit resonator
antenna with linear polar
ization.
tripline
� !/or,ncHOB~p
� o' "O~
Figure .9.2. A printed circuit resonator
antenna with a rotating
polarization field.
11. 1102
%
/~2 ~v?
a) (a) (b)dJ ~
Figure 9.3. Antenna excitation circuits.
One of the methods of exciting radiating elements is excitation using a system of
branched lines of the same electrical length. (Figure 9.3). If the excitation is
realized using a line with a characteristic impedance pl, then with ON branches
having a characteristic impedance of p2 (Figure 9.3), the relationship pi = Np2 is
observed. With a large number of radiators, it is exped3ent to insert a,trans
former ahead of each branch (Figure 9.3b). Such an excitation techniques is
realized especially convenientJ.y using striplines.
In another approach, the traveling wave excites the radiating elements which are
arranged along the transmission line. This technique is also realized quite well
using strip transmission lines. A drawback to it is the great dependence on
frequency. Other excitation met'hods are also possible, but they are used compara
t ively rarely. Almost all of the elements of a feed line channel which are used for coaxial and
waveguide transmission lines, as well as the feeder channel as a whole can be
constructed in a printed circuit design. However, as a rule, only individual
printed circuit assemblies are used in a feeder channel. For printed circuit
antennas, coaxial or waveguide lines are most frequently used as the main feed
line. Because of this, i.t becomes necessary to have elemertts for joining strip
lfnes to waveguide and coaxial lines. The major crnnponents of striplines,
'  192 
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. ~ = � ,
' ~ " FOR (r'FFICrAL US.1~,4NLY
.
including coaxialt^~ ~;nd k_veguide to striPline junctions, as well as
stripline connectors and splitters, are described in [07, 014, 021.
9.3. The Major Cha.racteristics and Design of Printed Circuit Resonator Antennas.
We shall single out among antennas of this type the antenna which is shown in
Figure 9.1 as the taasic antenna. The linearly Polarized field is produced by the
radiation of two slots, which form the walls of the resonator, which represents a
halfwave section of an asymmetrical stripline. Antennas of this type are usually
employed as receiving antennas.
It is assumed in the antenna design that the dimension h(Figure 9.1) satisf ies the
condition kh � 1, wtiere k= 21r/a, a is the working wavelength. It is also
~ assumed that the field distribution in the radiating slot corresponds to a T mode
f ield distribution in the crosssection of a regul.ar stripline. In this way, the
influence of higher modes on che radiation of the slot is neglected.
These presuppositions make it possible to represent the radiating slot of a resona
tor as a linear radiator, sinilar fio a narrow slot in a conducting shield
Figure 9.4). Thus, the analysis of a resonator antenna reduces to the analysis of
ordinlry slot antennas. The field in the radiating slot,of the antenna has the
form E= xpEX, ixl < h/2} Th4s field determines the magnetic~ current of an equiva
lent linear radiator as IM = z02EX, Izl < b/2, where xp and zp are unit vectors of
the coordinate system of Figure 9.4.
The Antenna Directional Pattern. The field of a linear magnetic radiator is lmown
(f or example, see 01). The electrical f ield of the radia.tor has component s of
Ee and Ee in the spherical system of coordinates of Figure 9.4. The antenna polari
 zation is determined by the projection of the Ee component on the normal to the
plane. of the slots (the Y axis). Then for Lhe indicated polarization, the direc
= tional pattern (DN) of the antenna. as a system of two equivalent linear radiators,
which are excited in phase, has the form:
(0, (p) sin ~~'h~ cos os cos A Cos SiI101. (9.1)
4p \ i
The first two factors in expression
(9.1) define the directional pattern
of an equivalent linear radietor for
the ind icatad polar izat ion, while
the last term is the directivity
characteristic of a system of two
identical radiators, spaced a distance
Z from each ather.
Figure 9.4. The coordinate system for the
radiating slot of a printed
circuit antenna.
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1~ 1
~o
,OA ~ jB IOA jB ~ Figure 9.5. The equivalent circuit of a
printed circuit resonator type
~ antenna.
taxa1h
>0
 10
10
Figure 9.6. Antenna slot conduct
ance G as a function
of the quantity b/a.
9, 7
0,4
0, 3
> >U G/9
Figure 9.7, The cu:itity lequiv/h as a func=
t ion of the rat io of the d imen
sions b/h and the dielectric
permittivity e of a printed
circuit antenna.
~ The Antenna Input Admittance. The equivalent circuit of the antenna as a trans
mission load is shown in Figure 9.5. The two radiating slots of the antenna,
 havi.ng an input admittance of Y= G+ jB are separated by a line section of length
 Z with a low cnaracteristic 3mpedance of pA. The input admittance of the antenna
 Yin is the result of camb3ning the slot admittance at the antenna input (the
terminals i1') and of the slot which is transformed to the input through the line
gection Z so that:
~
Y�x G .1_ i13 + yn (0f'l13)IYAtg P1 (9.2)
Y. .
ln YA1 (G +1 a) t6 P"
where a is the line propagation constant; YA = 1pA.
The radiation conductance G is calculated by the method usually employed in ~
slotted antenna theory. The conductance G is shown in Figure 9.6 as a function
, of b/a. For b/a > T, we have [2] :
G [Ohms1) = b/120a (9.3)
The reactive component B of the slot admittance is due to its capacitance and is
computed from the formula:
 194 
I ~ equiv~h
a=1,0
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B [Ohms1] = Zeqaiv/6011
(9.4)
where Zequiv is a quantity equivalent to the length of a stripline open at the end
having the same input admittance; A= a/seff Where A is the wavelength in the
stripline; eeff is the effective dielectric permittivity of the substrate, which is
determined in [014, p 621J. The curves for Zequiv/h for various values of e are
plotted in Figure 9.7 as a function of b/h. _ An antenna is tuned to resonance if its input admittance is areal quantity. The
_ resonance condition follows fram expression (9.2) :
1g (t/  27'A /3/(G= I /32. 11 A2), (9.5)
Expression (9.5) def ines the resonant length of a line section Z having a low
characteristic 3mpedance pA. In this case, the input admittance of the antenna is
Yin  2G. The quantity Z computed in this fashion is somewhat less than half of
a wavelength in the stripline.
The design of an antenna consists in computing the dimensions of its resonator and
selecting a stripline to obtain the specified width of the main lobe of the
directional pattern (or directio*_~al gain) of the atitenna. Additional requirements
are set which are related to the conditions for the 'placement and operation of the
antenna on board the vehicle. These requirements are important when selecting the
dimensions of the strip conductor and the dielectric substrate of the antenna which
are its major structural components. The design of an antenna is most eaElily
accomplished by means of trialanderror selection of its parameters..
The selection of antenna dimensions consists in the following. Based on a speci i
fied directivity characteristic, the dimension b is determined for the strip
conductor of the antenna (Figure 9.I). In this case, the dimension Z is assumed
to be equal to 0.0 to 0.5a. The stripline conductor can ha.vp either a square or
a rectangular shape. The characteristic impedance pA of an asymmetrical stripline
depends on the value of b(Figure 9.5), where this impedance should not be too
low and usually amounts to 10 to 15 ohms. Then the h dimension of the antenna
is chosen, usually h< 0.1X, as well as the material of the dielectric substrate
[0I4]. The dielectric permittivity of the substrate is most frequently chosen
equal to e= 2.252.5. In individual cases, a ceramic (e = 10) can be chosen as
the substrate.
The selected antenna parameters make it possible to calculate the characteristic
3mpedance pA of a low impedance asymmetrical stripline as well as the input admit
tance of the radiating slot of the antenna Y= G+JB using farmulas (9.3) and
(9. 4) , taking into account the function shown in Figure 9.7. The propagation
constant s of a low impedance line is determined from [014]. The resanant length
of a low impedance stripline section Z and the 3nput admittance of thb antenna
Yin are determined fran formula (9.5). An asymmetrical stripline with a character
istic impedance of po = 50 ohms is usually chosen as the transmission line. A
matching element in the form of a quarterwave transformer is used to match the
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ii Plane n
vi. nh
17,qoch'49c na f tf pp
E Plane
Key: 1. Printed circuit antenna.
antenna to the stripline. Matching is an extremely labor intensive operation.
It is accomplished by the trial and error design of the matching element and is
more successful, the closer the c3aaracteristic 3mpedance of the antenna is to the
characteristic impedance of the line. Where there is a substantial difference in
these values, the antenna design procedure is repeated for its other parameters.
A linearly polarized antenna (Figure 9.1) with a sqare stripline cenductor
designed for a frequency of 9 GHz, has the following characteristics. The radi
ating slot admittance of the antenna is Y=(0.922 + j7.45) �'103 ohms 1. The
resonant length of the antenna is Z= 0.46a when pA = 15'ohms. The antenna is
matched to the stripline having a characteristic impedance of pl, = 50 ohms by
means of a quarterwave transformer. In a passband of Of /fp = 2%, the SWR is less
than two. The measured gain is 7.6 dB iqith losses in the line of 0.3 dB. The
typical directional pattern of the antenna in the E and H planes is shown in
,Figure 9.8. '
The antenna is extremely narrow band. To improve the bandwidth performance it is
recammend2d that+�.the characteristic impedance pA of the low impedance stripline
be increased, a dielectric substrate with a greater value of e be selected to
reduce the resonabor length, the inductance of the antenna be increased: by means
of making holes ar slotted cuts in the stripline conductor of the antenna, as well
196
(1
i
50 0 Input
FOR OFF[CIAL USE ONLY
Figure 9.8. The directional pattErn of a
printed circuit resonator type
antenna.
G
i.
/IC�/I!/r/lUA 2y71PNN.?
er �.�n t~ ~
4'^ K'0 ~f^ MC
(1 ~ /1r, vum.von ~/iime ya~
11 1 10 fl1 41 :/I �O~ l0 59
sa tin
, I.
~n ir. n+ ~n
Fxud .s/lllro
50 S2 Input
Figure 9.9. Excitation conf igurations for a
printec circuit resonator type antenna
having a large value of the dimension b.
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as that broE.dband techniques for matching the antenna to the transmission line be
employed. A11 of this will make it possible to increase the passband of the
antenna Af /f 0 up to 50%.
For an antenna with a rectangular strip conductor and dimensions of d> a, the
excitation system is built for the condition of equal electrical paths of the
branched *_ransmission lines (see 49.2). Several points are excited in the strip
conductor in this case. Two excitation configurations are shown in Figure 9.9 fcr
an antenna with a dimension b= 2X for a line with a characteristic impedance
p~ = 50 ohms. The proposed technique is also applicable to the design of rota
tionally polarized antennas.
9.4. Antenna Arrays with Resonator Elements
Antenna arrays with radiating resonator type elements are constructed in the f orm
of strings of radiators and sets of these strings. When designing a linear
antenna array, it is usually assumed that the radiators are arranged at equal
spac ings d from each other and are excited inphase or with a constant and small
phase difference. The analysis of such arrays is performed as an analysis of
inphase arrays, with subsequent accounting for the inclination of the main lobe
of the directional pattern if this is necessary. Such excitatin presupposes a
single transmission line for a linear array. It is also possible to excite array
elements where the electrical length of the transmission lines are equal (see
�9.2). .
Basic Relationships for a Linear Array. The directional pattern of a linear
system of identical radiators with inphase excitation has the form (see
Chapter 2) :
N
FN /I� exp [ jk (n1) d cos 0),
(9.6)
where An is the amplitude of the nth radiator; 9 is the angle read out from the
axis of the array; N is the number of radiators. It is assumed in this case
that the number of resonator type elements is N/2. If the spacing between the
radiators oi the array is d= a/2, then the directional gain of the array is:
N a
D ( A�}' rv~ A~,
1n=i / n=1
(9.7)
The greatest directivity of an array is achieved when all of the amplitudes are
equal: An = A. Then the directional gain of the array is D= N. This is the
case of uniform excitation of a linear array and it is of the greatest practical
interest.
The directional pattern of a uniform array, using the principle of directional
pattern multiplication of [Ol], can be written in the form:
(p, Fi (0, (P) rN (0), (9.8)
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where F1(0, is the directional pattern of a single radiator; FN is the group
directional pattern of the array. The directional pattern of a resonator type
element is described by expl�essior, (9.1). In the case of N/2 elements in the
array, the graup d irect ional pattern is:
FN = sin (N(D/4)l(N12) sin m, (9.9)
where (D _ (kd cosA 4)p) is the phase shift between the fields produced by
adjacent elements; ~Pp is the phase difference in the excitation of the adjacent
elements. In the case of inphase excitation of the elements of the array, (Dp = 0.
Ways of Exciting Array Elements. In the case of inphase excitation of re.sonator
type elements, one speaks of resonani excitation of an array. In order to avoid
the appearance of secondary main lobes in the directional pattern, the spacing
between array elements (taking the directional pattern of an element into account)
(Figure 9.8) should not exceed a/2. Resonant excitation of an array is character
ized by the fact that the main radiation is directed along a normal to the plane
of the array. The major drawback to such excitation is the poor matching of the
array to the transmission line. For an array of four series connected resonator
elements, designed for a frequency of 9 GHz, the matching passband f.)r a SWR'level
of no more than two amounts to 1.7 The resonant frequency, the ilput admittance
of the array, as follows from the schematic ghown in Figure 9.10, is Yin = NG,
where Y= G+ jB is the input admittance of the radiating slot of a resonator
element (see g9.3); N is the number of slots. An antenna array with the elements
excited "off of resonance" in a traveling wave mode is free of this deficiency.
With a large number of elements, the reflections fran each of them "on the average"
cancel out, which provides for good matching of the antenna array. A drawback to
this excitation hechnique is the devi.ation of the direction of the main lobe fram
a normal to the plane of the array, which changes with a change in frequency.
However, with a small phase difference for the excitation of adjacent elements
"close to resonance", this deviation is small.
+ * ~fAZ + Figure 9.10. The equivalent circuit
of a linear inphase array
Y using printed circuit
resonator type elements.
An example of an array excited in a traveling wave mode is shown in Figure 9.10.
One end of the array is connected to a coaxial feed line, while the other is
loaded into an absorbing load. The angle of inclination 6 of the main lobe of the
directional pattern to the antenna axis is computed from formula [3]:
cos 0 (l 0,5b)]l1, (9.10)
where Z and b are rhe dimensions of the array. It follows from formula
(9.10) that the inclination angle 6 changes with a change in frequency, where the
 198 
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main radiation is directed in a direction opposite to the direction of wave propa
gation in the transmission line. It follows fram the theory of periodic structures
that this is explained by the choice of the propagation constant S= k cos9 for
the spatial harmonic which is responsible for the primary radiation.
The characterist ic f eatures of antenna arrays with resonator type element s, when
they are excited in resonance and traveling wave modes are similar to the features
of slotted waveguide arrays which were treated in Chapter 5, with the same excita
tion modes.
The design of an antenna array consists iii selecting the number of elemPnts in it
and designing the elemei,ts for a specified directivity, i.e., main lobe width of
the directional pattern or directional gain. A uniform inphase array is taken as
the basis for the design calculations, for which expressions (9.7) (9.9) apply.
The calculation of array gain is extremely approximate, since it is necessary to
take transmission losses into account, and the calculation of the gain can serve
only as a qualitative estimate of the selected antenna circuit. The design proce
dure for a linear array is as follows.
Antenna
ANmeNHa
' Y
1(c:. ;=~l F
Baod ~ Hn~,ny~Ha
Input Load
Input o the
load ,0A1"
Figure 9.11. A linear traveling wave array Figure 9.12. A printed circuit
with printed circuit resonator antenna in the form of
type elements. a composition of
Key: 1. Resonabor antenna elements. linear traveling
wave arrays.
The number of resonator type array elements, N/2, is chosen for a specified direc
 tivity. This number is taken equal to the directional gain of the array. Then
the radiating element is designed using the procedure given in �9.3. The spacing
between the array elements d is chosen equal to the dimension Z, which is the
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11
resonant dimension of an elelnent. The dimension b of a radiating element is.
chosen equal to either Z or a for a rectanguiar stripline conductor. In the case
of a large v.Alue of b, the excitation of, an elemeitt is camplicated and takes on
the form shown in Figure 9.9. With inphase excitation of the antenna array, the
directional pattern is computed using formulas (9.8) and (9.9) for lop = 0.
When an array is exc3ted in a traveling wave mode, the angle of inclination of the
 main lobe of the directional pattern is calcula_ted using formula (9.10). This
makes it possible to determine the phase shift, ~Dp, and to employ formulas (9.8)
and (9.9) to calculate the directional pattern as well as the directional gain of
the antenna array. The gain of the array is determined bq the value of the eff ici
~ ency, which under conditions of weak coupling of the radiators to the transmission
line may be less than 50 percent. The radiation losses in a line loaded with an
antenna array are taken at a level of 10 dB, which makes the results of analyzing
traveling wave antnnas reliable and makes it possible to obtain the optimal gain.
The coupling of the radiators of a yagi antenna to a transmission line is governed
by the h dimension (Figure 9.11) and the characteristic impedance of the line
p. The smaller h is, the smaller the attenuation constant a for the traveling
wave in the line. It is assumed in this case that the propagation constant a
does not change aver the length of the line. The longer the antenna array, the
smaller the height h. For an antenna structure with a length of 20a, the height
h reaches 0.025a.
If the yagi antenna designed in this manner does not have the requisite directivity,
then its design calculations are repeated for a different number of radiating
elements. A traveling wave array, designed for a frequency of 635 MHz, has dimen
sions of: Z= 0.4X, b= a and h= 0.075a [3]. A set of strips is used to improve
ths directivity. An example of an antEnna of four strips is shown in Figure 9.12.
9.5. Printed Circuit Dipole Antennas
Dipole antennas and modifications of them are some of the most used radiators in
antenna engineering. They are used particularly as the radiatin,g elements of '
largP antenna arrays. This explains the ever greater use of printed circuit dipole
antennas. A stripline dipole takes the form of a strip conductor on a thin di
electric layer (Figure 9.13). When used as a part of an antenna array, a printed
circuit dipole is usually positioned above a flat conducting shield.
The design calculations for a printed circuit dipole can be performed as tM
calculations of a strip dipole, with the subsequent accounting for the impact of
the thin dielectric layer. In turn, a correspondence can be established between
the strip dipole and c , dipola with a circular crosssection (a wire dipole), which
has the same directional pattern and input impedance. In this case, the cross
sectional dimension of the wire dipole is half as great (Figure 9.13). Such a
camparison is experimentally conf irmed given the condition that the length of the
strip dipole 2L is substantially greater than its crosssectional size 2d where
2d � A. In this case, to calculate the cha.racteristics of a strip dipole, one
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an use the results of numerical and experimen+tal studies of fine wire antennas.
The influence of a dielectric layer consists in changing the length of a strip
dipole, in particular, in shortening the resonantllength of the dipole.
The Current Distribution and Overall Input Impedance of a Strip Dipole. The
surface current, (x, Y) Xo Y" (t, J), , induced in a narrow strip cariductor of
a dipole where d < y< d and L < x< L, can be characterized by the quantity:
d
1(X) _ _ .1 3t (x, y) dy,
n (9.11)
d
which is used in calculating the total iaznut impedance of a strip dipole. The
surface current v (x, i/) has a singularity at the corner edges of the strip
conductor, which is of a local nature and constant over its length. Taking this
singularity and expression (9.11) into account, the current ~J� (,r, ,y) has the
representation:
(z, J) ! (z)/ Vdzyz.
Printed Circuit Dipole
!levamyai~% Fu6,oainop
~
Equivalent
Wire Dipole.
~n~BaBane~s+irHeiv
npoBonoyHaiu Budpamop
~
d
Figure 9.13. A camparison of a printed circuit dipole with a
wire one.
(9.12)
Taken as the current I(x) in this expression is the current of an equivalent wire
dipole (Figure 9.13).
The results of a numerical investigation show that the current distribution over
the length of the dipole approaches a sine distribution, as is adopted in approx
imate dipole theory [OlJ, only for a dipole length.of 2L < 0.571: Examples of the
current distribution for other values of L are given in [Ol]. Resonant Tength
dipoles f ind the most widescale practical applications:
The resistive and reactive components of the input impedance, Zin = Rin + JXin, of
a strip dipole are shown as a function of its length L for various values of d
in Figure 9.14. The value of the input impedance of a strip dipole differs from
the input impedance of the inf initely f ine wire d ipole which is treated in
 201 
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approximate dipole antenna theory. We will also note that the resonant length of
a strip dipole is close to 0.23a, and practically does not change with a change in
the d dimension af a narrow strip.
,pgX, O,y R, ohms X9x, OM g, ohms
1000 ' in . 400  in de001.t
.
dac,o~.z
800 
200
g03.t
0,05.~
s~p
69,1 0,2 0,3 04 45
400
0,05.Z
z00 
200 
'400
`
~ ~
600
0 0,2
0, 4 L/.i
.rJ (a)
1,8
1,4
f,0
0
Figure 9.14. The input �Lmpedance of a strip dipole as a function
of the arm length L/a and the dimension d.
s4 I's 
~
0,2 0,4 t/2.Z
Figure 9.15. The retardation of the
surface wave as a function
of the dielectric layer
thickness.
1,2 L_
A 0, >5
7,01
O 0,2 0, 4, h/.:,
Figure 9.16. The retardation Y of a
surface wave as a function
of the heignt h/a of a di
electric layer of thickness
t above the surface of a
shield.
The directional pattern of a strip dipole where L/d > 5 is taken to be the same as
for an infinitely fine wire dipole. The dipole directional pattern is shown in
[01]. However, when L/d = 5, ins;:ead of nulls, minima at a level of approximately
12 dB appear. Such "swelling" of the directional pattern nulls i.s undesirable
when a dipole is used as an element in an antenna array, primari.ly because of the
increase in the coupling between elements, the appearance of crosspolarization
of the radiation and the reduction in the gain. The design method described here
for a strip dipole is applicable to a conductor with a dimenaion of 2d < O.la.
I
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t/.; =o,z
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Arms of the dipole
/l#evo Bu~,oMo,oa
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Arms of the dipole
/li~evu 6ud,oamn,or~
40eyu
bainopn
Ao,aomKOaa
MaikameV
' )hort
zircuiter
/I~OOBOO'HU~YU 0./~ Hm,o(3)
(2)
Cu~r~em,ou,oy~ou~v/i4 )
3/!ei`lEH/J! ~
B~
ir) (a)
Figure 9.17. Excitation configurations for a printed circuit dipole
using a balanced (a), balanced threeplate (b) and a
twowire (c) stripline.
Key: 1. Dielectric substrate;
2. Conductors;
3. Dielectrics;
4. Balancing element.
The Influence of the Dielectric Layer. The dielectric layer ')f a printed circuit
dipole is chosen to be extremely'thin t< O.la, since it is obly a structural
component with low losses. For this reason, as a rule, it does not influeace the
directional pattern of a dipole and is consi,dered primarily when calculatiag its
resonant length. The shortening of a dipole depends on the retardation of the
electrical wave propagating in the planar dielectric layer with a thickness t.
When t+ 0, these waves degenerate into a free space Ttype mode.
The retardation Y= c/uo of a lower mode is shown in Figure 9.15 as a function of
the layer thiclmess t[4]. The retardation y of the indicated mode is shown in
Figure 9.16 as the function of the thiclmess t of a dielectric layer positioned
above a conducting shield at a distance of h from the shield where the dielec~�
tric permitbivity of the layer is e= 4 [4]. The resonant length of the dipole
is taken equal to LreS 5 0.23A'/Y�
Excitation of a Printed Circuit Dipole. The transmission line can be tied into
a printed c ircuit dipole both perpendicularly to the strip conductar of the :1
dipole, and in the plane of the conductor. In the first case, a coax line with a
balancing device is usually employed, just as in the case of a wire dipole. In
the second, excitation by means of balanced stripline f inds the widest applica
tions (Figure 9.17a, b). Sometim es the excitation is acc)mplished by means of
a two conductor stripline (Figure 9.17c). As a rule, striplines connected to the
input of a dipole by means of transition couplers [07, 014] are connected to
other tynes of transmission lines (stripline and coaxial transmission lines, as
well as waveguides), which are more convenient in structural terms and have
better characteristics.
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9.6. Antenna Arrays with Printed Circuit Dipole Elements
.:s"J
Printed c�ircuit dipole radjators are successfully used as phased array elements
both for transmission ansi reception. The basic mode for the study of large planar
arrays is an infinite array, the radiating elements of which are excited by a
current having the same amplitude and a linearly changing phase. Such a model�can
yield satisfactory results for a d3pole array aver a plane shield where the number
of dipole el enents is ju st 10 x 10. The analysis of a dipole antenna array can
sists in analyzing the input impedances as a function of the scan angle. Knowing
these impedances, the influence of the latter on mismatching in the f eed system
of the antenna array can be minimized.
Printed circuit dipoles in a periodic antenna array are placed at its nodes,
usually above a conducting shield. Printed circuit dipoles incorporated in an
antenna array can be combined in quadrupole elements (quadrupoles), as shown in
Figure 9.18. By changing the interconnection of the dipoles iL3 a quadrupole, one
can substantially change the characteristics of the antenna array. Printed cir
cuit dipoles are usually assumed to be resonant and have a size of 2d �X. Under
these conditions, the analysis of an arr with dipole elements can be carried
out based on the existing literature [03, Vo? 2; 6]. For dipoles of arbitrary
length, a study of dipole arrays using integral equations is given in [7].
The Totai Input Impedance of a Dipole Element
of an Array Pasitioned Above a Shield. A
pxinted circuit dipole as an element in an
infinite array, depending on the numbers m, n,
has an exciting voltage at the input which
varies in accordance with tl'ae following law.
~
Y
Umn U"e_J,tiNdYe j�ry,in, (9.13)
where r: sin O cos y; (Z  ic sin 0 siit (p, n, 2n/X;
Figure 9.18. Quadrupole
elcments of
an.antenna
array.
dX and dy are the periods of the array.
Since an array is a periodic structure, the
surface current induced with such excitation in
the strip conductors oi the dipoles, can be
repr.esented by a Fourier series expansion:
~ ci> ~ vy)~1Rx eix~~ r~, ^ (9.14)
n!ooR= .M
where (i,,, r~ IZnnilti,�,.a,, a 'I 2niddrr ; i is the r.umber of the dipole which
combines the subscripts m' and n'. The coefficients r~n can be calculated if
the current distribution in the dipole is specified. This distribution is
imown for a resonant length dipole. Taking (9.12) into account, the surface
current of the ith dipole is defined as:
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cas "=Y ~ (9.15)
2t,
where I~i) ~;s t~:e current at the input to the ith dipole. Then the Fourier
coefficients in expansion (9.14) have the form:
2 l;; ) 2I. ~o n ~a ros (irn (9.16)
rnit dx d', 1_(20 m L/n)3 ~
where Jp is a zero order Bessel function.
The input impedance of a dipole is def ined as the ratio of twice the camplex
power P at the surface of the array within the bounds of its elementary cell to
the square of the absolute value of the current at the dipole input:
2!'
) n~L .
Z7 f '
. l/~ ~s 11)o d ~2t ll n LC7. ni n J mn~ X
m3oon:oo
x t(0�, /K)' ~9.17)
[1exP(JYmn 2/[)], Vmn /K
where y2 + s2 + a2 = k2; po = 120w ohms; h is the spacing from the array to t: a
shield.um m n
~ The series (9.17) converges, and when calculating the value of Z, one can limit
~ oneself to a f inite number of terms in the series. Rnowing the input impedance
{ of the dipole, it is not difficult to calculate the reflection factor in the
I transmission line which couples the dipole to the generator. It depends on the
; scan angle and is determined fram the formula:
~ _ _
~ P (a, (i)? Ipq, Z (0, 4j))Ji[P,J, 1 z (n, fi))1, (9.18)
~ where p0 is the characteristic impedance of the feed line. .
' Forcaulas (9.17) and (9.18) are easily sub3ected to numerical study:
When studying the 'influence of the input impedance of a dipole on the reflection
factor, which determines the conditions in the transmission line, a distinction
must be drawn between the behavior of the resistive R and the reactive X
components of the impedance. As studies of dipole arrays shown, these components
are different functions 9f the scan angle. For this reason, the convergence of
series (9.17) when calcula.cing the 'quantities R and X requires separate
treatment. When additional inain lobes are absene in the directional9 pattern of
 an array, one can lim{t r;aesE;lf to one term of the series (9.17), sahich corres
ponds to the numbet� m= 0 and n= 0, to calculate the resistive component R.
The calculation of the reactive component X requires taking a large number of
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terms of this series into account [03, Vol. 21. Ttxe detailed analysis of the
quanti.ties R and X depends on the specif ic dimensions of the antenna array.
However, there is no need in many cases to imow the true value of the total input
impedance Z, since the array elements are matched for a certain scan angle,
usually normal to the plane of the array. In this case, it is of interest to
change the input impedance when changing the scan angle, which reduces the volume
of canputational work.
The Total Input Impedance of a Quadrupole Eleznent of an Array. A systezn of twc>
coupled dipoles, which form a quadrupole element of an array (Figure 9.18), is
excited by the voltage Up of a generator which is connected to its center point.
Depen3ing on the number m, n of the quadrupole eleanent, the exciting voltage
var ies in accordance with (9.13). By representing the surface current induced in
the strip conductors of the dipoles with expansion (9.14), where i= 1, 2, we
obtain thE axpansion factors for the current in the following form on analogy with
(9.16) :
c~t___2 ~o~i2L lo(and cos~'mL)
~ rnn dx du ' ~ l (2Pm L/ n)' ~
c:~ 2 j03) 2L 1o (an d) cos (pm L) (9.19)
dx dy 1(20m Lln)
Taking (9.19) into account, the in,ternal and mutual impedances of the dipoles
comprising the quadrupole, Zuy, where u, v= 1,2, are determined by expression
[s]:
Zu x
u` 21(~a)�1(~o) m.._oo naao
~ (u~ d/~' 1 C2)vmn h r
X 1mn 7Ymn ( ,
where pp = 1207r
Because of the identical nature of the dipoles, we write Z11 �
(9.20) converges, and when calculiating the quantity ZuV, one can
a f inite number of terms in the series.
(9.20)
Z22. The series
limit oneself to
The input impedance of a quadrupole, Z= R+ jX, as a generator load, is composed
of the input impedances of the dipoles under conditions of their mutual coupling,
transf ormed to the po int where the generator is connected. Then, taking (9.20)
into account, we have [5]:
Z=l(of)+~~t~ I(T,z ZztZ11)COSS'plPASllla'vI
`l j7.11 Pn cos yl sin Yl1/IZ1z + Z21 2Z11(si na yl cosa yl)
2JAn' (Z1zZaLZi 1P2) cos yl sin yl),
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_ where pA, Y and Z are the chaLdcteristic impedance, propagation constant and
= length of the transmission line segment respectively (Figure 9.18).
~ Knowing the input impedance of a quadrupole, Z, we calculate the reflection faetor
r(e, from formula (9.18), where this faetor determines the coriditions in the
quadrupole transmission line as a function of the scan angle. The remarks made
for a dipole element of an array atso apply to the calculation of the quantities
Zuv and Z.
The design calculations for a printed circuit dipole array are carried out using
the procedure indicated in Chapter 2. The design of a dipole and a quadrupole
as elements of an array with a selected cell size for the array, consists in
choosing prined circu it dipoles at the resonant length (see 99.6) and the quadru
 pole dimensions, with the subsequent calaulation of thE input impedances using
�ormulas (9.17) and (9.21) respectivOly, as well as the reflection factor T in
the transmission line using formula (9.18). The array gain can be det.ermined
based on T(see Chapter 2). If the gain is less than the requisite value, the
design calculations are performed for other array dimensions.
The study of dipole arrays has shown that the size of an array cell is one of the
major parameters governing the input impedance of a dipole. Cell dimensions should
be chosen somewhat less than follows from the condition for the lack of additional
main lobes in the directional pattern. This makes,it possible to match the input
impedances of the d ipoles in the,array in a wider scan sector. Moreover, an
important parameter is the,spacing of the array dipoles, h, from the shield. It
has been determined tha.t one can select a value of h such tiiat the dipole mis
matching in the scan sector is the same in the E and H planes. In this case,
the maximum value of the SWR in the transmission.line is minimized and the best
matching results are obtained within the scanning sector. The initial value is
h= 0.25X. As a result of matching, one can obtain a SWR of no more than two in
a scan sector of 45�.
9.7. Other Printed Circuit Radiating Systems.
 Also to be singled out among printed circuit antennas are planar spirals (detailed
data on them are given in [03, Vol. 2], as well as other types of antennas, the
major difference in which is the manner of excitation. We shall consider a few of
t hem.
Dipoles 3ystems With InPhase Excitation. Dipole arrays with inphase excitation
find practical applications. The connection of the dipoles in a quadrupole (see
�9.7) makes it possible to produce inphase apertures, the effective area of which
is practically the same as the geometric area of the aperture. For this reason, in
composing apertures of different areas, the width of the antenna beam directed
along the normal to f.ts. surface can change. An example of a quadrupole composed
af triangular dipoles is shown in Figure 9.19. Another method of inphase excita
tion of dipoles is their series connection to the transmission line, similar to
the excitation of a system.of resonator type radiators (see �9.4)._ Series excita
tion is extremely narrow band.
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Input ~xod
(1) <
3nPMHm61
Figure 9.19. An inphase antenna agray
of four printed circuit
dipoles.
Key: 1. Quarterwavelength trans
f ormer s;
2. Matching slot.�
Figure 9:20. A five elanent dipole array
with resonant excitation.
eld Key: 1: Array elements;
2. Transmission line.
Input Bxod
Dielectric
hield
Dipole Systems with Resonance Excitation. Series excitation of dipole systems can
also be accomplished by the method realized in a Franklin antenna [02]. In this r
case, each dipole of the antenna system excite.s the next dipole so that� an in
phase radiating system is formed. An examplp, of the structural design of such
an antenna with five dipole elements which are capacitively coup7ed is ahown in
Figure 9.40. Planar arrays are put together using the same principle. Radiating
systems with resonant exciation are narrow band systems. The direction of the
radiation depends on the frequency.
Traveling Wave Radiating Systems. The principles employed in the design of
antennas for the long wave band are used in printed circuit radiating systems
made in the form of traveling wave antennas [Yagi antennas]. An example of such
an antenna (a "sandwich" type) is shown in Figure 9.21. The radiating structure
takes the form of zigzag strip conductor (wave shaped), through which the
traveling current wave propagates. The conductor is placed abov e the conducting
shield, whieh can be replaced by a resonator. The main radiation direction 9p is
camputed fram the formula: 
SII7 Oo  ~'~d/A. ~ .
where L is the length of the conductor from point A to point B; d is the
period of the structure. For L/a = 1, the angle 9 m 0, and we obtain a transverse
radiation antenna. If L/a = 2, then the angle Ap a 90�, i,e., the antenna radi
ates longitudinally.
Slotted Antennas, Fxcited by a Strip Transmission Line, are used in the same band
of frequencies as slotted waveguide antennas. In contrast to the latter, slotted
antennas have the advantage that the transmission has practically no dispersion.
For this reason, the frequency dependence of the characteristics of these slotted
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_ antennas is less than for slotted waveguide antennas. Drawbacks to the slot an
_ tennas are the increased requir anents placed on the transmission stripline for
antennas of great length and the necessity of experimentally working out its
d imen s ions .
Input
(1) 1I1OH//IOy/1b/P /7~I090dHUK(!11
(2),4fl3/7C'M?Ip(/4eC/fOA 70j"IIfHQ
~ S'o, a o mK o 3 a~ b i~ v a~ u~ue
Figure 9.211 A sandwich type traeling wave Figure 9.22. A slot antenna in a
antenna array. symmetrical stripl.ine.
Key: 1. Str ip conductors; Key: 1. Short circu iting pins;
2. Dielectr ic substrate. 2. Center conductor of
the stripline.
pi
Slot radiators for an antenna are cut in the outer conductor of a balanced stripline.
The presence of the slot causes higher modes to appear in the stripline, where a
combination of pins is used to suppress these modes (Figure 9.22). The slot length
is computed from the formula Z= 0.5a ( [08] and is made more piecise
~t/>0
Slots
ri..
90
u /
~r50r
^~90 Ohms
300H ~
Slote
!!(cnu
. ,900ry
7.2,SOM ~ /(eHm,oa~bHbia
nn~,9odHU~ nuviiu k1)
aJ (a) d) (b) .
 Figure 9.23. Excitation.configurations for a multislot antenna using
a threeplate symmetrical stripline.
Key: 1. Center conductor of the stripline.
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experimentally. The coupling of the slot to the transmission stripline is adjusted
by shifting the slot relative to the center conductor of the line.
T?ao excitation circuits, which are shown in Figure 9.23, are used for comparatively
small slot arrays. The circuit which realizes series excitation of the slot is
shown in Figure 9.23a. The dimensions indicated in the schematic were worked out
experimentally. The circuit which provides for excitation of the slots with
identical electrical paths is shown in Figure 9.23b. In long arrays, the slots are
excited by traveling waves in the feed line. It is also possible to have slot
excitation in a standing wave mode. The directional characteristics of slot arrays
are determined just as for slotted waveguide antennas (see Chapter 5).
A slot antenna excited by a stripline is convenient for frequency scanning. To
increase the phase difference between ad3acent slots with a change in frequency,
one can place devices in the stripline which increase its electricaZ length, in
particular, employ a zigzag center conductor for the stripline. The electrical
length between the slots can amount to several wavelengths. Thus, one can obtain
wide angle scanning. A scanning angle of up to 60� has been obtained in the 3cm
band when the frequency is changed by 5%.
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10. YAGI RADIATORS FOR PLANAR PHASED ANTENNA ARRAYS
10.1. Phased Arrays of Yagi Radiators
A design procedure is given in this chapter for antenna arrays of radiators in
the form of director antennas, or as they are sti11 callefl, "wave channel"
antennas [yagi antennas] [05] (Figure 10.1). The technique is realized in the
form of a computer program which makes it possible to calculate the main charac
teristics and optimal geometric dimensions of the array radiators.
The set of the directors of the antenna array radiators form an interacting
structure, which can be treated as a layer of an artificial dielectric, covering
the array [08]. By varying the parameters of this dielectric, ane can improve
the matching of the array radiators to the exciting feedlines in a specified
scan sector, which is an important merit of yagi radiators [2]. Such antenna
arrays can be used in the traditional meter and decimeter bands for yagi antennas.
The development of stripline technology has made it possible to use yagi radia
tors in the centimeter band.
Ik
N 4
k ^0
h ~
i~
/ .
.1'(
~ X
 Figure 10.1. Schematic of a yagi Figure 10.2. The geometry of a phased
radiator. yagi array.
 During antenna array beam scanning, becavse of the interaction o�f the radiators,
_there is a change in the input impedances which leads to their mismatching.
Therefore, when designing antenna arrays, it is necessary to assure those geo
metric dimensions of the radiators, shape and dimensions of a cell in the
array [03].and parameters of the radiator input circuit for which the best
matching of the radiators of the array to the exciting feedlines is provided in
, the specified scan sector in the working band of frequencies. Since mismatchi.ng
during scanning is due to the interaction of the radiators, which occurs only
, in arrays, the design of a yagi radiator should be based on the analysis of i.ts
characteristics as a part of an array of identical elements.
10.2. Analysis of the Electromagnetic r'ield of a Phased Antenna Array of Yagi
Radiators
 The properties of an antenna array of yagi radiators (Figure 10.2) can be des
cribed most completely by means of solving the electrodynamic boundary problem
for Maxwell's equations in the case of boundary conclitions ;for the tangential
components of the field vectors at the separatiori boundary of' the different media.
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' In the case of phased arrays with large dimensions (more tlian 10 x 10 radiators),
the mismatchir.g of the majoriCy of its radiators, located in the central region
of the array, can be studied using a simpler model in the form of an infinite
antenna array with a uniform amplitude distribution [08].
 An infinite planar array is a periodic structure, the study of the electromagnetic
field of which can be reduced to the solution of the electrodynamic boundary
problem in one eell of the structure [08]. This boundary problem is solved based
on the formulation of an integral equation for the currents in the dipoles of a
yagi radiator and solving it by the method of moments [2]. As a result, the
following expressions are derived for the directional pattern of a radiator in
the array when the dipoles are oriented along the y axis (Figure 10.2):
1'm (0, (D) � cos (p1 (Y, u) ~i v bmk (Y, u) gmh (7, !1) Sh SI[i i`Ilik:
' ko0mo0
K M
Fo ((l, ~p) cos 0 sin fp! (1', u) Z; I b,~1t`(Y, u) k`T.. n En (0) sin OIl.,
kmOm l (10.1)
,1l
where b,,,it Xiiin/ ~ ~~,~X~~~~ are the expansion coefficients for the current
i
distribution in the kth dipole of a yagi radiator for the sinusoidal harmonics
of the current: 
141(k) (J) sin I ~h ( !z ~~1J' ~rt = l, 2,..., M
(lk is the length and hk is the mounting height of the kth dipole) (Figure 10.1).
The quantities xmk are determined from the solution of a system of linear
algebraic equations:  
h ni
!J y �Ym' h' 7m' h' mh (Yr 1 m SkOt
~1~.2~
k'=0m' 1
where Zmlk'mk are the mutual impedances for the m'th current harmonic in the
k'th dipole and the mth current harmonic in the kth dipole of a radiator when
the entire array is excited. Expressions for the mutual impedances are given
 in [2]: m IIEqCT(iUC, odd
~ 0 , 111 11tCT110e, even
where dkk is Kronecker's delta: dkk = 1, dkk+ = 0 when k� k'; I(y, u) is the
current in the gap of the active dipole;
y:icsin0cosm; ii=ksin0sincp; 0.iccos0. (10.3)
The directional pattern of a radiator in an array is influenced by the parameters
of the equivalent circuit of its input circuit (Figure 10.3). In this circuit,
the characteristic impedance of the transmission feedline p, the transformation
ratio of the ideal transformer n and the reactive component JX are the equivalent
parameters of its input fourpole network;
/ K M hf
Zux (T, ll) _i N' ~~Jm � h' Um0 Zm' h', m0 (79 U) (10.4)
 k'=Om=1m'=1 '
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is the input impedance of the active dipole, taking into account the influence of
the directors and the adjacent radiators (when the entire array is excited with
a uniform amplitude and a linear phase distribution).
This impedance (in contrast to the input impedance of an individually exr_ited
radiator) is frequently called the effective input impedance. It follows from
the equivalent circuit that the dipole current is:
I u)2n I/ 20/fZin(~~1 u)~ ZtL ("10.5)
where zl 112n j t .
It is usually necessary in practice to
~X turn to the experimental alignment of
.
~ Zex(y,~i) the input circuit for good matching of
ttie radiators, for example, using a wave
guide model of the phased array [08],
which corresponds to matching of the
Figure 10.3. The equivalent circuit of phased array for radiation in a certain
a yagi radiator. direction Ao, ~0. It is not difficult
to determine from the equivalent circuit
that for the condition of matching, the equivalent parameters of the i.nput four
_ pole ne*.work are defined by the e�,pressions:
n ==V ~Ze Z,~X ~Yo~ uo)~!: X IIri Zox ~'Yu~ �o)~
(10.6)
where K sin 0� cos (p,,; tt�  ic sin O,, 5111 lp..
For this reason, when calculating the characteristics of phased arrays, it is
expedient to assume that the equivalent p;irameters of the input circuit corres
pond to (10.6), and when designing the circujt, it is necessary to provide for
the selection of its equivalent parameter..~ in accordince with the calculated
or measured value of the effective input impedance zin(YO, u0) �
Based on the effective input impedance of a radiator, one can determine the
effective reflection factor From the radiator input. We have from the equivalent
circuit and formulas (10.6):
I.it  ' 7nx (Y~ Znx (1'n, un) (10.7)
'7nx (1' . I,*x (Yn, t/n)
[ZBX  Zin]
Information on a program written in the algorithmic Fortran language which
realizes the calculation of the indicated characteristics of a yagi radiator in
an array using the BESM6 computer is given in [1].
10.3. The Characteristics of a Yagi Radiator in a Planar Phased Antenna Array
It is essential to know the number of the current harmonics in the dipoles, M,
and the number of spatial Floquet harmonics [08], which must be taken into
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account when calculating the mutual impedances in (10.2) to obtain satisfactory
precision in practice in calculations of the characteristics of a radiator in
an array. Computations show that for a dipole length of 0.2 to 0.7 X, to assure
 a precision of 0.5 to 1% it is sufficient to retain three to five harmonics
when caTciiiating the Eplane directional pattern und one to three harmonice
when calculating the Hplane directional pattern. In this case, the requisite
number of Floquet spatial harmonics amounts 60100 [2]. A slightly greater
error, running up to a few percent, will be observed in this case in a narrow
region of sharp resonance changes in the directional pattern. To illustrate the
convergence of the solution, the directional patterns of a yagi radiator in an
array'are shown in Figure 10.4a in the Eplane where the different numbers of
current harmonics considered are M= 1, 3, 5.
F(9, ~t/Y~~irdz dy~.t 2 F(B, 0~4~rd; dy .i
>,0 E ~~ocHOCma 1,0. H~nocNOCm~ plane
0,8  E plane 08  Ke3 ~'~1k~03~t '
06  4k�O,ri'.t
Q4  5 O,G  I II K.3~`\� '
0,2 0.2  ~k
~ 20 40 '60 BO B� ~ 20 40 B_t 60 BO B� .
W (a) J) (b) ,
Figure 10.4. The directional pattern of a yagi radiator in an array
. with a rectangular grid.
t~/4 j rdsdy/�Z2  ,
1,0
0,8
0,6
0,4 i''~ �I ~I ~ ~
. ~
0, 2 ~ )r/3 ~ I I 1~
0
Figure 10.5. The directional pattern of a yagi radiator in an array
with a triangular grid.
An important feature of a yagi radiator is the possible presence of sharp reso
nance "dips" of a finite depth in its directional pattern in the Hplane (and in
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other close planes where Ifl < 3045�) (Figure 10.4b). In particular, in an
array with a rectangular grid, a dip occurs in the Hplane in directions close to
the angle 8, 6_1 = aresin(a/d  1) on the part of the smaller values. The angle
0_1 is frequently called the ~grazing" diffraction lobe occurrence angle. In
the general case, the directions of the dips are also close to the directions
of the "grazing" diffraction lobes, which are determined from the equation:
(sinOcosip} s pla~(sinOsinq)~. ~ q pctgS Z=1, (10.8)
\ / \ , u x )
where p, q= 0, + 1 are the numbers of the diffraction lobes (pZ + q2 # 0). The
dip in the directional pattern of a yagi radiator is due to the retarding pra
perties of the aggregate of array directors where the dipole length is less than
resonant (about a/2) [3]. If the retarding interacting director structure is
treated as a layer of an artificial dielectric [07], then as follows from a
comparison with an array covered with a dielectric plate [08, 09], the existence
of a dip is to be anticipated if the retardation of the yagi structure is
sufficiently great. The greater this retardation and the coating thickness,
the closer the dip should be shifted to the transverse direction to the array.
This shift actually occurs when the r.etardation increases in a director structure,
in particular, with a reduction in the spacing between the directors and with
an increase in their length (but no greater than the resonance length) [3], and
amounts to a few degrees (Figure 10.4b).
I ~ildzdy/x4 . f/4~rd=dy~.it
1,0 h10 1,0
~
Q6 H
. . K�3
Plane ~
0, 4
qnoC~vocme N
\
0,1 
~
/I~oc~ocma E
0,1 
ds
ZO 40, 6qa~ BO B'
~
0'
~b~ BO 49�
.29 9M 40
_
a)
d,
Figure 10.6. The directional pattern of an optimized yagi radiator
; in an array.
_ a. Rectangular grid:
b. Triangular grid:
Since the analogy with the case of an array covered with a dielectric layer is
not complete, there can also be no dip in the directional pattern of a yagi
radiator at certain values of the radiator parameters (for K= 1 in Figure 10.4b).
In this case, there is a sharp rolloff in the directional pattern at angles of
e > e_1. Since with an increase in the wavelength, the direction of a dip moves away from
the transverse direction to the array, then the array step in the Nplane, dX,
is to be chosen from the condition for single beam scanni.ng at the upper working
frequency in an angular sector which exceeds the specified scan sector by the
width of the dip region in the directional pattern of a radiatur in the array.
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 By virtue of the fact that dipoles do not radiate along their own axis, a dip
does not occur in the directional pattern of a radiator, as a rule, in the E
plane and in planes close to it (1�I =n/2) (Figure 10.4a). Since the directional
pattern of a yagi radiator in the Eplane is of a smooth monotonic nature and
takes on small values at angles of A close to 90�, then the spacing between
radiators in the Eplane, dy, can be chosen somewhat greater than the value
which follows from the condition for single beam scanning. The step dy is
chosen depending on the permissible decrease in the gain at the edge of the
scan sector and the permissible diffraction lobe level at the highest frequency.
Arrays with a triangular grid and the dipoles oriented along one of the sides
of a triangular cell are an exception. In this case, a dip also occurs in tYie
Eplane (Figure 10.5). For this reascn, the use of such a grid is not expedient
in a number of cases.� A grid with ar.i orthpgonal orienzation of the dipoles is
preferable, in which there is no dip in the Eplane (Figure 10.6b).
However, it must be remembered that all of the directional patterns cited here
belong to an infinite array.
The finite dimensions of an actual antenna array has an impact first of all on
the directional pattern of a radiator in the region of the dip. The finite
nature of a phased array is not felt if the dimensions of an array are so great
that the beam width does not exceed the region of the dip. With a decrease in
array dimensions, the depth of the dip will fall off, while its width will
increase in proportion to the beam width of the array. With a further reduction
in antenna dimensions, the dip completely disappears. In this case, a model in
the form of an infinite array can be considered justified only for directions
falling outside the region of the dip in an infinite antenna array.
10.4. The Optimization of a Yagi Radiator in an Array
We shall now consider questions of designing the geometry of a yagi radiator:
the choice of the number of directors, the length, the mounting height, etc.
The existence of a program for calculating radiator characteristics on a computer
makes it possible to automate this portion of the design work to a certain
extent. The mathematical tools for this are numerical optimization techniques
[4]. Where these techniques are used, by working from the requiremenCs placed
on the antenna array characteristics, a socalled quality indicator is put ,
together, which depends on the radiator parameters. Numerical optimization
algorithms provide for searching out the optimal values of the parameters which
a.ttain the extremal value of the quality indicator.
The average array gain in the scanning sector can frequently be chosen as the
quality indicator for the phased array, which by virtue of (2.13), is propor
tional to the quantity: ( r .
i�= .u ,1 I1' (0, (P) I'sin OdOd(p, (10.9)
ctscan
where SlCK is the scan sector.
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The optimization of a yagi radiator based on this quality indicator is carried
out in accordance with the program of [1]. Computation of the double integral
of (10.9) in the program is replaced by summing using Gauss' formula (n = 4)
for the inner integral with respect to 8 and using the rectangle formula for the
outside integral with respect to Since the range of variation in the radiator
parameters is limited: the spacing between the dipoles is always greater than
their width, the length of the dipoles is always positive, etc., it is necessary
 to employ optimization techniques with limitations [4]. Since the quality
indicator (10.9) is always positive, one can employ the following variant of
the external penalty function technique: set f= 0 outside the range of permis
sible values of the parameters. To find the extremum of the resulting function
f, defined in an unlimited range of values of the arguments, the method of local
variations is employed in the program [4].
The major parameters which characterize the properties of the yagi structure
(Figure 10.1) are taken as the parameters to be optimized in the program. These
are the mounting height for the layer of directors hl, the spacing between the
dipoles Ah = hk+l  hk (k = 1, 2, K 1) which is assumed to be constant,
_ the length of the first director 1 and the shortening of the directors A 
k+l  k(k = 1, 2, K 1), which is also taken to be constant. As a
result of this, the number of variables is curtailed so much that it is now
possible to optimize a radiator in a comparatively small amount of machine time.
In this case, the following approach to the design of a yagi radiator can be
proposed. The optimization with respect to the selected main parameters is
carried out in a first approximation (M = 1) in the first stage. Then, treating
 the resulting geometry of a radiator as the starting point, a more precise
selection of these parameters is made for M= 3...5. In the third stage, the
' radiator can be optimized with respect to the remaining parameters, for example,
one can choose the best 80, �0 matching direction. Such an approach to the
solution makes it possible to choose parameters for a yagi radiator, expending
J' no more than a few hours of BESM6 camputer time on each step. Results of
' calculations show that even after the first optimization step, sufficiently
good matching of t� ra::iator to space is ac;,ieye3, so that the subSequent stegs
may prove to be superfluous.
Since a quality indicator usually has several local extrema, the choice of the
' starting point for the optimization program is of considerable importance.
Calculations show that such parameters as the mounting height hl and the length
of the first dipole 11, can be arbitrarily chosen in a range of hl = 0.250.4 a
and 11 = 0.30.4 X. At the same time, depending on the choice of the initial
values cf the parameters Ah and A1, one can obtain different "optimal" values .
of the parameters. For this reason, it is necessary to take somewhat different
starting sets of values for Ah and A1. Usually, these quantities fall in a
range of Oh = 0.10.35 a and O1 =0.050.15 X. The initial direction for
the matching can be arbitrary, just so the condition 90 < 6_1 is met, for
example, 00 = 0. Experience with the calculations shows that the number of
directors in a radiator is expediently chosen larger than K= 23.
The directional patterns of an optim3.zed yagi radiator with three c;irectors in
a scan sector of + 40� in the Hplane snd + 60� in the Eplane are shown in
Figure 10.6a for a rectangular grid with steps of dX = 0.6 a and dy 0.54 X.
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The directional patterns of a dipole radiator (K = 0) are also shown in this
same figure for comparison. As can be seen from these curves, the. reduction in
the gain af the array for optimized radiators as compared to the maximum possible
value amounts to 0.3 dB overall in the scan sector. This is considerably less
than for an array of radiators consisting only of one active dipole (K = 0).
Better matching to space in the main planes can be achieved in an antenna array
with a triangular grid for the configuration of the radiators, since in this
case, the dip in the directional pattern of a radiator in the Hplane is
removed considerably from the transverse direction. However, the dip in the plane
I~1 = 30� is brought closer to the direction o� the normal in this case. The
directional patterns of an optimized yagi radiator in an array with an equilateral
triangular grid for a specified scan sector of 191 < 32� and array steps of dX =
0.7453 a and dy = 0.6415 a are shown in Figure 10.6b. As can be seen, practically
ideal matching of the phased array in the single beam sector is achieved, with
the exception of a narrow dip region. ,
It must be noted that the quality indicator in the cases cited here has yet
another maximum at Ah = 0.030.05 X and A1 =0.050.06 a, which corresponds
to a more compact structural design of the yagi radiator. However, the maximum
gain losses of the phased array in this case because of mismatching amount to
about 0.5 dB. The calculation of the characteristics of optimized radiators in a band of fre
quencies shows that an array matched to space at the high frequency remains
well matched with a reduction of 20% and more in the frequency, given the
condition that the parameters of the input circuit conform to (10.6).
10.5. Designing the Input Circuit of a Yagi Radiator
The conditions for matching the radiators of an array during scanning in a
chosen direction 9o, ~0 (10.6) mean that the input circuit accomplishes the
matching of the characteristic impedance of the transmission line p to the
impedance of the load Zin(yo, uo) in a specified frequency band. Such an input
circuit is designed using the methods of microwave network theory.
We shall consider the procedure for designing the simplest input circuit. The
sCructural design of a linear array of yagi radiators for the centimeter band
 using striplines is shown in Figure 10.7a [6]. In this figure: 1 are the
directors of the radiator; 2 is the active dipole; 35 are the balancing device
elements for the excitation of the dipole; 6 is the exciting stripline radiator;
7 is a phase shifter; 8 is a directional coupler; 9 is the matched load for
the free arm of the coupler; 10 is the distribution stripline exciting the
phased array; 11 is the dielectric substrate. The dashed lines show the confi
guration of the conductors on the back side of the substrate.
The strip transmission line section 4 is a quarterwave transformer [06] which
matches the load impedance connected to the balancing device in the active dipole
gap to the characteristic impedance of exciting line 6. The short circuited loop
5 using a slotted transmission line provides for symmetrical excitation of the
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dipole. At the center frequency, its length is = a5/4, where a5 is the wave
length in the slotted line. In the case of a purely resistive load, the length
of an opencircuited section of stripline 3 is also equal to a4/4 (a4 is the
waVelength in the stripline). In the general case, the length '3 is chosen from
the condition for the compensation of the reactive component of the input
impedance of the radiator:
ctg (Znt3/,k0 = xox (va, uo)/P,. (lo. io)
3
TL J 2~
J ~
149 8
a)
~
i�a
~j
Ar '0o zEx As
11 l_I ,14/4
6J
Figure 10.7. The structural design of a linear stripline array of
yagi radiators (a) and the equivalent circuit of the
exciter (b).
The characteristic impedance of the quarterwave transformer p4 = p3 is deter
mined by the values of the impedances being matched [06]:
Pa = YRd: ('yo, tio) Pe�
(10.11)
The calculation of the wavelength in stripline and slotted line, as well as the
calculation of the geometric dimensions of the lines based on a specified value
of the characteristic impedance can be carried out using the techniques given
in [5].
Since the input circuit configuration cited here can provide for only narrow
band matching of the radiator to the transmission line, the entire calculation
is carried out at the center frequency. The passband of such a radiator amounts
to a few percent. Since the methods of calculating the input circuitry are
rather approximate, while the mathematical model for the yagi array considered
here is idealized, the results obtained from calculating the parameters of the
input network require an experimental improvement in the precision. As has
already been noted, this is conveniently done using a waveguide model of the
phased array, which simulates the radiation in the direction A0, ~0�
If the requisite passband of a radiator is more than 10%, then a more complex
microwave network is to be used instead of the quarterwave transformer (4),
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where broadband matching techniques must be used for the design of this network
[7]�
10.6. A Design Procedure for a Yagi Radiator for Phased Antenna Arrays
Various sets of initial data are possible for the design of a yagi radiator.
Typical is the specification of the scan sector, the permissible reduction in
the gain during scanning and the permissible level of the sidelobes of the
phased array. The parameters of a unit cell in the array dXidy and f can be
chosen based on these initial data (see Chapter 2). In partcular, with a
rectangular grid for the layout of the radiators, the steps are chosen in
accordance with the specified scan sector using formula (2.3), and in the case
of a triangular grid, using formula (2.4).
The mounting height of the active dipole, h0, is ordinarily chosen equal 0.2
0.25 X and the length 10 = 0.450.5 X. The thickness of the dipoles is chosen
in a range of 0.02 to 0.05 a and the matching direction Ao < 9_1. The number
of directors of a radiator is chosen as K= 1 and the initial values of the
radiator parameters being varied, Z 1 and hl are chosen in accordance with the
recommendations given in � 10.4; the initial cptimization of the radiator para
meters is accomplished on a computer fnr the case where M= 1. It is necessary
for the optimization program to specify the precision in the determination of
the extremum and the error in the determination of the optimal dimensions of
the dipoles. It is usually sufficient to take the former as 0.005  0.01, and
the precision in the determination of the geometric dimensions as 0.005  0.01 X.
The optimization results are evaluated in the sense of attaining the specified
radiator characteristics. The array steps are made more precise in accordance
with the recommendations given in � 10.3. In particular, the array step can
be slightly increased in the Eplane, given the condition of assuring a,specified
gain and diffraction lobe level. Where necessary, the number of directors is
increased and the the initial value of the parameters A1 and Ah is specified
(see � 10.4). The optimal dimensions of an array cell and the radiators are
found as a result of several trial and error calculations, which are then made
more precise using optimization programs where M= 35.
Based on the value of the input impedance Zin(yO, uo) obtained with the computer,
the input network is designed and the structural design of the antenna array
is worked out.
The calculation of the directional pattern characteristics of the antenna ar'ray
is then carried out on the whole in accordance with the general procedure (see
Chapter 2).
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11. APPROXIMATE DESIGN CALCULATIONS FOR PHASED WAVEGUIDE ANTENNA ARRAYS TAKING
MUTUAL COUPLING INTO ACCOUNT
11.1. General Considerations
The open ends of waveguides are the most widespread radiators for antenna arrays
in zhe centimeter uand. Various modifications of waveguide radiators, realized
by means of dielectric inserts, stops and other devices, are described in Chapter
12. A design procedure for phased array geometry is given in � 2.12 without
taking mutual coupling into account. The results of such design calculations
can be used as the initial approximation in drawing up the mathematical model
of a phased array, in which the interaction between radiators, edge effect,
excitation circuit configuration, etc. should be taken into account in the
general case.
An approximate design procedure for phased arrays is proposed in this chapter
using graphs, calculated taking mutual coupling into effect. The graphs are
plotted for planar wavegLide arrays with a rectangular grid for the arrangement
of the radiators in the case of small crosssectiona of the waveguide radiators
and small thickness of their walls. In such arrays, the mutual coupling is due
primarily to the dominant mode, however, the ma3ority of the graphs in this
 chapter were plotted taking into account the existence of higher modes also.
Thus, the material of this chapter makes it possible to improve on the precision
of the design procedure adopted in Chapter 2.
11.2. Design Graphs
The concept of'the gain of an element in an array was introduced in Chapter 2
(formula (2.16)). The power transmission gain of 1 r2(e, o, incorporated
in (2.16) is a function of the position of the main lobe of the directional
pattern (6maX, ~max), since as is well known, the ~input~admittance of the
radiators and the reflection factor r(e, change during the scanning process.
iIrI1 arwE
dy=0,7.i
~0.6.i dr= 0,65.t
0,6
0,4
0,2
0
Bmot E
al (a)
Figure 11.1. The power transmission gain in the c4se of Eplane (a)
and Hplane (b) scanning for various spacings between
the array radiators.
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Curves are plotted in Figure 11.1 which characterize the change in the power
transmission gain during scanning in the E and Hplanes for an infinite wave
guide array with a rectangular grid for the radiator layout with various
spacings dX and d between the radiators (see Figure 2.1) [1]. The curves in
the Eplane were Zbtained experimentally; they were calculated for the Hplane
assuming infinitely thin walls (a = dX and b= dy). The arrows on the abscissa
indicate the values of the angles 9maX, which cahen the main lobe deviates by
these amounts, a grazing (at an angle of 90�) diffraction maximum appears.
Corresponding to each spacing between the radiators (dX or dy), as is well
known, is its own value of emax' With the deflection of the main lobe through
an angle approximately equalto 9maX, a sharp mismatching of the radiators to
the feeders is observed, the reflection increases while the power transmission
gain falls off. The sharp drop in the power transmission gain limits the scan
sector escan' In the case of Eplane scanning, the permissible scan sector is
less than the ultimate angle 6maxE:
escan E  0�7emaxE
, Oc.aE 0,70ma:E.
(11.1)
When scanning in the Hplane, 9maX H practically coincides with the angle escan H�
When taking only dominant mode mutual coupling into account, the reflection
factor changes monotonically within the bounds of the scan sector Ascan�
BQNscan The maximum permissible beam deflection
SO � H PYane angles escan H and escan E is shown in
1117ocHOCmaH Figure 11.2 as a function of the spacing
40 between the array radiators for the
30 dominant mode.
14 /lnoc~rocme E~'`~`~
~p E P ane The reflection factor can be determined
055 06 0,66 d d as a function of the aperture dimensions
. ,
of the radiators and the spacings between
them taking mutual coupling via higher
Figure 11.2. The permissible scan sec modes into account using the results
tor as a function of the found in the literature [08]. Also
spacing between array studied there is the reflection factor
radiators. as a function of the waveguide wall
thickness t in an infinite array. It is
shown that changiis6 the thickness of the waveguide walls with a constant spacing
between waveguides has no impact on the position of the minimum gain in the
scan sector which is due to the considerable mismatching at the moment of the
appearance of the highest, the first maximum in the array factor; on the other
hand, changing the thickness of the walls has a substantial influence on the
absolute value of the reflection factor.
The change in the absolute value and phase of the reflection factor in the E and
Hplanes for various thicknesses of the waveguide walls is shown in Figures 11.3
and 11.4 by way of example. In accordance with these figures, as well as based
on similar curves available in the literature [08] for other array dimensions,
one can plot generalizing graphs for the maximum possible reflection factor in
 222 
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 the scan sector as a function of the dimensions of one radiator, a and.b, for the
case of a constant spacing between radiators (Figure 11.5). One can draw the
following conclusions based on what has been presented.
Irl  
dy=Q5i14.Z
0, 8 dy b
t = 
%
0,6 
0, 4
t 0
4,2 0,063 002
0
0,1 0,3 sinBE
Figure 11.3. The absolute value of the
reflection factor for E
plane scanning as a func
tion of waveguide wall
thickness.
1. The maximum value of the absolute Value
of the reflection factor occurs with
radiation along a normal when scanning in
the Hplane in a pl.anar array of rectan
gular smooth waveguides with a small crosssection, which are placed at the
junction nodes of a rectangular grid.
The absalute value of the reflection fac
tor Irl is greater, the smaller the
radiator aperture a for a constant dX,
or what is the same thing, the thicker
the waveguide wall.
Using the curves of Figure 11.5b, one
can approximate the absolute value of
the reflection factor for the case of
radiation along the normal based on the
selected spacing between the radiators
dX/a and the aperture dimensions of a
single radiator.

