MICROWAVE CIRCUITS & COMPONENTS
GP Srivastava
MICROWAVE CIRCUITS & COMPONENTS
GP Srivastava
Defence Research & Development Organisation
Ministry of Defence
New Delhi – 110 011
2004
DRDO MONOGRAPH SERIES
MICROWAVE CIRCUITS AND COMPONENTS
GP SRIVASTAVA
Series Editors
Editor-in-Chief
Dr Mohinder Singh
Editors
Dr JP Singh, A Saravanan
Coordinator
Ashok Kumar
Cover Design
Anjan Das
Asst. Editors
Kumar Amar Nath
Sanjay Kumar
Editorial Asst.
AK Sen
Production
Printing
JV Ramakrishna, SK Tyagi
Marketing
Dr Rajeev Vij, RK Dua, Rajpal Singh
Cataloguing in Publication
SRIVASTAVA, G.P.
Microwave circuits and components.
DRDO monograph series.
Includes index and bibliography.
ISBN 81-86514-14-7
1. Microwave circuits 2. Transmission line 3. Electronic warfare I. Title (Series)
621.38.04
© 2004, Defence Scientific Information & Documentation Centre (DESIDOC), Defence R&D
Organisation, Delhi-110 054.
All rights reserved. Except as permitted under the Indian Copyright Act 1957, no part of this
publication may be reproduced, distributed or transmitted, stored in a database or a retrieval
system, in any form or by any means, electronic, mechanical, photocopying, recording, or
otherwise, without the prior written permission of the publisher.
The views expressed in the book are those of the author only. The editors or publisher do not
assume responsibility for the statements/opinions expressed by the author.
Printed and published by Director, DESIDOC, Metcalfe House, Delhi-110 054.
CONTENTS
Preface
xi
CHAPTER 1
INTRODUCTORY BACKGROUND OF MICROWAVE ENGINEERING 1
1.1
1.2
1.3
1.4
1.5
1.5.1
1.5.2
1.5.3
1.