G
G
!
arqr�
~,~=0,5714.i,
d,~o 60 �
f= 
d,~ �
f00
0,
140 64R,
180 Q 1
0,3 47,5 0,7 sinB,y
Figure 11.4. The absolute value of I' and the phase, argT, of the
reflection factor for the case of Hplane scanning as
a function of waveguide wall thickness.
2. In the case of Eplane scanning, the reflection factor function is more
comples. Its absolute value in the case of radiation along the normal depends
not only on b and dy, but also on the a and dX dimensions of the array in tlie
Hplane. When the main lobe of the direction pattern is deflected from the
normal, IPl initially falls off to a certain minimum value, and then rises
rather sharply. Corresponding to each waveguide wall thickness is its own main
lobe deflection angle for which lI'I is minimal. The maximum value of Irlmax
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~l'~mor
�
~r~maz
ts0,12
.
'
~
O6 ' 0,1
~ 0, 063
0,02
0
0
~ 0,3 o, s
0, ~ B _ 38�
.5
,
0~~
0, 3 ~
10�
0,2
t =
150
O'Z
~
dy,
,
0,1
250
0,5 0,6
0,6 � a/~t
(a) u1
. (b) dJ '
Figure 11.5. The maximum reflection factor of a radiating waveguide
for various wall thicknesses in the case of Eplane (a)
and Hplane (b) scanning.
(Figure 11.5a) is obtained in the majority of cases at the edge of the scan
sector (thus, for example, for b/X = 0.5714 when A= 38� and b/a = 0.6724 when
9= 25�, where the angles 38� and 25� bound the scan sector for the correspond
ing array dimensions), and lTl rises considerably outside the bounds of the
scan sector.
The graphs of Figure 11.5a were plotted
with respect to two points, and can
0,5 therefore be used only in rough calcula
0,4 tions. The resulting graph (Figure 11.6)
was plotted based on the curves of Figure
0,3 D3 OOti 11.5, which shows what maximum mismatch
0,20 30 40 eg� can be anticipated in an array when
cK scanning throughout the entire permissible
Figure 11.6. The maximum reflection gector Ascan in the E and Iiplanes.
factor as a function of The maximum permissible mismatching in the
the selected scan sector, exciting waveguides II'maXl may be stipu
eCK [escan] in the H lated in the technical specifications
plane (solid curves) and When designing the antenna array. Then
Eplane (dashed curves). the permissible scan sector will be limit
ed by the specif ied value of I I' I max and
can be determined for the waveguide array without the matching devices using
the graph of Figure 11.6. As can be seen from Figure 11.6, the reflection
factor cannot be less than 0.2 for any wall thickness or dimensions of the
waveguide aperture when scanning in a sector of more than 30�.
If the requisite values of the maximum permissible reflection facror and scan
sector are not assured,.then a provision should be made for matching the
radiators to the exciting waveguides. Impedance transformers, dielectric inserts
inside the waveguides and dielectric coatings in the antenna aperture can be
employed as the matching devices. The presence of a dielectric can substantially
improve the matching thoughout the entire scan sector, but at the same time, it
leads to the appearance of anamalous nulls in the gain.  224 
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Questions of matching waveguide radiators in scanning arrays are treated in
detail in Chapter 12.
The graphs shown in Figures 11.211.6 can be successfully used to determine
the reflection factor in waveguide phased arrays where the waveguide wall
thickness is small and where their crosssection dimensions satisfy the condi
tions:
a < 0,75k; h < UA. (11.2)
In this case, the interaction of higher modes changes the reflection factor by
no more than 10% as compared to the value calculated when taking only the
 dominant mode into account.
The direct method of determining the reflection factor r(e, o, taking mutual
coupling via higher modes into account for any array structure, consists in
the following. By treating a large multielement phased array as a infinite
periodic structure, the field in the exterior region (where z> 0) can be broken
down in terms of the spatial harmonics of this structure. The field in the
interior region (where z< 0) can be represented in the form of the superposition
of the dominant mode and higher modes, of which only the H10 mode tnay propagate
through the waveguide [3  5].
The condition of field equality at the boundary of the internal and external
regions (when z= 0) leads to an integral Fredholm equation of the first (or
second) kind. For the numerical solution of a Fredholm equation, it is necessary
to make a transition from the integral equation to a system of linear algebraic
equations, by sel.ecting the appropriate system of base functions. In the case
of a waveguide phased array, it is convenient to take the set of modes in the
waveguide as the base functions. Only a limited number of modes in the waveguide
and spatial harmonics in the external space, needed to obtain a good approxima
tionr are used in the calculations. The computational program, compiled using
the algorithm described here, is given in Chapter 12.
11.3. Design Recommendations
1. When designing phased waveguide
coupling of the radiators can have
the exciting waveguides and on the
2. The geometric dimensions of the
mined without taking mutual couplii
Chapter 2.
arrays, it must be kept in mind that mutual
a substantial impact on their matching to
antenna gain in the scan sector.
array and its elements can be roughly deter
1g into account using the formulas given in
The initial values for the design calculations are the width of the main lobe
 of the directional pattern, the level of the first sidelobe, the scan sectors
Ascan E and Ascan H, the permissible reflection factor II'Imax and the permissible
nonuniformity in the antenna gain within the scan sector. In accordance with the
design procedure recommended in � 2.12, the overall dimensions of the array LX
and Ly are determined, as well as the amplitude distribution (see Table 2.1) and
_ the spacing between the radiators and the number of them.
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 The results obtained are to be treated only as an initial approximation.
 3. Taking the mutual coupling ef the radiators in the array into account makes
it possible to specify its dimensions more precisely. In particular, the spacing
between the radiators is to be chosen from the graphs of Figure 11.2, taking into
account the fact that ir. the Eplane, the angle 6maX E included in formulas (2.3)
(2.6) must be determined using formula (11.1). Increasing the spacing between
the radiators to a value greater than the design figure is not permissible, since
this leads to the appearance of a dip in the gain within the scan sector. Reduc
ing the spacing between the radiators as compared to the calculated value is not
expedient in the majority of cases, since this leads to an increase in the reflec
tion factor II'maxl when scanning in the Hplane, although Irimax [sic] decreases
slightly in the case of Eplane scanning. Moreover, with a decrease in the
spacing between the radiators, it is necessary to increase the overall number
of them in the array to maintain the previous overall dimensions LX and Ly.
The anticipated Maximum value of the absolute value of the reflection factor in
_ a given scan sector can be roughly determined from the curves of Figure 11.6.
If II'ImaX exceeds the reflection factor permitted by the operational conditions
of the entire antenna and feed system, then the scan sector should be reduced
 or provisions should be made for matching devices in the structural design
of the radiators.
4.' The maximum aperture size*of a single radiator is determined by the permissi
ble spacing between the radiators in the array; the minimum*size amin > a/2 is
limited by the propagation conditions of the H10 mode. Moreover, it is necessary
to keep the following in mind when selecting the dimensions of the aperture of
a radiator. With a decrease in the dimensions a and b, the reflection factor
I I'I~X increases. The value of I I'I maX can be estimated by means of Figure 11.5.
On the other hand, an increase in the a and b dimensions can lead to the
appearance of anomalous nulls in the scan sector [3]. If the aperture dimensions
do not exceed those recommended by the conditions of � 11.2, then anomalous nulls
will not appear in the entire sectiir of + 90�; if the indicated conditions are
not met, then it is necessary to compietely calculated the input admittances
and reflection factors.
5. The recommended procedure, which was drawn up based on the results of analyz
ing infinite arrays, can also be used to choose all of the dimensions of
sufficient large finite arrays. This is justified by the fact that the direc
tions in which there are dips in the gain do not depend on the overall dimen
sions of the array. However, with a decrease in the array dimensions, a dip
becomes wider (occupies a greater angle) while its depth decreases. If it
is assumed that the edge effect is manifest in five radiators on each side of
the array, then arrays where the number of radiators is more than a thousand
may be considered large.
6. The electrical parameters of a phased array can lie calculated after its
geometric dimensions have been selected.
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To precisely determine such antenna parameters as the directional pattern, gain
and reflection factor, it is neces5ary to obtain a complete solution of the
problem, i.e., find the input admittances of all the radiators (the central and
edge ones) as well as the amplitudephase distribution of the fields in the
aperture. This calculation is extremely cumbersome and requires the use of
high speed computers.
The electrical parameters of antennas are approximately estimated as follows.
The normalized ,:urves for the change in the gain
b'lb'max=cos0[1I'Z(0, (p)] (11.3)
_ (see (2.16)) for various values of dX/a approxmately match each other up to
angles at which diffraction maxima appear, and within this range of angles, are
well approximated by the function:
f (011) = (cos 011 + Ycos Orr)/2. (11.4)
The gain along the normal is determined with respect to the width of the main
lobe of the directional patterns in the two planes 26g and 29E:
Gm,,x = 33 000,q/(2811� 20c),
(11.5)
(where n is the efficiency of the array), or based on the radiating surface of
the array:
Gma= 23tLx L y v71/A.2,
(11.6)
where v is the surface utilization factor for the array, which depends on the
amplitude distribution in the array. The directional pattern is approximately calculated from the formulas for a
continuous radiating aperture as a function of the amplitude distribution of
the field in the aperture (see Table 2.1).
Mutual coupling of the radiators somewhat changes the structure of the sidelobes
of the directional pattern, which in this case, cannot be described by a suffic
iently simple analytical expression. 7. The procedure considered here can be used for the approximate design of
phased antenna arrays. It yields more precise results than calculations without
considering mutual coupling of the radiators using the �ormulas of Chapter 2.
Using this same procedure, the initial approximation can be calculated when
constructing an algorithm for the more precise computer design of phased antenna
arrays.
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12. WIDE ANGLE MATCHING OF THE WAVEGUIDE RADIATORS OF PLANAR PHASED ANTENNA
ARRAYS
In a multielement planar phased array, the radiators, in the form of open ends
of rectangular waveguides, are placed at the intersection points of a generalized
triangular coordinate grid (Figure 12.1). The following symbols are used in the
figure: a and b are the waveguide dimensions; a' and b' are the dimensions of
the window of a stop, placed in the radiator aperture; dX and dy are the spacings
between the rows of radiators in an array along the X and Y axes respectively;
a is an angle which defines the mutual arrangement of the rows of radiators in
the arxay. In particular, when a= 90�, we obtain a rectangular grid and when
a= 60�, a hexagonal grid.
d ~ X
; ; o ;
11 #
,
~ o~
LJ Y L
j1 i
I~1 1~I I~I
L j ~J ~ J
i ' ar ~ t
II i I It
L
a
The end goal of designing the radiating
element of a phased array is the wide
angle matching of the radiator, i.e.,
finding those geometric dimensions of the
array and characteristics of the matching
devices for which the maximum reflectian
factor in the feeders does not exceed a
certain specified value within the scan
sector.
1 r The most effective method of designing
I I I I j waveguide radiators for phased arrays,
L ___J L_ __jdy~_ J matched in a wide range of angles, is a
technique based on the calculation of the
Figure 12.1. The layout of radiators radiator characteristics taking into
in an array with a account the matching devices both within
generalized triangular the feeder elements and outside of them,
grid. with the subsequent variation of the para
meters in the problem of designing phased
arrays until obtaining the requisite results [013]. The time and cost for the
development of multielement phased arrays with this method are significantly
curtailed as compared to methods based on the experimental development of the
radiators. The utilization of this method presupposes the presence of computer
programs with which one can calculate the characteristics of a radiator based
on the solution of the corresponding electrodynamic problem for a waveguide
array with the matching devices for subsequent system optimization.
12.1. Methods of Matching Waveguide Radiators in Planar Phased Antenna Arrays
We shall treat the most widespread methods of matching the radiators of planar
waveguide phased arrays.
The Utilization of Dielectrics [08]. The use of dielectrics in antenna arrays
leads to the appearance of additional parameters in the design problem. The
presence of dielectric elements exerts a substantial influence on the distribu
tion of the fields in the waveguide apertures. For this reason, the choice of
the parameters of dielectric elements such as the dielectric permittivity
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and the thickness of the dielectric has a strong influence on the characteristics
of an antenna array.
For optimal matching, the choice of the parameters in an antenna array with
dielectrics is beat of all accomplished using the method of parameter variation.
_ In this case, all of the parameters, with the exception of one, are fixed and
the calculations are performed while changing this parameter in a specified range.
The method of parameter variation is most effectively realized in a"mancom
puter" system, which makes it possible to narrow the range of values of several
parameters.
irl
0,8
49,4
?0
aryr�
Figure 12.2. The absolute value and the phase of the reflection
factor as a function of the scan angle in ttte Hplane
for an array of waveguides completely filled with a dielectric (dx = 0.5714 a; a= 0.5354 a).
Figure 12.3. A phased waveguide array with dielectric inserts.
We shall consider some design data to illustrate the influence of dielectrics
on antenna array characteristics.
Typical results are given in Figure 12.2 for a wave;,:ide array in the case of
 Hplane scanning when the waveguides are filled with a dielectric. We will note
that the absolute value II'I and the phase arg I' of the reflection factor change
little with a change in the scan angle, which makes it possible to have good
matching of the antenna array in a wide range of angles (at least at one fre
quency), even in those cases where considerable reflection is present. The
~ break in the curves considered here is due to the occurrence of a diffraction
beam.
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Ir~ tlrgr, .
, 0,8 t 03531.~ Z~~ t 03531.Z
06 150 . r
 ' 0,5BB5.t
Q4 0,59852
, f00
0,2 69
40 90 f10 160 40 80 120 160
2W r sinB
Figure 12.4. The absolute value and phase of the reflection factor as
a function of the Hplane scan angle for an array of
waveguides with dielectric inserts (e = 2; dX = 0.5714 a;
a = 0.5354 a).
Figure 12.5. A phased waveguide antenna array with a dielectric
coating.
In the case where the antenna array waveguides have dielectric inserts (Figure
12.3), an additional airdielectric separation boundary appears. In this case,
to control the characteristics of the radiators, two new parameters are added:
the dielectric permittivity of the dielectric s and the insert thickness t. For
each value of the dielectric permittivity, one can find that insert thickness
for which the absolute value and phase of the reflection factor change little
practically throughout the entire working scan range, i.e., in the region where
only one main beam exists (Figure 12.4). However, the presence of a supplemental
separation boundary leads to the fact that the dependence of the reflection
factor on the scan angle becomes more sensitive to a change in frequency. More
over, if the absolute value of the reflection factor has greater values (see,
for example, the curves for t= 0.3531 a), then the problem of matching the �
antenna array in the passband becomes complicated. With an increase in the
dielectric permittivity of the insert, the task of broadband matching becomes
even more difficult. Moreover, the presence of dielectric inserts can lead to
the propagation.of higher modes in the region of the waveguide filled with the
dielectric, where these modesz are excited in the antenna aperture and disappear
in a region not filled with the dielectric, something which at certain values
of t can produce resonance peaks in the curves of the reflection factor.
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I!'I  Figure 12.6. The absolute value of the
fthe reflection factor as a
y'�~,Z_: function of the acan angle
for an array with a single
lAyer dielectric coating.
4E/�' FT j ' dg = 8 = 0.5714a; e =
3.0625: aE is the wave
0,2
_ length in the dielectric.
~ 40 90 ' 120 ~
. . Zjr ~ 3cn9
In the case where a dielectric coating is used in the aperture of a phased array
(Figure 12.5), the reflection from the "coatingfree space" separation boundary
is used to partially eluninate the reflection from`the aperture. Just as in the
case of an antenna array with dielectric inserLS, ::ith the appropriate choice of
coating parameters, one can make the reflection factor in the working band only
slightly dependent on the scan angles. However, because of the fact that at
rather large values of E, the beam deflection from the normal leada to the
occurrence of a wave iri the antenna array similar to a surface wave, which pro
pagates inside the dielectric, but decays in free space; an increase in the
dielectric coating thicknese above a certain critical value cauaes a resonance
peak to appear in the curve of the reflectinn factor, the mgximum value of which
is practically equal to 1 and which, with an increase in the coating thickness,
shifts in the direction of the normal to thearray. A further increase in the
coating thickness leads to the appearance of two and more peaks in the reflection
factor curve (Figure 12,6), ,
We will note that the curves of the reflection factor plotted as a function of
the scan angle for an antenna array with dielectric inserts usually are of a
more continuous nature than the corresponding characteristics of an antenna array
with a dielectric coating. This is'of considerable practical importance when
match ing an array. Thus, the use of dielectrics makes it posaible to improve the antenna array
matching during scanning. However, wide angle matching by means of dielectric
inserts or coatings degrades the frequency response of the phased antenna array
parameters as the reault of the appearance of an additional separation surface.
It was shown in the literature [08] that matching can.be improved in a wide range
of scanning angles at a single frequency. The use of multilayer dielectric
inserts or coatings makes it possible to improve the frequency properties of an
element.~
Stops in a Waveguide Aperture [1]. An advantage of matching by means of a stop
is the lack af a finite spacing between it and the radiating aperture, which leads
to a lower frequency senaitivity of the array.
A waveguide radiator loaded with aAtop is ahown in Figure 12.7, in which a dielec
tric is incorporated to improve the matching. Here, e' is the dielectric per.mit
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tivity of the material filling the waveguide; el is the dielectric permittivity
of the insert material; E2 and e3 are the dielectric constants of the dielectric
, coating of the array surface; cl and c2 are the distances from the center of the
waveguide to the edges of the stop along the X axis: dl and d2 are the distattCea
from the center of the waveguide to the edges of the stop along the X axis.
Fz(dl, d6
c
Figure 12.7. A waveguide radiator
loaded with a dielectric
and a stop in the aper
ture.
Figure 12.8. The power directional
pattern in the Hplane
for a waveguide radiator
loaded with a stop.
The power directional pattern of the waveguide radiator depicted in Figure
 12.7 and arranged in a triangular grid is shown in Figure 12.8 for the following
parameter values: dX = 1.008 a, dy = 0.504 a, a= 45�, a= 0.905 a, b= 0.4 a,
e' = el = e2"= e3 = 1(the radiator is not loaded with a dielectric), dl = d2 =
0.2 a, the parameters cl and c2 are equal to each other and vary from 0.4525 a
to 0.226 a(the stop covers half of the waveguide aperture). The solid curve
corresponds to the lack of a stop, while the dashed and dotted as well as the
dashed curvea correspond to the presence of stops which cover 25 and 50% of the
aperture area of the waveguide respectively. In the absence of a stop, a
sharp dip is observed in the directional pattern at an angLe of A= 34� (although
the diffraction beam apgearance angle ia approximately 60�). The introduction
of a stop shifts the dip from the direction normal to the antenna aperture. A
further increase in the area occupied by the stop leads to a reduction in the
radiation along the normal to the aurface and to overcompensation for the mis
matching o� the array. In this case, the presence of the stop does not degrade
the characteristics of the array in the Eplane and Dplane (diagonal plane,
i.e., at an angle of 45� to the E and H planes). Eliminating the dip in the directional pattern of an element is an important
feature of the method of matching by means of stops, which makes it possible
to simply and effectively solve the problem of combatting anomalous "blinding"
of the array.
The reflection factor is shown in Figure 12.9 as a function of the scanning
angle in the H, E and D planes for the wavaguide radiator depicted in Figure
12.7 and arranged in a triangular grid, for rhe following parameter values:
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VI
~ E
U6 y
40  H
D . 8b
E 120
p 20 40 60 B� 0 20 40 60 B�
Figure 12.9. The reflection factor as a function of the scan angle
for a waveguide radiator loaded with a stop and a
dielectric.
dX = 0.9225 a, dy = 0.27 a,. a = 30�, a= 0.905 a, b= 0.187 X, e' = el = 2.569
e2 = 11 e3 = 2.69 tl = 0.984 a, t2 = 0.061 X, t3 = 0.101 a, dl = d2 = 0.0935 a
and cl = c2 = 0.2715 X.
One can note slight deviations of hhe reflection factor in the scan aector, which
makes it possible to effectively match the antenna array.
Thus, the use of stops as matching elements can aubatantially improve phased
array matching in a wide sector of scan angles in a rather broad bandwidth,
as well as significantly shift the resonance dip in the directional pattern of
a radiating element fram the direction normal to the array aperture, or even
eliminate it. The simplicity of fabricating atops is also to be noted.
12.2. Matching With a Fixed Scanning Angle
The radiating aperture of a waveguide element in a phased array represents a
complex load for the exciting waveguide where this load changes during scanning.
The conventional matching fourpole network inserted in the feed channel for
each element can match the feeder to the load for a certain scanning angle,
however, a considerable mismatch;will be retained for the other angles because
of the fact that the conventional matching fourpole network does not change
its parameters with a change in the scanning angle. However, if one can before
hand manage ta have the reflection factor change in a relative small range
within the scan sector (for exampYe, by uaing a dielectric or stops in the
waveguide aperture), then the use of matching for eome of the scan angles will
make it possible to achieve better matching of the phased array throughout the
entire sector. As a rule, matching consists in introducing an additional
inhomogeneity into the radiator waveguide, where thia inhomogeneity creates a
reflected wave equal in amplitude and opposite in sign to th3 wave existing in
the line wh ich is reflected from the load.
An equivaleat circuit for the connection of a feeder to the waveguide radiator
of a phased array is shown in Figure 12.10 [2]. The insertion admittance of the
radiating aperture which 3epends on the scanning angles is expresaed in terms
of the reflection factor I' in a given crosssection by the well knownrelation
ship: 1r(~~ 1
Y. (0 (P)  GR,t0, V~) J Ba (0, (P) = 1+ I` (0, (p) Po ~ (12.1)
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where Ba(A0, ~p) is the reactance of the matching device which compensates for
the reactance introduced by the radiating aperture at the specified scan angle;
nfl is the transformation ratio of the ideal transformer which serves for match
ing the characteristic impedance of the feeder to the resistive component of
the radiator input impedance; pl and pQ are the characteristic impedances of
the feeder and the waveguide radiator respectively.
Inductive and capacitive stops as well as inductive rods are used most often as
the compensating reactances in waveguides.
~ ~ ^ Coanocy~o~~~~e r, .
~ ycmpouc~nBo
I F igure 12.10. The equivalent circuit
~ I for matching a feedline
eo"00) i~o to a radiator.
Key: 1. Matching device.
 
L~no        j
4~�.__},
Figure 12.11. An inductive stop in a
waveguide.
~o
a
E
Figure 12.12. An inductive rod in a
waveguide.
An inductive stop (Figure 12.11) can be asymnetrical in the general case and
is characterized by the width of the window d and the thickness b, as well as
the spacing br..ween the centers of the waveguide and the atop window, c.
The following approximate formula can be used for the calculation of the normal
ized susceptance of a very thin (S � d) stop:
~ N a ,{g? 2a (1{sec~ 2a tga a c 1.
where:
71 
'v 1 (X/2a)'
is the wavelength in the waveguide.
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(12.2)
(12.3)
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For a symaetrical stop (c = 0):
B~ a ctg''2 , (12.4)
while for a onesided stop [c =(a  d)/2]:
g....._ a cig' 2a (11 sec~ ~ tg, n a 2ad (12.5)
\
The finite thickness of a stop can be approximately taken into account by substi
tuting the quantity d d in place of the quantity d. For inductive stops, the
influence of the finite thickness is comparatively small.
The Inductive Rod (Figure 12.12). The normalized susceptance of an inductive
rod is calculated from the approximate formula: .
~     ,
) C 2  r cos 2) 2](12.6)
B l sec a [ Z( ac ln
which yielda adequate precision in the cases of practical importance.
The Capacitive Stop (Figure 12.13). An approximate formula for the calculation
of the normalized susceptance of a capacitive stop, assuming that its thickness
is infinitely small, has the form: .
 4b nd nc (12.7)
B= A ln (cosec ~ sec 6 1.
Taking the finite thickness into account is accomplished by adding the follow
ing correction factor to B: eB =
2rc8 b d
A (_T b (12.8)
d
Figure 12.13. A capacitive stop in a
waveguide.
Capacitive stops reduce the electrical
strength of the waveguide channel and.
thereby decrease the power wh ich can be
transmitted through the waveguide. For
this reason, they are rarely used as
match irig elements. A more precise calculation of the
matching reactances can be made using
the grapha given in [3].
Quarterwave transformers, continuous or steppe3 tranaitions, etc. can be used
as the ideal transformers, the detailed design procedure for which is also given
in [3].  235 
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 Thus, the design of a waveguide radiator for a planar phased antenna array,
matched in a wide range of scan angles, is carried.)out in the following order:
1. The array geometry and dimensions of a radiator are selected (see Chaptere
2 and 11).
2. The matching device in the radiator aperture (dielectric, stops, etc.) is
seleCted using the method of parameter variation on a computer to obtain the
minimum change in tlie reflection factor within the scan sector. In this case,
a certain correction of the array geometry and radiator dimensions is possible.
3. The matching device in the radiator waveguide ia seleated to obtain a reflec
tion factor in the feeder,for all scanning directions no greater than the
permissible factor.
There are a program and description of an algorithm for the cal.culation of the
directional pattern and reflection factor for a waveguide radiator in a planar
array in the library of algorithms and programs of Moscow Aviation Institute,
where this program and algorithm can be used to calculate the characteristics
of a waveguide radiator for a specified array geometry and parameters of the
matching devices as well as to optimize the radiator characteristics by mear.s
of dynamic programming. .
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CHAPTER 13. SLOTTED RESONATOR RADIATORS FOR PLANAR ANTENNA ARRAYS
Slotted resonator antennas are used in the microwave band as independent antennas
as well as in the form of radiators for antenna arrays (AR) with linear, ellip
tical and controlled polarization of the radiated field. It is most expedienC
to use them at wavelengths of 10 to 60 cm.
A merit of slotted resonator radiators is the possibility of combining them with
the metallic surface of the objects in which they are installed.
A single slotted resonator radiator takes the form of a slot cut in a conducting
shield, where the slot closes a metal cavity (the resonator) and is excited at
one or mDre points by means of coaxial or striplines. Excitation directly in
the plane of the slot makes :.t possible to not only tune to resonance with a
small resonator depth, but to match the input impedance of the slot in a
structurally simple manner to the characteristic impedance of the exciting feed
line, by displacing the connection point of the feeder relative to the center
of the slot. The major characteristics in the design of antenna arrays made of slotted
resonator radiators are: the input impedance of the radiator incorporated in
the antenna array as a function of the scanning direction; the geometry of the
array and radiator; the partial directional pattern of a radiator; the polari
zation characteristics (for elliptically polarized radiators). In the case
where a slotted resonator radiator is used as an independent antenna (for
example, in telemetry, coumunications, etc.), the major characteristics are:
the input admittance within the passband [as a function of frequency], the
directional pattern and the polarization characteristic.
A complete analysis and the optimization of the indicated characteristics can
be made only by means of mathematical models close to the actual devices and
the study of these models by rigorous methods of electrodynamics.
13.1. Analysis of the Characteristics of a Slotted Resonator Radiator
The analysis is based on the solution of Maxwell's equation taking into account
the boundary conditions at the appropriate surfaces.
The following model has been adopted for Che antenna arrays (Figure 13.1). Each
radiator is excited by a system of N sources at the points r1i Q = 1, 2,
ej
Each source takes the form of an electrical current sheet with a density of ]ri
(r~i  y), directed along the OX axis. With the action of the field produced
by the sources, such a distribution of the magnetic current density ju(x, y)
arises in the slot that the tangential c6mponent of the electrical field inten
sity vector is equal to zero at the surface of the shield and is continuous in
the slot, while the tangentnal component of the magnetic vector in the slot
satisfies the condition [lJ: . .
Xo, (13.1)
r=t
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where H(1) and HQI) are the magnetic field intenaity vectors for the regions
under consideration; region I is the dielectric coating (0 < z< x8), region
II is the resonator (h < z< 0) (see Figure 13.1); n is a unit normal; xo is
a unit vector along the OX axis.
Taking (13.1) into account, one can derive the following integral equation for
the unknown electrical field distribution EX(x, y) in the slot [2] both in the
case of an antenna array [4, 6] and in the case of an individual radiator [3, 5]:
,
dx' y
d
I Ex (z', y') [GA0)+G`(l1)+29!!2lzz'O
dZ Spt
1 '
'JU=t SinKZiyglat = c1coSKZy+CZSinKZy
+ ~
2k,
i .
 N i ' (13.2)
~
2 Vl jnt ~t f(~lt y') sin xz I 9 J'l dy' . 14.11
Here:
e%P(1Kir) , r=jl(xx')a}(Jy')a+(zz')', �
r for an individual radiator,
A.nR onIiFlowHOro Hsnyuare,nn,
W. QXP {j [Kxrn (XX')+kpn (yy')1} X
m L ao n`oo j�2dxdy sin ayin i
X[ f mn (z, z') d 2Amn COS zj A,rtH N311yttaTeaR B AP for a radiator in an
antenna array
are the diagonal elements of Green's tensor function;
0 Ann onnrrrnIIroro ns.nyltaTe,nn, for an individual
f( r .I ~ \ C%~1 (KYm (x._x')I ti~~n X radiator
, Fa / 2dY i1,, Silt ap~n i
)rn=x n=ro
= 1
~in rt (.r:: ~ z' ) I 2/I m rt co; Tir i�rz
Xsinym~,~z   IGnsi I{aaiyva for a radiator in an
cOs xQ antenna array
irtri 1 ne~t ' Y nu~ 1 rn~t
renm n t1P
are the nandiagonal elements of Green's tensor function,
u JCz 2h~:
1C~ = Ep Fi 11.o ~ /CZ W f0 P2 I I Ymn ~ . ~~xrr~'. " l~rn
~ F:1Y~nn ALYntn'I
mmn fmn (Xa+ Z ) Fi Y ~I1i) COST(I!I) (1) (
.YfiIRx Ymn 51~ ynui a~N~
2( mff
mtt
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2rcm*x    ..._(d,, n m  _ _  
~ /1 0 1
ICYn, d, ; Kyn = 2Ji sin a dx 1~* ac) d~~ . ,
=L 2,...; m== 0, f 1, j 2,
1Px=KodxsinOcosT; %=aodysinflsincPr KO'WFOI40;
x a
YmJn1) h2
a Kxni K pn; K.1 = GJ F,p E3 110 ;
(10 o0
�(11) r 0, ~ _2F�~ / r ~
Gy .d :..1 q~ ~ vmn (Xr Y) Um n lx X
msOn1
COS7(1l)Z COS T0f) (h z')r z C z',
X mm m~
COS 7n~n) Z' COS }'iieq) Z>Z';
a R r b 1 ~tnt nt
vmn / lx, J) =COSfix JIC.. l lSlilt'pIf/2 I, ps' a+ fjy~~ +
1 ~ i
1, n = 0,
T'/lt
i11 *O,
where SL4 is the slot area; (x, y) and (x', y') are the coordinates of the obser
vation point and the integration point respectively; A, ~ are the angles in
spherical coordinates; ep is the absolute dielectric permittivity; sl, e2 and
e3 are the relative dielectric permittivity of the dielectric coating, the
resonator and the space above the coating respectirvely; C1 and C2 are complex
constants of integration; f(ni  y) is a function which takes into account the
specified features of the excitation of the antenna (one usually takes f(ni  y)
= d(ni  y), where d(ni  y) is a S function); 7.ei is the complex amplitude of
the exciting current;
for a single radiator
~ Y ll~~) 2 lt(')
~(ti2 K~) n~ k, for a radiator in an antenna array
Ay`1'  E,;(�r' y') G',i(`)dx' dy1
lt(>> _ ~ , , .
~x J) r) ~3r~~`z d.e cl~ .
n? J LY
5lit
To find EX(x', y') from integral equation (13.2), it is necessary to employ
regularizing methods. This is related to the fact that the solution of equation
(13.2) is an improper problem: as great a change in the solution as desired
can correspond to a small change in the right side of (13.2). One of the
regularizing methods which makes it possible to obtain a soiution of (13.2)
with a sufficient degree of precisian is the sutoregularization technique of
[2]. The basis for the method is the hypothesis of the smoothness of the
_ solution and apriori information on the type 1/4nr integrable singularity in
the nucleus of equation (13.2). By employing a piecewise constant approximation
 239 
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 f~
   
Z~///
.i , i.~.
~ J ~�J ~ �

_ 
I
(b)d) ~ BI (a)
Figure 13.1. Slotted resonator structures.
Key: a. Antenna array with a dielectric coating (rectangular
grid);
b. Triangular grid for radiator'layout;
c. Single radiator.
for the desired function and segregating the singularity from the nucleus of
(13.2) when the observation (x, y) and integration (x', y') points coincide,
one can derive a system of linear algebraic equations wh.ich have a stable
solution.
Equation (13.2) was reducAd to an algorithm on a crnnputer in papers [36] b,y
the autoregularizing technique and the diatribution EX(x', y') was found which
is needed to determine the characteristica of a single radiator and a radiator
incorporated in an antenna array. The program written in Algol60 is in the
library of algorithms and programs of Moscow Aviation Institute. The input
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PesoHamoP Resonator
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admittance Yni is defined as the ratio of the current Ini to the voltage at the
 points rti:
. d/2
U,l; = j E. (x, y) rlx,
'd12 1Y=T1j
Yn! = G 1 jR / Un1. (13.3)
The directivity function g(9, 0) of a slotted resonator radiator incorporated in
an antenna array, in a single beam scanning mode, which coincides with the
normalized partial directional pattern of a radiator with respect to powar, F2
with a precision of within the constant factor 4?rS/a~, is defined as:
b' (0, V') Xa cos 0[ 1  ~ I' (0, (13.4)
Here, S = dXdysin a;
(13.5)
P (0, ~r)  = li Y (0, (1))1/I1 Y (0, (1)1
is the reflection �actor, Y(9,�0) is the input admittaxYCe at the excitatian
point normalized with respect to the exciting feeder. When a radiator'is
 excited at several points: N
(0, (P) }'+i1 (0,.
The directional pattern of a slotted resonator radiator, used as an independent.
antenna; is found from the known field distribution in the slot EX(x', y').
13.2, The Ctiaraceeristics of a Slotted Resonator Radiator as a Independent
Antenna
Functions were derived based on the program for the solution of integral equation
(13.2), where these functions are recommended for the calculation of the reso
nant* operating mode of an antenna. The results of the numerical calculation;
are given in Figures 13.213.4.
Figure 13.2 illustrates the conductance component G of the input admittance Y
as a function of the position of the antenna excitation point relative to the
center of the slot in the xesonant operating mode (B = 0) for various relative
widths U4/0 I(dslot/X)l and lengths (21/0 of the slot. Curves for the
relative width of the resonator (a/a) are shown in Figure*13.3 as a function of
its relative depth (h/X) when B= 0. The voltage standing wave ratio Kst U
[VSWR] curves for the antenna as a function of frequency are shawn in Figure
13.4 for various slot widths.
*The resonant operating mode of an antenna is understood to be that mode in
which the reactive component of the input admittance of the antenna is Yn0
0.
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4
3
2
1
C�10; il r G 103 ottm 1
i i ~
16'
' 12
s
4
O
FOR OFFICIAL USE ONLY
n.> Q1S 0,7 Ih,/'J
a/,t
0,3
YK /?�i=1e
d,~/1~2=f02
0,1
d/�t = 2!/.t
0,2
~ _
zt/~t=aa
O, 3 h/.t
Figure 13.2. The conductance of a Figure 13.3. Resonator width a/a
slot (B = 0) as a function as a function of the
of the position of the depth h/a when B= 0.
excitation point relative tio the center of the slot. We shall consider a specific example
 of the determination of antenna para
kcT S~ meters which assure a matched operating
A
\
x
`
~�~o~.~f
. ~
\
~
~ ~
. ~ .
/
mode of a slotted resonator antenna
with an exciting feeder having a
characteristic impedance of p= 100
ohms with a bandwidth of about 12% for
a VSWR of 2 with slot dimensions of
21,71 = 0.6 and dslot/2a � 102. Since
the antenna and the feeder are matched
when G= 1/p, we find the point corres
ponding to G= 10 � 10'3 ohms'1 on the
curve for the specified slot dimensions
(21/a = 0.6 and dslot/2X  10 The
, abscissa of this point, which defines
0,.911 ~1981,P0 1,0; 1,06 ;1 ~~ro the displacement of the excitation
Figure 13.4. The SWR as a function of Point relative to the center of the
frequency for various slot is equal to 0.249 X. The resona
slot widths. tor dimensions which assure a resonant
 operating mode of the antenna are
 chosen from the graphs of Figure 13.3. For example, if it is important because
of structural.considerations to keep the depth of the resonator h small (approxi
mately 0.1 a), then for a slot of the same dimensions, the width a and length b
of the resonator should be 0.375 a and 0.6 71 respectively.
13.3. The Characteristics of a Slotted Resonator Radiator in a Planar Antenna
Array
Curves for the conductance component G and susceptance component B of the input
admittance Y of a slot antenna incorporated in an antenna array with various
dielectric coatings (el = 3, 2.35) are shown in Figure 13.5 as a func�ion of the
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B 103 Ot1ID 1
E~�3 a~~!~175.i~
15 I H n706'K01:m6 �z6:h,25,tE 25 � �0051e Hplane
` plane
H
,c ~.,,,f p0 ~�TSO,?�f~E f ~
Eplane
` ~  srr7,3u ~~nnocKOime
4
\4 L~~ ane
S 1�. V 1 f nnocKOCme i
5
0,05.1E 9_
~1__._.~~�~
10 20 30 GO 6�
8..~
10 30 40 B�
Figure 13.5. The conductance and reactive component of the input
admittance of a radiator incorporated in an antenna
array as a function of the scan angle in the E and
H planes: rectangular grid, dy = 0.6a and 2Z/a = 0.5.
I i/1,r1 I
&I I
=0,7 ,50,166 �0,0~933  0,0833 0,/66 y/~t
J
`
..00
1
 � 90 
?i1 i7166 Q/I93; U 0,0833 4166,04
ul (a) di ~b)
Figure13.6,. The amplitude (a) and phase (b) distributions of the
electricgl field over the slot.
dX = dy = 0.6a; 2Z/X = 0.5; the dashed curves are for E
plane scanning; A= 42�; the dashed and dotted curves
are for Hplane scanning, 6= 40�; the solid curves are
the characteristic distribution.
scanning angle in the E and H planes when each radiator is excited at a single
 point (r11 = 0). The dielectric coatings have a substantial influence on the
distribution of the fields and the mutual coupling of the radiatora in the array.
For this reason, the correct selection of the parameters E1 and xs is quite
important in the degign work. It can be seen from Figure 13.5 that with an
increase in el, the absolute values of G and B change significantly, and with
an increase in the thickness of the dielectric coating, the range of variation
in these parameters increases (the angle 9_1 for the occurrence of the first
' diffraction lobe is noted).
Curves for the amplitude and phase distributions of the electrical field in the
plane of the slot are shown in Figure 13,6 for various operating modes of a
radiator incorporated in an antenna array. As can be seen from Figure 13.6, when
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B� 10 3
12
8
4
O
8
>2
C�10~`~~ ~
>2 
oof
ss=0,05.iF ~x
B  ~xooox /
4 �~''i~
~ 10 40 60 B�
e,~4
x e,�3,0
E'igure 13.7. The influence of the quantity el on the total input
admittance.
9=42� (the occurrence of the first diffraction lobe), the field distribution
along the slot differs sharply from a sinusoidal distribution when scanning
in .tihe Eplane and is asytmaetrical and outofphase in the Hplane. ^.l'he most
characteristic amplitudephase distribution of the field in an array with a
dielectric coating where ei = 3 is shown in Figure 13.6 with the dashed curve.
Figure 13.7 illuatrates the influence of the relative dielectric permittivity
el on the .^.omponents of the input admittance of a slotted resonator radiator
incorporated in anantenna array with a triangular�grid where the coating thick
ness is xs = 0.05aE and for the case of Eplane scanning.
13.4. The Optimization of the Characteristics of a Slotted Resonator Radiator
in an Antenna Array
When designing a phased antenna array, it is necessary to assure a minimal
 reflection factor I'(8, in the specified scan sector and frequency band. The
optimization of a radi.ator incorporated in an antenna array at a fixed frequency
is accamplished by means of minimizing the function [08]:
! = f ~1'(0)1'dO~.JJ I 1Y(0) I'dO, (13.6)
I+Y (e)
cK or,n
wher.e 9CK [ASCan] i$ t11e specified scanning sector.
The optimization was carried out using the method of local variations in a
"user computer" dialog mode, which made it possible to narrow the range of
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values of the parameters needed to obtain the requisite charscteristics. The cal
culation of the double integral in formula (13.6) was carried out for three sec
tions: 0, 7/2 and 7r/4. The optimization was accomplished in the specified
sector Ascan through the choice of the geometry of the array and radiator, and the
parameters of the dielectric coating ei and xs.
0, S
Q�,
a
Figure 13.8. Optimization of the Eplane Figure 13.9. The frequency curves for a
directional pattern of a radiator within the sector
radiator. of scan angles.
Curves fur G(0) and B(9) in the E and H planes of an optimized radiator for a
rectangular grid are shown in Figure 13.5. It can be seen that by selecting the
parameters el and s, one can achieve a smooth change in these curves in an angular
sector of 6= 0 to 45� (el = 2.35; x$ = 0.05a$). Similar opti.mization results in
the Eplane for an arrangement of the radiators in a triangular grid are ahown in
Figure 13.7 (see the curves for el = 2.25).
The normalized power directional pattern of a radiator F2(9, 0) _(a2/41rs)g(e,
is shown in Figure 13.8 for a nonoptimized (ei = 1 and el = 3 and an optimiaed
(el = 2.35) slotted resonator radiator incbrporated in an antenna array.
The frequency characteristics of Y(6) for a rectangular grid in the Eplane of a
radiator optimized at the center frequency f0 are shown in Figure 13.9. The
working bandwidth at a level where the SWR is 2 is about 10 percent. We will note
that for complete optimization of the radiator, working from the specified sector
and frequency coverage, it is necessary to minimize (13.6) within the passband.
Matching of the radiators incorporated in an antenna array to the excited device is
usually achieved for 6= 0. This can be achieved for the radiators considered
here by means of shifting the excitatiom point relative to the center of the slot
(see Figure 13.2) for each radiator. To find the precise position of the excita
tion point ni in the plane of the slot, it is necessary to use the program for
calculating the characteristics of a radiator by the sutoregularization method;
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during the design stage, one can use the functione shown in Figure 13.2 with a
high degree of confidence:
13.5. Examples of the Realization of Slotted Resonator Radiators
Structural designs of slotted resonator radiators excited by a miniature coaxial
cable (Figure 13.10a) or by a stripline or a system of atriplinea in the case of
excitation at several points (Figure 13.10b) are ahown in Figure 13.10.
The stripline conductor (3) is run inside the resonator until it intersets the slot
1 and is shorted at a certain spacing from the slot by the jumper 4. The stripline
conductor is coupleii to the exciting feeder by means of the coaxial to stripline
transition (5). The choice of the dimensions of components 35 for matching is
accomplished experimentally. The structure shown in Figure 13.10b is used to
obtain radiation with circular polarization. A schematic of the excitation is
shown in Figure 13.11a and its stripline realization is shown in Figure 13.11b.
The circuit provides for the excitation of each slot at two points and a phase
shift of 90� between the slots. Radiator inputs 14 in Figure 13.10b are con
nected directly to outputs 14 of the excitation circuit (see Figure 3.11c [sic])
by means of the coaxial to stripline transitions (5). The stripline excitation
device consists of a 3 dB divider and two 3 dB directional couplers; the charac
teristics and parameters of the directional couplers can be determinQd uaing the
procedureset forth in Chapter 22 and 23. The excitation of each slot at two
points provides for 20 to 25 dB of isolation between the slots with respect to the
feed.
Ihput 4 .,Oxvd4 ti. .,CBxndilput 1
1.
7
~ .
�
;
/~Iv/~ /5
! ~ ~ . ~ .t.~ ~ ,,~F.'~ ~ / ~�`,T~~
j /
2
i ' Bxod ?  z� ` ~
jI � 6'~.%
1~.~:�:,~.~:{,~,. A* l3aUd3
fiJ (b) .
Figure 13.10. Ways of exciting a slotted resonator radiator with a
coaxial cable (a):
Key: 1. Rectangular slot;
2. Resonator;
3. RF connector;
4. Coaxial cable;
With a stripline (b): 1. Orthogonal slots;
2. Crossshaped resonator;
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[Key to Figure 13.10, continued]:
3. Stripline;
4. Shorting jumper;
~ 5. Coaxial transition.
_ Output
Bxad /
Input
Z    Output 2 Matched
Baand 2 C~a~ Lo~od
B&Aod 4 Ce; / y~a,oy3Ya
G
(AltrIlp"ut
(a)
L tA_Z'"/
b/XOd 1 C_.. ~b~ 00'3
\ utput \3
4 ? 7
Figure 13.11. Circuit configuration for the excitation of a alotted
resonator with rotating polarization of the field.
Key: a. Electrical circuit;
b. Topology; 1. 3 dB divider;
1. 3 dB divider';
2. Directional couplers with front coupling.
13.6. The Design Procedure
The design procedure for a slotted resonator radiator as an independent antenna
where the characteristic impedance of the exciting feeder, the dimensions and the
bandwidth are specified is accomplished using the functions ahown in Figures 13.2
~ to 13.4. The directional pattern and gain of the antenna are found from the
formulas derived, for example, in [0.1, 02]. .
The design procedure for phased arrays of alotted resonator radiators is similar
to the general procedure for the design of antenna arrays based on specified
technical requirements (see Chapter 2).
The characteristics of an individual radiator as part of an antenna array can be
computed uaing the program which makes it posaible to assure a resonant mode (the
 selection of the dimensions a,.b, h and 2Z), achieve matching (the choice of ei
when 9= 0) and optimize the radiator characteristics within the scan sector.
However, the following procedure for working with the program in a usercomputer
dialog mode is expedient for the cfficient utilization of the machine time:
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1. Working from the value found for the array step h, determined by Ascan (see
Chapter 2), we choose the slot length 21 and the resonator b> 2Z (Figure 13.1).
2. Working fram the specified value si for the dielectric coating.and the struc
tural requirements for the dimensions a, h and xs, we find the resonator oper
ating mode of the radiator (B = 0) and optimize it (the most expedient range of
~ change for a/2 and h/71 are respectively: 0.1 < a/a < 0.5 and 0.1 < h/X).
3. We achieve matching to the radiator excitation circuit by shifting the point
Ei (in a first approximation, this displacement can be determined from the graph
in Figure 13.2).
4. The gain of a radiator incorporated in an antenna array, g(6, ia determined
from (13.4), while the radiator directional pattern is determined using the
formula.: F(S, _ (X2/4ffS)g(9,
5. Where necessary, the bandwidth is calculated for tne radiator incorporated in
the antenna array (see Figure 13.9). The bandwidth of an optimized radiator is
approximately 10 percent for a VSWR of 2.
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14. RADIATING WAVEGUIDE MODULES WITH REFLECTIVE PHASE SHIFTERS
14.1. The Modular Design of a Phased Antenna Array
Regardless of the structural configuration of a phased array, homogeneous units can
be singled out in it which consist of a number of microwave elements and devices.
These units are joined together by a distribution feeder. Units of a modular '
design are emploqed to provide for production suitability of phased antenna arrays
and to standardize their structural design.
The basic elements of a module are the radiator (or a group of them), phase shifter
and an element for coupling the distribution feed line. The presence of isolation,
tuning and other auxilia.ry assemblies is also possible.
i
I
i~ ~
f(opomKO
3ar~p1Kamend
t 1 . B)
Ca~ (b) (c)
Figure 14.1. Conf igurations of radiating modules.
a. With a feedthrough phase shifter and a
matched load;
b. Without it;
c. With a reflective phase shifter.
Key: 1. Shortcircuiter.
The basic configurations of modules are depicted in Figure 14.1. A module with a
coupling element in the form of directional coupler is depicted in Figure 14.1a,
where arm of the coupler f orms a distr3but ion f eeder section, while the other is
loaded into a feedthrough phase shifter with a radiator and an absorbing load.
The absorbing load is provided to compensate for rereflections which occur 3n the
module. A module is depicted in Figure 14.1b in which the coupling element is a tee.
A module is depicted in Figure 14.1c which contains a radiator and a directional
coupler with a reflective phase shifter in one of the arms. The paths for the
distribution of the electromagnetic wave to the radiator are indicated by the
dashed arrow lines in these f igures.
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The greateest value of the directional gain of an array is achieved in the phased
array shown in Figure 14.2. The impact of various kinds of distortions and mis
matching on the directional gain is substantially attenuated in this circuit be
cause of the use of directional coupler with absorbing loads.
Variants of waveguide modules have been considered, however, the theoretical
results and circuits of the modules are also applicable to other types of micro
wave lines.
14.2. Multiposition PHase Shifter for a Module
The basis for a multiposition pbase shifter is an extremely simple series produced
pbase shifter using semiconductor nipin diodes with four phase positions
(Figure 14.3) [1]. It has the following main parameters: working frequency of
7.7 GHz, Discrete phase step of A = n/2, average thermal losses of from 1.2 to 1.6
dB, switching currp*_�r, of 100 + 10 mA, switching voltage of 1 volt and an ultimate
microwave through power of from 10 to 15 KW and an average power of up to 10 watts.
0
,
�
Figure 14.2.. Schematic of an antenna
array with a series distri
bution of the energy.
3
Figure 14.3. A reflective phase shifter
with nipin diodes.
Key: 1. Waveguide; 2; Partition with
the"gvitched slot;
3. nipin doide.
_ The thermal losses introduced by phase
shifters govern the efficiency of an
antenna and its gain. These losses
are determined by the quality of the
diodes used in the phase shifters [1] :
K = rrev loss/rfor loss'
where rrev loss is the reverse loss
resistance of the diode; rfor loss is
the forward loss resistance of the
diode. The relative gain of an
antenna, G, iN plotted in Figure 14.4
as a function of thd discrete pbase
control step, A. The quality of the
switchers is the parameter in these
graphs. In centimer band series
produced diodes, the quality f igure
fluetuates from 300 to 1,000. In the
case of large values of the discrete
phase control step (A > n/2), the
antenna gain is low because of the
large switching errors in beam steer
ing, i. e. , because of the low direc
tional gain. With a decrease in A,
the thermal losses increase, but the
directional gain rises more rapidly,
right up to a certain value of A.
which dependa on the quality of the
switchers.
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r,
O,;
/;6
,
0,4
0,3
,v,1717o
~
'u'V
4 ~ i� ^;y
1 2 N1 N
Figure 14.4. The antenna gain.:as a func Figure 14.5. Schematic of a through
tion of the discrete step transfer bridge phase
af the phase shifter and shifter with N and
the quality of the ninin 2N discrete phase s[ate
d iode. steps.
Thus, the maximum value of the coefficient is aehieved when 7r/4 < p sin (nM)  I 
~ I uYl',, cos (4/4) I (lil',) 1
uz pI'�) sin (A/4) cos (d/4)1: (14.12)
cos (4/4). '
The maximum phase error in the excitation is:
~ &1 aresin ~ AA (14.13)
Knowing the relative amplitude and phase errors of a module makes it possible to
determine the minimal values of the directional gain and gain G in a real
phased array (the inverse problem also frequently occurs in practice: having speci
fied the minimal directional and gain G of a phased array, determine the permis
sible scatter in the module parameters). The efficiency of a module is:
n IKM/ M id 12
(14.14)
In order to obtain KM id, it is necessary to substitute KM for the case of ideal
elements of a module in expression (14.10).
A module withoitt an absorbing load (Figure 14.10) is atructurally simpler than
that depicted in Figure 14.9. It consi$ts of two orthogonally arranged:waveguides
(1 and 2), which are directional;.y coupled together by whole (3) in the wide wall.
A reflective phase shifter (4) with a discrete step of A and a radiator (5) are connected to the upper waveguide. In such a module, the coefficient Y character
izes the energy passing through the coupling element from the input of waveguide
1 to thb radiator, bypassing the phase shifter.
/ ~  s
Figure 14.10. A radiating module without an
absorbing load.
v
Input
The resulting transmission gain fram the input of waveguide 1 to the aperture of `
the radiator is defined by the expression:
a(SQ  '1% 1 I'P3 c Jne l I S
_ I K. I  1 I aaQ1 rpeJnA (r D aQtclriA (14.15)
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In this module, even in the case of ideal elements, the input of waveguide 1 is
always mismatched because of the presence of a wave which propagates through the
coupling pole of the phase shifter to the radiator.
~
The reflection factor of the radiating module without an absorbing load, taking
into account the nonideal nature of the elements incorporated in it, is:
pa Q1 elrte r 1 2aYr�
Irr,Is (14.16)
I+aaQ_lrPelns
The maximum amplitude and phase errors of a module without an absorbing ?oad are:
~ Ail I I',, (14.17)
I I n.L ccl >o IQU i,ri,;
Figure 15.4. The power of semiconductor
devices as a function of
. the working frequency.
Key: 1. Permissible radiation
power for a single
element in an array
(2e~ 5 = 30�);
2. IMPATT diodes;
3. Tunnel diodes.
2fi9
Key: 1. IMPATT diodes;
2. Tunnel diodes.
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PEmax, W
Transistors
/03f I/uii~.~uc�.ni;~ni
/0''
/n
IO`
~o
~ L _  GHz
0,1 f, //i(
Figure 15.5. The total radiated power
as a function of fre
quency.
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The total radiation power of an active phased array is governed by the output
power of a module, the layout density of the radiators and the size of the
array. In turn, the array size depends on the width of its directional pattern,
290.5 and can be defined as L= 60a/280.5�
As has already been noted, the array step is approximately a/2. It is difficult
in practice to reduce the step, since the transverse dimensions of the radiators
are usually close to a halfwavelength (for example, a/3 for spirals, about X/2
for symmetrical dipoles, etc.).
Thus, the maximum number of radiators in, for example, a square array is N2 =
(2L/a)2 or taking into account the relationship between L and 260.5, NZ = 1.44 �
104/2AO.;�
Then the maximum value of the total radiation power is:
1'11 m~x I,94� 10' ~'ni/10~, s,
j where PM is the radiation power of a single array element.
An estimate of the total radiation power for arrays built using transistors,
IMPATT diodes and tunnel diodes which was obtained in this fashion in shown in
Figure 15.5. The dashed lines show the possible limitation of the power in the
long wave region because of the unacceptably large array dimensions.
The prohlem of minimizing the number of radiators for the purpose of simplifying
and reducing the cost of the structural design of an array and the control devices
usually comes up when developing a phased array. The same problem can also occur
in the case of active phased antenna arrays. We shall determine the reduction in
the radiation power with such minimization.
The maximum array step for which there are no spurious maxima is:
%/(1 + Slil 0mnx),
� where emax is the maximum b!aam deflection angle from the normal.
Correspondingly, the number of radiators in a square array is:
N3 I.Z (1 J sitl Uruax)/Xa,
and the total radiation power is:
pz...3,6,10aPM( iPsin0 maxl
\ 20o.n
The reduction in the power as compared to the maximum value is:
sin O1unx lZ .
2 1
This ratio is stiuwn in Figure 15.6 as a function of a specified scan angle 8. It
can be seen fiom this figure, for example, that with the minimization of the
A
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iVPZ mos
>n I
0
TpIysucmope
Transistors
60 :
40
/7 ~~,r (1)
(2)
Figure 15.6. The relative reduction in
power as a function of the
scanning angle.
J> 0,5 > 6 f0 f, %ri~
Figure 15.7. The efficie:icy of semi
conductor oscillators
as a function of
frequency.
number of radiators, and consequently also Key� 1. IMPATT diodes;
the number of array modules, for a scan ~ 2, Tunnel diodes.
sector of 8max < 15�, the total radiated
power amounts to approximately a quarter
of the maximum value determined by the graphs of Figure 15.5.
The Efficiency of Active Elements and the Thermal Conditions in an Array. The
maximum values of the efficiency which can be attained at the present time for
the semiconductor devices considered here are shown in Figure 15.7 for various
frequencies, from which it can be seen that the efficiency of the active devices
falls off with a rise in frequency, and at frequencies above 5 GHz, can drop
down to 10 to 25%. Along with this, the average radiofrequency power flux
density through the radiating surface of the array, S, for an array step close
to X/2, is: '
1'11S= PM Na' \ 2 N)' q PMI ~a
and, as can be seen, increases fo'r a set power of an array element in proportion
to the square of the frequency. The thermal flux density through the surfaces
bounding the array structure, by virtue of the decrease in the efficiency with
increasing frequency, rises even more rapidly,.something which can ledd to the
establishing of severe and even unacceptable thermal conditions in the modules.
The use of effective forced cooling methods though detracts to some extent from
one of the advantages of a semiconductor active phased array: its compactness.
We shall estimate the thermal limitations in the case of natural cooling of a
structure by the ambient air, keeping in mind that the thermal mode of an array
depends on the specific features of its scructural design, and for this reason,
a preliminary estimate can only be a very approximate one.
We shall represent the structural design of an array in simplified terms: in
the form of a compact planar unit with solid walls. We shall initially treat
the case where the area S of the radiating surface is considerably greater than
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the area of the sicle surfaces of the structure, while the heat output through
the surface opposite to the radiacing surface is made difficult because of the
presence of the distribution device which is close to it. It can be assumed
that the thermal output in this case, which occurs with the generation of narrow
directional patterns, takes place primarily through the radiating surface of the
array, i.e., the heat output surface can be considered as coinciding with the
radiating surface S.
Then, in a steadystate thermal mode, the heat flux density is:
f'M noT 1'nt
P /S   t I  II \ PM 1 `
M lossM 'S~~  S~~ 71n1 / 1�2 where PM loss is the power loss in the module; nM is the module efficiency; SM
is the array area allocated for one module.
The heat flux density determines the temperature gradient between the cooled
surface and the air in accordance with the relationship:
Pnr nOr / ~
S :aAl unn 4 ~ I lin` i)7
\ a is the heat transfer coefficient.
where aAt,
We shall set a permissible temperature gradient of At er = 50� C, for which in
the case of cooling by natural air convection and thermal radiation, a= 8 W�
m 2 deg 1. Tlie minimum permissible value of the module efficiency is:
1/nM Per = 0. 25 aAtperX2 /PM + 1 1 hlni uo11 ==0,25aAljt0jj 1a/PriI 1,
and for the values adopted for a and At:
1/r1M Per = 100X2 /PM + 1. 1/TIn+ rcnn 100%2 /PM4. 1.
The calculation performed with this formula for a transistor module shows that
the quantity nM er changes with frequency, as shown in Figure 15.7. It can�
be seen that at ~requencies above 0.6 GHz, the actual efficiencies of devices
is less than the minimum permissible, and this means (for the assumed value of
a), that the use of transistors in this frequency range in maximum power modes,
determined by the corresponding curve (Figure 15.4), leads to unacceptable
thermal conditions in the array.
Also shown in Figure 15.7 are the attainable values of the efficiency for
IMPATT and tunnel diodes. For IMPATT diode modules, the minimum permissible
efficiency amounts to no less than 90%, which significantly exceeds the actual
efficiency throughout the entire range of utilization of these devices. For
modules using tunnel diodes, the values of nM er approximately coincide with
the feasible values. Consequently, these devices can be used in maximum power
modes.
To establish the permissible thermal mode of an array in frequency regions where
the actual efficiencies of a module are less than the minimum permissible values,
the power of an array element should be reduced down to:
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PM per = nl,qaAtpera2 /4 (1  nM), pMnon=rlM (zAtnou 12/4 0 T1M),
while for the values adopted for a and At, it should be reduced down to:
PM Per = 100rtMa2 (1  nM).
In some usage ranges for transistors, the permissible power is more than an
order of magnitude lower than the possible power of the device (see the curve
for PM er for 260 .5 = 3� in Figure 15.4. The maximum value of the total
radiate~d power because of thermal limitations does not exceed:
,~11 :
~'E max =1.44 � !06 ~ ~m ~ 20o.s / ~
In the case of a broad directional pattern, the radiating surface of the array
proves to be reduced and the structural configuration of the array approaches
a cube. The cooling conditions are significantly improved in this case by
virtue of heat transfer from the side surfaces of the structure. Assuming the
shape of the structure to be close to a cube, one can figure that the thermal
flux density is reduced by a factor of three to five. Then, we obtain the
following for the minimum permissible efficiency and power:
I/ilnt nort  40OXZ/Pt 1.1;
PAt non  � 10011M )a/(I 11m)�
In this case, the gap between the maximum possible module power and the permis
sible power rroves to be smaller (see the curve for PM per for 2A0.5 = 30� in
Figure 15.4).
nEmaa.�m W
\
1/I ~
\
~%n11~~ 1)
~
1oa
. .
,i"� LA
,30�) \ ~/1nj
10
~
Transistors
~
.l _ _ i. ..1
o,> n,,li to Gfiz,~ 116, ;Yrff
Figure 15.8. The total radiated power
 as a function of frequency,
taking thermal limitations
into account.
Key: l. IMPATT diodes;
2. Tunnel diodes.
Graphs of the maximum values PE max are
shown in Figure 15.8 for values of
nM per determined from the graphs of
Figure 15.7 for narrow (28p 5= 3�) and
wide (2A0.5 = 30�) array directional
patterns for three types of active
devices used in modules. It can be
seen from Figure 15.8 in particular that
when using tunnel diodes, because of
the absence of thermal limitations, the
radiated power is increased while the
directional pattern narrows much more
rapidly than, for example, when using
IMPATT diodes, for which there are ther
mal limitations.
The estimate made here for the powers
and working frequencies of active phased
arrays with microwave amplifiers or
oscillators using semiconductor active
devices is approximate. The limitations
which have been ascertained are not to
be treated as the impossibility of
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_ of designing active phased arrays for certain values of transmitted powers and
frequencies. These limitations point only towards certain difficulties in con
structing an array. The thermal limitations which arise because of the
necessity of a rather dense layout of the modules, as well as because of the.
insufficient efficiency of semiconductor oscillators and amplifiers in certain
frequency bands are obviously the most important.
There are the following possibilities for reducing or eliminating thermal
limitations: increasing the efficiency of the active devices or developing
effective methods of cooling them, which do not substantially increase the size
and weight of the array structure.
15.5. Active Phased Antenna Array Efficiency
As has already been noted, an active antenna array can have a greater efficiency
than'a passive one as a result of the decrease in the losses in the distribution
system. The possibility of realizing a gain in the efficiency depends on the
efficiency of the array distribution system, the efficiency of a module and its
active element gain.
We shall estimate the efficiency of an active array designed in the configuration
of Figure 15.2a, and determine the advantage gained in the efficiency as compared
to a passive array. We shall introduce the following symbols: nr is the overall
efficiency of the transmitter working into a passive array; r1'~ and n', are the
efficiencies of the distribution units for the passive and active arrays [res
pectively]; rlg and rlM are the efficiencies of the exciter and a module of the
active array; KpM is the power gain of an active array module; P is the RF power
delivered to a radiator; we assume the efficiencies of the radiators to be close
to unity in both systems.
The efficiency of an array is nA = 1 Pv/Pp , where PD is the power consumed from
the active element power supplies; P,~ = Pn f P~ are the total power losses in
the array, which are composed of the RF losses P,~ in the distribution device and
the conversion losses P~ in the active elements.
For an active array, when figured on a per radiator basis, we have a coriversion
ioss power in the generator Pr,~ and the exciter PBTr:
 prn (1Tjr)Phlr; PAn = (1q0)P/KPM Tlfi 'qn.
The power losses in the distribution unit are:
p' (1 r41) P/KnM T14)�
The power of the power supplies for the modules Por and the exciter POg, when
figured on a per module basis, amount to Pp = POT, + POB  P/nM + P'Kp,~tn(D'nB.
By summing all of the losses, in accordance with the definition introduced for
the efficiency, we obtain the expression for the efficiency of an active array:
'lAg YI�Ilm YIn ('`PM I 1)IC'IM+ I\f AI 'I"b 'IA~�
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The efficiency of a passive array is nAw = nrno. Thus, the gain in the
efficiency of an active array as compared to a passive one is:
SIAA TIM yl~ _ Kp4t1" I
Ylep rICTId tIMIYItlKpm
The graphs of the advantage gained as a function of no plotted from the formula:
M (KrM I1)/(1I KrM +1(b),
are justified for n(b = n$; nr = nM = nB, are shown in Figure 15.9. To determine
the advantage gained, M, using these graphs, the scale on the or.dinate is to be
changed by a factor of nM/nr time5 in the case where nM # nr, which is.frequently
encountered in practice.
3 >U 70
y
~ .
~
\
, I '~n,,  Z
I
11.
>
0 0,7 ,n, 1 obtains if the efficiency
of the modules used in the active array satisfy the condition:
14t >
or when KpM � 1, the condition npf > nrn4~. For example, if in a passive array
nO = 0.3nr = 0.5, then a transition to an active array is expedient in power
terms where the efficiency of a module in such an array is no less than 0.15.
It should be noted that the increase in the radiation power in the direction of
the main lobe of the directional pattern when changing over from a passive to
an active array, with a constant power of the power supplies, will be less than
the advantage gained in the efficiency. This is due to the presence of additional
amplitude, and primarily phase errors in the active elements of the modules. The
influence of amplitude errors in the output signals of the modules can be dis
regarded, however, the appearance of additional phase errors leads to a reduction
in the directional gain of an active as compared to a passive array.
15.6. Recommendations for the Selection of Module Circuits and Parameters
The circuit configuration of a module is selected by working from the necessity
of obtaining the requisite microwave power level at its outputs assuring as
high an efficiency as possible for a module as well as a power gain sufficient
to reduce the power in the distribution system (for example, by an order of
magnitude as compared to the radiated power) for the purpose of increasing
the efficiency of the active array. Moreover, the point of insertion of the
phase shifter and the modulation method are to be determined, if modulation is
provided for the signals specifically in an array module.
The selection of the structural configuration of an active antenna array module
should start with the estimate of the module output power. For a known value
of the total radiated power PE and a specified directional pattern width of the
array, the power required for each radiator, for a square array with a step
_ close to a/2, is PM = 7. 105(200.5)2PS'
The power of the output stage of a module is P= 1.2P1. By knowing P, we choose
the semiconductor device for the output stage, which, in providing the requisite
power at the working frequency, has the greatest efficie,ncy. For a comparative
estimate of the power and efficiency of various semiconductor devices, one can
make use of the graphs of Figures 15.4 and 15.7. In accordance with these
graphs, it is preferable to employ transistors in the decimeter band up to
frequencies below 1 3 GHz. We will note that in this band, the output power
of a module can be increased as compared to the values defined by the graphs of
Figure 15.4 through adding the powers of several transistors in a module.
At frequencies of 1 3 GHz, one can recommend the use of a multiplier stage
using a varactor with a multiplication factor of 2 to 4, excited by a transistor
oscillator, at the output of a module. The output power of such a transistor
varactor network runs to a few watts with an overall efficiency of 20 to 40%.
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At frequencies of 1 3 GHz, one can also use TRAPATT diode amplifiers and
selfexcited oscillators.
Both transistorvaractor chains as well as microwave diode oscillators, IMPATT
selfexcited oscillators and tunnel diodes can be used in a frequency range of
3  10 GHz.
One can obtain powers of units and fractions uf a watt witl_i efficiencies of up
to 10% in the output stages of a module. Additional considerations may be taken
into account in the final selection of the module output stage. For example,
the use of transistorvaractor chains makes it possible have phase control at
zi reduced frequency, which makes it.possible in a number of cases to reduce the
losses in the phase shifters. On the other hand, diode oscillators and amplifiers
are simpler and more compact.
Ttke choice of the semiconductor device for the module output stage determines the
eritire structural configuration of the module to a considerable extent, since
a11 of the remaining module stages are chosen by working from the necessity of
obtaining a definite module power gain.
Transistorized oscillators and amplifiers in the decimeter band, as a rule, have
low power gains (2 to 4), and for this reason, to obtain an overall module gain
of about 10, it is necessary to use 2 to 3 stages of amplification.
Then one can estimate the overall module efficiency, taking into account the fact
that at low gainS per stage, it is determined by not just the efficiency of the
output stage, but also the preceding stages. Assuming the efficiency and gains
of all of the stages to be approximately the same, the overall efficiency of
a module can be estimated from the following formula:
ni=n ~
11 ~ ' .
Klnt
m I~ 'ic ~lt
where nl is the efficiency of a stage; KpK is the stage power gain; n is the
number of stages in a module. The losses in the phase shifter have not been
taken into account in the formula cited here, assuming that the phase shifter is
inserted at the i�YUt to the module. The insertion of the phase shifter at
another point leads to a drop in the efficiency, and this becomes greater, the
higher the power level at which the phase shifter operates. The value of the
module efficiency obtained in this manner can be taken as the basis for the
estimation of the thermal mode of the array (see � 15.4). Such an estimate should
ascertain the necessity of forced cooling of a module.
In the case of active phased arrays using diode generators in a module, as a rule,
, is a single stage design and consists of an amplifier or a selfexcited oscilla
tor and a phase shifter separated by an isolating element. The output of the
diode generator is fed to the input of the phase shifter, where the generator is
of the same type as the module generator. Taking into account the fact that to
obtain a sta'ole gain mode or reliable synchronization of a diode generator, the
ratio of its output power to the excitation or synchronization power should be
approximately 10, the oscillator or amplifier of the preceding stage can drive
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5 to 10 modules. Thus, in the case where diode amplifiers or oscillators are
used, an active phased array is designed in the circuit configuration of Figure
15.2b, which makes it possible to standardize the active elements of the array.
When choosing the modulation method, as well as the moduleted stage and number
of stages, one is to be governed by the same considerations as for multistage
transmitters. Here, we shall only note some of the special features which
are related to the fact that a large number of modules are modulated simultan
eously. Tao techniques can be used for the simultaneous modulation of the
modules. The first, incorporated in each module is its own modulator, while
the modulating signal is fed to the inputs of the modulatars at a low power
level. In the second, one rather high power modulator services all of the array
modules. With the second approach, the modulator power proves to be increased,
since a portion of it is lost in the distribution device for the modulatiug
signal. In the f irst, the module size is increased and its thermal conditions
can be degraded because of unavoidable losses in the modulator. In both cases,
the distribution unit for the modulating signals should be carefully designed,
since with broadband modulation (for example, using short pulses), various
modulation distortions can appear in a complex channelizing system.
We shall also note that any amplitude modulation in an active phased array using
synchronized diode amplifiers or oscillators is possible only in the output stages, while frequency modulation is possible only by means of synchronizing
 the output stages with frequency modulated signals. In all cases, the spectral
width of the modulating signals should not exceed the synchronization bandwidth.
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CHAPTER 16. EXTERNALLY EXCITED OSCILLATORS AND AMPLIFIERS USING POWER TRANSISTORS
16.1. General Information
Microwave power transistors are widely used in externally excited oscillators and
amplifiers (Figure 16.1), used in the modules of transmitting active phased
antenna arrays as the driver or the output atages, where these have been given the
name of power amplifiers. A specific feature of these amplifiers is the compara
tively high ovtput power level (more than a watt) with a relatively high efficiency
(more than 30 to 40 percent). The power gain Kp is of no less importance for pre
amplifier stages. Power amplifier usually operate in a transistor collector cur ~
rent cutoff made. They are frequently structured in the form of hybrid integrated
circuits (GIS). Questions of the theory and design of microwave power amplifiers
are treated in the literature [15].
~
C,
OA ~6n2
Cy ~ba1 ~ C4
_1
~Ho
Figure 16.1. Basic schematic of
a microwave trans
istor power amplitier.
A number of problems must be solved when
designing a microwave power amplifier,
one of which is assuring a transistor oper
ating mode which makes it possible to
obtain sufficiently high values of the
efficiency and Kp for a specified output
power. Because of the difficulty of an
analytical solution of such a problem,
experimental techniques are frequently
used in practice to determine the optimum
operational mode of�a transistor [6].
The procedure presented here for the design calculations of a microwave power
amplifier is based on the utilization of the "piecewiselinear" transistor model.
This thoery makes it possible to analyze the major processes in a transistor in
a collector current cutoff mode, ascertain the influence of transistor equivalent
circuit parameters of its onerational mode, develop an engineer procedure for
design calculations of high power amplifier stag2s and compare two transistor cir
cuit configurations: common emitter (OE) and common base (OB).
Tt:e amplifier mode depends in many respects on the proper design of its external
microwave circuits. In this regard, quest ions of, the electrical and structural
design of microwave circuits for transistor power amplifiera are treated in
Chapters 17 and 20..
16.2. The Equivalent Circuit of a Microwave Transistor
The equivalent circuit of a microwave power transistor is shown in Figure 16.2 for
the collector current cutoff mode. Taken as its basis is the physical equivalent
circuit of Giacoletto, supplemented witil certain elements of importance for the
t,icrowave band. We shall explain the elements of the circuit of Figure 16.2: Lb2,
Le2 and Lc2 are the external lead inductances of the base, emitter and collector
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[respectively], usually macfie in the form of strips or stubs; Lbl, Lel and Lc1 are
the corresponding inductances of the internal leads; CbO, Ce0 and Ccp are the
transistor lead capacitances to the package; CKK is the collector metallization
capacitance to the package; rb, rc.and re are the resistances of the base, collec
tor and emitter material (re also incorporates stabilizing reistance which is a
structural component of a number of microwave transistors); r is the recombina
tion resistance; Ce is the barrier capacitance of a cutoff emitter junction; Cdif
is the diffusion capacitance of a turnedon emitter junction; Cca and Ccp are the
components of the collector junetion capacitance, called the active and passive
component capacitances respectively of CC; CCe is the through capacitance of the
emitter contact region;
 Iu� > U"
~
lg r. U, iill' : U,
is the equivalent controlled current generator. The junction transconductance is
a complex quantity: S. CxI) j(,)T,,, , where wtn = 0.4i.o/wult is the phase
determined by the charge carr.ier transit time (wult  2nfult, where fult is the
ultimate current gain frequency in a cammon emitter configuration); u7 is the
instantaneous voltage across theemitter junction; U' is the voZtage shift for the
approximated static characteristic ic (uv) of the transistor with the piecewise
linear approximation. The capacitances between the leads of the transistor are
not shown in the schematic of Figure 16.2, which can be neglected in the case of
power devices.
The use of the relatively cumbersome equivalent circuit of Figure 16.2 is justi
f ied for practical calculations at frequencies w for which 11rwCVp < 10wLB2. At
lower frequencies, one may disregard the capacitance between the le3ds and the
package Cv0 (Cb0, Ce0 and Ccp) and the capacitance CKK, while the inductances of
leads I.v1 (Lbl, Lel and Lci) and I.v2 (Lb2, Le2, Lc2) are replaced by their sum
Lv = Ivl + Lv2�
The parameters ot the equivalent circuit depend on the currents flowing through
it and the applied voltages. Because of this, a rigorous calculation of a trans
istor operating mode is difficult, even on a computer. However, one can make a
rather sunple analysis of the processes in a transistor, taking into account the
major phenomenon in a cutoff mode, if a simplified model is employed. The para
meters of this model are the result of linearization of the actual transistor
parameters individually for the active operating regions and the cutoff region.
Linearized parameters depend on the transistor operating conditions, for a sel
ected mode, they are considered constant within the range of each region.
Reference data, which are usually the following quantities, can be used to
 estimate the parameters of a transistor equivalent circuit. h21e is the static
current gain in a common emitter configuration, Tk is the time constant of the
internal feedback c:ircuit, also designated as "rb, Cc, where Cc is the capacitance
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i    ____~CNw I
~h2 ~ L6/ r6 Citn CicO nH ~ 42
~ tH,
c6o ~ ~ cxo
1 V^ _r
I CA0 !�3 CN3 ~
~ransistor Package I
h'o,onyc 1a 1
LmpdH3UCmOpa
~
rao
172
Figure 16.2. The equivalent circuit of a microwave transistor.
(The closed position of the switch corresponds to
the turnedon state of the transistor; an open
switch corresponds to the cutoff state).
of the collector junction, Ce is the capacitance of the emitter junction, fult i$
the ultimate current gain frequency in a common emitter configuration. Moreover,
the collector current at which the value of fult decreases by F2 as compared to
the maximum value fult at a certain frequency f for a specified collector voltage
is also indicated for power transistors. This current is called the critical
current icr 111�
The following relationships exist between the data sheet parameters for a transis
tor and the equivalent circuit parameters:
Cd = Ce + C dif' Cc = CC8 + CcP, Cca = C�/(2...4);
Tk = rbCca' h21e S,r' fult STr/2nCd.
Ln Ca1 Ln11111; Cic: Cun' IAm, (%ita Ci,/('l..A;
rG (,3m+ /t213 Su I'; frP Sn/27LCn'
S.ff = qic /kT. = 42.5ic /(1 + 3.66 � 103t
J J
S� yi�/kT� 42,~iK/(1~3,GG � 10~ t�),
 28], 
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= TABLE 16,1
7'Hn
TPl II:IIICTOr:t l
Transistor
Tvpe
I'T3A7
 1CT(i0G
I("1'(i I UEi
hT9n4 n
!C"1'907A
 1CT909A
IC1'9()9li
I(7'91 3A
I(T91 3G
K'rgl 3T3
KT91 8r
I(T919A
I(T91 9G
KT919i3
 KT937A2
Aona W~Da~ta~
Operat
man e
Ult~
~
ac k
r
~
ge 
u

cY1
CI
.r
�r,
H
~
V
U
au
� N
p.
a
,
't
:4
`5
o
a 6
o
p7
a .
`.=8
1+ C~
W
.
N
12
0,I6
160.1
l0U
~~3
[[I
GO
4
0,8
0,4
:>0,1
44
1'l0
85
?,fi
11
26
4
11,3
��0,4
l50
1,5
11
60
(ill
4
4
1, i
3
O,H
I
0,8
1,8
ICi
7,J
120
120
80
8.)
r)
I:i,S
(iU
3,5
4
2
4
� 3,8
120
85
2i
7E
(A
3,5
g
q
8
1,9
120
&i
50
iri
;;,~i
I
(1,5
20
150
125
~
4,7
1'iI
J)
3,i
3,5
2
'L
I
I
1,2
2
10
II)
150
IJO
125
125
8
12
~i
30
2.5
0.2
0,2
50
IJO
HJ
2,5
fi
45
3,5
1,5
0,7
l,:'i
12
150
10
fi B
45
3,5
0,7
0,35
0,8
25
150
5,3
G
li
95
3,5
2,5
0,4
0,2
0,25
0,4
0,2
40
34,ri
150
150
3
3,6
Para~meter
e
d
M?
a1
Ty
n
3ne1 (1': (If.f~~
when u. > U'
u.  U' = r~~~:m~;t~(1' Ul U'
J
(16.8)
The complex amplitude of the first haraionic Ig1 and the constant component of the
equivalent generator current, Ic, can be f,1nd using the expasion coefficients For
the cosine pulse Y1 and yp (see Table 16.2):
Igl = 11.1 1 (16.9)
I = /K : _ /i�n /uirll Yn/co. . (16.10)
C
! The cutofi angle is governed by the balance equation for the DC voltages at the
; transistor input, which taking expressions (16.6), (16.7) and (16.9) into account
as well as the relationships between the transistor parameters, can be reduced to
the form:
(LvO  U.I)W u1tCe/Igi = cosA/Y1 + (1wultc e/S)Y0/Y1 =
 WnnU')Glrn Cj/Iri cos 0/yl+(I uirn ~o/s) YolYi�
Here, Uv0 is the baseemitter bias voltage.
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r� /
I
a1
d)
Figure 16.6. The equivalent circuit of a common base amplifier
for the first harnaonic current and voltages without
taking the package capacitances into account (a)
and taking them into account (b).
Knowing the transistor parameters S, wult, Ce and U', one can determine the
expansion coefficient yl for the specified current Igl and the bias Uvp using the
graph of Figure 16.4, which is plotted in accordance with equation (16.11).
The peak inverse voltage across the emitter junction, in accordance with (16.7)
and (16.9), is equal to:
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u
eb peak (t3r nNK 
(16.12)
The first harmonic of the voltage at the emitter junction, averaged over one
high frequency period, in accordance with (16.7), is equal to:
U
junction1 ti. Ujunction 1 off � Unt ' Un t aniap'= Jr (I Yt)/WC~� (16. 13)
1
In a common base circuit configuration, the voltage across a cutoff emitter junc
tion and the collector current pulse are somewhat asymmetrical in the general
case with excitation by a harmonic emitter current. This leads primarily to a
change in the phase of the, fundamental of the emitter output current relative to
the input current. However, at the higher operating frequencies of a transistor
(above frP [feutoff]), this change is comparatively small (about a few degrees)
and it can be disregarded because of the small gai� in the calculation precision.
The harmonic analysis ma.de here makes it possibie to move on from the transistor
model shown in Figure 16.2 to the equivalent circuita of gsnerators with common
emitter and common base configurations for the fundamental current and voltages.
For frequencies at the which the capacitance CBO can be disregarded, these cir
cuits are shown in Figures 16.5 and 16.6a, while for higher frequencies at which
a common emitter circuit is usually employed, see the circuit of Figure.16.6b.
An equivalent current generator with a switch is replaced with a fundamental
harmonic current generator IF [Ign 1], defined by formula (16.9), while the
emitter junction is represented by a capacitance averaged over a period of the
radio frequency, which in accordance with (16.13) is equal to ~ = Ce(l  Y1)'1.
The resistance rK represents the losses in the material of the collector in the
parallel equivalent circuit.
The system of equations which relate the complexing amplitudes of the currents
and voltages in the circuits of 2igures 16.5 and 16.6a is given in 516.5.
16.4. The Properties flf Common Emitter and Common Base Generator Configurations
In analyzing and comparing the main properties of common emitter and common base
oscillators/amplifiers, it is expedient to treat two cases separately:'
1. A low inductive reactance for the comnon lead wLcom, for which the fundamental
harmonic voltage across the inductance Lcom does not exceed 3 to 5 percent, of
, the voltage amplitude across the collector. The following inequalitycorresponds
to this case: for a common emitter circuit:
wLe = wLe < 0903R,,,
while for a common base configuration:
 '  289 
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wLb = wLr, < 0, 1(W,�1, 1'i/w) RK (t  I (0rp C~.c Rec Yl)i�
where RK = CRc1 =(0.25...0.35)UKQ/Pout, Y1 = 0.3...0.6.
2. A large inducfci.ve reactaiice of the camiaon electrade lead.
(16.15)
In the first casQ, all of tne design calculation equations prove to be simpler and
they can be used to e3si?y illustrate and explain the basic properties of ampli
fiers and simply execute the aesign o� an osci].latar/amplifier for a specified
power in a load. For this reasan, we xnitially turz to the first case. We shall
assume that f< fcutoff� Then, as a rule, one can disregard the resistance rk' and
consider the junction transconductarce ta be a real quantity. Moreover, in this
case, one can neglect the capacitances CKE [CCe] snd Ckk (Cccl�
~ By solving the eystem of equations which relate the complex amplitudes of the
voltages and currents introduced in the equivalent circuits of Figures 16.5 and
16.6a, we find the current transmission gain:
Ki com.em. Igenlo~ibl
Ki cam. base
Kio7  .1rt/Ar. J~~lrp Y~/('~ (1 ~ ~ c~~~�i, Vi),
Ntor~ =  (1 ~ ~ j~~/~o,.n 1'i)~;
And the input impedance for the fundamental frequency current is:
jXnsl; 711xi u9" U111161; 7ns116' Vi~~~r~l+
where
r,, (I I tol�p(%itnRicVi)1ffirpl.nVt+ r,'
rns o~ = I 1� ~ c% ~
Xux 1 (1:) a~L~
~rp �u ~~y~
I~ o~,.P Cic 12 u Vi
� ~orn Y~
rQ (I ~ a~rp CI(n ~lu yi) oi~.p LG 1'i �1 ~~ir~, Cic ~~ia 1'i) I' orC;, o~
~
~ I (lnrn Yi /(o)a
rnxt ol; . u1,3 , 1
11 ....Yi
uiL`~ (I Lot ~~ic1'i) I r'~ 0 ~ u~rp Lnn ~l~a Yi) o~ aiC..)
~ ! ~ 1�~(u~,.i,yi/u~)a ,
and the power gains are:
(16.16)
(16.17)
(16.18)
g
(16.19)
(16.20)
(16.21)
R,c
~ G~I ~I�~co,.p C,c~~'icYi) ~
l r
(16.22) (16.22)
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K r us Yi/o~)a Ru ( rrl,(1  ~ ui~�~, C~~n R~c 1'i) 
C,c RK Yi)  I (I Yi) (01�1) 1'1/(.)2C...f
(16.23)
_ The properties of microwave transistor oscillators/amplifiers are determined to
a significant extent by the high level inter.nal feedback loops in the transistors,
the nature of which differs,for the comnon emitter and common base configurations.
In a common base generator configuration with a resistive load, these are negative
feedback loops through the inductance of the emitter.lead Le and the collector Ck.
The governing factor in a common base configuration oscillator/amplifier is the,
positive feedback through the base lead inductance.
_ It follows from formula 16.16 that in a common emitter circuit, the coupling
through Ck leads to a reduction in the current transmission gain by a factor of
. (1 + wcutCkRkYi) as compared to a shortcircuited load. In accordance with (16.17),
the transmission gain Ki com.base depends on the load impedance. In both cir
cuits, the current transmission gain at microwave frequencies is usually less than
unity. Only at freuencies several times lower than fcut in a comnon emitter con.
figuration does Ki > 1. .
In a common emitter circuit, the real part of the input imgedance rinl OE (16.18)
is positive in the case of a resistive load and is independent of frequency. The
quantity rin pB [common base rnput resistance] (16.20) depends greatly on the
frequency and with an increase in the inductance Lb can become negative. This.
 means that an externally excited generator is potentially unstable beaause of the
positive coupling through Lb. _ For both configurations, the quantity rinl~. proves to be small: units or fractions
of an ohm. An increase in the maximum power of a single transistor up to hundreds
of watts is accompanied by a reduction in rinl down to hundredths of an ohm. In
this case, the efficiency of the input matching cirauit proves to be poor and this
is one of the reasons which limit the increase in transistor power.
The reactive component of the input impedance, xinl, close to the upper cutoff
_ frequer.cy of a transistor, is, as a rule, of an inductive nature which is due to
the inductances of the base and emitter leads. Usually, xinl component is consid
 ers:Dly greater than rinl and is a component part of the input matching network of
an amplifier.
It follows from formula (16.22) that the power gain in a common emitter configura�
tion is inversely proportional to the square of the working frequency. It is
governed to a considerable extent by the values of Ck [Ccoll] 'and Le. It can be
shown that if wcut Leyl > 3rb and ubutCcolRcolY1 > 3, then: .
Kp com . em. z 1/W 2C c L e h'r oa ^.1 /wz C,c La�
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 An amplifier/oscillator is little sensitive in this case to the scatter in the
transistor parameters and the change in the cutoff angle, but the gain proves to
be low. The upper working frequency of a common emitter amplif ier, corresponding
to a reduction Kp down to 2 to 3 usually does not excEed fcut
Positive feedback loops act through the inductance Lb anc3 the capacitance CCe in
a common base amplifier. The feedback through Cke is of secondary importance for
power transistors, just as the feedback through the capacitances Cka and Ckff. The
positive feedback through Lb explains!the.!comparatively high sensitivity of a
common base configuration to changes in transistor parameters and cutoff angle.
 This coupling can cause parasitic oscillation or strong gain in stability. It is
necessary to take special steps to prevent this: incorporate emitt:er degeneration
bias, insert neutralizing capacitances in the base circuit, or use an unblocked
resistance in the emitter circuit. A special resistor can be used for this re
sistance, and sometimes, the internal resistance of the exciting generator, the
output resistance of the power divider bridge, etc.
The upper working frequency in a common base configuration can run to approx4mately
3fcutoff�
Formulas (16.16) (16.23) were derived with the assumption that w Lcom. is sma.ll.
This assumption leads to a marked understatemcnt of the output power and efficiency
in a common emitter configuration with large values of wLcom and to an exaggera
tion of these quantities in the couanon base configuration. Because of this, we
shall consider the impact of Lcom on the power relationships in an amplifier/
' /oscilla~or.
 In studying an externally excited generator, we are dealing with a system of two
generators (the input signal generator and the equivalent controlled generator
of the active device), which are connected through the elements of the equivalent
circuit of the transistor to its load. It is well known from the theory of tt.e
joint operation of generators driving the common load that depending on the
amplitude and phase relationships in the circuit, both power addition in the load
as well as the transition of any of the generators to a power consumption mode are
possible.
In the common emitter configuration, the voltage across the load, Uk, and conse
quently also the power are determined by the voltage difference between Ug and Ue.
The voltage Ue i: partially produced by the excitation generator current. The
phase relationships for the currents ana voltages are such that there is an in
_ crease in the power in the load as compared to the case where Le = 0("straight
through"). With a short circuit of the output or with a low load resistance, the
portion of the ekcitation generator power related to the voltage Ue is dissipatod
in the collector of the transistor.
If the generator effir_iency is defined as the ratio of the power in the load, Pput)
to the power consumed from the collector supply, Pp, because of the straight
through flow, the generator efficiency increases simultaneously with the increase
in Pout. It should te noted that with this definition, the efficiency can prove
to be greater than unity.
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In a common base configuratio;i, the voltage across the load is primkrily determined
by the sum of U and ULb, since the voltage across the capacitance C is extremely
smaZl. Just asgin the preceding case, the voltage across the inductance of the
common electrode is produced partially by the input current, however, in contrast
to the cornnon emitter circuit, the phase relationships in a common base configura
tion are such that a portion of the power Pg is transmitted to the input network.
Because of this, the usEtul power in ttie load of an externally excited generator
and the generator efficiency are reduced as compared to the case where Lb = 0, but
nonetheless, less power is required from the excitation generator and consequently
the gain is increased and sel�excitation can even,occur.
16.5. The Procedure and Sequence for the Design Calculations of the Operating Mode
of an Oscillator;Amplifier
In the course of designing a generator with external excitation, one is to first
c'L'ioose the transistor and determine:its circuit cor.figuration based on the speci
fied power and frequency. If the requisite transistor type is not present in
Table 16.1, one can estimate the parameters of its equivalent circuit, using
reference data and the estimates given in this section. Then the design calcula
tions are performed for the.,electrical and thermal operating conditions of the
transistor.
The type of transistor is selected taking into account the specified requirements
for the output power and frequency from the reference handbook data. The para
meters of the typical operating condition, corresponding to the maximum utiliza
tion of the device both with respect to power and frequency are specified in the
reference data for microwave power devices. The indicated output power corres
ponds to a transistor package temperature of about 20 �C. The useful power falls
off with an increase in temperature, since the perrnissible power dissipation is
reduced. With a reduction in frequency, the maximum useful power of a transistor
increases.
It is expedient to use microwave power transistors at powers of no less than 40 to
50% of the power in the typical mode indicated in the handbook. Considerable
underutilization of a device with respect to pewer leads to a sut,stantial degrada
tion of its amplification properties.
The range of operating frequencies reconanended for a given transistor is also
frequently indicated in the handbook. The lower working frequency is usually
recommended at no less than 20 to 30 percent of fcutoff, while the upper frequency
is close to fcutoff for a contmon emitter circuit and reaches 2 to 3 times fcutoff
for a common base configuration. At the lower operating frequency of this range,
the maximum output power can be approximately twice as great as the power at the
upper frequency limit.
It sometimes turns out that the requisite power at a specified frequency can be
obtained with different transistors. Where a choice is possible, it is preferable
to use transistors with z higher gain, however, it is not desirable ta use devices,
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the lower frequency limit of which is higher than the specified working frequency,
since in this case, operational reliability will be reduced, and the probability of
selfexcitation will increase. Moreover, higher frequency devices also cost more.
The circuit configuraVion (common emitter or connon base) is determined in a num
ber of cases by the package structure of the selected transistor. For example,
KT907 and KT909 transistors can be used only in a common emitter configuration,
since they have the emitter connected to the package. The KT918 and KT919 trans
istors, on the other hand, are used only in a common base configuration: they have
the package connected to the base. The KT606 and KT904 devices can operate in
either configuration, since they have leads which are insulata.d from the package.
The KT911, KT913 and KT916 devices, aithough they also have leads insulated from
the nackage, are more conveniently used in a comnon emitter configuration, since
twr of their emitter leads should be inserted in the circuit in a balanced
fashion because of structural design considerations. The common base configura
tion is a higher frequency circuit and is:used considerably more often frequencies
above 1 GHz.
The parameters of traiisistors needed for operational mode design are given in
Table 16.1. If the selected transistor is not present in Table 16.1, its para
meters may be estimated by knowing the data sheet val.ues for fcutoff, rbCk and Ck.
Moreover, one must know the inductance of the common lead. Transistors which are
specially intended for common emitter or common base circuits have a minima.l
comnon lead inductance (0.1 to 0.4 nHy) while the inductance of the collector and
input leads are several times higher. The capacitattce Ce is usually 5'to 10 times
greater than Ck; the resistance rk is close to rb and re does not exceed 0.3rt.
The data sheet value of fcutoff is usually 1.5 to 2 times less than the actual
value, while the data sheet value of Ck is overstated by a few tens of percent.
The time constant 4Ck, which is indicated in the data sheet, can sometimes exceed
the actual value by an order of magnitude. It must be kept in mind that the
parameter rtCk is the product of rt times Cka, and not times Ck. The parameter
h21e is not critical in the design calculations for microwave amplifiers and
oscillators. The static characteristic shift voltage U' for silicon transistors
falls in a range of 0.6 to 0.9 volts. The parameter Srn [Scutoff] can be taken as
approximately equal to 15Pout/Uio, where Ukp and Pout correspond to the typical
mode (1'out in watts and Ukp is in vol,ts).
If the design calculations using the typical mode power and frequency yield a
value of Kp which differs from the data sheet value by no more than +20% for a
common emitter cnfiguration, one can assume that the equivalent circuit parameCers
have been correr.tly est.i,mated. If the absolute value of the peak inverse voltage
at the emitter Iueb peakl is greater than the permissible value or almost equal to
it according to the clesign calculations, this means that the calculated value of
Ce is understated.
We shall move on the design procedure for the transistor operating mode at a
specified power into a load Pout� The initial data for the design cAlculations
are: power.delivered by the transistor, Pout; the working frequency f; the
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ambient temperature, tcp; the transistor type and the circuit configuration (common
emitter or common base).
If the requisite power is close to the level which the transistor can deliver (but
does not exceed it), then the standard supply v;,ltage for this transistor is to be
used: most often 28 voirs. When a transistor is underutilized in terms of power, it
it expedient to lower thz supply voltage to improve the reliability. However, one
must take into account the fact that cutting Ukp in half leads to a reduction in
fcutoff by approximately 5 to 15% and to an increase in Ck by approximately 20 to
25%. The bias voltage Ugp in power stages is usually taken as zero. This simpli
fies the circuit and makes it possible to obtain. a cutuff angle close to 90�, fcr
which the ratio between Pout, the efficiency and Kp is cloaz to optimal.
The transistor package temperature can be taken equal to pk  tambient +(10...20)
�C, taking into account the extra heating of the heat sink relative to the ambient
medium.
If the influence of, .wLcom can be disregarded in accordance with inequality
(16.14) and (16.15), then in the design calculations one can employ the simpli
fied equations (16.16)(16.23). The procedure for such design calculations is
set forth in [56J.
We shall give a design calculation procedure for the more general case, where
inequalities (16.14) and (16.15) may not be observed. In this case, however, it
is difficult to accomplish the calculations directly for the specified power in
the load. If is considerably easier to carry out the calculations by specifying
the power Pg developed by an equi.valent generator. This power in a common
emitter configuration is to be taken as 10 to 20 percent less than Pout, since in
this circuit, the transistor output power has an increnaent because of the
straight flow through of a portion of the input power. On the other hand, Pg is
to be taken greater than Pout in a common base configuration, since a consider
able portion of the power developed by the current generator, Igl, is fed to the
input circuit of the amplifier. At frequencies above fcutoff, Pg is to be taken
at 20 to 20%.higher than Pout in a common base configuration; at frequencies
below fcutoff, this difference is less.
Initially, the calculation is carried out in the following order regardleas of
the circuit configuration (cotrQnon emitter or common base).
1. We determine the collector voltage utilization factor, specifyirg Pg and Uk0
taking what has been presented above into account:
~cutoff trr =:p+~ [1{ 11GP;/(5~~, UKo)
2. We find the current and voltage amplitude of the fundamental frequency of the
equivalent generator:
= U,. i~ _`lP,./U,..
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3. The peav collector voltage, uk peak, should not exceed the permissible value of
UKE max:
u  jJ jJ < jJ = /llCwn: UicU Ur< uI{3ninx�
coll.peak c0 gen CE max
This inequality is not observed, the operational mode is to be changes or another
type of transistor is to be selected.
4. We determine the transistor parameters:
5it=: 42,5L,,/~l ~3,GG� 1O' r== l~zi~lS~~, S=1~1i ~ Ir~..~
f r I r9 (1+ /1219)1'1�
The value of tn (tjunction] can be taken equal to the ultimate permissible value
(see Table 16,1).
5. Having cal.culated the values of the parameters (UgO  U' )wcutCe/igenl and
WcutCe/s, We find the expansion coefficient yl from the graphs of Figures 16.4 for
the fundamental frequency of the equivalent generator current. Then, for the
value found for yl, we determine the values of cos6 and the coefficient of the
_ form gi = Y1/Yp from Table 16.2.
6. We determine the peak inverse voltage at the emitter, ueb peak, from formula
(16.12). The absolute value of ueb peak shodld not exceed Ueb max�
Then in paragraphs 7 through 22 we calculate the complex amplitudes of the currents
and voltages in the element.s of the equivalent circuit of Figures 16.5 and 16.6a.
The current Igl is taken as the vector with the zero phase. In this case, the
vector Igl is equal to its own scalar value Igl found in paragraph 2.
co
1. 1.: j/t,1 (COS ULn  J Slll bYCu~, 171E (J'CI, 0,4(,)/(,),.p.
o0j.p yi
8. Uln 9. Ucea'= = Ul.l Uiil� 10. Ic:Ka..
j~~(: li:i Ur:r,~ � 1 I./rG' li:~ca� l2. UrG' rG Ir'C~ � ~3, U~:~cn
UiG' ~ UtNoi� 14. /f,K117 .~(~1~.unUCiui� 15. /'ic`..'(fil(.,c)2rN~
Ili. /n; flc:icn/ric� 17. 11;1�..: JrG' ICKn I. 1nc� 18. Ut.f /GI �
19� l:11 f Ilo� 20. Ua:_.~ j:>i ('3 j(,)L:.l� 21. T�  U~ I  Uu; ;
i vf~i' I�v,,t� 22. /h, /~lICKa ' fCKll._'//N�
23. We calculate the voltage amplitude across the load and the input impedance of
the tr.ansistor for the fundamental frequency: Ucol com.em.  Ugen  Ue;
Zin 1 com.em.  UB/Ibase 1; Uc com.base = UC col. g?n. + jJL base; Zin 1 com.base =
Ug/Ic1. . UKOa 
Ur zns 1 OR Un~IG1, UK 06=" UCKn1 UI,G+ Znxl OF, UdIaI �
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24..The excitation power and the power delivered to the loaa are:
~ Pexc P. ! 0,5 [Re UII Re 1,,1 Im U� Im I�x11;
pou,W = PANz;...U,)IRCUKiZC/KI I �IIIIUKIt11IKI].
For the common emitter configuration, Iin 1 Ibase 1; Ucol = Ucol com.em.; and for
the common base configuration, Iin 1 Iel, Ucol = Ucol com.base�
If the power in the load Pout found as a result of the calculations differs consid
erably from the specified value, the calculations are to be repeated, correcting
the value of Pgen, taking the deviation into account.
25. The DC componegt of the collector current, the power consumed from the.supply
and the efficiency are equal to the following regardless of the circuit configura
tion:
1K /rj/gi Po Iic UKn; 71`1',1LTx/Pa.
26. The power gain, the power dissipated by the transistor, and the permissible
power dissipation for a given transistor package temperature are determined from
the following formulas, regardless of the circuit configuration:
 KP" PnraxlPn; Ppar, P0Pll1.1X I PR; ' .
pmnx (fn wns~u~~.Rwc� .
The maximum value of tw max [maximum junction temperature] is the maximum permis
sible value of t7r from Table 16.1. '
It must be demonstrateu that Ppac [Pdiss] c Pmax�
27. The equivalent load impedance at the external leads of the transistor is:
Zload 1 Ucol/Ico1 1 j w Leol' iill Vicl 1xi jrol
where Ucol  Ucol com.em. for a common emitter circuit and Ucol = Ucol com.base
for a common base circuit.
In some cases, zero bias is nct optimal. For example, when a transistor is con
siderably underutilized in terms o` power, the cutoff angle in a zero bias mode
is too small as compared to.the optimal value. On the other hand, in a comnon
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base amplifier, a ren'action in the cutoff angles may be needed to stabilize the
operating mode. For this reason, it is necessary in the first case to introduce
~ unblocking bias to increase the cutoff angle, and ?n the second case, to use block
ing bias, for example, selfbiasing to reduce the cutoff angle. In these cases,
the design calculations should be performed for the specified cutoff angle. The
procedure for such design calculations differs somewhat from that given in para
graphs 1 and 5. A more pre.cise formula is used in paragraph 1:
Ecutoff. ~rn`O,ri I1 ~~1 8P.AS'Pat MUKo)
where a1W) is determined for the specified angle 0 from Table 16.2. The bias
voltage UB4 is found in paragraph 5 from formula (16.11), where this bias assures
the spzcified cutoff angle. If the bias is blocking bias, it can be realized by
means of a resistance Re =Ug0/Ic, which is bypassed with a capacitor.
The calculation procedure cited here for power amplifiers is given for frequencies
at which one can disregard the capacitance CBO. For the 3 to 5 GHz band, a more
complete equivalent circuit of a transistor is to be used (Figure 16.2). Common
base amplifiers operate in this band. The equivalent circuit of a common base
amplifier is shown in Figure 16.6b. The design calculations are carried out ini
tially in accordance with paragraph s 118. Then, the following currents, voltages
and resistances are calculated in paragraphs 1937:
19. IC KS1(oCi(,) Ur, 2Q. ILai =/~li�~~f~�K~~
21. UL91 �1(uL"tIL91~ 22. UC90�UL6~UrGI'UniIUL91
(We neglect the voltage across re because it is small).
13. 1C,o j`)C10 "C10 . 24. 1,t II sl ~c~0 �
~5. ll~,~ 2f>. U~ Uc,o'1 Ul.9z �
27. zns (Jg~1~1� 28. UCILK__ UCKII'I' 1ILG. 29. IrNK Jb1Ciflt UCKH'
1III. IIdtl I11K''CKfIlCKa/CK9~ICRK' 31. 32. UCKo "c.cx UtKI� 33. /CHU jotCcn UCKO' 34. 11 .__lCKO �1 1l.ic 1�
3>. (ILK2.,j(tLta:ln� 3fi. Uu=UCicO._'U[.e2. 37. 2n=U01tt�
38. The excitation power and the power delivered to the load are:
PexC  pn:0,5 (Rc Un Rc 1.91 1Itn Unitn
P = p.uax''O,5(PeU�Re/,j�j IinU,, iml0�
out
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The quantities Ic, Po, n, Kp and Pdiss are found just as in paragraphs 25 and 26
of the prec:eding design procedure.
As experience shows, at frequencies on the order ot hundreds of inegahertz the
experimental and calculated resulta for the averaged parameters of a given c;pe
of transistor are sufficiently close, and for this reason, there is no necessity
 of a subs*_antial reworkiizg of the circuitry and structural design as compared to
the calculated values. The influence of the imprecision in the knowledge of the
transistor parameters and their scatter is easily eliminatPd by means of using
fine tuning elements without changing the circuitry.
I The design of amplifiers for frequencies on the order of several gigahertz based
on the handbook data for transistors cai; apparently not be accomplished at the
present time without experimental breadboarding to realize possible changes in the
cixcuitry and structural design. This is explained by the approximate nature and
complexity of the equivalent circuits of tranaistors and the inadequate precision
in determining their parameters. The latter is related, on one hand, to measure
ment difficulties, and on the other, to the scatter in.the parameters. In a
wavelength range of 10 cm and shorter, even a comparatively slight scatter in the
' inductances of the leads and capacitances of the package can lead to sharp changes
_ in the input impedance of a transistor because of resonarce phenomena. This is
, explained by the fact that the input circuit of a transistor, which includes these
reactive elements, forms a resonant system with a hig,h Q(of about 10), which
resonates::within the working passband of the transisL�ox. For example, in a range
of 3 to 5 GHz, the calculated values of the resistive and reactive components of
the input impedance of a KT937 transistor change by two orders of magni.tude. The
resona*_or nature of the input circuit is also responsible for the high sensitivity
of the input impedance at a fixed frequency to small changes in the input circuit
parameters. For example, an error of 20% in determining the inductance Lel close
to resonance changes the values of the resistive and reactive components of the
input impedance of KT937 transistpr by an order of magnitude. Imprecision in
the fabrication of passive networks can have si.milar consequences. Moreover,
when designing coupling networks, certain parasitic coupling circuits which in
fluence the operating mode of an amplifier are not taken into account. Many of
these factors can be disregardea at lower frequencies. The measurement of the matrix parameters of a transistor does not eliminate the
indicated difficult. Such measurements are also have a great deal of ambiguity,
since a s.light change in the length of transistor leads can greatly change the
measurement result.
The conclusion that calculations where one fixed set of any parameters are used,
either "physical" or matrix, are inadequate follows from what has been said above.
It is recommended that calculations be perfbrmed with variations in the parameters.
A series of such calculations will assist in ascertaining the most critical para
meters, predicting possible changes in the operating mode of a transistor which
are related to the scatter in different specimens of the transistor, the change
in frequency, etc., as well as in selecting the kind of coupling network and
providing for the requisite means of fine tuning. .
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17. EXTERNALLY EXCITED MICROWAVE CIRCUITS FOR TRANSISTOR OSCIL]LATORS AND
AtfPLIFIERS
17.1. General Information
In externally excited generators, designed in common emitter (Figure 17.1a) or
common base (Figure 17.1b) confiourations, the microwave networks can be repre
sented in the form of fourpole networks of linear reactive elements, the power
losses in which are neglectably small.
To obtain a selected power operating mode for a transistor, it is necessary to
provide the requisite impedances with respect to the fundamental frequency,
Zin 1 and Zload 1(Figure 17.1), at its input and output. These impedances can
in principle be determined by calculating the operational mode of the transistor
based on its physical equivalent circuit (see Chapter 16 or [1  5]). At the
present time, the calculation of the operational mode of a microwave power trans
istor is an approximate ca'culation, and as a rule, requires in addition that
the electrical parameters of the transistor be found more precisely experimental
ly. Because of this, the method of experimentally determining the total input
Zin 1 and output Zout 1 impedances of the transistor with respect to the first
harmonic at some specified frequency and a definite electrical operating mode
has become widespread in practice along with the analytical technique. A general
ized schematic of a generator in which the transistor is replaced by the
equivalent circuit taking these impedances into account is shown in Figure 17.2.
,
(2)
Cb12
=8x1 Zw1 C6n2
zAx1 Znf /'6n2
e
'
O
vei
,
, ~bn2
Z
4ei
O~
~
~
vei
o~
6n1
~b'a1
i
Z Q
0~
bv
bnf
11
o~
~V
~c ~
b V
~O
6'dnf
(p
(a) I�
~12
Uxo ~27
(b) 61
Figure 17.1.
General schematic of
an
amplifier/oscillator with
external eacitation.
Key: 1. Input microwave networks.
2. Output microwave network.
TpnN3ucmop Txansistor
i t
~Zinl (1) I~~ b iZuBh ~1~ lOad
Bxodiia.v yenh input ~ 6a.rnd~ion qNne OtitpUt C rCUit
ciraui.t
Figure 17.2. Generalized radiofrequency circuit of an amplifier/
oscillator with the transistor equivalent circuit.
It is presupposed that Zout 1 corresponds to the complex conjugate of the
load impedance of the transistor Zload 1, i.e., Zout 112 Zload 1'
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LQX 'L' C6xC+'I'L
Z
3~ 1 rex~
o) (Zi '
z ~ 
ewril '  ~Awx
ZO~l~ ~ ReaK1 C~ur
Figure 17.3. The equivalent circuits of transistor input (a) and
output (b) networks.
TABLE 17.1.
(31
(4) (5)
(6) TaG.n it ua 17.1
Transist
r o
e ~
~
l71
p011911tT0~1H
~q
x
2
a)
;n
a
2
4
KP
(gY
L't
Type
~
a a O
v ~
i
h1'607 G 1,3 129 ~
9 180
2 0
hl'612 4
1,2 l
2,2 1
1 3 45
0,3 6 GQ
28
7
,
,
1~7'!)p4 5,5 6,5 85
5 3 105
h'I'9011� 4
B 0,4
4,9 0,4
3 3' 50 �
3 4 GU
28
28
,
' Kl'!I I A 3 5 36
~ hT!I I A' 5 3 35
G 1
5 l
l,5 4 40
1~5 5 40
28
28
F 1913A 3 3 ,19 52,5
2 2,55 32,9
' 0913G 1
2,7 1
4 1
3 3 45
5 2,8...3 55...G0
28
28
,
i KT913I3 1,2 2,23 16,0
4,4 1
10 2,6...2,8 60i6, Gb
4
28
28
y IiT918 6,5 1,5 200
11 25,0
KT919A 1
6 1
2,4 2
12,7 1
1
4 3 45...b0
25
,
,
KT919r) B 2,3 1.27 64,1
t hT9198'y 4,5 0,955 118,6
~
7,1 I
5,8 1
2,6 5 45
2,6 10 55
20
22
` � Fl`CKUj)qyCliAf1 KOIICTPyK1(IIA
*Unpackaged structure. ,
Key: 1. rin 1, ohms;
2. Lin, nanohenries;
~ 3. Rout 1, ohms;
4: Cout, Picofarads;
5. Working frequency,
GHz;
6. Pout, watts;
7. Power gain;
8� Ucoll. 0, volts.
Calculations and measurements of the
impedanc
es Zin 1 and Zout 1 have shown
[3, 6, 81 that the input circuit impedance,of
a trarAsistor can be
approximated
by the overall impedance of a series
circuit
consisting of a resistance rin 1,
an inductance Lin and a capacitance Cin (Figure 17.3a), the resonant frequency
_
of which can be higher or lower than
the work
ing frequency of the
amplifier/
oscillator, while the impedance of the output
circuit is quite well aaproximated
� by the impedance of a parallel circuit consis
ting of Rout 1, Lout
and Cout, which
is shown in Figure 17.3b. The parameters of
the transistor input
and output
circuits depend on its operating power condit
ions and frequency.
For this
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reason, the impedances Zin 1 and Zout 1 are determined at the working frequency
for the selected operating mode. When an amplifier or oscillator operates in a
certain band of frequencies, it is necessary to determine Zin 1 and Zout 1 for
the transistor, taking its power operating mode throughout the entire specified
bandwidth into account. The reactive component of the impedance Zin 1 or Zout 1
~ can be of in inductive or capacitive nature, depending on the working frequency
of the transistor. Experimental values are given in Table 17.1 for the elements
~ of the equivalent circuit of the input and output networks of some microwave
transistors, with the operating mode parameters indicated for which they were
measured.
When designing microwave circuits for oscillators/amplifiers, we shall assume
 that the impedances Zin and Zload 1(or Zout l) are known. Then the fourpole
network in the input circuit of a generator (Figure 17.2) should transform the
impedance at the generator input Zi to the impedance Zin 1, while the fourpole
network in the output circuit should transform the load impedance Zload to the
impedance Zload 1(or Zout 1)� Consequently, the microwave fourpole network
 in this case plays the part of an impedance transformer and for this reason is
called a transforming network. Since the transformation of the impedances in
the input (or output) microwave network is usually accomplished for matching in
this circuit, the fourpole network is also called a matching circuit. It is
understood in this case that when matching is achieved in the microwave input
circuit of a generator, the greatest power will be transmitted from the stage
driving the generator to the input circuit of the transistor. In this case, the
impedance Zin 1 and the impedance of the fourpole network at the connection
points, Zin 1, Will be complex conjugate quantities. In the output microwave
network when matched, Zout 1 is the complex con3ugate of the input impedance
Zload 1 of the fourpole network on the transistor side, which will deliver the
specified power to the load. Taking that presented above into account, we shall
call the microwave network formed by the transforming fourpole network of
linear reactive elements a matching network [7].
The major electrical requirements placed on the microwave networks of an exter
nally excited generator are providing for the requisite impedance transformation,
as low as possib].e power losses during power transmission, the specified band
width, the requisite filtration level of the higher harmonics and suppression
of spurious frequencies.
A specific feature of high power amplifier transistors as compared to low and
intermediate gower transistors is the small values of the resistive compunents
rin 1 and Rout 1 of the impedances Zin 1 and Zout 1 respectively, which are fre
quently substantially less than the resistive components of the impedances at the
generator input and output. In this regard, a microwave network should provide
a reiatively high transformation ratio (from a few units up to 10). In this
case, the power losses increase markedly in the networks and the passband is
narrowed.
The efficiency of power transmission to the load is estimated in terms of the
circuit efficiency, ncir, defined as the ratio of the power Pload, dissipated in
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in the load, to the oscillatory power P delivered to the microwave network:
ncir  Pload/p
(17.1)
In modern transistorized transmitters, including integrated circuit stages,
externally excited generators are usuaily not tuned. The requisite bandwidth of
the amplifier stage is governed by the conditions necessary for normal trans
mitter operation (for example, the kind of modulation, the range of frequencies
covered without tuning the stage, the requisite phase stability of the signals
at the output).
The requirement of filtration of the higher harmonics basically applies to the
output microwave circuit of an amplifier/oscillator. This is explained by the
fact that microwave power transistors usually operate in modes in which the
voltage waveform at the collector differs substantially from a sine wave. For
this reason, to obtain a voltage close to a sine wave at tlie output of an ampli
fier/oscillator, the output microwave circuit should filter out the higher
harmonics as much as possible.
Matching microwave fourpole networks of the coupled resonant parallel circuit
type or individual I', T and II section filters (or two to three series stages
of such filters) meet the electrical requirements considered here to a sufficient
extent. The use of one to two sucb sections makes it possible to obtain a
rather high impedance transfortnation ratio, provide fr,r a comparatively wide
passband and filter the higher harmonics. In the case of elevated requirements
placed on the passband and the suppressicn of spurious and outof=band signals,
complex filters are enplayed.
When designing the microwave networks for amplifier/oscillators used in the
modules of active phased arrays, it is desirable to use the simplest microwave
circuits which are convenient for integrated circuit technology. Microwave circuits for transistorized amplifier/oscillators using integrated
circuit technology can be constructed from elements with lumped parameters, such
as inductance coils, capacitors and resistors. These components have small
dimensions and a sufficiently high Q in a frequency range of from hundreds of
megahertz up to 1 GHz*. In circuits intended for operatior~ at frequencies above
1 GHz, elements with distributed parameters are used in the form of sections of
unbalanced striplines. Making a stripline on a'substrate of a dielectric
material with a high relative dielectric permittivity (e > 7) makes it possible
to substantially reduce the dimensions of a circuit. With the present state of
the art in microwave microelectronics technology, integrated circuits with
distriUuted and to a greater extent, with lumped parameters have relatively high
losses, which are primarily due to the significant reduction in the perimeter of
the conductors in 4tep with the deciease in the element size. Because of this,
microwave circuits should not be especially complex and contain a large number
of elements. �
See Chapter 20 of this book.
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17.2. The Design of the Microwave Networks of Amplifiers and Oscillators
The Input Microwave Matching Network (Figure 17.2). If the excitation power
is delivered to the generator input by means of a matched transmission line vith
a characteristic impedance of p, then one can assume that the internal impedance
. of the driving sourceZi is equal to p.
In accordance with the equivalent circuit of the input circuit of a transistor,
shown in Figure.17.3a:
Zin 1  Znz1 ~nxl'I' ~ ((Znx I A)Cnx) = rnxi I1x.:i.
The reactive component xin 1 of this impedance can be both of an inductive nature
(at a working frequency higher than the resonant frequency of the transistor
input circuit), and a capacitive nature (at a working frequency lower than the
resonant frequency of the input circuit). For many modern intermediate and high
power transistors, operating in the decimeter band, the values 1/wCin are sub
stantially less than wLin [3.6], and in this case, one can approximately assume
that:
Zin 1 rin 1f" JwI'itz 7ntit r,,i 1 j(,)! ilx�
Because of the fact that the inductance Lin cannot be less than a certain value
governed by the dimensions and structural design of the package and length of
the leads (where a package is absent) of a transistor, while rin 1 decreases
with increasing transistor power [3.8], the quality factor Qin of its equivalent
input circuit at the working frequency f, which is defined as:
' Qxx Q7Lr/ nx/rDx1,
proves to be rather high, something which makes it consi_derably more difficult
to design broadband input microwave circuits for an amplifier or oscillator.
An input microwave circuit using lumped elements is simplest when a I'section
reactive fourpole network is used. Examples of such networks are shown in
Figures 17.4a and b. These circuits are feasible if the resistance p(or the
resistive component of the impedance Zi) is greater than rin 1� In the circuit
of Figure 17.4a, the inductance Lin can be incorporated in the inductance L1, and
then the total series inductance of the I'network is Lseries  L1 + Lin� In tha
ci.rcuit of Figure 17.4b, the inductive reactance wLin can be partially compensated
by the reactance 1/wC2, if Lin is greater than the requisite value of Lseries of
the I'network. The T and II section networks (Figure 17.4, ce) make it possible
to provide for impedance transformations in greater ranges for a specified
frequency band than does a I' section circuit. Moreover, with rather large
parallel capacitances of these circuits, the filtering of the higher harmonics
at the generator input is improved.
When it is necessary to match impedances which differ significantly in value in
a certain range of frequencies, stepped transformation is employed. Circuits
are used for this which consist of several I' or II sections with low transforma
tion ratios.
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CZ ~Qn lQn C~ ~6n � ~8n
C2 02 o, I~z
(a) aj (b) fj (cI9) e~ (e)el
Figure 17.4. Circuit configurations of input microwave matching
networks for an amplifier/oscillator using lumped
elements.
!wr tw2
r o) ta)
l, p (11
~sh ,
ii
a) (Jb)
.
!wt lw2 C6M
15h,, CW H
H
Figure 17.5. Circuit configurations for input microwave matching
networks of an oscillator/amplifier using asymmetrical
stripline sections.
[lsh = shunt inductance].
At frequencies above 1 GHz, microwave networks are des:igned around asymmetrical
_ stripline sections (Figure 17.5), in which lumped noninductive capacitors are
frequently inserted, which make it possible to additionally create an isolating
capacitance in the circuit for the DC. In the circuit of Figure 17.5a, matching
is achieved by using a single loop transformer (1, lsi, 1). In the circuit of
Figure 17.5b, the matching network is made in the form of an irregular stripline
1 with a changing characteristic impedance p(1). The circuit of Figure 17.5c
differs from the circuit of Figure 17.5a only in the presence of capacitsnce
C1. The loops lsh 2 in the circuits of Figures 17.5a and c and lsh in the cir
cuit of Figure 17.5b play the part of RF blocking chokes. The loops are
structurally made in the form of short circuited line sections with a length
close to a/4 (where a is the working wavelength in the line), having a high
characteristic impedance (of about 100 ohms). The radiofrequency short circuit
ing of the loop lsh 2 in the circuit of Figure 17.5c is achieved by connecting
capacitor Cbl.l to it which has a rather high capacitance.
Naturally, the examples cited here do not exhaust the possible circuit configu
rations for these networks. When selecting a microwave network configuration
which meets the electrical requirements placed on it, one must remember that
the use of simpler circuits with low power losses makes it possible to simplify
the structural execution of the microwave network and reduce the overall area
occupied by the circuit on the substrate of a hybrid IC.
The Output Matching Microwave Network (Figure 17.2). In the general case, the
loacl impedance is Zg = rH.+ JxH, where rH and xH are the resistive and reactive
components of this impedance respectively. In the case where the generator load
is the inpue impedance of a matched transmission line with a characteristic
imgedance of p, ZH = p.  3 QS 
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 In the ilesign calculations of a generatoi: output circuit, it is more convenient
to use the admittance instead of the_impedance Zout 1(See Figure 17.3b):
,.Auaxi gn~axt f Jb~~,xi = J Rntr:t (*LaM:
1
Here gout 1 andAbout 1 are the conductance and the reactive components of the
admittance Yout 1�
For the majority of modern transistors in the decimeter band, the reactive com
ponent of the output admittance has a capacitive character (see Table 17.1) and:
Yout 1 1/ROUt 1+JwCOUt' y"~.~xt N 11Rui''xt+ JWCowx�
The Q of the equivalent output circuit of a transistor in this case is:
Q,ll.lX R,il~Xi wC,il,: Rili wCAl,, .
Here, w= 2wf (f is the working frequency of the generator); RH 1 ls the resistive
component of the iII,pedance ZH 1 of a parallel circuit consisting of RH 1 and
XH 1.
Microwave power transistors usually have a low quality factor Qout, Which is
substantially less than the quality factor of their input circuit Qin. .In this
respect, it is easier to design a generator output microwave network of suffi
cient bandw:.dth than an input circuit.
Besides the impedance transformation, requirements are also placed on a matching
fourpole network in the output microwave circuit of a generator to provide for
a high efficiency, rtc , a definite bandwidth and a requisite higher harmonic
filtration level. The meeting of these requirements depends in many on the
correct choice of the microwave network output circuit, the electrical character
istics of which are governed to a considerable extent by its quality factor Q,
taking the load into account. With a small Q in the circuit, it is easier to
obtain a high efficiency and a relatively wide passband, but in this case, it
is more difficult to meet the requirement for good filtration of the higher har
monics. For this reason, such a value of Q should be assured in the design of
a microwave output network that certain compromise requirements are satisfied.
CEnf=Cbn2 ~em=Ce112 L~ 01 ~~No ~ = 2
pwn pKo
~rtn I C6n,t 161 Cp''// L2 ICp �~6n L1 L2
CBeix 1 C~
II~' I~Z //HO G6n 1rAeix IC2 I~J
ol (a) 61 ~b~ BJ (c). a1 (d)
Figure 17.6. Circuit configurations of output microwave matching
networtcs of an amplifier/oscillator using lumped
elements.
3Qfi
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J~, p C6a~
Hce~:
H
Ux0
al la)
FOR OFE TIAL USE ONLY
10
n3 T
1
i FHAIC2"
L H H //xo UKo
~l (b) Bl cC) al (d)
Figure 17.7. Circuit configurations for the output microwave matching
. networks of an amplifier/oscillator using asymmetrical .
stripline sections.
I' and II section networks are frequently used in the matching output microwave
circuit of transistor amplifier/oscillators. The simpleat of them (the I' section)
can be used in cases where increased requirements are not placed on the filtering
of higher harmonics at the generator output and it is necessary to match imped
ances'which are close in value in a narrow band of frequencies. It is expedient
to have a Q of such a circuit of no more than two to three. IInetworks have become widespread in the circuit configurations of output micro
_ wave networks. To improve the filtering properties of microwave networks with
respect to the higher harmonics, capacitances are inserted in the parallel
branches of the IInetwork. For this purpose, IInetworks are used which contain
an additional series tuned circuit in the series branch, which is tuned to the
fundamental frequency of the oscillator/amplifier (Figure 17.6bd). The presence
of such a filter makes it possible to substantially reduce the impedance of the
series circuit (L1, C1 in Figure 17.6c, d) for the fundamental as compared to
its impedance for higher harmonics, and thereby improve the filtering properties
of the microwave network. IInetworks which start with an inductance are used in
a number of cases'to improve transistor efficiency. Such microwave networks,
~ because of the presence of the inductance, create a considerable resistance to
~ higher harmonics, and a relatively large voltage level of these harmonics appears.
at the transistor collector, something which produces a substantially nonsinusoid
al waveform of the collector voltage. The collector voltage is small during
that portion o� the signal period when the majority of the resistive collector
current is flowing, something which leads to an improvement in transistor
_ efficiency.
The circuit configurations for the output microwave network of anoscillator/
amplifier, depending on their operating frequency and structural requirements,
are designed around components witti either lumped or distributed parameters.
Examples of output microwave circuit designs for a transistor oscillator/ampli
fier with external excitation and using lumped.elements are shown in Figure 17.6.
The circuit of 17.6a contains a IInetwork, starting with a capacitor C1. Fre
quently, capacitor C1 is absent from the circuit and its role is played by the
capacitance Cout of the transistor.. Capacitor Cbl 3 is a blocking capacitor.
The circuit of Figure 17.6b with a series resonant circuit, ths inductances of
which are a part of the inductance L, has better filtering properties with respect
to higher harmonics. In the circuit of Figure 17.6c, the I!network starts with
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' the inductance L2, the function of which has already been stated. The circuit
of Figure 17.6d makes it possible to sati.sfy higher requirements placed on the
matching of impedances in a rather wide band of frequencies as well as the
filtering of higher harmonics at the generator output.
Examples of output microwave network configurations are given in Figure 17.7
using asymmetrical stripline elements. Lumped isolating and blocking noninduc
tive capaeitors are also used in these circuits.
In the circuitof Figure 17.7a, matching is achieved by a single loop transformer
(11, lsh 2)� The characteristic impedance p of the line 11 is equal to the load
impedance. The shortcircuited loop lsh 1 With a length of a/4 performs the
function of a radiofrequency blocking choke. A quarterwave transformer is used
in the circuit of Figure 17.7b which matches the resistances Rout 1 of the
transistor and p bf the load. The reactive component of the output impedance
of the transistor output circuit is compensated by the impedance of the short
circuited loop lsh. In the circuit of Flgure 17.7c, line section 1, capacitance
cout and capacitors C1 and CZ form a microwave network close to a IInetwork. In
the circuit of Figure 17.7d, the microwave network consisting of line section
, loaded into capacitance C1, is tuned to resonance at the fundamental frequency.
The necessary coupling to the load is assured by connecting the load resistance
p through an isolating capacitor C2 to a part of the line section 1.
When designing the circuit configuration for a microwave output network which
 meets the electrical requirements placed on it, one must strive to see that the
circuit is as simple and as convenient as possible for its execution in the
form of a hybrid integrated circuit.
17.3. Oscillator/Amplifier Power Supply Circuits
The power supply circuit for an oscillator/amplif ier should be designed so that
it does not disrupt the operation of its microwave circuitry. A parallel supply
circuit is most frequently used (Figure 17.8), since the usual microwave circuit
configuration does not allow for the use of a series supply circuit. In the
case of a parallel supply circuit, the DC source is connected to the transistor
terminals through a blocking choke, Lbl 1, Which has a high resistance to the
alternating component of the amplifier/oscillator current, so that the supply
source has no influence on the operation of the microwave circuitry. Improved
blocking of the voltage supply is achieved by inserting a capacitor which has
a low resistance to alternating current (capacitors Cbl 3 and Cbl 4 in Figure
17.8a and b). To prevent the direct current component of the oscillator/ampli
fier from flowing into the load networks (or into the network of the preceding
stage), isolating capacitors are inserted in the circuit (Cbl 1 and Cbl 2 in
Figure 17.8a and b). A series inserted microwave circuit capacitor (C1 in Figure
17.6c, d) frequently performs the function of an isolating capacitor. The
choice of the choke inductance and capacitance of the blocking capacitc,rs is
made by working from the requirements for normal operation of the oscillator/
amplifier circuit and the possibilities for realizing the blocking elements.
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pM~ Cdn2
C6n1
T f nt
Ui;O
Ic~bl4
al (a)
FOR. OFFICIAL USE ONLY
z C6nt
C6nf
,P 463 CRe4
~
pz V"o ~ec
Ql (b)
b1
CAnt rdnJ
Uwo
C6Af L
el CcI
Figure 17.8. Parallel pocaer supply configurations for an amplifier/
oscillator.
In order that the blocking choke (Figure 17.8a) does not exert any marked
influence on the operation of the transistor output circuit, its inductance
Lbl 2 is chosen by using the approximate relationship:
wLbl 2'_ lORload 1 co1 6�z %i 10Rm (17.2)
�
The capacitance of capacitor Cbl 4 is determined from the relationship:
_ Cfi,i4 > 50� 1 0"/(u"Lr,n21 (17.3)
derived fr.om the condition that the resonant frequency for series resonance of
the circuit Lbl 2, Cbl 4(Figure 17.8a) should be considerably lower than the
working frequency of the oscillator or amplifier*.
The upper limit for the values of the inductance I,bl and the capacitance Chl is
basically limited by the production process capabilities. To reduce the requi
site value of Lbl in the case where Rload 1' rload, it is expedient to connect
the power supply circuit closer to the load, for example, as shown in xigure
17.8b. The value of Lbl with this circuit configuration can be chosen from the
condition wLbl 2 > lOrioad�
To estimate the approximate values of the parameters of the blocking elements
inserted in the input circuit of an amplifier/oscillator (Figure 17.8a), one
can derive relationships similar to (17.2) and.(17.3):
i 50� 10'/6P Lfnl+
G)Lfint ~ I OZnxt+ Cfi,i,
where z~x~ fu , the nonlinear properties of the diode are
manifest only weakly, somethingpii1ch also leads to a substantial reduction of
the conversion gain. For diodes being produced by our industry, fuPper does
not exceed GHz. For this reason, in multipliers with an output frequency below
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10 GHz, a diode can operate in both modea, while multipliers where fout ' 10
GHz, it can operate only in a cutoff pn junction mode.
In a block pn junction mode, the instantaneous voltage, u, across it in the
absence of a breakdown and with cutoff should satisfy the condition:
0 < u < Uper
0 < U < UIln �
(18.9)
However, in a partial cutoff mode, the voltage u should satisfy only the condi
tion for the absence of junction breakdown: .
u < Uper
tl < Unnn.
(18.10)
It follows from (18.9) and (18.10) that in a cutoff pn junction mode, in con
trast to the partially turnedon state, limitations are placed on the maximum
amplitude of the oscillations. This is also due to the greater working powers
of frequency multipliers using nonlinear capacitance diodes which operate in
a partial cutoff mode [7]. An advantage of partial cutoff is also the higher
multiplier conversion gain given the same multiplication factor and diode Q.
In this case, when operating in ajunction cutoff mode, the conversion gain falls
off so sharply with an increase in the multiplication factor n, that n> 3 is
not used in practice.
It follows from what has been presented here that in multipliers with an output
frequency of fout < 10 GHz, it is most expedient to employ varactors in a partial
junction cutoff mode, and especially, charge storage diodes, the nonlinear pro
perties of which are manifest to the greatest extent in this mode.
The power'parameters of diodes are the following: the normalized power pnorm
and the permissible diode power dissipation P er. The power Pnorm characterizes
the maximum output power without breakdown [2J: Pnorm = Uper/RS, where:
RS = 1/21TfultC(Un) RS = 1/27ij�pPnC(Ur) (18.11)
is the diode loss resistance.
The power Pper characterizes the maximum output power without thermal breakdown
of the junction, since
Rdp _1 Ppernd/(l  nd)� PnnCPnonT1nl(11lu)�
(18.12)
It is well known [6, 7] that with an increase in the diode Q, the conversion. gain
increases, tending to unity, while the output power of the multiplier falls off,
tending to zero. For this reason, when selecting a diode, one must be governed
by the conditions for assuring the specified power with a relatively high
conversion gain.
For a parallel varactor type multiplier operating in a cutoff pn junction mode,
a preliminary selection can be made by means of the expression:
Pdp/Pnorm < 1/4a
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where a is a certain coefficient which depends on the type of the pn junction
and the multiplication factor. A number of values of a are given in Table 18.1.
TABLE 18.1
VOmax
n
q
(1 ~ VAon
~
a
v
2 O,G5
, 0,94
16,25103
231 � 1/2
2 0,732
0.466
9,15�102
927 1/3
3 0,82
0,476
0.7� lU2
112� l04 1/3
Key: 1. UO maX!Uper�
The parameters of the selected varactor should satisfy the expressions:
 4min < fult/f in < Qmax Qmin < tnpen/l e: 1:
I 2oJC . (20.8)
Po w1h2.420,441t1tv J(Ih/w)n
01 0, 2 9,4 0,8 l,Il ? 4 rv/h
The error in calculating po using these
Figure 20.8. The characteristic imped formulas amounts to + 0.25% when w/h =
ance of an asyuunetrical 0...10 and + 1% when w/h > 10 [2].
stripline as a function
of the ratio w/h for Curves for the characteristic impedance
various values of E. p are shown in Figure 20.8 as a function
of w/h for various values of the relative
dielectric permittivity of tiie substrate, where these curves were calculated in
accordance with formula (20.6), using formulas (20.4), (20.7) and (20.8). It
was assumed in the calculations that the thickness of the conducting strip was
t= 0. In reality, t is a finite quantity. It is sufficient in practical cases
for the thickness t to be 3 to 5 times greater than the penetration depth 0
(A is the distance from the conductor surface at which the amplitude of the
current density falls off by a factor of e= 2.718 times). The values of A for
various metals are given in Table 20.1, where the major characteristics of the
conductors are, also shown. When w/h > 0.1, this thickness has little influence
on the characteristic impedance of a line, and for this reason, one can assume
t= 0 when computing it.
The characteristic impedance of an asymmetrical stripline (if it is not specified
in the design plan) is frequently chosen equal to 50 ohms for convenience in
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TABLE 20.1.
M�1ni,n I Q~ 107 CwJw IA. 104111f. IRJR. 1~~ % I TKr, 1 io0
11K I AJII'C3IIA
~ r11N1:1~1:1 MKM 011
c Adhesioa
Uaromete
s
Ag
6.17
6,41
2,5
21
(Inoxau PQOr
01
5,K
G,fi
'l,(i
iR
Uicnb nnonst'Ve7cx ppor
A
4,1
7,86
:i
IS
Ta Hcc The se6me �
AI
;1.72
8,24
3.3
26
llnoxag Poor
W
1,78
11,88
4,7
4,6
Xupoluaa Gppd
Alu
I,7t;
12
4.7
(i
s N
Ni
1,14
1,38
55,0
15
r
Cr
0.77
18,07
7,2
9
Olienh xapou,aA Very good
Ta
0,64
19,78
7,2
fi,f,
Ta Wc The same
Note: f is the frequency in Hz.
Key: 1. Conductor material;
2. Thermal coefficient of expansion, 10'6 �C1
[Conductivity v, 107 mno/m].
connecting the line to radiofrequency connectors and individual microwave units.
The use of line sections having a high characteristic impedance, and this means
with very narrow conducting strips, is not desirable because of the technological
diff iculties in their fabrication and the increase in the attenuation in the
line.
, The attenuati.on in an asymmetrical stripline, a(in dB per unit of length) is
composed of the aCtenuation due to power losses in the conductor at radiofre
quencies am, in the dielectric ad and the losses to radiation arad� The attenua
'tion in a conductor, assuming that the radiofrequency current flows primarily
in a surface layer of thickness A, can be approximately determined from formula
[014, 3]:
a� ct, 8,68Ri1/pw, (20.9)
while the attenuation in the dielectric is:
2,73 tg S. (20.10)
Here, R11 is the specific surface resiatance of the current conducting layer in
ohms per unit area; tand is the dielectric loss angle tangent.
It is difficult to estimate the attenuation in a line due to radiation and it is
frequently determined experimentally. In asymmetrical str;plines where e> 7 and with low losses, the major source of
attenuation is the losses am. An analysis of fornaula (20.9) shows that to
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reduce the losses in a line, one must choose sufficiently thick substrates and
wide conducting strips. However, the thickness h in this case should not come
close to hmaX. Moreover, increasing h and w leads to an increase in the line
dimensions, something which is undesirable in the structural design of hybrid
IC's. To reduce the impact of the thickness of a conducting strip on the
value of am, it is recommended that t be chosen equal to or greater than 3A to
SA.
TABLE 20.2.
: Substrate (1 ~
Ma�ropnain Ku ~~M~~uttu~�irr 7�t�nnn� 7~KJI V. l u~ ~
' unn.nuHU:n upoito/luac�ru. 10�t1 I
Mat.1"la~. ! ~IOG~ (2oMC Ibr/(wM��(:~ ~2~ c:

POl].ICOZ' flnnjnJ1AH16 16 15�104 250
Key: 1. Coefficient of thermal conductivity, 103 W/~mm ��C);
2. Thermal coefficient of linear expansion, 10 �C1.
When applying a conducting stripline to a dielectric substrate, the adhesion of
 the metal to tiie dielectric is taken into account. Because of the fact that
copper, aluminum, gold and silver, which have a poor adhesion (see Table 20.1)
are most frequently used for the conductors, a thin film of inetal having a
high specific resistance is initially applied to the dielectric substrate to
improve the adhesion. The presence of such a film (sublayer) of thickness tl,
which is comparable to the penetration depth A1 in this film, leads to an
increase in the radiofrequency resistance in the conducting strip. If tl 3O op, where j0 oP is the DC current density in the conducting stripline in
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TABLE 20.3.
Method of Fabricating the
Conducting Strip
Vacuum depostion
Electrolytic buildup
Foil application
Permissible Direct Current Density A/mm2
Sitall
30
30
50
Ceramic
200
200
400
[Sitall: ceramic glass similar to pyroceram]
the operating mode. The value of JD peY depends on the substrate material and
the method of fabricating the line. In order to increase J0 peril materials are
to be used fcr the substrates which have sufficiently good thermal conductivity.
Some approximate data on the permissible DC current density, jp per, are given
in Table 20.3 and in the literature [6].
20.3. Printed Circuit Inductance Coils
 When designing microwave networks around elements with lumped parameters, the
requisite inductances of the circuits can be obtained using sections of inetal
strips with a rectangular crosssection: socalled strip single turn inductance
coils (Figure 20.9) or strips bent in the shape of.a meander (Figure 20.10) and
in the shape of a spiral (Figure 20.11).
. L
~
QIa) ~
61 (b) _ r~
(a)
Figure 20.9. Stripline inductance coils.
S
u .o
D
aI
i
~ t7) (a)
.b
dl (b)
Figur.e 20.10. A meander type inductance Figure 20.11. Spiral coil.
coil.
Stripline single turn inductance coils (Figure 20.9b) have inductances from 0.5
to 4 nanohenries [1]. Flat spiral coils provide for greater inductances (up to
100 nHy), where square spiral coils (Figure 20.11b) make it possible to obta?n
a greater inductance than in the case of circular coils (Figure 20.11a) for a
specified area of the coils on a printed circuit board. The inductance of a
coil in the shape of a meander (Figure 20.10) reaches 100 nHy. However, parasitic
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resonances are observed in these coils at frequencies substantially higher than
the working frequency because of the linear sections s and b of a coil, which
at high frequencies then behave as line sections with distributed parameters.
The Q of stripllne single turn and spiral coils at freauencies above 1 GHz
amoun.ts to 50100 (see Figure 20.1). Spiral inductance coils have a higher Q
than single turn coils, but also a greater interturn capacitance. The Q of
coils for a fixed inductance value increases in proportion to Ff up to frequencies
of 5 to 6 GHz, and then falls off with an increase in frequency.
The inductance and Q of a coil depend on its geometric dimensions, as well as
on the presence of inetallization on the bottom side of the dielectric substrate,
even when the metallized side of the dielectric substrate is a considerable
distance from the plane of the coil. To preclude the influence of the metalli
zation on coil inductance, the spacing to the metallized surface under the coil
for'a substrate with e= 10 should exceed the width of the coil conductor w by
a factor of more than 20 times [1]. In those practical cases where this require
ment is not met for technological reasons, the calculation of coil inductance
must be made taking into account the presence of the metallized surface. Metalli
zation in the same plane that the inductance coil is in has little impact on
its inductance,, and it is sufficient in practice to make the distance from the
coil to the adjacent metallized layer equal to 5 times the width of the coil
conductor [l].

1o t=o~._W NO~� 
L
J,8 ~ *N~p~
O,/i
10 20 40 60 f00} w
Figure 20.12. The per unit length
inductance L1 as a
function of the ratio
1/w for a single turn
inductance coil without
taking the strip thick
ness into account.
Key: 1. L1, nHy/mm;
2. Lower scale;
3. Upper scale.
nanohenries amounts to + 2%, while it
80 to 100 nHy.
As a result of calculating a coil for
a specified inductance, it is necessary
to select its geometric dimensions such
that they permit obtaining the requisite
inductance and which are technologically
convenient to realize.
Design Calculations of Coil Inductance.
Formulas are given in Table 20.4 for
calculating the inductance L or the per
unit length inductance Ll of coils, :shape and designation of the dimensions
of which are given in Figures 20.9 
20.11.
The curves for the per unit length ind.uc
tance L1 are shown in Figure 20.12 as a
function of the ratio 1/w when t= 0 for
a single turn coil, calculated using the
formula given in Table 20.4. Values of
the coefficients Cn used in the calcula
tion of the inductance of ineander type
coils are given in Table 20.5. The error
in the determination of the 3nductances
for these coils on the order of tens of
runs up to 6% for inductances of about
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TABLE 20.4
E
op(1M3 N37YII110I
I P8C4CTIIi1N tbOPMY4A
JIMIR8 OpOOnAI1NN8
Coil Shape
Design Formu].a
T,erigtiYT6W"dot],
Single Turn
I 1
Conductor
Lt =0.2(in f1,193
OnItnnurron:isi
1 w {t
l L
(luc. 20.9, a, G)
o 20,9a,
Fig
, I0,21;15
/
1)
2(!n ~ f'
I
~1)
h
,
32
t. \ / J
flpAmnyroni.nan
w ! 2, f  0,
unn(N:Ka uan nuTan
h
.vcsnponaiuwii uo�
)%IWCTLIO
t
U.6`1R
1  
cI
I
(puc. 20.9, o)
rv/2/i}0,9 10,318 In (w/2h{
~
N
. t
'
h
~
�I
J
)
o,
Meander
L 0, Ih (tn In 2(;/u, C�), t 0,
(2)
~~aunl~
(
20
10
.
n tmcno 3ne&icuT01% McauJtporu,Ci nli
l=nG (uw)
pnc.
.
)
ueN nnnwA b
F].go 20.10
CnCM. T1(JI. 20,6
3)
flnrnKan Kpyi:~an
L.5(D ~d)2 n=/(15L~7d), l ^ '0,
rmipani.
U=d1(2n1)sf�2w,
1_nn[d{0,5s(2nI)1
({nc. 20. 11, n)
.
niNCno nwrKOi n is number
oP turns
flnuc~4)
Koa~l�
=6(D}I)7 n2 i(I5D7J). t ^0.
1.
1=4n [d
r,TIM ~~~HPanh
IpNI. 20. I I,()
D=d+(2nI)s 12uu,
 0,5s 1,5
(2n ~I
tt micno sNrKOn
n is the number of turns
fl p n M et a n n e. I3ce nimeAnwc pa3Mrpt,i KaTywcK nWpaNCawTCSt n MHnniiMcT
PIx. minyK�rIInuocrb L ii naHorcupn, f10i'Of1118N HIIJ4yKTNBHOI."Cb Lle naumrcupii
m n+i+nnnMe'rp.
:1ote: All of the linear dimensions of the coils are expressed in millimeters;
the inductance L is in nanohenries and the per unit length inductance L1
is in nanohenries per millimeter.
Key: 1. Fectangular strip above a metalized surface (Figure 20.9c);
2, n is the number of elements of a meander line of length b; Cn  see
Table 20.5;
3. A flat circular spiral (Figure 20.11a);
4. A flat square spiral (Figure 20.11b).
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TABLE 20.5.
n I 2 3 I 4 I 5 G 71 8 I 9 I 10 I I1 I 12
1 I I I ~
l:1. 2,76 3,92 I 6,23 I 7,60 9,70 10,921 13,38114,92116,86 I18.4fiI20,36
I I I (
The determination of the geometric dimensions of flat spiral coils for a
specified inductance L is made using successive approximations, in which certain
geometric dimensions of the coil are specified based on structural design and
production process considerations and the missing dimensions are determined using
the formulas for L and A. For example, having specified the ratio D/d and using
the formula for L, the number of turns n is determined. Then the conductor
width w is chosen based on production process considerations and the requisite
coil pitch s is found by using the formula for D. If it is convenient to realize
this pi.tch, then the design calculation is terminated at this point.
In order to be able to change the inductance of a coil, part of the coil conduc
tor is subdivided into sections having cantact pads for the connection of tap
conductors to them (Figure 20.11b).
Design Calculations of the Q of an Inductance Coil. The quality factor of a coil
Q= 27fL/r for a specified frequency f and inductance L is determined by its
resistance r, which reflects the actual radiofrequency power losses in the induc
tance coil. This resistance is composed of the coil conductor resistance for
the radiofrequency current, rm, the resistance introduced by power losses in
the dielectric substrate rd, the resistance introduced by radiation power losses,
etc. It can be assumed in practice that in a properly designed typical structure,
the power losses in a coil are determined primartly by the resistance rm. In
this case, the coil Q is:
(20.11)
Q y
rM kRul
Here, k is a coefficient which takes into account the degree of nonuniformity in
the current distri'nution at the edges of the conducting strip. The value of k
is determined from the graph shown in Figure 20.13.
. .
?,17 ~ 1
1,6
1, ~i   
1,2
1 2
t0 20
Figure 20.13. The curves for the
coefficient k which takes
into account the degree
of nonuniformity of the
current distribution at
the edges of the conducting
strip.
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Expression (20.11) can be used to calculate flat inductance coils of various
shapes made from a conductor in the form of a metal strip with a rectan.gular
crosssection (Figure 20.9).
The Q of a single turn coil (Figure 20.9) increases with an increase in the
ratio w/1, while the per unit length inductance L1 decreases. It is frequently
desirable wnen structurally designing such a coil to obtain a sufficiently large
value of Ll at a quality factor of Q> 50...100. For coils intended for opera
tion at 4'requencies up to a few gigahertz, this condition can be met when w/1 =
15...20.
When designing spiral inductance coils, one must consider the fact that the
increase in the conductor width w leads to an increase in the coil Q. If it is
desirable that the external diameter of the coil D be rather small with a high
value of the Q, then it is necessary to reduce the spacing between the turns.
This leads to an increase in the interwinding capacitance of the coil. An
analysis of the formulas for the Q of flat spiral coils shows that the maximum
Q is obtained when D/d = 5.
20.4. Capacitors
Primarily film plate capacitors (Figure 20.14), capacitors formed by a short
section of an asymmetrical stripline with a low characteristic impedance (Figure
20.15), comb capacitors (Figure 20,16) and outboard miniature ceramic capacitors
find applications in hybrid integrated circuit structures.
To tune a circuit by means of varying the capacitance, a block of parallel low
capacitance capacitors is made instead of a single capacitor of the requisite
nominal value. A struetural design of a tunable film capacitor is shown in
Figure 20.14b. The uppe:r plate is fabricated in the form of strips of different
sizesy the resoldering of wb:tch makes it possible to change the capacitance of
the capacitor [5].
9
��rti r ~
,lju3~e~mpuk
o) (a)
,!f u3r,CiF~mO~iy Dielectri.c ~
T .
/lu3neNmnu,r Die lectric
6J (b)
Bl (c)
Figure 20.14. Film plate capacitors.
The typical structure of a film capacitor, which is s:.:own in Figure 20.14c, takes
the form of two metal plates, separated by a dielectric layer. The film capaci
tors have a weak external electromagnetic field, and for this reason, can be
placed close to other microwave components.
The capacitance of film capacitors used in microwave circuits at frequencies of
up to about 2 GHz amounts from a few picofarads to hundreds of picofarads [4].
The Q of such capacitors changes depending on the nominal value and the quality
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~w oo
oo�o o ~
o~ (a) 6J (b) a) ta) 61 (b)
Figure 20.15. Capacitors formed by a Figure 20.16. A comb capacitor to
section of an asymmetrical obtain a series capaci
st'ripline. tance in a microwave
_ circuit.
Key: a. With a f ixed capaci
_ tance;
b. With a variable capa
citance (only the
conducting strip is
shown).
of the materials of the conductor and dielectric. To reduce losses in a capaci
tor, metals with a low specific surface resistivity (see Table 20.1) and a low
loss dielectric are used .for the plates. Aluminum is mos*_ frequently used for
the plates of capacitors. The dielectrics of film capacitors, besides a small
loss angle tangent, should have a high relative dielectric permittivity and
electrical strength. Silicon monoxide is most frequently used in the fabrication
of film capacitors (see Table 20.6).
A gap in a conducting strip (Figure 20.16a) produces a series capacitance in an
asymmetrical stripline. To obtain a considerably capacitance (more than a few
picofarads), the gap d should be quite small, something which is difficult to
. execute in practice. Greater capacitances (up to 10 to 20 pFd) can be obtained
if a comb capacitor is used (Figure 20.16b). The capacitor formed by a gap in
a strip is a special case of this. Another capacitor structure intended for
creating a series capacitance in an asymmetrical stripline is shown in Figure
20.14a. The capacitor takes the form of two short sections of a stripline conduc
tor which overlap lengthwise, where the sections are separated by a dielectric
layer. The overlap area of the plates in such capacitor structures does not
usually exceed 10 mm2.
To create a capacitance which is connected in parallel to an asymmetrical strip,
_ line, one can use a capacitor in the form of a short section of asymmetrical
striplines (1 � a) with a relatively low characterista.c impedance (less than
20 ohms) (Figure 20.15). The sectional structure of the capacitor shown in
Figure 20.15b makes it possible to change its capacitance.
Outboard miniature capacitors are convenient for applications in hybrid IC's
intended for operation at frequencies up to a few gigahertz, since the fabrication
of a circuit with such components does not require a complex technology. The
Q of miniature outboard capacitors is sufficient for their use in microwave
circuits of hybrid IC's in the indicated band.
When designing capacitors, it is necessary to know the relationship of the
capacitance of a capacitor to its geometric dimensions and the relative dielectric
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' TABLE 20.6.
(1)
JAiinoK
7'hllK K0II I
/tcnca�ropa
6 I
tg 8 If{N ~ I Krll~
2~
Clip. ! 0l
C
Si0 (
5...G I
0,002...0,01 I
1...2
Si02
I3,6 ...4,2I
0,0007...0,005I
3...5 ,
G c0
I 10...12
( 0,005...0,01
I0,5...0,8
A 120;,
I 8.. I 9
I 0,003
I 8. 10
permittivity of the dielectric used in
its construction. A rigorous calculation
of the capacitance of plate capacitors
taking the edge effect into account is
difficult, and far this reason, we shall
limit ourselves to an approximate calcu
lation.
Fi1m Plate Capacitor (Figure 20.14c).
The capacitance of a capacitor is deter
mined from the well known formula for
a plate capacitor:
C [pFd] _
(20.12)
Key� 1. Capacitor dielectric; C fn(~'] _ 8'8'~ '10~ ~'S/~t.
� 2. Tan d when f= 1 KHz; Here, e is the relative dielectric per
3. Ep [electrical strength], mittivity; S is the overlap area of the
105 V/mm. plates in =2; d is the dielectric
thickness in mm.
Dielectrics with large values of e are used and d is reduced to increase the
capacitance of a capacitor. The minimum thickness dmin i;; determined by the
permissible electrical strength of the dielectric.
Ci pFd 1//b^0,1,
..t ,  l%~SNH
0,9 ' a,~6
m5
C, pFa
~ L
0,5 0 10 ~ 60 80 C,n0
~ h/b 0,7,
Figure 20.17. On the determination of
the area of capacitor U
plates takin; the edge ~S.YN
effect into account. 1 A1~=~~ ~a 0 92 04 M b//,d/h
For normal operation of a capacitor, Figure 20.18. The capacitance of a comb
the thickness of its dielectric capacitor as a function
should satisfy the condition: of the geometric dimen
d > d /NE (20.13) sions.
= min  uwork Pr
Here, uwork.is the working voltage between the capacitor plates in volts; Epr
is the electrical strength in V/mm; N is a safety factor, taken equal to 0.5
0.7.
When designing a capacitor, the dielectric material is chosen first (see Table
20.6) and then the dielectric thickness d is determined from formula (20.13).
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Then, working from the specified value of the capacitance, the requisite overlap
area of the plates S is found from formula (20.12). Formula (20.12) yields a
somewhat overstated value of S, since it does not take edge effects into account.
Because of this, a correction is to introduced into the calculated value in
accordance with the approximate graph shown in Figure 20.17, in which Sc is the
area taking the correction into account.
It is recommended that the piates be made wide and short so as to reduce losses
in the metal plates of a capacitor as well as losses to radiation.
A Capacitor in the Form of a Short Section of Asymmetrical Stripline (Figure
20.15a). The capacitance of such a capacitor can be calculated by working from
_ the easily determined per unit length capacitance of an asymmetrical stripline:
C1 [pFd/mm] = 3.33 e/p C~ ~n(D/MMj=3,33Vea,~,/p. (20.14)
For a specified capacitance of a capacitor C, the requisite length of a line
section is 1 = C/C1.
CombCapacitor (Figure 20.16b). The capacitance of a capacitor formed by two
"combs", arranged on a dielectric substrate (e > 1) can be computed from the
approximate formula:
C [n(D] c_3,6� 10$ (e}1)1 x
 )C 1 h h (2rn1) (l~dd )0,25 ( h )0.4388]
' (20.1
5)
where m is the number of protruding lugs on one side of the capacitor; 1 is the
length of a protruding lug in mm. The error in calculating a capacitance using
formula (20.15) does not exceed + 5%. An example of the capacitance plotted as
a frinction of the geometric dimensions of a comb capacitor is shown in Figure
20.18.
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CHAPTER 21. MICROWAVE PHASING DEVICES (PHASE SHIFTERS)
General Remarks
The development of microwave engineering is tied to success in the development of
h:.gh speed electrically controlled microwave devices. Thus, controlling the phase
of microwave signals in antenna equipment is accomplished by means of phase shif
ters which are controlled by magnetic or electrical fields.
~ A conditional classification of phase shifters which make it possible to continu
ously or discretely change the *;Iiase of microwave signals ra*+ he made using the
following criteria: the operational principle and function; the permissible micro
wave power level (pulse, CW); the working frequency range (wavelength); the struc
tural design (waveguide, coaxial, stripline or microstripline,
The following requirements are placed on the parameters of phase shifters [1] a
working bandwidth of no less than 5 to 15 percent of the carrier frequency; a
pulsed transmission power of S to 220 KW and an average value of 5 to 50 W; a
switching time of 0.1 to 100 usec; losses of no more than 0.5 to 1.5 dB and good
matching (SWR < 1.5).
_ Electrically contralled phase shifters can be designed using diverse controlled
elements: semiconductor diodes with various structures (pn, pin and nipin),
~ ferrites, ferroelectrics, etc. [14]. This is due to the function of the phase
shifters and the requirements placed on them: providing a high efficiency, high
_ electrical strength, stability of the characteristica, low control power and suf
ficients operational speed. There arP three methods of phase control: continuous (analog), digital and sw;*cr.zd.
~ In the first, the phase shift changes continuously. However, this method is dif
ficult to implement because of the necessity of supplying continuously changing
contnol signals. Moreover, time and temperature instabilities exert a marked
influence on the phase characteristics of the phase shifters. This deficiency is
also preserved in the case of digital phase control, when a number of operating
points are used on the operational characteristias of analog phase shifters, and
 for ttiis reason, the phase change takes place in a jump by an smount Al~ (discreCe
step). The influence of instabilities is p:actically elimi:,ated in digitally
switched phase shifters [OTiO], the phase of the electromagnetic oscillations at the
output of which can assume fixed values. The stability of such phase shifters is
governed by the fact that the controlled elements (ferrite rings or semiconductor
diodes) operate in a mode in which only the extreme regions of their operating
characteris+.ics are used. This makes it posaible to ease the requirements placed
on the time and temperature stability of the switchers of digitally switched phase
shifters and the controllers, since the requisite phase ahift is not governed by
the value of the control voltage, but rather by its presence at particular
switchers.
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Various controlled elements can be used in the constructioiz of phase shifters
regardless of the manner of phase control. However, pin diode phase shifters
with a continuous phase variation which make use of the change in the conductance
component of the diode admittance are of no interest because of the large conduc
tance losses. For this reason, pin diodes are used primarilv for switched phase
 control, for example, by means of turning transmission li.ne sections on or off
which change the overall length of the channel. A characteristic feature of phase
shifters with a continous phase change is the use of controlled veractors: elements
_ with a controllable capacitance [3]. Ferr4':es are used both in phase shifters with
a continuous phase change and in discretely switched phase shifters [2, 41.
The major parameters of an electrically controlled phase: shifter are: the phase
control range 4~min... Dmax; the losses introduced by the: shifter L; the traveling
wave ratio at the input (or the absolute value of the reflection factor T).
Moreover, specific requirements can be imposed, for example, on the shape of the
phasefrequency response (its linearity).
It is convenient to introduce a"phase ahifter quality" parameter for the compara
tive evaluation of pliase shifters:
Kit [deg/dB] _ O/L K4, CrraNn61 mli..
Digitally switched phase shifters are completely characterized by maximum phase
shift values 0 and L as well as the smallest phase jump (discrete step).
In phase shifters using ferrites, controlled by an external magnetic field,
electromagnets must be used (in the majority of cases of considerable size and
weight), which have an operating speed of 106... 10'~ sec, something which limits;
their application. In this respect, microwave phase shifters designed around
semiconductor devices are more promising, in which the phase shift is controlled
with the action of an electric field. For this reason, we shall consider the
operational principle, major types and characteristics of semiconductor phase
shifters, as well as the procedure for determining their major parameters.
21.1. Semiconductor Phase Shifters
The change in the input impedance of semiconductor devices with the action of a
control voltage is used in semiconductor phase shifters. In this case, the semi
conductor device can be inserted in the channel in series or parallel, as shbwn in
Figure 21.1, where Z and Y are the normalized impedance and admittance of the
semiconductor device for a series and parallel configuration respectively:
7 R +j X r=1 jx�, y  �+l'_"~ =g{Jb, (21.1)
no Po . Yo Yo
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where R= 1/G and X= 1/B are the resistive and reactive components of the imped
ance of the semiconductor device; pO = 1/YD is the characteristic impedance of the
line in which the semiconductor device is inserted.
V, d'z ~i z ~z
.
1...
G6n
y L Gbn UCOn
6n U
COA "D1 Cbn UynD
d) (AI
Figure 21.1. Schematic of the insertion of a semiconductor device
in a line.
a. Parallel; b. Series.
The impedance of a semiconductor device can change with the action of the control
voltage Ucon of the source (Figure 21.1). Decoupling the control circuits and
transmission channel is accomplished by network Lbl and Cbl. If the lower fre
 quency wf.l in the transmieted signal spectrum is considerably higher than the
maximum frequency Stcon in the control voltagespectrum (which is usually the case),
then the values of Lbl and Cbl are chosen from the relationship:
(21.2)
cun L6n f) o 1 ~4)u Cfin� .
If the frequencies uH and SZcon are commensurate, then the control circuit and the
transmission channel should be decoupled by a filter network with a cutoff fre
quency falling a_bove Stcon and having an attenuation (i.e., a decoupling) no worse
than the specified value at the frequency wg. In this sense, the network Lbl and
Cbl which is shown in Figure 21.1 takes the form of a very simple low pass filter
(FNCh) and can be designed not only liy working from expression (21.2), but also
using microwave fiZter theory. In this case, one can provide for the guaranteed
decoupling during phase shifter operation within the frequency band and in many
cases, reduce the dimensions of the circuit Lbl and Cbl, by appropriately choosing
the cutoff frequency of the low pass filter.
Inserting the semiconductor device (a varactor or pin diode) in the line causes
both a reflection of a portion of the microwave power by virtue of the mismatch
at the insertion point and its partiaL absorption in the semiconductor device
(ohmic losses).
Using wave transmission matrices, we write the resulting transmission matrix
[t] for the circuit of Figure 21.1a:
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rt~ ~ ~11 ~12 _ It1~It21ltJJ
l
~ ~ �
zl 221
[(I~ y12) et M +D.) Y/2 .
Y/2 . (1Y12)et(0 , +M) (21.3)
where [tl] and [t3] are the transmission matrices of line sections having an
electrical length of 01~2 = 2w11,2/Xline, where aline is the working wavelength
in the line.; 1 is the geometric length of the line; [t2] is the transmission
matrix of a fourpole network with an admittance y. The approximate sign in
(21.3) is due to the fact that we neglected the losaes in the line itself as well
as the dimenaions of the semiconductor device as compared to X.
We determine the losses L introduced by the semiconductor device into the
channel from expression (21.3):
L = P in /Pout = L = PeXI1'Brax  =1U Ig I ttt ~'=141g[(1{ 0,5g)1+(0,56)zJ. ( 21.4 i
In this case, the absolute value of the reflection factor is:
I, = v(,'a _1_ b2)/(4  g, b2). ( 21. 5)
Similar expressions can also be derived for the circuit of Figure 21.1b; in this
case, g is replaced with r and b is replaced with x.
21.2. Semiconductor Phase Shifters with a Continuous Phase Change
In pbase shifters of this type, both the resistive and reactive components of the
_ i.mpedance of the semiconductor device change with the action of the controlling
voltage. For varactors, the change in Ucon within the range of permisaible values
(with the pn junction cut off) leads to a change in primarily the reactive compo
nent of the impedance. In this case, the change occurs in a relatively narrow
, range and rather smoothly. This is responsible for the uae of varactors primarily
_ in phase shifters with a continuous phase bhange. At the same time, the resistivP
component of the impedance changes in pin diodes with the action of Ucon in a
wide range (changes in almost a jump), which limits their application in phase
shifters with a continuous phase change.
Semiconductor phase shifters w ith cantinuous phase change can be both transmissive
4nd reflective types.
A transmissive semiconductor phase shifter can be designed in the circuit config
urations of Figure 21.1. Its operational principles consists in the fact that
with a change in the capacitive susceptance (Figure 21.1a), the electrical length
 of the line in which this susceptance is inserted also changes. NegleCting the
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resistive losses of the diode, one can write the following for the phase shift
intr.oduced by the phase shifter:
cI (I)0 arctg (G/2), 0 = 'Dp + arctan(b/2) (21.6)
where 00 = 01 + 02. . IN = (1)l + cUZ.
For the circuit of Figure 21.1b, it is neceasary to substitute :x for b in this
expression. A drawback to this circuit is the fact that in the process of control
ling the phase, the phase shifter introduces considerable loases, which are caused
by reflection from the controlled element [see (21.4)]. When g< 1, the reflec
tion losses in a single element phase shifter are substantially greater than the
resistive losses. For this reason, the quality of a simple one element phase
shifter is poor: KD < 15 deg/dB.
An improvement in the parameters af a phase shifter is achieved by introducing
adc;itional devices into the circuitry (two and four pole networks), as well as by
increasing tt.e number of controlled elements.
A phase shifter circuit configuration with a campensating reactance is also pos
sible, where an equivalent inductance in the form of a shortcircuited line section
is inserted in parallel with the controlled capacitance C. This line section,
ls,c,, can cancel, at a particular frequency, either the initial value of the
susceptance b, due to the minimwn capacitance of the element (primarily the capaci
tance of the package, Ck), or the value of b corresponding to the average value
of the controlled capacitance. In bbth cases, the result is an expanaion of the
phase control raTge without increasing the inaertion losses L, something which
leads to an incr.�ease in 4. The length 18,c, is determined from the cond ition:
wC/YD = cot(2wls.c./ x line~ (oC/yo=cig(2nl,,,l%�),.
where C is the controlled capacitance for the control voitage selected on the
voltfarad characteristic. To design a phase shifter with such compensation at a
fixed frequency, the resulting susceptance b, cot(2w1g.c,/Xline) is to be sub
stituted in p.lace ot the susceptance b in the formulas for calculating the losses,
phase shifC and reflection factor.
Multiple element phase shifters based on controlled capacitances represent a
cascade configuration of single element phase shifter circuits.
Increasing the number of controlled elements considerably complicates the calcu
lation of the phase shifter parameters: the abaolute value of the transmission
gain and the phase shift. In this case, it is expedient to use a computer employ
ing the tools of wave transmission and scattering matrices.
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L dB
6
4
2
o�
Figure 21.2. Schematic of a multiple element phase ahifter with a
continuous phase change (a) and its amplitude (solid curves)
and phase (dashed curves) characteristics (b).
dB
8
6
4
2
~a oo L'dddA
2 3 6
o (a)
b~1,8 10
� I .12,p5 e
. ~ . 6
1.2 4
UP ~ 2
I
0 J ~ ~
5
10 �5 0 5 Aw,k~o % U 0,05 Q1 41
61 (bY DJ (Q)
Figure 21.3. The amplitudephase (a) and phase.frequency (b) character
istics of a nine element phase ahifter; the influence of
the resistive losses on the insertion attenuation and the
phase shift (c).
A general equivalent circuit of a phase shifter with a continiious phase change
and an arbitrary number of controlled elements is ahown in Figure 21.2a. The
task consists in finding the insertion losses as the coefficient of the firsC line
of the first row of the resulting transmission matrix [t] of the entire device:
[t] _(tIJ [t2] [tn], where [ti] are the transmiesion matrices of the secti.ons
(the line sectiona 11, 12 and the controlled capacitance c(u) in the case of
ideal isolating networks).
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The reflection factor from the phase shifter is also determined from the values
found for the coefficients of the resulting transmissioii matrix [t]. Thus, the
characteristics of multiple element phase shifters are: a range of phase change of
(Dmin ~Pmax, the insertion losses L and the absolute value of the reflection
~ factor I'; these are defined as functions of the normalized admittance of the con
trolled capacitance C(U).
In a phase shifter with n identical equidistantly spaced varactors, by virtue of
the change in their capacitance from Cmin to Cmax; 40Nn (arctg G2 mux__arct~* �2Y inl
~ o ol
~
Here, the influence of multiple reflections between varactors on the phase shift
was not taken into account, which is permissible in a first approximation if their
reflection factors with respect to the absolute value of I' 1, n< 0.25.
The formula cited he.re can serve as the basis for selecting the number of elements
in the design of a phase shifter. Characteri.stics of phase shiftPrs with a contin
uous phase change for various numbers of equidistantly spaced control elements
[3] are available at the present time which have been calculated on a computer and
' plotted.
The case where the resistive losses in the controlled elements can be neglected
(g = 0) is of practical interest, and then the losses in a phase shifter are
determined only by the reflection losses. ihe amplitude and phase characteristics
of phase shifter with different numbers of elements are showr in Figure 21.2b for
this case. It follows from the figures that with an increase in the number of
elements, the phase shift is practically proportional tc the reactive component;
the nonuniformity in the characteristics of the insertion losses increases with an
increase in the nuffiber of elements.
The influence of the spacing between elements on phase shifter psrameters 4s
illustrated in fi&ure 21.3a. Depending on this spacing, the slope af the~phase
characteristic (the dashe(i lines) also changes as does the nonuniformity of the
insertion losses (the sdlid lines). Based on the curves of Figure 21.3a, one can
determine the attainable minimal insertion losses and their nonuniformity for a
 specified (D in a specified range of frequencies. The inverse problem can also be
solved: find the range of frequencies within which the permissible insertion ~
losses L are preserved with the attainable valiie of ~D. (In this case, it is
necessary to keep in mind the feasible range of change in b.) One can also
estimate the bandwidth of a phase shifter, eomparing its characteristics for
various spacings between the elements, i.e., for different electrical lengths of
the line sections.
It can be seen from Figure 21.3b that in step with an increase in the normalized
capacitive admittance, the nonlinearity of the phase characteristic increases
within the passband.
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Where there are resistive losses in the controlled elements, the phase relation
ships in amultiple element phase shifter practieally do not change. A practically
i linear relationship exists between the insertion losses of a phase shifter and the
conductance component of the admittance of the controlled capacitance (Figure
 21.3c).
~ Some general requirements for the parameters of controlled elements can be formu
lated based on the characteristics treated here and optimal circuit design
approaches can be fonnd. These requirements can be reduced to the following. The
maximum capacitances of controlled phase shifter elements with a continuous phase
_ change should not have a normalized susceptance of b> 2.5 to 3. Otherwise, it
is impossible to obtain operationally acceptable values of the amplitude modula
tion level during the phase contral process in the frequency passband. If it is
necessary to have low values of the SWR for a specified change in the phase shift
0, then the number o� controlled elements is to be increased and the maximum
 capacitive susceptance b reduced. However, with a significant increa3e in the
nrmmber of elements (n > 9 to 10), the resistive losse,� corresponding, for example,
to g= 0.1, yield too mucx attenuation.
With certain requirements placed on the working frequency range, a positioning af
the'elements where 1= J+line/4 can prove to be optimal. Then the quality of the
phase shifter increases because of the increase in the slope of the phase charac
teristic and the possibility of reducing the number of controlled elements.
Thus, while the quality of a single element phase shifter is primarily determined
by its reflection losses, the quality of a multiple element phase shifter depends
on tre value of the normalized conductance g of the controlled element.
As a rule, the major components of retlective phase shifters with a continuous
phase change are shortcircuited linetsections with varactors; reflecting sections.
They can be connected to the comnon channel either directly or through multipole
networks (Figure 21.4). The controlled elements regulate the signal phase on the
path to the shortcircuiter and back. The characteristics of a reflective multi
element phase shifter;are calculatLd using the same method as for a transmissive
one, but the values of the parameters obtained as a result of the calculation are
doubled (with the exception of the phase shifter quality, K~ = 20/20.
A 3 dB directional coupler (slotted and loop bridges) or some other multipole
network which poeses similar characteristics can be used to segregate the incident
and reflected waves. The insertion of the controlled elements by means of the
indicated multipole network on the whole forms a transmissive phase shifter con
figuration (see Figure 21.4). In the circuit of Figure 21.4a., the directional
coupler is loaded at the outputs (2 and 3) into reflective phase shifters, which
in the general case each contain one or more controlled elemer.ts.
In the case of identical reflecting sections with a single controlled element
without resistive losses and an ideal directional coupler, the insertion phase
shift is equal to twice the phase ahift pre,vided by a shortcircuited line section
and the controlled element [3]:
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(U = arctg '2 lhctg (2nt/11�))
i (b'ctg (2rc1J).n)]2 ~
where b= wC/Yp is the normalized capacitive susceptance of the varactor; 1 is
the distance from the element to the shortcircuiter of the reflecting section.
,p1, ~  
C(U) 1 2 ipf
~
Of 3
4 r C(Ul
C(U,~
` al (a) (bl ~J
Figure 21.4. Circuits of reflective phase shifters using a 3 dB
bridge (a) and a circulator (b).
Circulators can be used to separate the incident and
phase shifters (Figure 21.4b). Such a phase shifter
numbers of controlled elements. The insertion shift
controlled element is determined in a manner similar
with a bridge circuit. T~ao controlled elements (var,
':ion are rufficient to change the phase shift from 0
reflected waves in reflective
can also contain different
for a phase shifter with one
to that for a phase ahifter
actors) in a reflecting sec
to 360�.
~ It.follows from the analysis of semiconductor phase shifter operation that they
have a comparatively poor quality Ko and considerable nonuniformity of the inser
tion losses within the range of phase change. The indicated drawbacks limit the
range of applications of these phase shifters.
21.3. Discret2ly Switched Semiconductor Phase Shifters
As is well known [1] pin diodes can sharply change (with a jump) the resistive
component of the impedance in a wide range with the action of a control voltage
Ucon; however, the reactive compo^ent is small and almost does not change at all.
The sharp change in the diode impadance is used in discretely switched phase
shifters*. In this case, the ohmic losses in th e diode are small, since the fol
lowing conditions are met (for the parallel insertion of the diode in the line):
~
= In the following, we shall call a discretely switched phase shifter simply a
discrete [digital] phase sh ifter for the sake of simplicity.
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Y1  gl po/rfor � 1~_ Y2_ g? _PO/r?nv � 1 (21.7)
Yi gi Polrnp > 1, Yz ga = Po/ro6r � 1,
where Y1 and Y2 are the pin diode admittances for the forward and inverse bias
modes respectively; pp is the characteristic impedance of the line (see Figure
21.1a) in which the pin diode is inserted.
In expressions (21.7)9 the reactive components of the diode admittance bl and b2
are taken equal to zero. When 92 fram (217) is substituted in (21.4) and (21.5),
it is not difficult to convince onself that the losses in the diode and absolute
value of the reflection factor are small. This is explained by the fact that the
inverse resistance of the diode rinv is high and it practically does not shunt the
transmission line.
If the value of gl from (21.7) is.substituted in the same equations (21.4) and
(21.5), then we obtain greater losses and a higher value of I': practically all of
the power is reflected [this follows from (21.5) when gl � 1 and bl = 0 are
substituted].
The nearly total reflection of the power from the diode in the case of forward
bias can be employed, for example, to design a reflective phase shifter. If a
radio frequency signal is fed to.th e input of the circuit of Figure 21.1a and the
output is shortcircuited then such a reflective phase shifter provides for two
insertion phase delays: with forward biasing of the diode, the wave is reflected
from the diode, and with inverse biasing, it is reflected frrnn the shortcircuited
endof the line. The difference between the inaertion phase delays is a discrete
step (jump) in the phase of such a phase ahifter.
The series insertion of a diode can be treated in a similar fashion (Figure 21.1b).
In this case, the diode opens the line in the case of invrerse biasing and allows
a wave to pass through low insertion losses in the case of forward biasing. By
using two or more diodes and the appropriate circuit designs, une can provide for
switching the microwave power from one line to another. This circumstance is also
utilized in digital semicrnductor phaae shifters.
Digital semiconductor phase shifters make it possible to eliminate the majority of
deficiencies inherent in continuous semiconductor phase:;shifters, specifically:
improve the quality factor M4~ = 200 deg/dB), reduce the SWR (Kst [SWR] < 1.5),
equalize losses for various phase shifts and control'Aarge microwave power levels
(especially in a reflection mode). The possiblity of obtaining the characteris
tics enumerated above in a wide frequency band (up to an octave and more) is also
important, something which makes it possible to use such phase s'hifters in phased
antenna arrays. The possibility of obtaining the requisite phasefrequency
characteristics and assuring stability also .promotes this to no small extent.
The design calculations for discrete semiconductor phase shifters are carried out
using matrix analysis tools.
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The operational principle of digital semiconductor phase shifters is rather simple.
The large nnmber of circuit designs which have been realized at the present time is
due to the widescale use of phse shifters of this type. For this reason, primarily
the specific features of the circuit design solutions for digital phase shifters of
various types to obtain a requisite phase characteristic and the specific opera
tonal features with such a characteristic are treated in the following, and where
necessary, the design equations are given. The treatment takes into account the
predominant application of digital phase shifters in phased arrays.
We shall initially consider digital phase shifters which make it possible to obtain
only one discrete phase sbep, and then we will show how by using them as camponent
elements in multiple element discrete phase shifters, one can obtain the requisite
number of discrete phase steps.
Semiconductor digital phase shifters can be broken down into three main groups
according to the operational principle: with switched line sections (Figure 21.5a),
reflective with incident and reflected wave isolators (Figure 21.5b, c) and the
periodically loaded line type (Figure 21.5d, e) [5].
Phase shifters using switched line sections are the simplest and most obvious in
terms of the operational principle. The difference in the electrical length of the
line section corresponds to the phase shift of AO _02 01 (Figure 21.5a). I
The following can be numbered among the advantages of phase shifters of this group:
tne diodes h3ve practically the same insertion losses for both values of the phase
. delay (slight deviations are possible only by virtue of the change in the length of
the switched line sections); the circuit is convenient for microstripline fabrica
tion; it is compact (especially for small phase shifts). The drawbacks are: the
relatively large number of diodes (up to four per phase shifter element); the
necessity of supplying control signals of different polarities; phase shifter losses
do not depend on the phase shift, while in all other groups of phase shifters, the
losses fall off with a decrease in the phase shift [5].
Phase shifters of the second group have become widespread (Figure 21.5b, c). Both
_ reciprocal multipole networks (directn,onal couplers, bridges) and nonreciprocal
_ (most often circulators) are used as the device to segregrte the incidant and
reflected waves. In this case, the energy reflected from the diodes falls entirely
in the output arm of the multipole network. The phase shift itself (discrete phase
step) at the output of the phase shifters of the second group is formed by virtue
 of the phase change in the reflection factor when the diodes are switched in the
appropriate line section, which are connected to the separating device: A(D _
= arg(I'1/I'2).
Merits of phase shifters in this group are the minimum number of diodes which are
used (down to one per element) for any phase shift, as well as the possibility of
separate optimization of the isolation device (with respect to decoupling and match
ing) and the manner of inserting the diodes (based on the requisite phase Eunction
within the passband, the balance of the insertion loeses in the two phase states,
etc.).
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w2
.
of
a1
~
>
3
,
~
2
Pl 10,1iDp /'p
B) W..
 y ~ ~
~
Z
r r2 r,3 r?
(d,
, . d) (b)
'
st
aZ
d) (e)
fi
02
of
Figure 21.5. Circuits of discretely switched phase shifters.
0~
~
_
b
1 >+Au/c~o
Figure 21.6. The phase characteristics of switched
nondispersive line sections.
The operational principle of periodically loaded line type phase shifters consists
in the fact that the electrical length of a line increases when a shunting capaci
tance is inserted and decreases when an inductance is inserted. To reduce reflec
tions from inhomogeneities, represented by the shunting capacitance or inductance,
a pair of identical reactive elements is used, spaced at a distance apart approx
imately equal to a quarter wavelength. For good matching, the shunting reactances
should be rather small, but this leads to small phase shifts (usually, no more
than 45 degrees), which limits the application of the phase shifter. The phase
shift in the phase shifters shown in Figures 21.5d and e is determined by the
following relationships respectively: AO = arctan(bl/2)  arctan(b2/2) and AO _
= arctan(xl/2)  arctan(x2/2).
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Phase Shifters with Switched Line Sections. The great diversity of the phase
shifters in this group is due primarily to the requirements placed on the form of
the phasefrequency characteristic and the minimum phase shifter dimensions for
large discrete phase steps. We shall consider the operati.on of the phase shifter
shown in Figure 21.a within the passband. The electrical length of the switched
nondispersive line sections are:
0i.z 2jTll,z/Xo,
(21.8)
where ap is the wavelength correspondiMg tothe center frequency of the specified
bandwidth (Figure 21.6).
The phase characteristics of the switched line sections, wh ich take the form of
straight lines runnin.g through the origin, are depicted in Figure 21.6. In the case
of tuning off of the center frequency, as follows from Figure 21.6 there appears an
increment in the phase jump, ft, related to the frequency difference Aw/wp by the
ratio (we consider the diodes to be ideal switches):
.Zm = A(U n (dco�,
(21.9)
 where A(D _02 01 is the phase shift at the center frequency. It follows fram,
this that a phase shifter of the type shown in Figure 21.5a provides for a phase
 shift which changes linearly with frequency, and consequently, a time delay which
is independent of frequency. For this reason, such a phase shifter is convenient
 for use in wideband devices with a constant time delay. However, the bandwidth
and the maximum phase shiEt are limited by resonance phenomena which occur when the
length of a disconnected line section becomes a multiple a/2. In this case, the
disconnected line section becomes in essence a high Q resonator, which is weakly
coupled to the connected line section by virtue of the capacitance of the diode
cutoff switches (Figure 21.7a). Because of this, the insertion losses at the
resonant frequency increase, and moreover, phase errors appear.
To increase the decoupling between line sectiona and the channels, one can use the
circuit shown in Figure 21.7b with a permanent structural connection of both
channels to the incoming and outgoing lines. The disconnection of one of the chan
nels is accomplished by shorting its input and output to ground. In this case,
the length of the line sections from branch point A to the points of diode
insertion is ap/4, where ap is the wavelength corresponding to resonance in the
disconnected channel. When a forward bias is supplied, the upper diodes (in
accordance with the schematic of Figure 21.7b) are turned on. In this case, the
quarterwave shortcircuited line sections have an infinitely high input impedance
' at branch point A, which also creates increased isolation. The lower diodes are
cut off, and consequently have no influence on the operation of the channel with
an electrical length (D1 (the channel length is determined by the.length of the
line section between branch points A A).
Phase shifters of the type of Figure 21.5a can also be used in systems where it is
necessary to have the phase shift independent of frequency. In this case,
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expression (21.9) is the phase error 60 introduced by such a phase shifter. A con
stant phase shift in a wide frequency band (up to an octave and more) can be
obtained by using a dispersion line in a channel where this line has a sharter
electrical length and takes the form of coupled lines connectdd to each other at
one end as shown in Figure 21.8a. The length of the coupling region is ap/4, where
ap is the average wavelength of the working band. The phasefrequency character
istic of such a coupled line is [5]:
(ll=0, arccos( 0 Pa)ffla)tie lDocl ~
L (I I(x)/(l a)ItB'(Doc J
(21.10)
where a= 10C/20 ; C is the crosstalk attenuation in dB; ~pC = O.Swap/a is the
electrical length of the coupling region.
0? 02
~ Of
i
~
Q) (a) ~
~
~ A A _
� ' ~bn +'7. ~
R
!/y"p
cb7.  
_ Ucon
dl
Figure 21.7. Variants of phase shifters with switchable channels and
increased isolation between them.
The phasefrequency characteristic corresponding to expression (21.10) is shown in
Figure 21.8b by the curve 01. The phasefrequency characteristic 02 of a non
dispersive line section is also shown in this same figure. A schematic of a phase
shifter using lines having the characteristics 01 and 02 is shown in Figure 21.8c.
If the length of a nondispersive line section is chosen so' that its phasefrequency
characteristic 02 is parallel to a straight line passing through a point with
abscissas of ~D1 and 02 which are equally spaced from the point OpC _7r/2 (Figure
21.8b), then at frequencies corresponding to the point 01, 7/2 and 02, tre phase
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shift introduced by the phase shifter ia equal to a certain quantity 0O, while at
the remaining points which fall outside the range 01 0;
Z  P� , Y2= j 2n>~>n� (21.15)
� ~ ~ tB (0/2) po sin m
Similarly, for a T network:

Z, = JPo tg (0/2), YZ _ j sinm o , rc 0;
P
. (21.16)
Yl ~ ZZ j Po ~ 2n'>'~ > n.
Po t6(0/2) sin m
Expressions (21.15) and (21.16), taking into account the symbols adopted in
 Figures".21.9b and c, make it poasible to draw the conclusion that II and T four
pole networka, which are equivalent to a line section of length w >0 > 0 at the
working wavelength J1p, have the structure of a low pass filter (FNCh) section; for
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oo '
7Z2 Zl . Zl Z2 Zf
oo
' Ql ~ � 6l , B1
Figure 21.9. The equivalent represcntation of a section of
uniform transmission line.
~
a TT' Tc' Tc0
~
Ca)o;
2C,
~.T T I 0
(b)
T 11C1
4 Z2 C,
H H
Lee ~le
Uy"p U9^P
~aor~ S~e" Ucon
e~ (eX
C
a) (d)
Figure 21.10. Examples of circu'Lts of discrete phase shifters with
lumped reactive elements.
,
2n >~D >n, these fourpole networks have the structure of a high pass filter
(FVCh) section. Moreover, it followa from expreasions (21.15) and (21.10 that
one of these f4urpole networks cannot successrully realize an equivalent line
section of length 0 _w. Possible circuits for constructing a network.:for a line
section of electrical length 0 _7r are sriown in Figure 21.10, T'ne series connec
tion (Figure 21.10a) presupposes the use of two II fourpole networks, each of
which is equivalent to a section of line with a length of 0 _w/2. It is obvious
::hat this networks reduces to the form of the network in Figure 21.10b. A phasing
section of a phase shifter for a diacrete phase atep of AO _ir using a network of
the kind shown in Figure 21.10b is shown in Figure 21.10c.
Another structural variant of a phasing section for a diacrete phase step of AO=7r
is shown in Flgure 21.10d. Tvo II fourpole networks are also used tiere, one of
 which, having the.structure of a low pass�filter aection (C1, L2) is equivalent to
al.Iine section with a length of 0 _7r/2, while the other, having Tche structure of
_ a high pass filter section (L1, C2) is equivalent to a line section with the
length of 0 = 37r/2. .
One element of a phase shifter fox other discrete phase stepis can be made in a
similar manner. We will note that the circuit configurations treated here for
the elements of discrete phase ahifters can made uaing a T section fourpole
network (Figure 21.9c).
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Reflective phase shifters with isolators are differentiated according to the type
of isolating devices used and the methods of obtaining the specified discrete
phase step. In contrast to phase shifters with switched channels, in reflective
. phase shifters it is necessary to equalize the losses in both phase states, whrch
is achieved through different circuit design solutions.
The major requirement which should be satisfied by the mutual isolation device of
such phase shifters is that of assuring a 3 dB power division among the two arms
with a phase shift of 90 degrees. In line with a T mode, primarily the following
isolating deviees are employed: a loop bridge (Figure 21.11a), a ring bridge
(Figure 21.11b) and a coupler with electromagnetic coupling (Figure 21.110. In
waveguide phase shifters, primarily a 3 dB bridge is used for these purposes, since
the sizes of other devices with similar characteristics are considerably greater.
We shall consider the methods of obtaining a discrete phase step in reflective
phase shifters. The first method is similar to that used in phase shiftera with a
continuous phase shift change (see Figure 21.4a) with the only difference that the
diode resistance can assume only two values: either close to the resistance which
provides for a short circuit at the point ot diode installation (in this case,
power is reflected from the diode) or close to a resistance which provides for a
noload mode (in this case, the reflection takes place from the short circuited
end ofthe line in which the diode is inserted). The discrete phase step is &P._
= 24~1, where 4~1 is the electrical length of the line from the point of diode in
sertion to the short circuited end of the line. With this method, it is difficult
to achieve identical insertion losses in both phase states, since the reflection
occurs in one case from the diode and in the other from the shortcircuiter. To
equalize the losses in both phase states and expand the working bandwidth, a loop
is inserted in the line coupling the diode and the isolating device [5]. In this
case, the requisite values of the reflection factors are determined by the point of
insertion of the loop, i.ts length and characteristic impedance.
Ashas already been noted, elements with multiple discrete steps, which are shown
in Figure 21.12, can be realized in reflective phase shifters. In this structural
design, two loops 401 and ~D2 are connected at a common point. The characteristic
impedances (pl and P2).larid the length of these loops are chosen so that when
switching the bias voltage of the diodes, the susceptance of the first loop at Che
cammon point is equal to +jbl, and +jb2 at the aecond point. The combination of
these susceptances yields four values of the total susceptatace at the comnon point:
j(bl + b2), j(bl  b2), j(b2  bl), j(bl + b2). Four values of the discrete
phase step are provided in this case. The dimensions of such a phase shifter are
slightly larger than the dimensions of a single digit phase shifter with one dis
crete step, since only one separating device is used.
A phase shifter with a nonreciprocal isolator  a circulator  operates just as the
phase shifter shown in Figure 21.5c does. The use of circulators providPS for
smaller dimensions and a smaller numbPr of diodes, which is responsible for lower
losses.
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  Z 4
2 Po112 y
4 ~/4 3
.~/4
al (a) A/4 aJ ~b) Cl(c)
� ,
Figure 21.11. Isolators for digital phase ahifters.
� ' Uy n p �
U
con
+ 'Ni C6nc
�e~� bl ,
To the ieolator '
Kaosdenume~aNO~u '
~
U
� con
. ~
 CQA
. . Cbl
Figure 21.12. An example of the
realization of a
multiply diacrete
element of a phase
ahifter.
.
. .
. .  
a� n0 /O/r Oj � ~0
~ .
,
'04, 04 C6n
t7a".1.4 ~Q~~ Fb)
p=
b 1 ,
, . .a~ (a). , .
Figure 21.13. Circuits for obtaining the necessary susceptances
(a) and reactances (b) in a phase ehifter of the
periodically loaded line t.ype.
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Phase shifters of the periodic loaded line type differ in ttie methods of realizing
the reactance inserted in the line to change the line's electrical length. The
reactive elements inserted in the transmission line can be made in the form of
loops. The length and characteristic impedance of tfie loops are chosen from the
condition for obtaining the requisite input susceptanee; usually bl = b2 = b,
where b is the shunting susceptance (Figure 21,5d). A shcematic of such a phase
shifter with parallel loops is shown in Figure 21.13a. Phase ahiftera of the
periodically loaded line type with a series configuration of distributed reac
tances also find application (see Figures 21.5e and 21.13b).
Multiple Element Discrete Phase Shifter. The major requirement placed on them
is the requirement of assuring a phase change with a diserete step AO in a partic
ular range of values from Omin to OmaX (in the general case from 0 to 27r). The
discrete step AO is determined by working from the requirement placed on the
characteristics of the device in which the given aultiple place phase shifter will
_ operate. Usually, a multiple digit phase ahifter eontains nl digits. Each dig,it
can exist in only one of two phase states (a single diacrete step digit): there is
no phase delay (or the insertion delay is taken as zero); or the insertion phase
delay is AOi, where i is the number of the digits.
The minimum number of diRits nl in this case is assured through the choice of the
following ialues of AOi:
  
. e01= e(v, .
aFi)z  nOl +n(v 2e(n, .
e(n;,  o0z i o0l  f n(b 4n(i,
. . . . . . . . . . . . . c21.17>
n1
A0n = A(U 1 ~ A(D 2" A0.
The range of phase change from 0 to 21r will be covered if the overall phase delay
introduced by all of the digits is:
(v n(r), e(rZ 4 n(Pc2i. is
� 2n  nm.
By using equations (21.17) and (21.18), we obtain:
logx (n/&D), (21.19)
It follows from (21.19) that nl is an integer with the condition eO _7r/21A is met,
where m are also integers.
Digital phase shifters which have already been described are used to realize the
discrete digits. The selection of the type of digital phase shifter is made
based on various criteria. For example, if minimal average inaertion losses are
required, then one can employ phase shiftera of the periodically loaded line type
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as the lowest order digits (i.e., the digits with the small discrete phase steps),
and for digits with larger discrete phase steps, one can use phase shifters with
switched channels. In this case, a gain is obtained in the average insertion
losses both as compared to the case where all digits are realized using phase
shifters with switched lines as well as the periodically loaded line type.
Figure 21.14. The structural design of a three element microstripline
phase shifter.
Key: 1. Phase shifter housing;
2. Stripline conductor;
3. Dielectric plate;
4. PIN diodea;
5. Power aupply terminals for the PIN diodes;
6. Blocking capacitors;
7. Choke;
8. Coaxial to stripline transition.
This is explained by the fact that phase shiftera using switched lines introduce
approximately the same losses for any diacrete ateps, while phase shifters with
periodically loaded lines have low losses for amall discrete ateps. At the same
time, the loases in phase shifters of this type increase with large discrete phase
ateps. One of the poasible structural designa of a phase shifter using awitched line
section is shown in Figure 21.14,
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CHAPTER 22. MICROWAVE FILTERS.
22.1. The Classification of Microwave Filters
Electrical filter is the term for a passive linear network with a sharply pronounced
frequency selectivity. Filters are very widely used in radio systems for the fre
quency selection of the requisite signal against a background of other signals or
interference. A filter is frequently used to suppress interfering signals.
In the microwave band, a filter takes the form of a tranamisaion line which includes
inhomogeneities, matched in a definite frequency band and aharply mismatched outside
of this band. In this sense, filter operation is similar to the operation of a
broadband matching device. (A filter is sometimes used for.broadband matching). It
is apparent that to reduce losses within a passband, a filter should be made of
reactive elements.
At the present time, the most widespread procedure for microwave filter design is
the procedure in accordance with wh=ch the low frequency prototype of the filter ie
designed initially, in this case determining the inductances and capacitances for
the loaded Q's of the resonant circuits of the prototype, Then the queation of
the realiaation of the calculated elements with the appropr;.ats inhomogeneities or
resonant systems in the selected transmission line is resolved. Thus, it is neces
sary to have an equivalent circuit of the microwave filter for design calculations
based on this procedure.
The equivalent circuit imparts clarity to the design calculations and makes it
possible to use techniques which have been well worked out in the theory of low
frequency filters for the design of a microwave filter. However, it must be remem
bered that the equivalent circuit reflects the actual microwave device with only a
certain degree of precision. It frequently does nQt take into account various
parasitic.;scattering fields, equivalent to additional capacitances and inductances.
It is also necessary to remember that resonant microwave sqstems (volumetric
reaonators, line sections) are multiple resonance systems, something which is not
at all taken into account in the equivalent circuit. The tranaient processes in
the equivalent circuit and actual device will also be different.
The main parameter of a filter is its frequency characteriatic: the working attenu
ation L(f) or the reflection factor r(f) as a function of frequency. We recall .
that L = 1/(1T2).
Filters are broken down into low pass (FNCh), high pasa (FVCh), bandpass (PPF) and
bandstop or rejection (PZF) filters.
Bandpass and bandstop filters are most frequently used in the microwave band,
although, for example, low pass filters are used to filter the higher harmonics of
oscillators and frequency.multipliers. Bandpass filters are sometimes used both as
low pass and as high pass filters. The right side of the frequency response is
usdd for a low pass filter and the left side for a high pass filter.
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The following are usually apecified in the design calculationa for a bandpass
filter: the cutoff frequencies for the passband fPr and f_pr, the mismatch toler
ances (I'pr) or the insertion losses.Lpr w ithin the passband, the stopband cutoff
frequency is fZ and f_Z and the minimum permissible losses within the stopband Lz
or I'Z.
It is obvious that the optimal shape of the frequency response would be a rectangu
lar form, in which the ftaquencies pass through and the blocked frequencies coin
cide: fpr = tZ and fpr = fZ. However, such a response shape is obtained only
witih an infinite number of filter sections. In actual devices, the slope of the
frequency response curve is determined by the kind of function L(f), which in
turn, depends on the number of sections and the Q's of the tuned circuits in the
sections.
a
With respect to the passband, bandpass filters are broken down into narrow band
for which the relative passband is leas than 5% ([2Afpr/fp] � 100 < 5), average
bandwidth filters (5 < 100 2Af r/fp < 20) and broadband filters ([2Afpr/fp] �
� 100 > 20), Here, fo = fP rf_pr~ is the center frequency of the passband.
In low frequency filtera, the filter sections are connected directly to each other
and there is strong mutual coupling between the sections. In microwave filters,
the sections can be coupled directly to each other by means of coupling elements
(such microwave filters are called indirectly coupled filters), or through quarter
wave line sections (quarterwave coupled filters), where the aeries resonant
circuits are transformed by line sections into parallel reaonant circuits.
Microwave filters can also be classified according to the type of line which is
used to construct the filter: waveguide, coaxial and stripline filters.
22.2. The Design of the Low Frequency Filter Prototype
The determination of the parameters of a filter prototype is a problem of para
metric analysis, i.e., the filter elements must be found based on the known fre
quency response of the f ilter. In order to make the design procedure more general,
in which the nunerical calculations for a specific sample are minimal, all of the
quantities are normalized. Nornielizing impedancea consists in the fact that the
load impedances at both ends of the filter are considered equal to unity. For a
load resistance of R, all of the prototype reaistances are increased by a factor
of R times; the frequency response of the filter does not change in thia case.
If the filter is not matched within the pasaband at e.ither end, then an ideal
transformer which provides for matching ahould be used.
Then the frequencies are normalized so that the normalized frequency at the edge
of the passband ia equal to unity. We make the substitution:
v = klw
(22.1)
If kl is chosen from the condition that kl = 1/wpr, then at the boundary of the
passband, the equality v= 1 will be observed. In this case, all of the filter
reactiances should be multiplied by the actual cutoff frequency apr.
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When solving a problem of parametric synthesis of filters, all of the types of
filters are reduced to a single prototype. Such a prototype is most often a low
pass filter.
The transition fram one filter to another is made by substitution of a frequency
variable. Thus, the transition from a frequenc}r w to a frequency defined by the
equality:
y = vw. (22.2)
will transform a low pass filter into a high paes filter.
92 9a 
90
9> qi n 9n�
A aubstitution of variablea of the kind:
v = k8 too (w/6u0 (22.3)
transforms a low pass filter into a bandpass filter. The values of the cutoff
frequencies for the bandpass filter and its passband can easily be derived fram
formula (22.3): .
WaAWnP= WOi 2A w=wQv'w_np~voa/k,~l/ke, (22.4)
Here wp is the center frequency of the passband.
To derive a stopband filter from a low pass filter, two conversions, (22.2) and
(22.3), must be applied aequentially.
Thus, any of the filters can be designed on the basis of a single low pass filter
prototype in the form of a ladder circuit (Figure 22.1).
When designing a filter, it is first of all necessary to have the frequency
characteristic apecified, L(f), such that the filter can be realized, i.e., the
 design calculations should not lead to quantities which are not phyaically
feasible.
Three types of filters have become the most wideapread, categorized according to
the type of frequency response: ,
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Figure 22.1. Schematic of a
prototype filter.
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1. Filtexs with a Chebyshev characteristic, the function of
of which is described by means of Chebyshev polynomials of
L� =1 + /o Tn (v/S).
Here, v= f/fp  fp/f is the frequency variable;
h ,  : rap
h
the working attenuation
the first kind:
(22,5)
is the amplitude coefficient; S is a scale factor which normalizea the cutoff
frequency; n is the degree of the Chebyahev polynomial; vpr/S = 1.
The frequency response of a three aection filter is shown in Figure 22.2. A fitter
with a Chebyshev frequency response (a Chebyahev filter) ia optimal in the aense
that in the case of identical starting data, of all of the filters which can be
described, it Yias the smallest number of aections. The slope of the frequency
response is the maximum of all of the filtere which can be uaed. A drawback to
the filter is the pulsation of the insertion loases within the passband and the
nonlinearity of the phasefrequency characteristic.
2. Filters with a maximally flat response (Figure 22.3):
L =1} !i' (v%S) zn,
(22.6)
The insertion losses within the passband vary frrnn the maximum values at the edge
of the band to zero at the center freq.uency. A merit of the filter is the linear
ity of the phasefrequency response. 3. Filters made of identical resonators are the simplest to fabricate and align.
The frequeacy response of the filter is described by a Chebyshev polynomial of the
second lcind and has greater operations within the passband, especially at its
boundarita. To reduce the oscillations, the Q's of the end sections are cut in
half. However, a major advantage of the filter is lost in this case: the identi
cal nature of the sections. The phase response of the filter is nonlinear. The
filter finds fewer applications as compared to filters of the firat two types.
We will note that a filter which has a frequency responae described by a function
using a Zolotarev fraction has the greateat slope of the frequency characteristic.
The response has oscillations within the passband and within the stopband. The
filter has not found widescale applicatii,on aince special sections with mutual
inductances are required to make it. ,f
In formulas (22.5) and (22.6), the degree of the polynomials n is equal to the
number of filter sections, and one can derive the expreasions to calculate the
number of sections from theae formulas.
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40 L. dB
Figure 22.2. Frequency response of
a bandpass filter
described by a
Chebyshev polynomial.
Z` 6� L, dB
t
L1
lop
Figure 22.3. Maximally flat response
of a bandpass filter.
For a Chebyshev filter, the number of:�sections is:
n> Archv([�aI)/(LTlp 1) . (22.7)
i . � Arch (ve/vnp) . .
For a filter with a maximumally flat response:
i~' V(La I)I(LnP  1) (22 . 8)
n > .
l8 (va/ynp)
If in calculations using formulas (22.17) and (22.18) [sic] the number 'n proves
to be fractional,,it is rounded off to the nearest whole value (usually the
greater one).
After determining the number of filter sections, the components of the ladder
circuit are found (Figure 22.1) as well as the loaded Q's of the bandpass filter
sections. This is the most labor intensive part of the problem.
There are two kno�an methods of overcoming the computational difficulties. In the
first, the general laws governing the distribution of the parameters of the
circuit components are ascertained. These governing laws are studied and then
generalized. In the second approach, tables and graphs are drawn up for the most
frequently encountered cases of filter design.
The.simplest distribution of the values of g can be successfully eatablished for
a filter with a maximally flat responae, fior which:
gi, = 2 sin [n (2k  1)/2n], (22.9)
where k is the number of the branch reckoned from the filter input (Figure 22.1).
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fg fnp f0 fnD . fI f
f_i f_op fp fop fa f
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It follows from formula (22.9) that when n= 1, gl = 2, when n= 2, gl = 92 =r2,
when n= 3, gl = 93 = 1, 92 = 2, etc. The filter is symmetrical.
The transition to a bandpass filter is made by aubatitufing variables in accordance
with formula (22.3). In this case, the loaded Q's of the resonant circuits are
determined fram the formula:
Q�=Qogh/2=Q$ sin[n (2k1)/2rz]� (22.10)
Here:
' Q4, _ ;/'/t /S
r
is the loaded Q of the entire filter at the three decibel level.
(22.11)
For a Chebyshev filter, there is no formula as simple as (22.9). The coefficients
g can be calculated from the following formulas:
b'i = 2ai/1'r gh = 4Uk1 ahI bp1 9R1+ ( 22 .12)
where
aa = sin [ic (2k  1)/2n1; bk = y' JI sin' (rck/n), y= sti 0/2ir,
~ = ln [coth(L [dB]/17.37)]
In [cth (L (AB)/I7,37)1.
The transition to a bandpass filter is also made by substituting variables in.
accordance with formula (22.3). The loaded Q's of the filter resonant circuits
are determined from the formula:
QN = Sn/2S.
(22.13)
In filters with quarterwave coupling, it is necessary to take into account the
influence of the frequency sensitivity and dispersive properties of the quarter
wave line sections. The initial loaded Q of the filter sectione is determined
from the formulas:
(center sections) (22.14)
QK = Q,< (2,o/Xao)'nl4 (cpeAHxe saeirbR),
Q� = QK (~o/Xbo)a J6 f 8(Kp2AHH8 3BEHbA): (end sections ) (22.15)
We shall briefly deal with the problem of synthesizing a bandatop filter. A band
pass filter is taken as the prototype here. Sy applying the transfozmation (22.2)
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to it, we obtain a bandstop filter. The Chebyshev frequency characteristica of the
bandpass and bandstop filters are shown in Figure 22.4. As can be seen from Figure
22.4, the cutoff frequencies for the bandwidth and the insertion losses are the
same for both filters:
vnp Tli[Q,= vnp ll3m =.S, v_,p nnm = v_npnsm; ( 22.16)
Lnp ilIlm = Lnp 113m, Ls nnm = L3 nsm. (22.17)
[nnO = bandpass filter; n3O = bandstop filter].
The conversion (22.2) transforms the cutoff frequencies of the stopbands and the
loaded Q's of the tuned circuits:
(22.18)
(va IIIIo/.S) (va tl3mI.S) = j,
'fQK nnm S) (QK nsm S) =1'. (22.19)
When equalities (22.18) and (22.19) are observed, the number of sections in both
filters is the same.
Yet another type of filter is used in the microwave band which does not have any
analog at lower frequencies: the stepped filter. It consits of line sectionaof
equal length and different input impedances. In contrast to a stepped matching
transformer (taper), the change in the characteristic impedance from step to atep
takes place nonmonotonically here. The design prodedure for a stepped filter is
based on the use of a stepped transition as the prototype. The frequency response
of a Chebyshev stepped f ilter is described by the formula:
L=' 1} h' Tn (sip (Dl s) (22 . 20)
where ~D is the electrical length of one step.
A comparison of the frequency responses of a filter and a transition show that
the frequency response of a filter is shifted by 7r/2, i.e., there where the transi
tion has a stopband the filter has a passband. The length of the step amounts to
half the resonant wavelength. The filter bandwidth is twiee as narrow as the
bandwidth of the transition.
Thus, the solution of the problem of prototype synthesis of a bandpass microwave
filter is completed with the determination of the number of filter sectione and
the calculation of the loaded Q's of the resonant systeme of the sections.
At the present time, tables and auxiliary computed graphs are given in the refer
ence literature for the parameters of filters which are encountered most
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frequently in practice, for example, [1, 4,=5, 014], because of which the design
calculations of a filter prototype are speeded up subatantially.
dB
, U070f jfnp fq nnm
f np fa nJm
~
Figure 22.4. The fretquency responses of
Chebyshev bandpass and bandstop
filters.
Key: 1. Bandpass filter;
2. Bandstop filter.
22.3. The Structural Execution of Microwave Filters
The execution of the structural deaign of the filters in the microwave band can
be extremely diverse.
A spatial resonator is used as a microwave reeonator. With careful fabrication,
it has an extremely high Q: up to 15.000  20,000 in Lhe centimeter band and
dimensions whieh are too large. For this reason, it is used in the short wave
portion of the centimeter band and the millimeter band as the resonant systems of
, a filter for very narrow band filters.
The major typea of reaonators of microwave filtera are the reaonant sections of
tranemission lines, which are open circuited, shortcircuited or loaded into
reactances.
As is well known, shortcircuited and open circuited line sections, the length of
which is a multiple of a whole number of quarterwavelengths possess resonatit
properties. Such systems, just as volumetric�resonators, are multiple resonance
systems. The inherent Q of a resonant line section with a Tmode is de�ined by
the formula: . Qo=nYe/~a, (22.21)
where n is the coefficient of attenuation in the line.
The natural Q of coaxial and stripline resonators, filled with a dielectric,
amounts to 250 to 400 in the decimeter band. For resonators filled with sir, the
Q is increased up to 500 to 600. For waveguide resonators, the natural Q can
b e calculated from the fornula:
a
QO  fxn Xn ( 1% / �
A waveguide resonator has an inherent Q of aeveral thousand at centimeter
wavelengths.
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(22.22)
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 The topology of a halfwave resonator using striplines is depicted in Figure 22.5a.
The equivalent circuit of the resonator is shown in Figure 22,5b. When the center
of the resonator is shifted (el), the amount of ita coupling to the line changes,
i.e., the loaded Q changes. The greater A1, the higher the loaded Q of the
resonator.
The structural design of a threesection atripline bandstop filter with quarter
wave coupling is depicted in Figure 22.6. The degree of coupling ia adjuated
by means of tihe gap between the main stripline and the end face of the resonators.
As a rule, microwave filters are transmissive devices. For this reason, through
transmission resonators find the greatest application in them. We shall deal in
more detail with two types of transmissive reaonators: a waveguide bounded on two
sides by reactive inhomogeneities, and a resonator made with coupled striplines.
These resonators find the greatest applications in microwave filters.
A waveguide resonator is depicted in Figure 22.7. It takes the form of a waveguide
bounded at the end faces by reactances, in this case, inductive stops with a nor
malized susceptance b. The reaonant length of the resonator for the case of
inductive susceptances of b< 0 ie determined from the formula:
l0 = ;12n (nn  arctg (bl n=1,2,3,... (22.23)
(a)
(b)bl ~
Figure 22.5. A stripline resonator. Figure 22.6. A bandstop filter.
ers
6 6 ,
! Figure 22.7. A reaonator in the form of a line
� limited by inhomogeneities.
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. ~ 
o
Figure 22.8. A waveguide bandpass filter.
~~o o ~,o� o0
p O O J O O
u
f~
Figure 22.9. A bandpass fi:ter resonator using
coupled striplines.
�
The loaded Q of the resonator is:.
Qu  V M4 462 `x. )'(n9 arctg b I . (22.24)
For large and amall valuea of b, formula (22.24) can be simplified. When lbl > 50,
nn 1 ~oo ' (22.25)
QH= 4 b C
and for emall values of lbi, the quantity arctan(2/lbl) ;W2 and:
Q,r = ~21 r ~0 Inn~ 2~1. (22.25)
l ll 1
For a filter with quarterwave couplings, the spacing between adjacent sections is
determined from the formula:
(22.27)
'lk. n+i 7~2(2m1) %�ol4k%Ao/2 I ln lh+l/2 m =1, 2,
An array of inductive stubs and inductive stops is uaed as the reactive inhomo
geneities in waveguide tranemissive resonators. The natural Q's of centimeter
band resonators with inductive atubs amount to 1,500 to 2,000, and with inductive
stops, 3,000 to 4,000.
A threesection bandpass filter with quarterwave couplinga is depicted in
Figure 22.8. The sueceptance here is formed by the array of three inductive
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_ stubs. The capacitive screws, which are placed in the center of each resonator,
are.intended for the experimental alignment of the filter.
 Po
p2 � ~i '
Figure 22.10. A bandpass filter.
Another widely used through transmission resonator is a resonator made with
coupled striplines. The resonant sections of line are coupled together with
distributed electrical and magnetic coupling. Filtera uaing such resonators are
small, structurally aimple;and thPir production can be automted. The resonator
of a bandpass filter using coupled lines is depicted in Figure 22.9. In it, 0 is
the electrical length of the coupling section, at the frequency equal to 00 =W/4.
The free arms of theline can be siiortcircuited as depicted in Figure 22.9, or
opencircuited. The loaded Q of a resonator circuit uaing coupled atriplines
is defined by the approximate formula:
Qload QH � nArs, (22.28)
' where r is the coupling resistance, determined by the atructural parameters of
the line: the width and thickness of the strip, the spacing between the bases and
the spacing between the coupled striplines. The gap between the atriplines
exerts the major influence on r. Formula (22.28) yields better precision, the
greater Qload is� The error in the calculation when Qload > 20 does not exceed
. 1%, and when Qload = 5, it increases up to 8%. The working attenuation function of a resonator uaing coupled striplines is:
n~r sin ml t~~1 Cdsa ~D. (22.29)
1" 1~ (;sl 1
It is convenient to calculate the coupling resistance r in terma of the cross
talk attenuation of a directional coupler made with coupled Iines:
, C=(1}r')/r'..
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By means of series connecting the sections, one can obtain a bandpass filter. The
direct connection of the filter sections is depicted in Figure 22.10, where only
the conducting strip is shown.
A variant of a stripline bandpass filter is a filter using opposing stubs.
22.4. A Design Procedure for Microwave Filters
The technical requirements placed on microwave filters can be extremely diverse.
First of all, the frequency characteristics of a filter are important. Additional
requirements can be placed on filters which follow from specific operational or
production features. It is not always possible to meet all of the technical re
quirements. The designer should have a clearcut idea of the entire complex in
which the filter will be used.
The frequency properties of a filter are usually specified in terms of the para
meters fpr, fpr, LPr, fZ, f_Z and LZ. Filter design begins with the selection of
the frequency response. We shall make one note: fo�r a Chebyshev filter with an
even number of sections, the normalized output resistance Rout is nat equal to
unity, but rather to tanh2(0/4), i.e., an ideal transformer is needed to match the
filter to the line. For this reason, Chebyshev filters with an even number of
sections are rarely used: it is simpler to add one section.
In the following design step, the number of sections is determined by using formulas
(22.7) and (22.8).
Then it is necessary to select the type of resonator coupling: direct or quarter
wave. The length of a filter with direct coupling is less, and therefore, if
strict limitations are placed on the filter length, then a direct connection of
the sections is selected. Striplina filters are also moat often direct coupled
filters, something which is explained by their structural compactness.
The length of quarterwave coupled filters increases somewhat because of the
connecting line section. A merit of such filters is the emaller amount of coupling
between the sections, which makes it possible to independently tune the filter
 section by section. The ohmic losses in tilters with quartercouplings is less,
and the calculated characteristics are in better agreement with the actual ones,
which is explained by the lower values of the loaded Q's of the individual reso
nators. The fabrication tolerances are leas stringent here. A drawback to quarter
wave coupled filters is the considerable length and limited bandwidth, which should
not exceed 15%.
The subsequent design oonsiats in finding the loaded Q's of the prototype filter
sections. The calculations can be.made using formulas (22.9) (22.13), or what
is simpler, by making tise of the extensive reference literature [1, 59 014]. Then
the problem of the practical realization of the resonators is solved. In this case,
first the type of line, resonant frequency and passband are selected. The design
calculation procedure is governed by the specific type�of reaonator and is not
given here.
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We shsll limit ourselves to some recammendations are general guidelines. Very
narrow band filters with a bandwidth of 0.5 to 1�6 can be realized only by using
high Q systems: spatial resonators, waveguides and air filled striplines.
Waveguide filters are used at frequencies 5 to 10 GHz. The majority of these
filCers are through transmissive resonators with quarterwave coupling (figure 22.8).
If the resonator is formed by inductive stubs, then the filter bandwidth is limited
to 20%. With a greater bandwidth, stubs are needed which prove to be too thin to
replace with inductive stops. The tuning screws of the sections make it possible
to change the resonant wavelength of a resonator by 3 to 5�6.
Stripline filters are used in very wide range of wavelengths from tens of deci
meters up to 3 cm. At longer wavelengths, the dimensions increase greatly and at
shorter wavelengths, the requisite fabrication precision increases. Stripline
filters using resonators with end coupling are desigaed for wavelengths of 60 to
4 cm and bandwidths ctif 0.5 to 5%. With a greater bandwidth, the gaps between the
sections prove to very small. Stripline filters using opposing stubs operate
well at wavelengths of 70 to 5 cm with bandwidths of 2 to 50%. They are quite
compact and well suited for production, and for this reason have found very wide
scale application.
,A couanon drawback to stripline filters is the difficulty of experimentFl alignment,
which is accomplished by changing the dimensions of the conducting stripline.
Coaxial filters are used at decimeter and meter wavelengths.
'The geometric dimensions of resonators and other filter components are calculated
based on their equivalent circuits and using reference literature [1, 3, 4, 71.
For transmissive waveguidel resonators and end coupled stripline resonators, the
length and loaded Q of a resonator can be determined form formulas (22.23) 
(22.27).
Microwave filters are manufactured in accordance with the third precision class
with a purity of the current carrying aurfaces of no worse than V6 V7.
If the ohmic losses a in a f ilter do not exceed 1 dB, then they have little
influence on the frequency response, shifting it along the ordinate.
We will note that literature has appeared in recent years which is devoted to
automated (camputer) filter design. The uee of a computer makes it possible to
vary the change in many filter parameters, optimizingits requisite characteristics
I6l� .
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CHAPTER 23. DIRECTIONAL COUPLERS AND DIRECTIONAL FILTERS USING COUPLED STRIPLINES
Stringent requirements for cost reduction, increasing reliability as well as
reducing size and weight are placed on microwave band radio equipment, including
antenna arrays, both with mechanical and electrical scanning.
These requirements can be met to acertain extent by using strip transmission lines
in the antenna arrays. They are used as the channelizing feeder system in the
decimeter band, and serve as the basis for the realization of individ+ual feeder
channel components in the decimeber.:and centimeter bands (power dividers, direc
tional couplers, filtera, etc.); the use of lumped reactive elements make it pos
sible to use striplines in the meter wavelength range also.
The use of striplines in antenna arrays makes it possible to realize structures
which are more suited for production and have low size, weight and cost.
To be numbered among the drriwbacks of stripline structures are primarily the high
losaes (especially in;:the centimeter band) as compared to waveguide and coaxial
transmiasion lines.
When comparing the posaibilities for using microstripline and stripline eleiaents
in antenna equipment, one must keep in mind the following. The use oft,microstrip
line elements is expedient and justified when fabricating individual components
and assemblies for both active and passive antenna arrays (phase ehifters, mixers,
converters, amplifiers, etc.). However, in excitation circuits for antenna
arrays (the simplest series and parallel circuits or more complex ones with a
large number of directional couplers and filtera), the advantagea of microstrip
line construct.ion are lost because of the considerable losses in a long feeder
channel.
Questions of designcalculations and planning of directional couplers (NO) using
coupled lines as well as loop type directional filters (NF) designed around direc
tional couplers are treated in this chapter.
Directional couplers are used in antenna arrays primarily for the following:
To obtain the requisite;.amplitudephase distribution in the radiators of an
array; ,
To decouple the radiators of an array, something which is especially important
for correct operation of a phased array; .
In compensating circuits to reduce the influence of the affect of'the change in
input impedances of radiators during electrical beam scanning of a phased
antenna array;
As elements of more complex radio frequency assemblies (phase shifters,
amplifiers, etc.).
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Directional filters find appiication in transceiving (reradiating) antenna arrays
for the segregation of the receive and transmit channels. '
23.1. The Classification of Directional Co.uplers and Filters and Their Operating
Characteristics
A directional coupler is an eightpole system. The directional coupler transmis
sion line through which the greatest power flows is called the primary line, while
the line in which a part nf the power is split off is called the secondary line.
Directional couplers with three types of directivity are shown in Figure 23.1.
The major characteristics of directional couplera are: the crosstalk attenuation,
the directivity, the decoupling, the matching of the arms of the coupler to the
input feed linea (SWR), the phase relationahips for the voltagea in the output ,
arms and the working attenuation in the primary line.
The crosstalk attenuation is defined as the ratmo of the primary line input power
to the output power of the working arm of the secondary line. For example, for
the coupler depicted in Figure 23.1a, the crosstalk attenuation is:
C� = 10 ig (P1/P2). ( 23 .1)
The directivity is the ratio of the powers at the output of the working and non
working arms of the secondary line. For example, for Figure 23.1a, the directivity
is:
Cu = lO lg (P2/P6).
(23.2)
The decoupling [isolation] is defined as the ratio of the primary line input
power to the output power of the secondary line nonworking arm. For the eight
pole network of (Figure 23.1a):
C,'J.= 101g (P,/P,).
(23.3)
The working attenuation of the primary line is defined as the ratio of the powers
at the input and output of the primary line. For Figure 23.1a:
C� = lO lg (Pl/P,).
(23.4)
The matching of the directional coupler arm with the input feed line is charac
terized by a atanding wave ratio which ia measured from the input arm of the
directional coupler, when matched loads are connected to the remaining arms.
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To determine the band coverage properties
~ pi ~ 3 3 of directional couplers, the major charac
teristics are determined as function of
Z pT P4 4 frequency (wavelength).
Q~ (a) De ending on the crosstalk attenuation
1 3 1 C~, directional couplers are broken down
Z 4 into devices with strong couplin (~C~ _
= 0,..10 dB) and weak coupling (fiCi> 10 W.
~ 6J Directional couplers which have different
1 3 power levels in the output arms (ICl _
� = 3.01 dB), fall in a special class of con
2 figurations: hybrid or 3 dB directional
01) (c) couplers.
Figure 23.1. Three tqpes of direc Of the small directional couplera used in
tional coupler direc . practice, the following have become the
tivity, moat widespread:
  1) Coupled line couplera are the most
1 i 3 compect broadband devices with respect to
. fo the frequency characteristics of the work
Z ' 4 ing parameters; they make it possible to
realize both strong and weak coupling;
Figure 23.2. A diredtional filter 2) Loop couplers are the simplest to fabri
in the form of an cate and provide for the simplest topo
eightpole device, logical configuration of the output netwnrks
in muxers, phase ahifters and switchers for
active phased antenna arrays;
3) Cascade coupled line couplers make it possible to increase tY.e bandwidth of the
device with a slight increase in structural complexity.
Directional filters are eightpole devices which are used for the frequency
segregation of eignals. If a microwave power aource is connected to one of the
_ filter arms, for example, to arm 1(Figure 23,2), then at a certain frequency fp,
almost all of the power will go to arm 2(the directional coupling arm). With
a change in frequency, a redistribution of the micrawave power flux takes place:
the power in arm 2 is reduced, while the power in arm 3(the direct coupled arm)
increases. If a matched load is connected to the arms of the filter, then with a
change in frequency, practically no power is split off to arm 4(the isolated arm).
Directional filtera are made as loopitypes, with capacitive coupling and with
_ quarterwave coupling lines. We ahall treat questions of the design calculations
and structural deaign of single loop directional filtera which use directional
couplers with coupled lines.
The main characteristics of.directional filtera are: the insertion loss factor
for the directional coupling circuit; the attenuation factor for the direct
coupling circuit, as well as these factora as a function of frequency.
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The insertion loss factor (attenuation coefficient) for the directional coupling
circuit is defined as the ratio of the primarq line input power to the output power
of the working arm of the secondary line (Figure 23.2):
Ldir.coup. LAc=101g(P,/Pz)� (23.5)
The ratio of the powere at the primary line input and output is called the attenua
tion factor for the direct coupling circuit:
Ldir . coup. =101g (Pl/Pe). (23.6)
The matching of a directional filter to the input feed line is characterized by the
SWR.
The definition of the directivity for directional couplers and directional filters
coincides with (23.2).
Direction3l couplers and filters can be designed around two types of striplines:
syametrical (Figure 23.3a0 and asymmetrical (Figure 23.3d). A drawback to an
asymmetrical stripline is the lack of shielding ~the impoasiblity of designing
"multistory" modules around them), and the elevated lossea due to radiation losses
in the line where e< 10. The expediency of using asymmetr ical striplines with a
high relative dielectric permittivity e> 10 (they are called microstriplines) was
discussed at the beginning of the chapter. .
Questions of the design of directional couplers and filters using coupled sym ,
metrical striplines will be treated in the following. The major parameters of such
lines (characteristic impedance, attenuation, Q and ultimate power) are related
to the geometric dimensions (the thickness t and width w of the conducting
strip, thickness b and width a of the substrate) as well as its type (configur
ation, dielectric permittivi.ty, specific conductivity of the material).
A detailed procedure for calculating the geometric dimensions of a stripline for
a specified characteristic impedance is given in [0k4, 015, 161. Also given
there are the types of substrates and the limitation3 on the dimensions of a
symmetrical stripline with a tranaverse electromagnetic wave are treated.
23.2. The Main Design Equations for Single Section T Mode Coupled Line
Directional Couplers
Parallel Coupled Lines. Lines of various configurations (Figure 23.4) can be used
in Tmode coupled line directional couplers.
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oJ (a).
x
By(c)
~
~o
Figure 23.3. Striplines.
o   .  _ . _ _ . _ .
_
w w
�t=4~ ~
0 (a) 61 (b) 4l (c) ~ (d)
P 4 1
f
d) (e) Bl ( f) , (g) sl (b)
Figure 23.4. Coupled symmetrical striplines of various
configurations.
Primarily the following are used in directional couplers with loose coupling:
striplines with thin conductors (Figure 23.4a); atriplinea with thin conductors
coupled through a slot (Figure 23.4b); striplinea with circular inner donductors
(Figure 23.4c). The following are uaed in directional couplers with strong
 coupling: striplinea with two thin inner coriductora, parallel to the outer plates
(Figure 23.4d); similar striplines with diaplaced conductors (F igure 23.4e); an
insert type configuration with thin conductors (Figure 23.4f); striplines with
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with two thin inner conductors perpendicular to the outer platea (Figures 23.4g)
and those with thick rectangular rods (Figures 23.4h).
The use of a line with a particular configuration depends on many factors. However,
conf igurations shown in Figure 23.4a is most frequently used for loose coupling,
directional couplers with sidewall coupling, while the figure of 23.4d for the
case of strong coupling is used for directional couplers with end face coupling.
+ .E +
f E + +  E +
o) (a) 0 (b) B) (c) z~ (a)
Figure 23.5. The electrical linescof force in coupled atriplines with
even and odd excitation.
Design Equations. Identical coupled lines represent a symmetrical eightpole .
network, the analysis and synthesis of which can be accomplished by means of a
 classical transmission matrix or using wave transmission and scattering matrices
with the symmetry analysis technique. In accordance with this method, the task
of studying a directional eoupler using identical coupled lines reduces to the
description of the processes in the two pairs of fourpole networks for the case
of inphase (even) and outofphase (odd) excitation (Figure 23.5).
The scattering matrix of an ideally matched directional coupler using coupled
lines with isolated arms (1 and 4 in Figure 23.1a), where this coupler represents
a symmetrical eightpole network, has the form:
U ~l~ , sls 0
0 ais [S1=~ al, p 0 s o ' (23.7)
i
0 s1B sls 0
where Six 9K sin
= . m
� .
1/1K2 cosm}jsin0
~
YIx2 ,
' K'cosmFjsinm
Sii= l~ 314 =0;
In this case: K PownoH ~ . _21c (23.8)
Puv1Poit A
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poy and poH are the normalized characteristic impedances for even and odd excYta
tion respectively; A= a/re  .is the wavelength in the stripline; 1 is the length
of the coupling line; a is the wavelength in air.
It must be kept in. mind that the following relationships hold true:
I Siz 1a 4 1 Sis I2 = 1 (23.9)
is the unitary condition for the scattering matrix;
 (23.40)
p01l n r'04 `c 1
is the condition for ideal matching and isolation.
It follows from formulas (23.7) and (23.8):
1. The power distribution between inputs 3 and 2 of the directional coupler
(Figure 23.1a) depends on the electrical length of the coupling line 4). As a
rule, we choose:
1 = Ao/4 ((D = n/2),
(23.11)
where Ap is the center wavelength of the working frequency band, defined in the
transmission line (ao is the center working wavelength; e is the relative.
dielectric permittivity of the subatrate). At this wavelength, the scattering
matrix element S12 (or the coupling) is maxi.mwn, while the absolute value of the
coupling coeffibients between the lines at the center frequency when 4~ _w/2 is
equal to K= IS12I and is defined by expression (23.8).
2. The phase difference in the signals at output 3 and 2 amounts to 90�, which is
easily established from (23.7): .
arg (Sla/s,a) .rc/29
where the signal phase leads at output 2, which follows from (23.7), (23.8) and
the inequalities pp even ~ PO odd and K> 0.
It is convenient to represent formulas (23.7) and (23.8) in the form:
Ksin cD' e1p
S12 'cos'Q) K ~ (23.12)
Sla = el(V+n/2),
_V t  K9 cos2 (D (23.13)
where
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arctg ~ � t 1
K' go
(23.14)
Po', =V(1+0(1Pon=.Y(1+K) (23.15)
.
At the center frequency of the passband (1 = Ap/4):
rr' (23,16)
142 l312=,` .
In accordance with expression (23.1)(23.4), we write the main characteristics
of coupled line directional couplers, assuming for the sake of definition that
arm 1 is the input arm (Figure 23.1a):
The voltage standing wave ratio at the input:
. 1+1811 I (23.17)
I~c:= 1_~aitl *
The crosstalk attenuation:   
1
C12=1019
1 allkl' ~ (23.18)
.10
The working attenuation
i
CU=101g 6 (23.19)
The isolation ~
Cl4 =.lO lg ~ Sl~ I' � (23.20)
The directivity: ,
t~ I
cuiotg a
is1.i'' " (23.21)
Bandwidth Properties. It follows fram expressiona (23.12) (23.15) that the
quadrature relationships between the voltages in the output arms of directional
couplers using identical coupled lines are preserved at all frequencies.
The elemente Sil and S14 of the acattering matrix are equal to zero at all frequen
cies if condition (23.10) ia met, i.e., in thia case the directional coupler is
ideally matched (SWR1 = 0) and posaesses ideal isolation C14 (the directivity CL�. 0�) . .
By substituting (23.12) and (23.13) in (23.18) and (23.19), weobtain:
Cl$ , C4 a F ; ACyz~
C12101g(1lKa) (23.22a)
(23.22b)
where
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AC121019(1 (11(') ctg�mj; (23. 22c )
CI,=cYg+eC,a; (23,23a)
C1g=101g (23.23b)
OCIa = 10 lg (11(2 cos' 0). (23.23c)
In the expressions given here, C?2 and C~3 are the directional coupler parameters
at the center frequency of the band; AC12 and AC13 are the deviations in the direc
tional coupler parameters from their average value within the working frequency
band. 
50 70 90 79'O
po
Cilzd6
,so
P6
1B
~
1f0
~ so~ itl 961no ~ 0 �
!{s ad' 48 > 42 f/t'e
Figure 23.6. The frequency responsea of the crosstalk attenuation of
a directional coupler ueing coupled lines.
'A
TABLE 23.1
Ci,, As I
3
5 I
10
I 20 I
30 I
40
p fo %
I 64 I
55 I
48
I 45 I
44 I
43
e~o=, gb
I 38
I 33
( 27 I 24
I 23
( 22
The frequency characteristics of the crnss
talk attenuation, calculated in accordance
with (23.22a)  (23.22c), are shown in
Figure 23.6 and together with Table 23.1
W0,5 and Afp,2 are the bandwidths in
pp:cent at the 0 5 and 0.2 dB levels of
deviation fraa C~2 respectively) make it
possible to eatimate the bandwidth proper
tiea of very aimple directional coupli;rs
using coupled lines. When C22 W, Gf0.5
tends to the ultimate value of 42%.
Directional Coupler Operation with Unmatched Loads. A detailed study of the
impact of urnnatched loads with complex reflection. factora I'1, I'2, I'3.and I'4,
connected to the corresponding arms 14 of a coupled line directional coupler
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on its major characteristics is made in [Ols].� A eonclusion of practical import
czce follows from [015]: that it is neceasary to carefully fabricate the coaxial
to stripline (or waveguide to stripline) junctions, the points where the direc
tional coupler joins other radio frequency assemblies, which are the major sources
of inhomogeneities, so that 1I'il < 0.05; in this case, all of the major character
istics of the directional coupler will differ to an insignificant extent from the
naminal values.
23.3. Extended Bandwidths Directional Couplers Using Coupled Lines
The bandwidth of coupled line directional couplers can be extended by increasing the
number of sections of equal electrical length 0 which are cascaded together. Such
multiple section directional couplers, although they make it possible to obtain
multiple octave bandwidths, are nonetheless larege in size as compared to the
extremely simple directional couplers described in the precious sections, something
which makes it difficult to use them in antenna arrays. Design relationships are
given in this paragraph for a compact tandem directional coupler, in which a cascade
configuration of two directional couplers, H01 and H02 (Figure 23.7), is used to
widen the bandwidth.
9
J
Figure.;23.7. The topology of a
tandem directional
coupler consisting of
two directional�
couplers.
'With auch a configuration, the production
proceas realization of a directional coupler
with strong coupling is also simplified and
the requirements placed on the tolerances
during coupler fabrication are reduced.
Generally speaking, several directional
couplera can also be connected in a similar
manner. When a signal is fed to arm 1, out
put signals appear in arms 4 and 3; arm 2 is
decouoled. A tandem directional coupler
[~Qj is a symmetrical eightpole network
with the coupling shown in Figure 23.1b; for
it, we shall use symbols with a subscript
"!T": the crosstalk attenuation is C14T, the
working attenuation is C13T, the directivity
is C42T and the decoupling is C12T (cf.
formulas (23.1)  (23.4)).
The parameters Cm T are defined on analogy with (23.18) (23.21), with the s.ub
stitution of Smn T for Srmt where Simn T are the elements of the scattering matrix
of the tandem directional coupler, which are found by known methods in terms of'the
scattering matrix smn for H01 and H02 (Both HO's [directional couplers] are assumed
to be identical: K1 = K2 = K, while the connecting aections are equal).
Then:
2 jr ain m
51~= ^(cos m{ J_V1+~ sin 4D)2 ~
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~ 1rs sins a
Q ,
(cos m+I"{/ 1fA sin m?
slar=0, Su:=4,
(23.25)
where r= K//lKZ; 0 is the electrical length of the coupling region of H01 and
H02; K is the coupling coefficient at the center frequency for H01 (or H02).
At the center frequencies (o _ ff/2):
Is14*I~K==2K (23.26)
I SI'sTI' = iI s14Tr. .
' . (23.26a)
Expression (23.26) defines the crosstalk attenuation of a.tandem directional
coupler (KT) based on the known coupling coefficient K for H01 (or H02); the
inverse relationship to (23.26), which takes into account the feasibility of the
directional couplers r.omprising the tandem configuration is:
SAS
S
4
_
3
3
.
2
1
.
~
C104T=0
S(I 7(/ yU i/u ioa w
~_l I 1 I I I 1 I t I 1 I
0,4 0,6 O,B >,0 f, 2 >,4 f/fa
Figure 23.8. The frequency character
iatics of the coupling
[coefficient] of tandem
directional couplers uaing
coupled lines.
(23.27)
K=Y1Y 1K* ~2.
The crosstalk attenuation as a function
of frequency is:
2K1/TK' slnm (23.28.)
�~~i~~. 1 K' K'sin2 iD '
Then: .
C14r = io ig c14T+ec 
,4T'
(23.29a)
where: C&T= lOlg aY4TP _
lOlg I (23.29b)
4K' (1.K')
determines the coupling at the center
frequency (0 _ w/2);
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11C~F1C~sin~~ (23.29c)
AC,tr=101g 
~ ~
stn m
is the deviation of the coupling of the tandem diroctional coupler from the
average value in the working frequency band.
TABLE 23,.2
C14T I 0( 1
( 2 I
3 I b I
IOI
18I
26I 30I 40
Af0, 6
1 90
I
160
I I
110
I 105
100
(
95
I
90
95 I 84
I I
83 I 82
r�
efo,, I
140 I
110
75 (
70 I
65 I
60 I
56 I
50
I 48 I
46 I 44
The frequencq char.acteristics of the coupling coefficient, calculated in accord�
ance with (23.28)(23.29), are shown in Figure 23.8 and in conjunction with
Tables 23.2 W0,5 and Afp,2 are the bandwidtha in percent at the 0.5 dB and 0.2
dB levels for the deviation fram C12T respectively) make it possible to estimate
the bandw'idth properties of extremely simple tandem directional couplers ueing
coupled lines. .
It must be kept in mind that the electricallengtlis of the sections 212" and 31411,
which join the directional couplers (Figure 23.7), are to be made identical.
23.4. The Chararteristic Impedances of Coupled Lines in the Case of inPhase and
OutofPhase Excitation
It should be noted that the expresaions cited in 923.2 are valid for directional
cnuplers using identical lines regor.dless of the configuration of the latter. 'The
structural (geometric) dimensions and the electircal characteristics of direational
couplere are relatdd by means of the characteristic impedances for the case of
inphase (even) pp even gnd outof~phase (odd) pp odd excitation.
The determination of the values pp gven $nd p0 odd represents a rather complex
mathematical problem, for the solution of which three main methods are used: the
solution of Laplace's equation with boundary conditions, a aolution using the
technique of conformal mapping and the precise calculation of the stripline capaci
tance.
Coupled Strinlines with Lateral Coupling (Figure:l23.40'. For a zero thicknesa
(t/b = 0), the precise value of the characteristic impedance p0 even is computed
from the formula: .30n (23.30a)
P~~ =
Ve K (ky) ~
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where K(ky) is a camplete elliptical integral of the first kind;
k9==th( 2 b)th r 2 ~'b S l; kq= Y�1kq. (23.30b)
The precise value of the characteristic i.mpedance is:
p 0 odd  Pon = 3on K (kp) ~ (23.31a)
y8 x ckH>
where: " ' �
kA  th ( 2 6 l~cth r�' b S 1: k H =~1kp.
~ i ~ 1
(23.31b)
One can employ the following formulas with a high degree of precision (an error of
less than ly for w/b < 0.35), when calculating pp even and PO odd for conducting
strips of zero thickness:
gq.lb/Ye_ (23. 32a)
Poa= m In2 1 In 1F(ibi)]
th S b +
+ ' 94,t6/Y8
Po,R = ~ `i�
b, � ~ ln f I{cth.~ 2b 1J. (23.32b)
L
The values of p0 even and pp odd as a function of the geometric dimensions of the
coupled lines are shown in Figure 23.9 in the forms of nomograms. A straight line
joining the specified values of PO even and p0 odd, Poaitioned on the outside
scales, will yield the value of w/b and S/b at the interaection of the center
acale. Striplines with Lateral Coupling.(Figure 23.4b, c and h). For coupled striplines
with conductors having a thickneae greater'than 0(t/b � 0) (Figure 23.4h), the
formulas for the calculation of p0 even and PO odd and Che corresponding tables
and grapha are given, for example, in [I, 6]. The valuea of pp even and p0 odd
are calculated for the configuration of Figure:s23.4b using the formulae in [1, 5],
and for the configuration in Figure 23.4c, uaing the formulas in [5].
 429 
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End Face Coupled Striplines (Figure 23.4d). The equations for pp even and p0 odd
of coupled striplines of zero thickness (t/b = 0) with end face coupling have the
form: pO even
188,3/~/a_
Poq = w/(bS)l(in 4)/n}C/rccea '
188,35/b I/a_
Pox = m1(6S)} C/neo ' .
(23.33a)
p0 odd 
where:
These equations are valid when
a= b SS ln Sln (1 b 1.
1
.
w/b > 0. 35 = w/b > 0,350
r~�ov, ~M ~/DOH~N ~~O~,~N ~d � ' ~~DONOH
15 . . d00 r
tD .o ~ 100 . � '
30 L~. . i20 ~s
f00 ~  ' f'~ s'D
� 70 12
40 S/6 6'p �60
11 d'p
(23.33b)
(23.33c)
50 ~a 50 , 10..
60 0,1. 10 09 ~p
70 4& ' .
'
80 0,01 ~ � 48
90  ' 0,002 80' 47  Bp 
>20 0,0~01' gp Q6. .~90
160 = 30 . ~p0 . ~pp
, 200 = . ,
300 15~ 3p0 300
Figure 23.9. Nomograms for the determination of the dilnensions S/b (a)
and wb (b) in coupled striplines for specified values of
poy [characteris.tic impedance with inpYiase.(even)�exciCa
tion] and~ppH [characteristic impedance with outofphase
(odd) excitation]
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~
2
1
O
v~ qZ 0,3 0,4 8/6
Figure 23.10. The geometric dimeneions of coupled striplines with endface
coupling.
If these conditions are not met, the preaision of the cdnformity of the dimensions
to the characterietic impedances decreases.
Thel dimensions of coupled striplines are shown in Figure 23.10 as a function of
the characteristic impedance pp. even for the case of inphase excitation. To take
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the finite thickness of a conductor into account (t/b # 0), one should use Kon's
corrections [5, 61.
Striplines with EndFace Coupling (Figure 23.4eg). To calculate pp odd and
pp yn for the configuration depicted in 23.4e, one must make use of the results
of g9~f; the characteristic impedance of the lines shown in Figure 23.4f, g were
treated in [5].
Coupled microstrip lines with endface coupling (a configuration similar to
Figure 23.4a) are treated in [014, 015].
23.5. The Relationship Between the Structural and Electrical Characteristics
A Stripline Directional Coupler with Lateral Caupling (Figure 23.4a). The rela
tionship between the structural and electrical characteristics is determined from
expressions (23.32a) and(23.32b) after substituting pp even gnd po odd from
(23.15) in them and after the appropriate transformations. Then:
S/b
0,3 S.
,
0,2 '
 0,>
� 6
u~/b
1,0
f1,8 
0,6
04 _
>,8
1,4
1,0
46
~
02
f0 15 ,
~f;d6 20 30 40 50 ii;d6
. w/6 6 f0 15 4.,o'6 20 3!! 40 50 .!;d6
Figure 23.11. On the determination of the geometric dimensions of directional
couplers with side coupling.  432 
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S 1 94''15rt1(
S. ln cth
 n PovevI K'
b 94.15 V 1+K  n ln [2(1 ~exp x)),
Po ~
where 
x _ 188,3rt1(/ao  
YW Yirt2.
g
0,
0,
!J,
' S/6 S/b
5
3
w 

t_~
~
� 
2
p
o=so oN
1
f
~
f I I I
I
I
I
i
1 2' 3 4 5 6 7� 8 9 K,d6
w/b ;1
O,9
4
q4
9,3
22
(23.34)
(23.35)
1 Z 3* 5 6 ? d 9/(d0 10 ZO 30 iS;d6
Figure 23.12. On the determination of the geometric dimensions of directional
couplers with endface coupling.
Values of the structural parameters S/b and w/b are shown in Figure 23.11 as a
function of the coupling coefficient at the center frequency C22 =.K for pp =
= 50 ohms at valueg of the relative dielectric permittivity of the subsrate E of
from 1 to 3.
It is useful to employ extensive material from graphs and tables (especially when
the ratio t/b 0 0) in the calculations [6].
For K< 10 dB, the dimension S can become infeasible either atructurally or in
terms of the production process,'and then it is preferable to employ end face
coupling (see Figure 23.4d). r 433 ~
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Stripline Directional Coupler with EndFace Coupling (Figure 23.4d). The relation
ship between the structural design and electrical characteristics is determined by
expressions (23.33a) and (23.33b):
 
_ . _    . ,
< (23.36)
S 1K 1K ~Poln4
6 V . 1~K ~ 1 K ^ 188.3n
6 1~,Po ItK l b!li 6 1]+ n l\1 6) X (23.37
1/8
~
: Xlnrl 6~( b lln( b (23.37)
\ \ 1 1 1J .
Curves for the parameters S/b and w/b are shown in Figure 23.12 as a function of
the coupling coefficient at the center frequency C?2 = K for pp = 50 ohms and for
various values of e. The limitations imposed on (23.36) and (23.37) as well as by
inequalities (23.33c) reduce the accuracy of the relationship of the geometric
dimensions and the coupling coefficient;.in the region falling below the dashed
lines.
To determine S from the ratio S/b found from (23.36), the following relationship
 should be used (see Figure 23.4d):
s_ 2dS/b  (23.38)
1S/6 *
w/6 S/b cu16 S/b 49,0
0,3
0,2
41
0
> 3 S 7, 9 e 1 2 3 4 F
Figure 23.13. On the calculation of the geometric dimensions of a
3 dB directional coupler with inphase coupling.
Stripline directional couplers with endface 3 dB coupling occupy a special place
in the design of antenna arrays; the curves for w/b and S/b are shown to these
couplers in Figure 23.13 as a function of the dielectric permittivity E for pp =
= 50 and 75 ohms.
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23.6. The Major Design Relationships for Single Loop Directional Filters Using
Striplines
A single loop directional filter using striplines can be obtained from two direc
tional filters by connecting terminals 2'  1" and 4'  3" (Figure 23.14).
The design procedure for such directional filters, which presuppoaes the design of
the microwave structure around a prototype filter with lumped parameters [5], has
a number of limitations and in some cases yields a perceptible error, and for this
reason, we shall give theoretical relationships, the basis for which is the tool
of wave matrices [7]. Such an approach makes it possible to use the basic expres
sions derived in �23.2.
The scattering matrix of a single loop directional filter with equilateral loops
and identical coupling (KI = K2 = K), consisting of identical directional couplers
with coupling secti_ons (HO1 and H02) of equal electrical length 0 (B'igure 23.14)
has the following form (for the sake of determinancy, we assume that the first arm
1' is the input):
O s120 SAt 0 (23.39)
I5OI ^ s120 0 0 Sia,0 ~
slSO 0 0 sl26
. 0 s130  sn,~ 0 .
where 'si2,p = s is/(1 si3)
(23.40)
is the transmission gain of the directional filter from one line to another through
the directional coupling channel;
M M
S1.10 S2 ~2 Si~~ll'S13)
(23.41)
is the transmission gain of the directional filter from one line to another through
the direct coupling channel;
512_= StzeJm, 1~. (23.42)
is = Sia e ;
S12 and S13 are determined by (23.12) (23.14).
The frequency properties of the various filter networka are definedby.the expres
sion for the input losses L as a function of frequency. For the directional
coupling channel (112" and 3'4") in accordance'with (23.5):
Ldir. coup. L�c lO lg . . (23.43)
Is,M,I ~
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Taking (23.39)  (23.42) and (23.12)  (23.14) into account:
L�, =101g I 1{ 2v~K' ctg~ (23.44)
\ ~VTKa / .l
There follows from the unitary nature of the matrix [S~] (23.39):
~%z.b Sl&b (23.45)
Then, taking (23.6) and (23.44) into account, we derive the following expression
for the attenuation of the directional filter in the forward circuit (1'3', 2"4"):
 _ ~ 1_ ~ _K'
Lo,, =101g I s~3~ 12 =101g~1( 2v1_K2 t~4'/J� (23.46)
~
Thus, the frequency charaGteristics of the channels of the filter considered here,
 without taking lossea into account, are governed by functiona (23.44) and (23.46).
By analyzing them, one can establish the frequency properties of directional
filters.
1. The directional coupling channel behaves as a bandpass filter with periodically
alternating passbands. The argument of the frequency characteristic is cot~p, and
for this reason, its zeros are located at the points 4~0 =(2n  1)w/2, while the
poles are located at the point 0,, = nw, n= 1, 2, 3...
2. The forward channel is similar in terms of its frequency properties to a stop
b and filter. The stopbands alternate periodically, and in this case, the argument
of the frequency characteristic is taniP. The poles of the curve Lwc [forward
insertion losses] are located at the points it� _(2n  1)w/2, while the zeros are
located at the points iDp = nw, n= 1, 2, 3,
3. With a decrease in the coupling
~ ~ 1 o coefficient K, the passbands and the
stopbands are narrowed, and viceversa,
with an increase in K both bands widen,
y where these functions are nonlinear.
w ; � Na> ~ .
Z ~ Specifically, the indicated relationships
11142 are established from the following
considerations. We introduce the approx
imating function:
2,� , 4�
6
,
Figure 23.14. Outline drawing of a
single loop directional L}�101g(1}ha(ctg(D/ctg (Dm)2], (23.47)
filter. �  436 ~
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where ~Pm is the electrical length corresponding to the edge of the specified
passband of the directional filter; h is a coefficient whicla dezines the nonuni
formity of the frequency response within the passband.
By equating the right sides of (23.44) and (23.47), and then solving the resultant
equality for K, taking into account the faet that K [0, 1], we obtain:
V i ( h )3 .
~ hF' 2 I ctK mm I (23 . 48)
_ At the resonant frequency fp, a11 of the energy should be transmitted vxa the
directional coupling channel; in this case, in accordance with (23.44), FO =
_(2n  1)7/2. To assure the minimum loop dimensions (ef the directional filter),
~ one is to set n= 1, and taking (23.8) into account, we have:
_   .
fio = a/2; fm = fo I At, (Dm = (Do fm/fo = (Do Xm/Xo1 (23.49)
l= A6/4 =X0/4 Ye , (23.50)
where fm(am) is the frequency (wavelength) corresponding to Che edge of the
specif ied passband; Af is half of the passband.
The design of a directional filter is completed with the plotting of the frequency
characteristics for Ldir.coup. (23.44) and Lwc (23.46), shown in Figure 23.15 for
one special case of the valuea of directional filter parametera.
23.7. The Influence of Tolerances on the Parameters of Directional Couplers
When �abricat�ing printed circuit directional couplers and filters, it is important
to es'imate the impacX of the structural (geometric) tolerances on such parameters
as the crosstalk attnuation, isolation ar.d standing wave ratio of the inputs
[014, 015, 8].
Curves for the change in the most important parameters of a aide coupled direc
tional coupler (Figure 23.4a) are shown in Figure 23.16 as a function of the
tolerances for the geometric dimensions: the width of the conducting strip in
the ceupling region w, the gaph S and the dimension b for various coupling
coefficients K. Increasing the width by Aw and the spacing between them by
AS reduces the coupling in a directional coupler by AC, while increasing the
dimension b causes the coupling to increase.
It follows from the graphs that with identical tolerances for Ow/w,and AS/S,
changing the gap S has a great effect in the case of weak coupling and chang
ing w does the same with strong coupling. �
It should also be noted that making the gap S in directional couplers with
strong coupling ICOI < 10 dB with a precision of even 10 percent (AC < 0.2 dB is
structurally difficult to do because of the small size of the gap itself.
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Curves for the crosstalk attenuation 012), isolation (C14),working attenuation of
the primary line(C13) and the SWR are shown in Figure 23.17 as a function of the
relative change in the gap (S/Sp) and the width of the conducting strip in the
coupling region (w/wnom) (wnom is the nominal width of the conducting strip in the
coupling region) for a directional coupler w ith endface coupling (Figure 23.4d);
the graphs are plotted taking into account the losses in the dielectric substrate.
It follows from the graphs that making the width of the strips in the coupling
in the coupling region with a precision of +10%, for example, and the width of the
gap with a precision of +5% changes the degree of coupling by AC12 + 0.2 dBi in
this case, C14 > 30 dB, while the SWR < 1.1. The influence of the losses for the
materials considered here primarily brought about a finite amount of isolation and
a change in the crosstalk and working attenuations of less than 0.1 dB.
~,dh
>0
Figure 23.15. Theoretical frequency responses.
5
f/fo = 1.0048 (2Af0 /f0 = 1%) ;
h = 0.5 dB.
0,95
0,6
0,4
42
1
~d
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6,0 10,0 ZO,OK,d6
+dC,d6
S/52096
10
02
& 0 6,0 10,0 20,0 h;d6
_ ec,a6
2,0
5
b�s ~
0,2
1 2,0 QO >QO 240 h;d6'
.C24~d6   /!cr
50
40 dm/fi~f03
30 5
>0 t
240 ef 0 140 10,0 h;d6 ~2,0 6,0 10,0 20,0 if;c
. ,
CT�.06 '
oor
50
�10
40 0
30 '
?0 6,010,0 ?G{OiN,dB
c24,ao,
so ea~e~x
~
' S 30 0
zo
2,9 40 1402qoHa6
1,0 F' 11"
2,0 . 40 100 240 l;d6
Asr
1,2 
db/610�~
1,1
1'o
2,0 449 >o,o zo,o h;a6
Figure 23.16. On the influence of tolerances on the dimensions of a directional
coupler with endface coupling (pp = 50 ohms, e= 2�5� [Kct SWR]�
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The curves sh wn
The curves shown in Figure 23.17 make it posaible to draw the conclusion which is
of practical importance that it is posaible to construct directional couplers
wifih a crosstalk attenuation of 2,to 8 dB only by changing the gap S(Figure
23.4d) with a constant width of the strip in the coupling region (designed for
C?2 = 5 M.
The tolerance for the coupling section length Z can be easily estimated, taking
(23.8) into account, by means of the curves shown in Figures 23.6.
We will note that in the realization of directional filters, it is necessary to
specify such tolerances for the geometric dimensions of the directional couplers
(HO1 and H02 in Figure 23.14) that they are equal in terms of their absolute
value and opposite in sign. Ci2.03d6  ~ A'cr,Cf4,d6
8
4
0
O,'f
4,12
i~
~
~
i
~
i ~
Ci3
�
0,9 1,2 .4/Sp
a~ .
C12,C1J,d6
8
6
~

2
0
1,0
B)
1,4 40
/
i
1, 2 20 _ .r
_1_ &r
1,0 L_
0,4
X6
jo4
72
~o
, ~ Kcr; ~i4, d6
0, B ' 1, 2 S/Sp
61
I
\
�
_
2
.
� /~'cr

w/weoM , '0,9
1,0 w/wNOM
a) ~
Eigure 23.17. On the influence of tolerances on the pa.rameters of a
directional coupler with end.face coupling (pp = 50 ohms):
Solid curves are for CF2A substrate material;
Dashed curves are for FAF4 substrate material.
On the Precision of the Realization of Printed Circuit Directional Coupler
Dimensions. In the fabrication of microwave printed circuits using striplines
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_ made of foil materials, better precision in the reproduction of the circuit
dimensions is realized using photochemical technology: down to +0.025 mm 0:1 mm
when the drawing is,,made with Whatman's paper and +0.05 mm when the drawing is
made on glass).
~ The process of cutting ouit the conductors with the requisite configuration can be
used successfully in the case of nonseries production of striplines; in this
case, the precision of the reproduction of the major dimensions of a directional
coupler can be kept within +0.1 mm, and for couplers with end coupling, dielectric
inserts are used which make it possible to change the gap size with a precision
of +0.03 mm and less.
23.8. The Structural Design of Directional Couplers and Filters Using Coupled
Striplines.
Some Recommendations for the 9tructural Design of Printed Circuit FJirectional
Couplers and Filters. The correctness of the structural design of printed circuit
directional couplers and filters using couple_, lines determines their electrical
characteristicstto a considerable extent. Besides the limitations imposed on the
dimensions of striplines with the dominant mode, the following recommendations
should be adhered to [0.14, 0.15, 16]:
1. The bend angle of a stripline a(see Figure 23.14) (the necessity of a
bend arises in the fabrication of a directianal filter for the sake of conven
ience in bringing the conducting stripline into a coaxialstripline jucntion or
to a matched load, etc.), is to be chosen equal to 30...45�. Fastening screws are to be provided for a tight contact between the upper and
lower circuit boards of directional couplers and filters, where these screws are
arranged at a distance of no closer than 2b to 3b from the conducting strips.
The fastening screws also serve to suppress higher modes at the points of con
nection of coaxial to stripline transitions and other inhomogeneities.
Figure 23.18. A directional coupler with
lateral coupling.
Key: 1, 6. Fastening boards;
2. Upper circuit board of
the directional
coupler;
3. RF plug connector;
4. Conducting strips;
5. Lower circuit board of
the directional
coupler;
7. Holes for the fasten
ing screws;
8. Fastening screwa.
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Figure 23.19. A directional coupler with
endface coupling:
Key:1,6. Fastening boards;
2. Upper board of the directional
filter;
3. Dielectric spacer;
4. Conducting stripa;
5. RF plug connector;
7. Conducting strips;
8. Lower board of the directional
filter;
9. Holes for the fastening screw$;
10. Fastening.screws.
C>q,d6 ~
.
22 ,
>8
>4 ~
k6B,g9 _
0,7
C12,d6 4 _
0,6 0,8 1,0 >,2 . f/fp
Y! �
Y
I
f,~v ~ I faM
f, ~ I y0
Dxoo'
Ao~o'yaA
Figure 23.20. Experimental characteristics of a
directional coupler with endface
. coupling.
.~O
ya Cnedy~ou~r,~f
A~Odf/17b
Figure 23.21: Block diagram of a
transceiving module of
a phased antenna array.
Key: 1. Radiator;
2. NF2 = directional filter
. filter 2;
3. Directional filter 1;
4. Directional coupler;
5. ftrans;
6* freceive;
7. Module,input;
8. To the next module.
3. When manufacturing a large batch of the elements considered here, it is expedient
to experimentally work out their corner sections uaing breadboarded models, since,
strictly speaking, corner inhomogeneities slightly change the equivalent length of
a line section: Another effective me.thod of aligning a directional filter is the'
use of four tuning screws, arranged about the perimeter of the loop at intervals of
Apf4, as shown in Figure 23.14 (the small dark circles).
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/4
Figure 23.22. A multilevel configuration of a transceiving module of a
phased antenna array.
Key: 1.
2.
3.
4.
5.
6.
7.
8.
9.
Matched load;
Upper and lower boarda of the directional couplEr;
Transmitting channel phase shifter;
Multilevel RF transition;
Dielectric spacer of directional filter 1;
Upper and lawer boards of directional filter 1;
Module output to the radiator;
Upper and lower circuit boards of directional
fil'ter 2; .
Receive channel phase shifter.
Structural designs of coaxial bo stripline and waveguide to stripline transitions
are treated in [014, 015, 16]. An example of the design of a lumped matched
ldad is shown in Figure 23.22, while a distributed load is shown in [014, 015].
Practical Structural Designs for Directional Couplers and Filters. Structural
designs of directional couplers and f ilters are shown in Figures 23.18 and 23.19'.
'The construction of tandem directional couplers, deaigned in accordance with
'Figures (23.7), is facilitated when the directional couplers are realized using
lines with endface coupling; in the case of directional couplers with side
coupling, it is necessary to use layer to layer transitions (Figure 23.22).
The correctness of the design calculations and structural design solution, arrived
at in the planning stage, is evaluated during the proceas of laboratory testa.
Special cases of the appropriate frequency responses are ahown in Figure 23.15
for directional filters and in Figure 23.20 for directional couplers.
A block diagram of a transceiving m.odule for a phased array which is used in
directional couplers and filters is shown in Figure 23.21, while its realization
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in a threelevel design is shown in Figure 23.22. The conductors of the syBUnetrical
striplines are made in the form of a meander to reduce the longitudinal dimensions;
the directional couplers and filters are made with endface coupling.
The module (Figure 23.21) operates as follows: the signal at the transmit frequency
ftrans from the input of module (I) is fed through the directional coupler with
the corresponding division to the next module (II) and through the level to level
transition (4) (Figure 23.22) to the input (III) of the directional filter NF1.
Then thesignal (ftrans) is further fed through the forward coupling channel of NF1
to the input ( N) of the phase shifter of the transmitting channel, and following
the appropriate phasing, to the input (V) of NF2 [directional filter 2], through
the forward coupling channel of which the signal (ftrans) is fed to the input (VI)
of the phased array radiator.
In the reception mode, the signal (frec) is fed from the output (VI) of the radia
tor via the directional coupling channel of directional filter 2 to the input (VII)
of the receive channel phase shifter, and following the appropriate phasing, is
fed to the input (VIII) of directional filter 1, and through the directional coup
ling channel to the input (III) of the directional coupler and through the level to
level transition (IV) (Figure 23.22) and the directional coupler to the input of
the module (I).
23.9. The Design Procedure
When designing printed circuit direcfional couplers and filters, besides the re
quirements placed on the major electrical characteristics, there are limitations
on the size and weight, temperature and radiation conditions, power handling
capacity, etc., which follow from the requirements placed on an antenna array.
Before setting about the calculation of the electrical characteristics.in the
general case, it is necessary to choose the type of stripline and its character
istic impedance, as well as the type of..coupling (side, endface, mixed) for the
directional couplers and filters, working from an entire series of contradictory
requirements, where one is governed by the requirements of �23.2 and �23.9 as well
as [014, 015 16], We shall limit ourselves to the treatment of the simplest
cases, introducing the following symbols to facilitate the presentation: P is the
power (CW or pulsed) transmitted through the d irectional coupler (or filter) in
KW; fp (or ap) is the center working frequency (or wavelength) of a directional
coupler or the resonant frequency (or wavelength) of a directional filter in MHz
(or cm); +Of (or +pa) is the wot king.bandwidth, in MHz (or cm); AC is the permis
sible deviation of th e crosstalk attenuation of a directional coup ler from the
average value within the passband, in dB; CQ2 is the crosstalk attenuation of a
directional coupler at the center frequency* in dB; ChT is the crosstalk attenu
ation of a tandem directional coupler at the center frequency**, in dB; Cmin is
For the sake of definition, we assume that arm 1 is the input (Figure 23.1a).
See Figure 23.1b.
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the minimum isolation (or directivity) in the working passband, in dB; KCT max
[SWRmaX] is the maximum value of the standing;,,wave ratio within the working pass
band at the input to a directional coupler (or directional filter); 2Af (or 2AX) is
the passband of a directional filter, in MHz (or cm); h= LgC (fm) is the attenu
ation factor at...the boundary.of the passband in the directional coupling channel,
dB; fnc (or ~ffc) is the frequency (or wavelength) of the signal transmitted through
the forward coupling channel of a directional filter, in MHz.(or cm); LHC (f7d is
the a*_tenuation factor at the frequency f7c in the directional co�pling channel,
in dB; Itil, i= 1, 2, 3, 4 is the absolute value of the reflection factors from
inhomogeneities in the inputs of a directional coupler (for example, coaxial to
stripline transitions); pp is the characteristic impedance of the supply feed lines,
ohms; b/2 is the dielectric substrate thickness, in mm; t is the thickness of a
conducting strip, in mm; and e is the relative dielectric permittivity of the
substrate.
We shall consider the variant of the specificationa for the calculation of the
structural and electrical parameters f a directional coupler using coupled lines.
The following are specified: P, fp, C$2 (or CQ4T, +Af (or OC), Cmin, SWRmax, the
line is a symmetrical stripline, pp, and the dielectric substrate: F, b/2, t and
the type of coupling in the directional coupling is either side or endface.
The following design calculation procedure is recomaended.
 1. Determi.ne the width of a conducting strip, wp (Figures 23.18 and 23.19), using
the procedure given in [16, 014, 0151, and then determine the a dimension of
the stripline (Figure 23.3a), taking into account the limitations in *_his dimen
sion [16].
2. Find the attenuation, Q as well as the ultimate power by using [014, 015, 161.
3. Using the graphs of Figures 23.6 and 23.8, and Tables 23.1 and 23.2, establish
ing the agreement between the passband +6f and the permissible coupling nonuniform
ity, giving preference to the simplest directional couplers because of their
structural simplicity.
Using formulas (23.18) and (23.16), determine the coupling coefficients K for
the specified C?2. However, if a tandem directional coupler is selected, then we
find the coupling coefficient of a tandem directional coupler, KT, for the specii.
fied C14 using (23.29b) an.a (23.25), and then we find the coupling coefficients
K from ~23.27).
4. Determine the coupling line length 1 of the directional coupler using formu
las (23.11).
5. Using expressions (23.15), find p0 even gnd PO odd'
6. Using the known values of pp even and pp odd, determine the dimension of the
conducting strip in the coupling region w, and the gap S, using formulas (23.30)
(23.32) or the graphs of Figure 23.9 for a directional coupler with side coupling
and formulas (23.33) or the graphs of Figure 23.10 for a directional coupler with
endface coupling. One can also employ formulas (23.34) (23.37).
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For directional couplers where pp = 50 ohms and e=.1, 1.5, 2, 2.5 and 3, the
dimensions S and w are most simply determined from the graphs of Figures 23.11
and 23.12, plotted in accordance with (23.34)(23.37).
It is expedient to use the curves shown in Figure 23.13 to determine the dimen
sions of a 3 dB directional coupler with inface coupling where pp = 50 and 75 ohms.
Equation (23.38) is to be used to determine S in the ratio S/b in a directional
coupler with endface coupling.
7. The frequency resonse Cj~ = C12(f) is plotted1for the simplest directional
couplers, or C14T = C14TM is plotted for tandem directional couplers in accord
ance with formulas (23.22) and (23.29), or the appropriate curves from Figures
23.6 and 23.8 are used.
8. Where necessary, find the phase relationships at the outputs from formulas
(23.12)  (23.14), (23.24) and (23.25) respectively.
9. Estimate the influence of the nonideal nature of the.matched loads and the
corresponding coaxial stripline transitions (or other inhomogeneities at the out
puts of the direcfionil coupler) on K, Cmin and SWRmaX from the specified values
of II'il, using �23.2. .
10. Working from the requirements placed on the values of C~2 (or AC12), Cnin 8nd
SWRmax, set the appropriate tolerances for the precision in the realization of'the
geometric dimensions of the dir.ectional coupler, as indicated in 923.7.
11. Draw the directional coupler (Figures 23.18, 23.19 and 23.22) taking into
account the recommendations for the structural design of printed circuit direc
tional couplers.
Notes: 1. If the type of stripline and coupling in the directional coupler are not
stipulated, then it is recomended that they be selected where is one is governed
in this case by the considerations indicated in the literature [014, 015, 16].
2, If the passband of the directional couplers considered here using coupled lines
do not satisfy the technical requirements, it can be widened by using multiple
section directional couplers or two or more tandem directional couplers [014, 015].
It must be kept in,,mind that the use of a tandem directional coupler aubstantially
improves the isolation (or directivity) as compared to the simpleat directional
couplers, for which because of structural and production process.factors,; the
decoupling does not exceed 20 to 30 dB.
We shall further consider the specffication variant for the calculation of the
structural design;and electrical parameters of single loop directional fi.lters
using coupled lines. Required: deaign a directional filter which segregates the
receive channel (arm 2" in Figure 23.14) and transmit channel (arm 3') when working
into a common antenna (arm 1'); a matched load is connected to the free arm of the
directional filter (49,
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The following are specified: P, fp = frec, 2Af, h= LHC (fm), frtc  ftranss
LgC (fnc), the line is a symmetrical stripline, the value of e for the substrate,
b/2, t, and the type of coupling in the directional coupler is either inphase or
eide coupling.
The following design calculation procedure is recommended:
1. Pertorm calculations similar to those in paragraphs 1 and 2 for directional
couplers.
2. Calculate the geometric length of one side of the loop Z(Figure 23.14) using
formul.a (23.50).
3. Using formulas (23.49), find fm and Om.
4. Determine the coupling coefficient K at the frequency fp using formula (23.48).
5. The dimensions of the conducting atrip in the coupling region and the gap are
determined using the procedure of paragraphs 5 and 6 for directional couplers.
6. Plot the frequency function LHC = LHC (f/f0) and Lwc = Lnc(f/fp) (see Figure
23.15) in accordance with expressiona (23.44) and (23.46).
7. Where necessary, f ind the phase relationships at the directional filter outputs
from expressions (23.40)  (23.42) and (23.12)  (23.14).
8. Make the drawing of the directional filter (Figures 23.14, 23.19 and 23.22),
taking into account the recommendationa for the structural design of printed
circuit directional couplers and filters (923.8 and [014, 015 and 16]). Notes: 1. The notes of paragraph 11 for directional couplers also remain valid for
directional filtera.
2. In single loop directional f ilters using coupled lines, the parameters h=
= LgC(fm) and LgC(f.ffc) are not completely independent, and for this reason, in
satisfying the rq.quirements for the attenuation at the boundary of the passband in
the directional coupling channel (h), one may not obtain an altogether satisfgctory
value of Lgc(f7rc). One can partially avoid the indicated correlation by using
single loop directional filters with different lengths of the loop sides [7], or
by completely using single loop two section directional filters [5]. It is
recomended that dual loop directional filers be used to increase the directional
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CHAPTER 24. STRIPLINE MICROWAVE POWER DISTRIBUTION SYSTEMS
24.1. The Functioa and Major Characteristics of Micrawave Power Distribution
Systems
In a hole series of radio engineering systems for the microwave band, devices are
needed which make it possible to divide the power of the source in a definite
ratio in several channels or to add the power into a common load. Such functiona
are performed by multichannel excitation systems for phased arrays which produce
the requisite amplitudephase distribution of the field in the antenna apertures,
as well as by power adders for several generators. Taochannel power adders (or
dividers) find widescale use in modulators, frequency converters and other radio
engineering equipment. .
As a rule, passive bidirectional (reciprocal)
1 2 3 N devices are used for micrawave power distribution.
A divider with a counnon input 0 and N outputs
(Figure 24.1) is a multiport network with 2(N+1)
poles and can be used as an adder with N outputs
~ and one coIIenon input zero by virtue of the reciproc
ity principle.
Figure 24.1. A microwave
power distribution debic~e The following requirements are placed on microwave
in the form of'a multi power distribution devices:
port network.
Providing for a definite distribution.of the
amplitudes and phases of the signals of the N out
puts (or inputs) in a specified frequency range;
Matching the common input of the divider or N inputs of the adder in the
working frequency band;
Providing for the isolation of the N outputs (inputs) within the passband to
reduce the mutual coupling of the channels;
A high system efficiency;
A sufficiently simple structural design, small overall dimensions, high
reliability and low cost.
Stripline distribution systems based on hybrid integrated circuits (GIS) satisfy
the requirements enumerated above to a certain extent. Because of the fact that
the possibility for experimental alignment is almost completely lacking in such
devices, the theoretical analysis of the circuits and computer methods of analyis
and optimization of the working characteristics become of great importance.
Tte working characteristics of power distribution systems are uniquely defined in
terms of the elements of their scattering matrices in the following fashion: the
standing wave ratio at the ith input is:
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Swtti ~ KC: i=(1} I Stt I)I(1I Sii I); .
(24.1)
the crosstalk attenuation between the central input (0) and the ith output is:
~ . Coi = 2019 1/1 Sac 1; (24.2)
The nonuniformity of the crosstalk attenuat3,on within the frequency band is:
(24.2a)
eco,  co,  co, .
(Cbi is the crosstalk attenuation at the centex frequency); the isolation between
the ith and jth channels is:
i (24.3)
C1j,= 201g ISeJI~
The phase relationships of the signals at the outputs are determ3ned by the
arguments of the elements o� the scattering matrix.
24.2. The Camparative Performance of Various Tqpes of Microwave Power Distribution
Systems
The stripline distribution systems used in microwave equipment are distinguished
by the number of channels, structural configurations, working frequenc3� band,
power handling capability and construction.
Nondirectional distributors fnrmed by branching transmission striplines are the
simplest in structural terms. A considerable drawback to euch circuits is the
impossibility of simultaneously completely matching all of the inputs and de
coupling the channels: Because of the finite amount of isolation, a change in the
load impedances of the distributor during scanning can lead to considerable devi
ations of the signal amplitudes and phases at the outputs. This limits the
application of nr,ndirectional devices. as excitation systems for phased antenna
arrays.
Directional distribution devices (Figure 24.2) provide for matching all of the
inputs at the center frequency and isolating the channels. Both directional and
nondirectional distribution systems are broken down into series (chain) (Figure
24.2a) and parallel (Figure 24.2b, c) types according to the principle for the
channelizing of the microwave power.
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N
N7
~ m~. � Af
` o FIA
. I m; 1
i ~
. ~
~ . a
OI
FOR OFFICIAL USE ONLY
1 2 1 > 2 3 4 N1 N
S ~ 4V
i
r t I ~ r+
A2 I Ai I ~ AN mfi ~tn I m,;.~~
L~~4 b
~ i 4
~i Pi mit mrt
I '
_.L_ : � .
~ mif
0 0
~1 � ' OI
Figure 24.2. Series (a) and parallel (b, c) microwave power directional
distributors.
~
O Ap � O AQon 0. ~
~ r
po~ ~2 /~p yA2 ~2 CZ Z .QFanii A6un3 R Artan/ 2
aJ /i Bl
0 A
� ~i ~
RQan 3 pZ ' '
02
eJ
=Co Z2 =c~o
0
L~ �~s
3 =~o 12 ~Co 2
d1 ,
Figure 24.3. Circuit configurations of. dual channel microwave
power dividers. Series systems are distinguished by their campactness, however, they have a number
of substantial drawbacks. First of all, the range of variation in the cross
talk attenuation of the dual channel dividers incorporated in the device increases
with an increase in the number of channels, which limits the possibility of using
certain types of dual channel dividers, and also generates definite technological
diff iculties in the realization of the device. The unequal electrical length of
the paths from the common input to each radiator leads to a different phase
frequency response of the transmission gains of the channels. Moreover, the
distributor units which are closest to the central input pass the maximum power,
and for this reason, they should possess �an intreased electrical strength.
The drawbacks enumerated above are inherent to a lesser extent in parallel
distributfion systems. Distzibutors using quarterwave transmission line sections,
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depicted in Figure 24.2b, find application in the case of a small number of out
puts (less.than 10). If resistances are used as the isolating fourport networks
[Ai], connected in a star configuration, the device realizes an inphase uniform
power distribution. The frequency propertiES of such distributors depends sub
stantially on the number of channels, where the SWR bf the common input as compared
to the SWR of the outputs is a more pronounced function of both the number of
channels and the frequenc.y. Thus, within an octave range, the maximum SWR where
the number of channels is N= 3 and N= 25 is 1.75 and 7.5 respectively, while the
SWR of the outputs under the same conditions does not exceed 1.1. The efficiency
of a distributor likewise falls off with increasing N and in an octave ainounts
to 0.93 and 0.41 when N= 3 and N= 25 respectively [lJ.
A comon drawback to such power distributors is the increase in the characteristic
impedances of the quarterwave line sections with an increase in the number of
channels, something which makes their technical realization difficult. Moreover,
the circuit topology cannot be represented in the form of a flat structure: inter
sections of the conductors are unavoidable, which likewise represents an incon
venience in the realization of stripline distributors.
A binary power distribution circuit ("christmas tree") has become the most wide
spread (Figure 24.2c). In this case, the device composed of 3 dB power dividers
realizes an inphase 2nchannel system with a uniform amplitude distribution of
the field at the outputs. When dividers are used which have a division factor
other than l, one can design a system with a specified power distribution for an
arbitrary number of channels.
In the low frequency portion of the microwave band, the dimensions of devices
using distributed transmission l:.ne sections become impermissibly large. One of
the ways of reducing the overall dimensions:;is replacing each line section with
its analog using lumped elements [4]. In this case, the working bandwidth of the
devices is narrowed, however, within a 10% passband, they can successfully replace
systems using distributed elements.
The following can be used .as the constituent assemblies of branched distributors:
ring configurations, loop quadratLre bridges and directional couplers using
caupled lines, the isolated outputs of which are loaded into matched impedances
(Figure 24.3).
24.3. Calculating the Electrical Parameters and Characteristics of Two Channel
PoWer nistributors
Recommendations are given in this section for the calculation of the electrical
parameters of the equivalent circuits of distribution devices (the characteristic
 impedances of quarterwave transmission line sections and the parameters of
lumped elements), as well as for the elements of their scattering matrices.
Single section ring configurations (Figure 24.3a) assure that the signals at
outputs 1 and 2 are in phase when power is fed to the 0 input or provide for
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summing inphase signals fed to outputs 1 and 2 in the common 0 channel. They are
formed by quarterwave line sections with characteristic impedances pl and P2 as
well as a decoupling fourport netork (Ap) between terminals 1 and 2. Dual channel
power dividers are inserted in the common circuit by means of connecting line
sections having characteristic impedanees of p'p and p'1 amd p'2.
~2 ~S Z
R6an R6on3 R6on5
R6an2 R64 . al
A/1 /0n
1 C
Rbnnti R6andi
Rdan2i
6)
Figure 24.4. Dual channel (a) and multichannel (b) power divideY circuits
with decoupling fourport networks.
In the simplest case, a ballast resistor Rbal inserted in series is used as the
isolating fourport network. If the length of all of the sections are equal to
a/4 at the center frequency, then to achieve complete matching and isolation, the
parameters of the device are calculated from the following formulas:
R'm r'12_.
P2=p i
Ri R1
; p, ' ` R, Rn m
Pa = pi nt; Po i
P nt I 1 ,
(24.4)
where mw= P1/P2 is the power division factor. The characteristic impedances pl
and p'1 are chosen arbitrarily in a range of 25 to 150 ohms.
The maximum attainable power divisian factor of such devices in a stripline
design is limited by the feasible values of the characteristic impedances of the
lines and does not exceed five.
Dividers with an isolating fourport network (Figure 24.4a), comprised of four
quarterwave line sections with characteristic impedances p" and resistances Rbal
which differ f:om the general case have a greater range of change in the coeffici
ent m. One can use modifications of an isolating fourport network whch are
obtained by excluding the parallel resistances or shorting the series resistances
(with the exception of the last one). For example., a divider with one resistor
Rbal 2(or Rbal 4) takes the form of a ring configuration with a length of 3a/2
with a matched load at one output. Devices with symmetrically arranged decoupling
resistors have a more uniform frequency response of the working characteriatics.
 451 
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It must be noted that the presence of the additional line sections p.1"p4.narrows
the working bandwidth of the devices as compared to a divider having only a resis
tor as the decoupling fourport network.
The matching of all inputs and the isolation at the center frequency are achieved
when the following equations obtain [2].
The characteristic impedances of the quarter wave sections:
P,=Y(mI 1)RoR,/m Pz ~Y(m+ i)RoRz. (24.5)
The elements of the normalized transmission matrix of the isolating fourport
network:
Rt Ra m ~I Rt Ra
niip~m Rs , A~zv =V mRi Aizp = po ~ m ~ A21p= 0. (24.6)
The parameters o� the isolating fourport network~ Rbal, and p" are determined by
means of equating the elements of its transmission matrix at the center frequency
to the corresponding elements Allp and A12p (24.6), In this case, there is the
possibility of a free choice of at least one of the parameters Rbal or p", which is
used in the optimization of the frequency properties of the device. For example,
the transmission matrix of the is.olating fourport network with one resistor Rbal 3
when pl = pI and p4 = py has the form:
( Pl'/ P, )2 R6en ~ .
. ~AP~ 0 (24.7)
Consequently:
R bal 3 R6an3(m.{_ 1)YR )?Z/m; (24.8)
i p~~=P2 mRI/R2.
The value of p2 can be chosen arbitrarily in a range of 25 to 150 ohms.
An analysis of the working characteristics of ring dividers without connecting
sections can be made using the expresasions given in [2]:
The reflection factor at the ith input is:
~ Atazr (Aii!4gj2) ,
r`A1,+_, cA,I+A,12>' (24.9)
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The working attenuation function is:
L 20 (Al'`Fzj(AiiFAu2) (24.10)
" 1J ~ ,g 2 (AAlt) zoz~I'.
where Zi and Zj are the normalized complex impedances of the load and generator
in the generalcase (i,'j = 0, 1,".2); [A], [A] and [X] are the transmission
matrices of the cascaded fouxport networks between the feed points and the load.
For example, in the case of the excitation of the 0 input
[A] = [Ao, ] [A:j [Ap] [A:z] [An2
When calculating Lpl:
When calculating L02:
where
[Aj = [Ant 1; ~AJ [AP] [Ae21IAp2 1�
LA1= IAot 1 [A:i1 [Apl; ~~1J = [Ap2 L
(Azl = 1 o
. Z ~ I J
; is the transmission matrix of the impedance inserted in parallel;
i
I chYl Pd1 7 1
I [Ap 1[ p 1 s11 y1 ' chy! l .
~
is the tranamission matrix of a line section with a length Z having a normalized
characteristic impedance of p; Y= a+ jg is the propagation constant.
The transmission matrices for the case where inputs 1 and 2 are driven are com
i puted in a similar manner. This algorithm is convenient for machine analysis of
; six and eight port bridge microwave circuits which are not ideal in the general
case (ring configuration, a dual loop bridge), which can be represented in the
~ form of a closed ring of elementary fourport networka with distributed orlumped
~ elements.
i
A multisection divider (Figure 24.3c) has a greater working bandwidth than the
circuits considered here. The number of sections in dividers which are used in
, practice does not exceed tour.
For a divider with uniform power division, the normalized characteristic imped
ances of the quarterwave line secti.ons with respect to pp can be determined from
Table 24.1 as well as the isolating resistors of a nsection device as a function
 453 
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of the requisite frequency coverage factor k= fg/fg [fupper/flowerl and the
maximum values of the working characteristics at the edge of the band: the SL:'R of
inputs 0, 1 and 2, and the isolation of the channels.
TABLE 24.1
i
a.
a
'
ee
~
a~
a
rs
a~
:
a
~
d
~
a
O
d
a
i
x
~
c
o
c
~
2


1,86.
5,32


1.67
1,2
36,6
1,007
1,036
1,8
2


1,96
4,82


1,64
1,22
27.3
1.021
1.106
2.a
3

1,9
3,75�
10,0

1,8
1,91
1,11
1
38,7
27
9
1,015
038
1
1,029
1
105
24
0
3
3

2,14
4,23
8,0

1,74
1;41
.15
,
,
,
,
4
~2,06
3,45
5.83
.
9,64
1,8
1,59
1,3
1,16
26,8
1,039
1,1
4,0
Key: 1.
2.
3.
4.
5.
6.
Rba1.4, ohms;
Rba1.3, ohme;
p4, ohms;
C121 dB;
KCT 1, 2= 5WR1, 2;
SWRp.
Dual loop directional couplers (Figure 24.3d) are quadrature bridge configurations.
When the 0 input is excited, the power is divided between outputs 1 and 2 in a
ratio of m, where at the center working frequency, the output 2 signal lags the
output 1 signal by p/2 in phase. When the device is used as a power adder, the
signal at input 2 should be fed in with a phase lead of p/2 with respect to input 1.
The characteristic impedances of the quarterwave line sections with loads equal to
pp are chosen from the relationship:
pt= Po I/ M ; Px= PoY mAm+l) . (24.11)
In practical circuits, m does not usually exceed 3 4 because of the rechnolog
ical difficulties of fabricating lines with a high characteristic impedance.
The analysis of the operating characteristics of a dual loop bridge is made using
formulas (24.9) and (24.10). Dual channel dividers based on loop bridges have
a snealler bandwidth than ring configurations.
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Dividers using lumped elements are anlogs of devices using quarterwave sections
(Figure 24.3b, e). Their electrical parameters, the capacitances and inductances
of I[ and T section filters are calculated from the formulas [4]:
L= p/2n f o; C= 1/2n f oP, (24.12)
where p is the characteristic impedance of an equivalent quarterwave transmission
line section; fp is the center frequency.
The analysis of the operating characteristics is also made using formulas (24.9)
and (24.10).
A directional coupler using coupled lines takes the form of a quadrature bridge,
the characteristics of which are determined by the parameters of the coupled lines.
The electrical design and analysis of the working characteristics of such devices
are given in Chapter 23.
Dual channel dividers, designed around directional couplers using coupled lines,
have the greatest working bandwidth of all of the devices considered here. How
ever, there are serious technological difficulties with the realization of
dividers with strong side coupling because of the strict tolerances for the
dimensions of the striplines.
24.4. The Calculation of the Electrical Parameters and Characteristics of Multi
Channel Power Distribution Systems
Nchannel distribution systems using quarterwave transmission line sections
(Figure 24.2b) realize the requisite power distribution at the outputs if the
characteristic impedances of the line sections are determined by the expression[2].
pt ='v Rt Ro/Pt ,
(24.13)
where Rp is the internal resistance of the generator; Ri is the load resistance of
the ith output; P.i is the normalized power at the ith output and N
~ Pj 1.
rm i
To achieve ideal matching of all inputs and isolation between the channels at a
fixed frequency, the transmisaion matrix [Ai] of the isolating fourport networks
should have the form:
l~
1. Pi /l t A� i R
/~2i 1 ~ I'i Rl pu P ' R~
(,qtl  V t
0 Azs iV Pi P,1P1 Rt (2.14)
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These fourport networks can be realized in the form of the circuit depicted in
Figure 24.4b, as well as its simpler variants, obtain by eliminating or shorting
one or two of the Rbal resistances. The impedances of the quarterwave sections,
pli and p2i, as well as the resistances Rbal i are determined by means of
equating the transmission matrix of the fourport network:; and the [Ail matrix
(Figure 24.14). For example, for a circuit with one resistor Rbal li:
Pl Pi._
R6an it; = Rt; Pii = Pze �
" Vpi R1
The i.mpedances p'1, p'2 and p2i can be chosen arbitrarily in a range of 25 to 150
ohms.
In the special case of uniform power distribution among the channels (P1 = P2 =
= PN = 11N) and identical loads Rp = R1 = Ri = RN = pp, the character
istics impedances of thb line sections and the isolating resistances are equal and
are determined by the equations: pl = Pi  PN  POvrN, Rbal 11 = Rbal 12 =
= Rbal 1N = Pp� In this case, the resistances Rbal 2i, Rbal 3i and the
line section impedances pli and P2i are absent and the circuit of the distributor
is the most wideband circ`uit of the multichannel dividera of this class.
It is expedient to perform a computer analysis of the operating characterstics
of Nchannel distribution systems using quarterwave impedance transformers by
means of reducing the multiport network to a sixport network and then make use
of expressions (24.9)  (24.10).
The calculation of the parameters of multichannel systems (Figure 24.2a, c) re
duces to the determination of the power division factors of each individual two
channel divider and the subsequent calculation of the parameters of its compo
nents in accordance with the recommendations of �24.3.
In series type dividers (Fi.gure 24.2a), the power division factor of the individ
ual branches are defined by the expression:
' ~ .
me =  ~ pk' (24.15)
P~ k.~t{1
In particular, with uniform power distribution among the N outputs, the division
factors of twochannel dividers incorporated in a chain circuit configuration are
equal to:
m;= N1, i= 2,..., N1. (24.16)
_Q~
Chain type circuits are conveniently realized using quadrature bridges. In this
case, the signals at the outputs are made iYrphase by inserting phasing sections
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with a length of 3a/4 between the adjacent chain components. Where dual loop
bridges with striplines are used, the overall number of circuit outputs is
limited: N< 10. When directional couplers with coupled lines are used, one can
construct circuits with a greater number of channels. However, difficulties drise
in this case with the realization of the last assemblies of the device hav ing a
factor m close to unity.
For 2nchannel parallel systems (Figure 24.2c), the power division factor of each
ith branch of the jth stage is determined by the expression:
a c!t/2>= �e
mIJ ?i Pk I }J . pk, (24.17)
ka (~~)+1 k�o (11/2)F.1
where
_ j I, 2.... ; n; t 1, 2,..., 211 ; a=2^1+1,
In the case of uniform power distribution, all dual channel dividers have a
division factor of m= 1 among the outputs of a.binary system and are quite well
realized using ring and dual loop configurations. .
The operating characteristics of Nchannel binary divider are defined in terms of
Sparameters in accordance with (24.1) (24.3). In this case, the scattering of
the device can be obtained by topological or matrix methods [3]. Ia the case of
large values of N, iterative methods of calculating complex microwave networks
are more optimal from the viewpoint of the efficient utilization of the'immediate
access memory of a computer, and sometimes also the machine time as well.
It should be noted that the recommendations given here for the electrical design
of microwave power distributors are valid for systems with low dissipative losses.
In the shortwave portion of the microwaue band, besides taking lossea into
account, it is also necessary to estimate the impact of inhrnnogeneities in T or
Y configuration lines, the bending of a line, etc. [014, 5].
24.5. An Approximate Design Procedure for Power Diatribution Systems
The design calculations for the parameters of a microwave power distribution
system can be broken down into the electrical and the structural design calcula
t xon s .
Initially, by working from the specific requirements placed on the number of
channels and the working frequency band, the structural configuration of the
device is selected taking the recommendations of �24.4 into account. Thereafter,
in the course of the electrical design calculations, the parameters of the
equivalent circuits are determined: the characteristic impedances of the line and
the lumped elements. For branched series and parallel type systems (Figure
24.2a, c), the power division factors of the constituent assemblies are calculated
beforehand, and then the parameters of the dual channel dividers.
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After calculating the electrical parametera of the circuit, the scattering matrix
is drawn up and the operating characteristies of the device within the frequency
barid are calculated. To establish the production process tolerances, the working
characteristics are studied where scatter is present in the parameters of the
circuit components: differences of the load and characteristic impedances from
the normal values, as well as in the line lengths and parameters of the lumped
elements. In the case of unsatisfactory results, the circuit parameters must be
optimiied, and possibly also the circuit structure.
The calculated parameters of equivalent circuits serve as the initial data for the
structural design. In this design stage, the material .and dimensions of the
circuit substrate are selected, and the structural dimensions of the transmission
lines and lumped film elements are calculated (aee [014, 015, 6], as well as
Chapter 8 in this book).
The concluding step in the design is that of working out the circuit topology.
,
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BIBLIOGRAPHY
Main Literature
01. Markov G.T., Sazonov D.M., "Antpnny" ["Antennas"], Moscow, Energiya Publishers,
1975.
02. Kyun R., "Mikrovolnovyye antenny" ["Microwave Antennas"], Translation from
the German. edited by M.D. Dolukhanova, Moscow, Sudostroyeniye Publishers,
1967. .
03. "Skaniruyushchiye antennyye sistemy SVCh in 3kh t." ["Microwave Scanning
Antenna Systems, in Three Volumes"], Translation from the English edited"by
R. Khansen, Moscow, Sovetskoye Radio Publishers, 19661970.
04. "Antennyye reshetki: Obzor zarubezhnykh rabot" ["Antenna Arrays: A Review of
Foreign Literature"], edited by L.S. Benenson, Moscow, Sovetskoye Radio Publi
shers, 1966.
05. "Antenny i usttoystva SVCh: Raschet i proyektirovaniye antennykh reshetok i
ikh izluchayushchikh elementov" ["Microwave Antennas and Devices: The Design
Calculations and Planning of Antenna Arrays and Their Radiating Elanents"],
Edited by D.I. Voskresenskiy, Moscow, Sovetskoye Radio Publishers, 1972. 06. Drabkin A. D. , Zuzenko V. L. , Kislov A. G. "Ant ennof idernyye u stroystva"
["Antennas and Feedlines"], Moscow, Sovetskoye Radio Publishers, 1974.
07. Zhuk M.S., Molochkov Yu.B., "Proyektirovaniye antennofidernykh ustroystv v
2kh t." ["The Design of Antennas and Feedlistes, in Tao Volumes"], Moscow,
Energiya Publishers, 1966, Vol 1; 1973, Vol 2.
08. Amitey N., Ga,lindo V., Vu Ch., "Teoriya i analiz fazirovannykh antennykh
reshetok" ["The Theory and Analysis of Phased Antenna Arrays"], Translation
fram the English edited by G.T. Markov, A.F. Chaplin, Moscow, Mir
Publishers, 1974.
09. TIIER [PROCEEDINGS OF THE IEEE], 1968, Vol 56, No 11, "Antennyye reshetki s
elektricheskim skanirovaniyem" ["Electrical Scanning Ahtenna Arrays"].
010. TRUDY MAI [PROCEEDINGS OF MOSCOW AVIATION INSTITUTE], 1964, No 159, "Skanir
uyushchiye antenny" ["Scanning Antennas"], Edited by L.N. Deryugin.
Oll. TRUDY MAI, 1973, No 2749 "Mikrovolnovyye skaniruyushchiye antenny" ["Microwave
Scanning Antennas"], Edited by D.I. Voskresenakiy.
012. Vendik O.G,, "Antenny s nemekhan iche skim dvizheniyem lucha" ["Antennas with
Nonmechanical Beam Steering"], Moscow, Sovetskoye Radio Publishers, 1955:
013. "Phased Array Antennas", Edited by A.A. Oliner, G.H. Knittel, Dedham, Artech
House, 1972.
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014. Maloratskiy L.G., "Milaaminiatyurizatsiya elementov i ustroystv SVCh" ["The
Microminiaturizat ion of Microwave Components and Devices"], Moscow, Sovetskoye
Rad io Publ isher s, 1976. '
015. Maloratskiy L.G.,Yavich L.R., "Proyektirovaniye i raschet SVCh elementov
na poloskovykh liniyakh" ["The Planning and 13.esign� Calculations of Microwave
Stripline Components"], Moscow, Sovetskoye Radio Publishers, 1972.
For Chapter 2
Voskresenskiy D.I., Ponomarev L.I., Filippov V.S., "Vypuklyye skaniruyushchiye
antenny" ["Convex Scanning Antennas"], Moscow, Sovetskoye Radio Publishers,
1978.
2. Vorob'yev V.V., "Ustroystva elektronnogo upravleniya luchom FAR" ["Electronic
Beam Steering Devices for Phased Antenna Arrays"], ZARUBEZHNAYA RADIOELEKTRON
IKA [FOREIGN RADIOELECTRONICS], 1976, No.l,. pp 68108.
For Chapter 3
1. "Skaniruyushchiye antennyye sistemy SVCh v 2kh t." ["Microwave Scanning
Antenna Systems, in 'Itao. Volumes"], Translation from the English edited by
G.T. Markov, A.F. Chaplin, Moscow, Sovetskoye Radio Publishers, 1966, Vol 1;
1969, Vol 2. 2. Shnikin H., "Electronically Scanned Antennas", MICROWAVE J., 19609 No 121,
pp 6772, 1961, No 1, pp 5764.
For Chapter 4
l. Voskresenskiy D.I., Ponamarev L.I., Filippav V.S., "Vypuklyye skaniruyushchiye
antenny" ["Comvex Scanning Antennas"], Moscow, Sovetskoye Radio Publishers,
1978.
2. Voskresenskiy D.I., "Kommutatsionnaya antenna s shirokugol'nym elektr icheskim
skanirovaniyem" .["A Switched Antenna with Wide Angle Electr ical'Scanning"],
IZV. VUZOV SSSR. RADIOTEKHNIKA [PROCEEDINGS OF THE HIGHER EDUCATIONAL INSTI
TUTES OF THE USSR. RADIQ ENGINEERING], 1963, Vo1 6, No 6, pp 688694.
3. Voskresenskiy D.I., Gudzenko A.I., "Diapazonnost.1 ostronapravlennykh dugovykh
antennykh reshetok" ["Bandwidth of Pencil Beam Arc Antenna Arrays"], IZV.
VUZOV SSSR. RADIOELEKTRONIKA [PROCEEDINGS OF THE HIGHER EDUCATIONAL INSTITUTES
OF THE USSR. RADIOELECTRONICS], 1968, Vol 11, No 5, pp 441451.
4. Stark J.L., Bell C.V., Notest R.A., et al., "Microwave Camponents for Wideband
Phased Arrays", PROC. IEEE, 1968, Vol 56, No 11, pp 1908,1923.
5. Bogolyubav V.N., Yeskin A.V., Karbovskiy S.A., "Upravlyayemyye f erritovyye
ustroystva SVCh" ["Controllable Microwave Ferrite Devices"], Moscow,
Sovetskoye Radio Publishers, 1972 (Elementy radioelektronnoy apparatury)
[(Rad ioelectronic Equ ipment Camponents)].
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6. Goebels F.G., Forman B.J., Nonnemaeker C.H., "Electronic Scanning of Linear
Slot Arraps Using Diode Iries [sic]", TRANS. IEEE, 1968, Vol AP16, No 1,
pp 8 14 .
7. Khardman, "Razvitiye RLS s fazirovannoy antennoy reshetkoy za posledneye
desyatiletiye" ["The Development of Radars with a Phased Antenna Array over
the Last Decade"], ZARUBEZANAYA RADIOELEKTRONIRA [FOREIGN RADIOELECTRONICS],
1971, No 1, pp 3958.
For Chapter 5
l. Pistol'kors A.A., "Obshchaya teoriya diffraktsionnqkh antenn" ["The General
Theory of Diffraction Antennas"], ZhTF [JOURNAL OF TECHNICAL PHYSIC3], 1944,
Vol 14, No 12, pp 693702; 1946, Vol 16, No 1, pp 310.
2. "Posobiye po kursovomy proyektirovaniyu antenn" ["Textbook on Course Required
Antenna Design Work"], VZEIS [AllUnion Correspondence Electrical Engineering
Inst itute f or Communicat ions] , Moscow, 1967.
3. YatsukL.P., Smirnwa N.V., "Vnutrenniye provod3mosti nerezonansnykh shcheley
v pryamougol'nom volnavode" ["Internal Admittances of Nonresonant S1ots in
a Rectangular Waveguide"], IZV. WZOV SSSR. RADIOTEKHNIKA [PROCEEDINGS OF THE
HIGHER IDUCATIONAL INSTITUTES OF THE USSR. RADIO ENGINEERING], 1967, Vol 40,
No 4, pp 359369.
4. Veshnikova I.Ye.,' Yevstrogov G.A., "Teoriya soglasovannykh shchelevykh
izluchatele:;r "["Matched Slotted Waveguide Theory"], RADIOTEKHNIKA I
ELEKTRONIKA [p,ADIO ENGINEERING AND ELECTRONICS], 1965, Vol 10, No 7, pp 1181
118 9.
5. Xevstropov G.A., Tsarapkin S.A., "Issledovaniye volnovodnoshchelevykh antenn s
s identichnymi rezonansnymi izluchatelyami" ["A Study of Slotted Waveguide
Antennas with Identical Resonant Radiators"], RADIOTEKHNIKA I ELEKTRONIKA,
1965, Vol 10, No 9, pp 16631671.
6. Yevstropov G.A., Tsarapkin S.A., "Raschet volnovodnoshchelevykh antenn s
uchetom vzaimodeystviya izluchateley po osnovnoy vo:l.ne" ["The Design of Slotted
Waveguide Antennas Taking into Account Daminant Mode Mutual Coupling of the
Radiators"], RADIOTERHNIKA I ELEKTRONIKA, 1966, Vol 11, No 5, pp 822830.
7. Shubarin Yu.V., "Antenny sverkhvysokikh chastot" ["Microwave Antennas"],
Khar'kov, State University, 1960.
8. Shirman Ya.D., "Radiovolnovody i ob"yemnyye rezonatory" ["Radio Wavegui,des and
Spatial Resonators"], Moscow, Svyaz' Publishers, 1959.
9. Reznikov G.B., "Samoletnyye antenny" ["Aircraft Antennas"], Moscow, Sovetskoye
Radio Publishers, 1962.
_45],..
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For Chapter 6.
1. Yershov L.I., Kremenetskiy S.D., Los' V.F., "Flektrodinamika vzaimovliyaniya
v nerezonansnykh volnovodnoshchelevykh reshetkakh" ["The Electrodynamics of
Mutual Coupling in Nonresonant Slotted Waveguide Arrays"], IZV. WZOV SSSR,
RADIOELEKTRONIKA, 1978, No 2, pp 4854.
2. Los' V.F., Kosmodamianskaya N.S., "Metod rascheta amplitudnofazovogo
raspredeleniya polya v raskryve volnovodnoshchelevykh reshetok s uchetom
vnutrennego vzaimodeystviya izluchateley" ["Method of Calculating the Ampli
tudePhaseDistribution of the Field in the Aperture of Slotted Waveguide
Arrays Taking Internal Mutual Coupling of the Radiators into Account"],
"Antenny" ["Antennas"], Edited by A.A. Pistol'kors, Moscow, Svyaz' Publishers,
1969, No 5, pp 2432.
3. Baktrakh L.D., Yershov L.I.,. Kremenetskiy S.D., Los' V.F., "Elektrodinamiches
kiye faktory vzaimovliyaniya i raschet volnovodnoshchelevykh reshetok" ["Elec
trodynamic Factors of Mutual Coupling and the Design of Slotted Waveguide .
Arrays"], DAN SSSR [REPORTS OF TfiE USSR ACADEMY OF SCIENCES], 1978, Vol 243,
No 2, pp 314317. 4. Repin V.M., "Difraktsiya elektramagnitnykh poley na sisteme shcheley" ["The.
Diffraction of Electromagnetic Fields in a System of Slots"], VYCHISLITEL'NYYE
METODY I PROGRAMMIROVANIYE [COMPUTER METHODS AND PROGRAMMING], Moscow State
University, 1968, No 16, pp 112121.
5. Yatsuk L.P., Zhironkina A.V., Katrich V.A., "Vozbuzhdeniye pryamougol'nogo
volnovoda naklonnoy i krestoobraznoy shchelyami" ["Excitation of a Rectangular
Waveguide with Oblique and CrossShaped Slats"], "Antenny", Edited by A.A.
Pistol'kors, Moscow, Svyaz' Publishers, 1975, No 22, pp 4660.
6. Fel'd A.N., Benenson L.S., "Antennofidernyye ustroystva v 2kh ch." ["Antennas
and Feedlines; in Two Parts"], Moscow, WIA [Air Force Engineering Academy],
1959, Part II.
7. Markov G.T., "Vozlxizhdeniye pryamougol'nogo volnovoda" ["Excitation of a
Rectangular Waveguide"J, TRUDY MEI [PROCEEDINGS OF MOSCOW POWER ENGINEERING
ISNTITUTE], 1956, No 21, pp 1634.
8. Fridberg P.Sh., Garb Kh.L., Levinson I.B., "Uchet tolshchiny stenki v
shchelevykh zadachakh elektrodinamiki" ["Taking Wall Thickness Into Account in
Slot Problems of Electrodynamics"], RADIOTEKHNIKA I ELEKTRONIKA, 1968, Vol 13,
No 12, pp 21522161.
9. Yevstropov G.A., Tsarapkin S.e.., "Raschet volnovodnoshchelevykh antenn s
uchetom vzaimodeystviya izluchateley po osnovnoy volne" ["The Design of Slotted
Waveguide Antennas Taking Dominant Mode Mutual Coupling of the Radiators into
Account"], RADIOTEKHNIKA I ELEKTRONIKA, 1966, Vol 2, No 5, pp 822830.
10. Breithaupt R.W., Ma.cormick G.T., "Traveling Wave Arrays of Mismatched Elements";
TRANS. IEEE, 1971, Vol AP19, No 1, pp 411.
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11. Bakhrakh L.D., Kremenetskiy S.D., "Sintez izlucbayushchikh sistem" ["The
Design of Radiating Systems"], Moscow, Sovetskoye Radio Publishers, 1974.
12. bayliz S.Yu., Akishin B.A., "Issledovaniye skhemy zameshcheniya naklonno
smeshchennogo volnavodnoshchelevogo izluchatelya" ["Study of the Equivalent
Circuit of an Obliquely Displaced Slotted Waveguide Radiator"], in the book,
"Antenny i SVCh uzly radiotekhnicheskikh ustroystv" ["Antennas and Microwave
Assemblies for Radio Electronic Equipment"], Sverdlovsk, 1976, pp 1623.
For Chapter 7.
1. Schwartzman L., Stangel, J., "The Dome Antenna", MICROWAVE J., 1975, Vol 18,
No 10, pp 3134.
2. Voskresenskiy D. I. , Ponomarev L. I. , Filippov V. S. ,"Vypuklyye skaniruyushchiye
antenny" ["Convex Scanning Antennas"], Moscow, Sovetskoye Radio Publsihers,
1978.
3. Voskresenskiy D.I., "Ostronapravlennoye izlucheniye s vypuklykh poverkhnostey"
["PencilBeam Radiation from Convex Surfaces"], IZV. VUZOV SSSR. RADIOTEKHNIKA,
1964, Vol 7, No 3, pp 276282.4. Yamaykin V.Ye., "Optimizatsiya amplitudnogo raspredeleniya na kruglom sinfaznom
raskryve s zatenennoy tsentral'noy oblast'yu" ["Optimization of the Amplitude
Distribution in a Circular Inphase Aperture with a Shaded Central Region"],
_ IZV. VUZOV SSSR, RADIOELEKTRONIKA, 1969, Vol 12, No 6, pp 578599.
5. Yamaykin V. Ye. ,"Opt imizats iya per ioda FAR" ["Opt 3mizat ion of the Per iod of a
Phased Antenna Array"], "Antenny", Edited by A.A. Pistol'kors, Moscow, Svyaz'
Publishers, 1975, Vol 22, pp 2035.
For Chapter 9.
1. Munson R.E., "Conformal Microstrip Antennas and Microstrip PYiased Arrays",
TRANS. IEEE, 1974, Vol AP22, pp 7448.
2. Anders G., Derneryd, "Linearly Polarized Microstrip Antennas", TRANS. IEEE,
1976) Vol AP24, No 11, gp 846851.
3. Tiuri M., Tallqir'st S., Urpo S., "Chain Antenna", Int. IEEE APs Symposium
Prograiren and Dig. [ sic] , Atlanta, Ga., New Youk, 1974, pp 274277.
4. 4la1ter K., "Antenny begushchey volny" ["Traveling Wave Antennas"], Translation
fram the English edited by A.F. Chaplin, Moscow, Energiya Publishers, 1970.
5. Tokumaru Shinobu, Shibacaki Taro, "Phased Arrays, Composed of Parallel Fed
Two Element Dipoles in a Rectangular Arrangement", TRANS. INST. ELECTR. AND
 COMM. ENG. JAP., 1976, Vol 156B, No. 11, pp 521528.
6. Stark L., "Radiation Impedance of a Dipole in an Infinite Planar Phased Array",
RADIO SCIENCE, 1966, Vol 3, pp 361375.
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7. Chang V.W.H., "Infinite Phased Dipole Array", PROC. IEEE, 1968, Vol 56, No 11,
pp 10681070.
8, Galejt T., "Excitation of Slots in a Conducting Screen Above a Lossy Dielectric
Half Space"], TRANS. IRE, 1962, Vol AP10, pp 443443 [sic].
For Chapter 10.
1. Indenbom M.V., "Algoritm analiza i optimizatsii direktornykh izluchateley v bes
konechnoy ploskoy antennoy reshetke" ["Algorithm for the Analysis and Optimization
of Yagi Radiators in an Infinite Planar Antenna Array"], INF. LISTOK/VIMI,
[INFORMATION SHEET OF THE VIMI], Series ILT91311, 1980, No 800599.
2. Indenbom M.V., Filippov V. S. ,"Analiz i optimizatsiya direktornykh izluchateley
* ploskoy antennoy reshetke" ["Analysis and Optimization of Yagi Radia.tors in
a Planar Antenna Array"], IZV WZOV SSSR. RADIOELEKTRONIKA, 1979, No 2, pp 3441.
3. Wa"Lter K., "Antenny begushchey volny" ["Traveling Wave Antennas"], Translation
fY om the English edited by A.F. Cha.plin, Moscow, Energiya Publishers, 1970.
4. Polak E., "Chislennyye metody optimizatsii" ["Numerical Optimization Methods"],
Moscow, Mir Publishers, 1974.
5. Ganston M. A. R. ,"Spravochnik po volnovym soprotivleniyam f idernykh liniy SVCh"
["Handbook on the Characteristic Impedances of Microwave Feedlines"], Moscow,
Svyaz' Publishers, 1976.
6. U.S. Patent 3845490, NKI 343821.
7. Vay Kaychen', "Teoriya i proyektirovaniye shirokopolosnykh soglasuyushchikh
tsepey" ["Theory and Design of Broadband Matching Networks"], Moscow, Svyaz'
~i Publishers, 1979. .
For Chapter 11.
1. Titov A.N., Sapsovich B.I., "Fazirovannaya re9hetka..kak antennaya sistema s
iskusstvennym dielektrikam" ["A Phased Array as an Antenna System with an
Artificial Dielectric"], "Antenny", Edited by A.A.Pistol'kors, Moscow, Svyaz'
Publishers, 1970, No 8, pp 6780.
2. Nittel' G., Khessel' A., Oliner A., "Nulevyye provaly v diagrarnne napravlennosti
elementa fazirovannoy antennoy antennoy reshetki i ikh svyaz' s napravlennymi
volnami" ["Null Dips in the Directional Pattern of an Element of a Phased
Anteniia Array and Their Relationship to Directed Waves"], TIIER [PROCEEDINGS OF
THE IEEE], 1968, Vol 56, No 11, pp 7188. . '
3. Elenberger A., Shvartsman L., Topper L., "Nekotoryye trebovaniya k geometrii
volnovodnykh reshetok s lineynoy polyarizatsiyey" ["Some Requirements Placed on
the Geometry of Waveguide Arrays with Linear Polarization"], TIIER, 1968, Vol
56, No 11, pp 116128.
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4, Borzhiotti G., "Analiz periodicheskoy ploskoy fazirovannoy reshetki metodom
sobstvennykh voln" ["Eigenmode AnaYysis of a Periodic Planar Phased Array"],
TIIER [PROCEEDINGS OF THE IEEE], 1968, Vol 56, No 11, pp 132150.
5. Cha.plin A.F., Khzmalyan A.D., Ryakovskaya M.L., "Priblizhennyy spektral'nyy
analiz bol'shikh antennykh reshetok" ["Approximate Spectral Analysis of Large
Antenna Arrays"], Moscow, Vysshaya Shkola Publishers, 1980, Issue 3, pp 101121.
For Chapter 12.
1. Lee S.W., Jones W.R., "On the Suppression of the Radiation Nulls and Broadband
Impedance Matching of Rectangular Waveguide Phased Arrays", TRANS. IEEE, 1971,
Vol AP19, No l, pp 4151.
~ 2. Sushkevich V.I., "Neregulyarnyye lineynyye volnovodnyye sistemy" ["Irregular
~ Linear Waveguide Systems"] , Moscow, Sovetskoye Radio Publishers, 1967.
3. Fel'dshteyn A.L., Yavich L.R., Smirnov V.P., "Spravochnik po elementam
volnovodnoy tekhniki" ["Handbook on Waveguide Equipment Camponents"], Moscow,
Sovetskoye Radio Publishers, 1967.
 For Chapter 13.
1. Fel'd Ya.N., "Shchelevyye antenny" ["Slot Antennas"], Moscow, Sovetskoye Radio
Publishers, 1948.
2. "Vychislitel'nyye Metody i Programmirovaniye" ["Computer Methods and Programm"
ing"], Moscow State University, Moscow, 1973, Issue 20.
3. I1'inskiy A.S., Grinev A.Yu. Kotov Yu.V., "Issledavaniye elektrodinamicheskikh
kharakteristik rezonatornoshchelevogo izluchatelya s istochnikami vozbuzhdeniya
v ploskosti shcheli" ["Study of the Electrodynamic Characteristics of a Slotted
Kesonator Radiatcr with the Excitation Sources in the Plane of the Slot"],
RADIOTEKHNIKA I ELEKTRONIKA, 1978, Vol 23, No 5, pp 922930.
4. Gr inev A.Yu., I1'inskiy A.S., Kotov Yu.V., "Kharakter istiki skanirovaniya
rezonatornoshchelevoy periodicheskoy antennoy struktury s dielektricheskim
pokrytiyem" ["The Scanning Characteristics of a Slot Resonator Periodic Antenna
Structure with a Dielectric Coating"], IZV WZOV SSSR. RADIOTEKHNIKA, 1978,
Vol 21, No 12, pp 18221833.
5. Grinev A.Yu. Kotov Yu.V., "Mashinnyy metod analiza i chastichnogo parametri
cheskogo sinteza rezonatornoshchelevykh antennykh struktur" ["Computer Method
for the Analysis and Partial Parametric Synthesis of Slotted Resonator Antenna
Structures"], IZV. WZOV SSSR. RADIOELEKTRONIKA, 1978, Vol 21, No 2, pp 3035.
6. Kotov Yu.V., "Issledovaniye elektrodinamicheskikh kharakteristik rezonatorno
shchelevykh struktur" ["Study of the Electrodynamic Characteristics of Slotted
Resonator Structures"], CHISLENNYYE METODY ELEKTRODINAMIKI [NUMERICAL METHODS
OF ELECTRODYNAMICS], Moscow State University, Moscow, 1978, Issue 3, pp 2640.
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For Chapter 14.
= 1. "SVCh ustroystva na poluprovodnikovykh diodakh: Proyektirovaniye i raschet"
["Microwave Devices Using Semiconductor Diodes: Design Calculations and
Planning"], Edited by I.V. Mal'skiy, , B.V. Sestroretskiy,. Moscow, Sovetskoye
Radio Publishers, 1969.
2. Vlasov V.I., Berman Ya.I., "Proyektirovaniye vysokochastotnykh uzlov radio
lokatsionnykh stantsiy" ["The Design of the Radio Frequency Assemblies of
Radars"], Lenino�~ad, Sudpromgiz Publishers, 1961.
 3. USSR Patent No. 358740, Published in Bulletin No. 34, 1972.
4. Voskresenskiy D.I., Mikheyev S.M., Popov V.V.,"Rommutatsionnaya skaniruyushchaya
poluprovodnikovaya antennaya reshetka" ["Switched Semiconductor Scanning Antenna
Array"], TRUDY MAI [PROCEEDINGS OF MOSCOW AVIATION INSTITUTE], 1973, No 274,
pp 515.
5. Popov V.V., "Issledovaniye razbrosa parametrov elementov izluchatelya antennoy
reshetki" ["Study of the Scatter in the Parameters of the Elanents of an
Antenna Array Radiator"], TRUDY MAI, 1973, No 274, pp 7990.
6. Kanareykin D.B., Pavlov N.F., Potekhin V.A., "Polyarizatsiya radiolokatsionnykh
 signalov" ["Radar Signal Polarization"], Moscow, Sovetskoye Radio Publishers,
1966.
7. Fel'dshteyn A.L., Smirnov V.P., Yavich L.R., "Spravochnik po elementam
volnovodnoy tekhniki" ["Handbook on Waveguide Equipment Camponents"] , Moscow,
Sovetskoye Radio Publishers, 1967.
For Chapter 16.
 1. Aronov V.L. , Mazel' Ye. Z. ,"Sovremennoye sostoyaniye v oblasti razrabotki
 moshchnykh VCh i SVCh tranzistorov" ["The State of the Art in the Development
of High Frequency and Microwave Transistors"], in the book, "Poluprovodnikovyye
pribory i ikh primeneniye" ["Semiconductor Devices and Their Applications"],
Edited by Ya.V. Fedotova, Moscow, Sovetskoye Radio Publishers, 1971, No 25,
: PP 729.
2. Kaganov V.I., "Tranzistornyye radioperedatchiki" ["Transistorized Radio Trans
mitters"]., Moscow, Energiya Publishers, 1976.
3. "Radioperedayushchiye ustroystva na poluprovodnikovykh pribarakh" ["Radio
Transraitting Equipment Using Semiconductor Devices"] , Edited by R.A. Valitov,
and I.A. Pooov, Moscow, Sovetskoye Radio Publishers, 1973.
4. Chelnokov O.A., "Tranzistornyye generatory sinusoidal'nykh kolebaniy" ["Trans
istorized Sine Wave Generators"], Moscow, Sovetskoye Radio Publishers, 1975.
5. "Proyektirovaniye radioperedayusychikh ustroystv SVCh"
Radio Transmitting Equipment"], Edited by G.M. Utkin,
Publ isher s, 1979.
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6. Kiyko G.I., Lib Yu.H., et al., "Issledovaniye shirokopolosnogo tranzistornogo
usilitelya moshchnosti s raspredelennymi parametrami" ["Study of a Broadband
Transistorized Power Amplifier with Distributed Parameters"], "Poluprovodni
kovyye pribory v tektmike elektrosvyazi" ["Semiconductor Devices in Electrical
Communications Equipment"], Moscow, Svyaz' Publishers, 1975, No 15, pp 1926.
For Chapter 17. '
1. "Radioperedayushchiye ustroystva na poluprovodnikovykh prihorakh" ["Radio Trans
mitting Equipment Using Semiconductor Devices"], Edited by R.A. Valitov, I.A.
Popov, Moscow, Sovetskoye Radio Publishers, 1973.
2. Koptev G.I., Panina T.A., "Raschet uslilitel'nykh i umnozhitel'nykh kaskadov
transzistornykh peredatchikov" ["The Design of Amplifier and Multiplier Stages
for Transistorized Transnitters"], Moscow, Moscow Power Engineering Institute,
1975.
3. Kaganov V.I., "Tranzistornyye radioperedatchiki" ["Transistorized Radio Trans
mitters"], Moscow, Energiya Publishers, 1976.
4. Petrov B.Ye., Tereshina G.N., "Transistornyye generatory" ["Transistor Oscilla
tors"] , Moscow, MEIT, 1975.
 5. Chelnokov O.A., "Tranzistornyye generatory sinusoidal'nykh kolebaniy" ["Trans
istorized Sine Wave Generators"], Moscow, Sovetskoye Radio Publishers, 1975.
6. Kiyko G.I., Limb Yu.N., et al., "Issledovaniye shirokopolosnogo tranzistornogo.
usilitelya moshchnosti s raspredelennymi parametrami", "Poluprovodnikovyye
pribory v tekhnike elektrosvyazi", Moscow, Svyaz' Publishers, 1975, No 15,
pp 1926.
7. Granovskaya R,A., Petrov S.B., "Proyektirovaniye SVCh tsepey tranzistornykh
~ generatorov s vneshnim vozbuzhdeniyem, vypolnyayemykh v vids gibridnykh
integral'nykh skhem" ["The Design of Microwave Networks of Transistorized,
, Externally Excited Oscillators/Amplifiers Made in the Form of Hybrid Integrated
Circuits"], Moscow, Moscow Avi.ation Institute, 1977.
8. Sobol G., "SVCh primeneniya tekhnologii integral'nykh skhem" ["Microwave
Applications of Integrated Circuit Technology"], in the book, "Poluprovodnikovyye
Pribory SVCh" ["Semiconductor Microwave Devices"], Edited by F. Brand, Moscow,
Mir Publishers, 1972, pp 8396.
9. Grey P., Grekhem R., "Radioperedatchiki" ["Radio Transmitters"], Moscow, Svyaz`
Publishers, 1965.
10. Atabekov G.I., "Osnovy teorii tsepey" ["Principles of Network Theory"] , Moscow,
Energiya Publishers, 1969.
11. Rayev M.D., Sttvarts N.Z., "Soglasovsniye kompleksnykh soprotivleniyy v SVCh
mikroelektronike" ["Matching Complex Impedances in Microwave Microelectronics"],
IZV. WZOV ESSR. RADIOELEKTRONIKA, 1972, Vol 11, No 6, pp 728737.
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12. Mattey D.L., Yang L., Dmhons Ye.M., "Fil'try SVCh, soglasuyushchiye tsepi
i tsepi svyazi, v 2kh t." ["Microwav e Filters, Matching Networks and Coup
ling Networks, in Two Volumes"], Moscow, Svyaz' Publisher s, 1971, Vol 1.
For Chapter 18.
 1. Granovskaya R.A., Shkalikov V.N., "Osobennosti pr imeneniya v peredayushchikh
aktivnykh antennykh reshetkakh moduley s umnozheniyem chastoty" ["Specific
Features of the Application of Frequency Multiplier Modules in Activ e Trans
m itting Antenna Arrays"], IZV. WZOV SSSR. RADIOELEKTRONIKA, 1978, No 2,
pp 6973.
2. Vizel' A.A., Pil'don V.N., "Metody.rascheta optimal'nykh parametrov umnozhi
teley chastoty na nelineynoy yemkosti poluprovodnikovykh diodov" ["Methods of
Calculating the Opt3mal Parameter s of Frequency Multipliers Using the Nonlin
ear Capacitance of Semiconductor Diodes"], ELEKTRONIKA I YEYE PRIMENENIYE
[ELECTRONICS AND ITS APPLICATIONS], 1974, Vol 5, No 7, pp 173213.
3. Kaganov V.I., "Tranxistornyye radioperedatchiki" ["Transistorized Radio
Transm itters"], Moscow, Energiya Publishers, 1976.
4. Shkalikov V.N., Iutin E.A., "0 fazovykh.kharakteristikakh varaktornykh
umnozhiteley chastoty" ["On the Phase Character istics of Varactor Frequ ency
Multipliers"], RADIOTEKHNIKA, 1973, Vol 28, No 10, pp 6066.
5. Lut3n E.A., Telyatnikov L.I., Shkalikov V.N., "Fazovyye kharakteristiki
I?vukhkonturnogo umnozhitelya chastoty na diode s nakopleniyem zaryada"
["The Phase Characteristics of a Two Zuned Section Frequency Multiplier Using
a Charge Storage Diode"], RADIOTEKHNIKA, 1975, Vol 30, No 10, pp 5260.
6. "Proyektirovaniye moduley SVCh: Diodnyye generatory, usiliteli i umnozhiteli
SVCh" ["The Design of Microwave Modules: Diode Microwave Os�illators, Ampli
f iers and Multipliers"], Edited by G.P. Zemtsov, Moscow, Moscow Aviation
Institute, 1973 (Summary of Lectures).
7. "Radioperedayushchiye ustroystva na poluprovodnikovykh priborakh" ["Radio
Transmitting Equipment Using Semiconductor Devices"], Edited by R.A.
Valitov and I.A. Popov, Moscow, Sovetskoye Radio Publishers, 1973.
8. Bob Weirather, "Good Microstrip Multipliers Don't Just Happen", ELECTRONIC
DESIGN, 1971, No 3, pp 3639.
For Chapter 19.
1. Tager A.S., Val'dPerlov V.M., "Lav innoproletnyye diody i ikh primeneniye
v tekhnike SVCh" ["Avalanche and Transit Diodes and Their Applications in
Microwave Engineering"], Moscow, Sovetskoye Radio Publishers, 1968.
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2. Kolosov M.V., Peregonov S.A., "SVCh generatory i usiliteli na poluprovodniko
I vykh priborakh" ["Microwave Oscillators and Amplif iers Using Semiconductor
; Devices"], Moscow, Sovetskoye Radio Publishers, 1974.
3. Khaddad G.G., "Printsipy raboty i osnovnyye svoystva LPD" ["Operational
Principles and Major Properties of IMPATT Diodes"], ZARUBEZHNAYA RADIO
ELEKTRONIKA [FOREIGN RADIOELECTRONICS], 1972, No l, pp 7592.
4. "Poluprovodnikovyye pribory SVCh" ["Microwave Semiconductor Devices"],
Edited by F. Brand, Translated from the English, Moscow, Mir Publishers,
1972.
5. "SVCh poluprovodnikovyye pr ibory i ikh pr imeneniye" ["Microwave Saniconductor
_ Devices and Their Applications"], Edited by G. Watson, Translated from the
English, edited by V.S. Etkin, Moscow, Mir Publishers, 1972.
6. "Mikroelel~'Gronika i poluprovodnikovyye pribory" ["Microelectronics and Semi
 conductor Devices"], Edited by A.A. Vasenkov and Ya.A. Fedotov, Moscow,
Sovetskoye Radio Publishers, 1976, No. 1.
7. Bouers H., Midford T., Plants S., "Impatt Diode Multistage Transmission
Amplif iers", TRANS. IEEE, 1970, V. MTT18, No 11, p 943948.
8. Kayl F.N., Midford T.A., "LPD v integral'nom ispolnenii" ["Integrated Cir
cuit IlKPATT Diodes"], TIIER [PROCEEDINGS OF THE IEEE], 1967, Vol 55, No 12,
pp 130132.
9. Magalkhayes F.M., K. Kurrokova, "Perestraivayemyy generator dlya izmereniya
kharakteristik IMPATT diodov" ["Tunable Generator for the Measurement of
IMPATT Diode Characteristics"], TIIER, 1970, Vol 58, No 6, pp 111113.
For Chapter 20.
l. Sobol G., "SVCh primeneniye tekhnologii integral'nykh skhem" ["Microwave
Applications of Integrated Circuit Technology"], in the book, "Poluprovod
nikovyye pribory SVCh" ["Microwave Semiconductor Devices"], Edited by F.
Brand, Moscow, Mir Publishers, 1972, pp 8386.
2. Schneider M.U., "Microstrip Lines for Microwave Integrated Circuits", BELL
SYSTEM TECHNICAL JOURNAL, 1969, Vol 48, No 5, pp 14211444.
3. Sobol G., "Ispol'zovaniye tekYuiiki integral'nykh skhem dlya sozdayniya SVCh
oborudovaniya" ["The Use of Integrated Circuit Hardware for the Design of
Microwave Equipment"], ELEKTRONIKA [ELECTRONICS], Vol 40, No 6, 1967,
pp 3346.
4. Colton M., et al., "SVCh integral'nyye skhemy na elementakh s sosredotochenymi
postoyannymi i perspektivy ikh primeneniya" ["Microwave Integrated Circuits
Using Elements with Lumped Constants and Prospects for Their Application"],
ZARUBEZflNAYA RADIOELEKTRONIKA, 1972, No 4, pp 104123.
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5. Demin V.V., Goreliicov N.I., Gotra Z.Yu., "Plenochnyye mikroskhemy i minia
tyurizatsiya" ["Film Integrated Circuits and Miniaturization"], L'vov,
Kamenyar, 1972.
6. Dolkart V.M., Novik G.Kh., "Konstruktivnyye i elektricheskiye kharakteristiki
mnogosloynykh pechatnykh plat" ["Structural and Electrical Characteristics
of Multilayer Printed Circuit Boards"], Moscow, Sovetskoye Radio Publishers,
(Biblioteka radiokonstruktoray [(Radio Designer's Library)].
For Chapter 21.
1. "SVCh ustroystva na poluprovodnikovykh diodakN' ["Microwave Devices Using
Semiconductor Diodes"], Edited by I.V. Mal'kiy,and B.V. Sestroretskiy, Moscow,
Sovetskoye Radio Publishers, 1969.
2. Mikaelyan A.L., "Teoriya i primeneniye ferritov na SVCh" ["Theory and Appli
cation of Ferrites at Microwave Frequencies"] , Moscow, Energiya PublishF.rs,
1963.
3. "Upravlyayushchiye ustroystva SVCh" ["Microwave Control Devices"], N.T. Bova,
et al., Kiev, Tekhnika Publishers, 1973.
4. Upravlyayemyye ferritovyye ustroystva SVCh" ["Controlled Microwave Ferrite
Devices"], V.N. Bogolyubov, et al., Moscow, Sovetskoye Radio Publishers; 1972.
5. Averbukh M. E. , Bkhlokhin V.N. [ sic] ,*tiroshnichenko A. S. ,"Diskretnyye
mikropoloskovyye fazovrashchateli na pi n diodakh" ["Digital Microstrip Line
Phase Shifters Using PIN Diodes"], "Elektronika" Central Scientific Research
Institute, Moscow, 1976, No. 1.
For Chapter 22.
1. Fel'dshteyn A.L., Yavich L.R., Smirnov V.P., "Spravochnik po elementam
volnavodnoy tekhniki" ["Handbook on Microwave Equipment Components"],
Moscow, Sovetskoye Radio Publishers, 1967.
2. Fel'dshteyn A.L., Yavich L.R., "Sintez chetyrekhpolyusnikov i vos'mipolyus
nikov na SVCh" ["Design of Four and Eight Port Networks for Microwave Fre
quencies"], Moscow, Svyaz' Publishers, 1971.
3. A1'tman D., "Ustroystva SVCh" ["Microwave Devices"], Translated from the
English, Edited by I.V. Lebedev, Moscow, Mir Publishers, 1968.
4. Khanzel L., "Spravochnik po raschety fil'trov" ["Filter Design Handbook"],
Translated from the English, edited by L.Ye. Znamenskiy, Moscow, Sovetskoye
Rad ia Publtshers, 1974.
5. Mattey D1., Yang L., Dzhons Ye.M., "Fil'try SVCh, soglasuyushchiye tsepi i
tsepi svyazi, v 2kh t." ["Microwave Filters, Matching Networks and Coupling
Networks, in Two Volumes"], Moscow, Svyaz' Publishers, 1974.
470
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6. Maloratskiy L.G., "Mikrominiatyurizatsiya elementov i ustroystv SVCh"
["Microminiaturization of Microwave Componeats and Devices"], Moscow,
Sovetskoye Radio Publisherc, 1976.
7. Kozlov V.I., Xufit G.A., "Proyektirovaniye SVCh ustroystv s pomoshch'yu
EVM" ["Computer Assisted Design of Microwave Devices"], Moscow, Sovetskoye
Rad io Publisher s, 1976.
For Chapter 23.
1. Ganston M. A. ."Spravochnik po volnovym soprotivleniyiyam f idernykh liniy
SVCh" ["Handbook on the Characteristic Impedances of Microwave Feedlines"],
Translated from the English, Edited by A.Z. Fradin, Moscow, Svyaz' Publishers,
1976.
2. "Konstruirovaniye i raschet poloskovykh ustroystv" ["Structural and Design
Calculations for Stripline Devices"], Edited by I.S. Kovalev, Moscow,
Sovetskoye Radio Publishers, 1974.
3. "Poloskovyye linii i ustroystva SVCh" ["Microwave Devices and Striplines"],
Edited by V.M. Sedov, Khar'kbv, Vysshaya Shkola Publishers, 1974.
4. Kovalev I. S. ,"Osnovy teorii �i'rasc~heta ustroystv SVCh" ["Principles of the
Theory and Design of Microwave Devices"] , Mfnsk, Nauka i Tekhnika Publishers,
1972.
5. Mattey D.L., Yang L., Dzhons YeM.T., "Fil'try SVCh, soglasuyushchiye tsepi i
tsepi svyazi v 2kh t" ["Microwave Filters, Matching Networks and Coupling
Networks; in Two Volumes"], 7.Yanslation from the English edited by A.V.
Alekseyev and F.V. Kushner, Moscow, Svyaz' Publishers, 1971, 1972 [sic].
6. Fel'dshteyn A.L., Yavich L.R., Smirnov V.P., "Spravoclmik po elementam
volnovodnoy tekhniki", Moscow, Swetskoye Radio Publishers, 1967.
7. Mashkovets B.M., Tkachenko K.A., "Volnovoy metod sinteza odnopetlevykh
napravlennykh f il'trov na poloskovykh" ["The Wave Method of Synthesizing
`Single Loop Directional Filters Using Striplines"], ELEKTROSVYAZ', 1969,
No 6, pp 2128.
8. Goyzhevskiy V.A., Levin A.F., Golovchenko V.G., "Vliyaniye dopuskov na
parametry pechatnykh napravlennykh otvetviteley" ["The Influence of Toleran
ces on the Parameters of Printed Circuit Directional Couplers"], IZV. VUZOV
SSSR. RADIOELEKTRONIKA, 1973, Vol 16, No 3, pp 8995.
9. Shelton J.P.,"Impedances of Off set Parallel Split Transmission Lines",
TRANS. IEEE, 1966, Vol MTT14, No 1, p 713.
10. Metcalf W. S. "Cascading FourPort Networks"], MICROWAVE T. [sic] , 1969, Vol
12, No 9, pp 14 17.
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For Chapter 24.
1. Kaganov V.I., "Tranzistornyye radioperedatchiki" ["Transistorized Radio
Transmitters"], Moscow, Energiya Publishers, 1976.
2. Myakishev .BYa., Solovtsov P.A., "Mnogokanal'nyy SVCh delitel' moshchnosti s
proizvol'nym amplitudnym raspredeleniyem na vykhodakh" ["Multichannel
Microwave Power Divider with an Arbitrary Amplitude Distribution at the
Outputs"], IZV. WZOV SSSR. RADIOELEKTRONIRA, 1978, No 2, pp 118121.
3. Silayev M.A., Bryantsev C.F., "Prilozheniye matrits i grafov k analizu
SVCh ustroystv" .["The Application of Matrices and Graphs to the Analysis of
Microwave Devices"], Moscow, Sovetskoye Radio Publishers, 1970.
4. Tsarenkov V.S., "Mnogoplechiye deliteli (summator.y) moshchnosti SVCh na
sosredotochennykh elementakh" ["Multiloop Microwave Power Dividers (and
Adders) Using Lumped Elements"], RADIOTEKHNIKA I ELEKTRONIKA, 1975, No. 5,
Vol 16, pp 943948.
5. Nefedov Ye.I., Fialkovskiy A.T., "Poloskovyye linii peredachi: Teoriya i
raschet tipichnykh neodnorodnostey" ["Strip Transmission Limes: Theory and
Design of Typical Inhomogeneities"], Moscow, Nauka Publishers, 1974.
6. "Osnovy proyektirovaniya mikroelektronnoy apparatury" ["Fundamentals of the
Design of Microelectronic Equipment"], Edited by B.F. Vysotskiy, Moscow,
Sovetskoye Radio Publishers, 1977.
COPYRIGHT: Izdatel'stvo "Radio i svyaz"', 1981
8225
CSO: 8144/0181
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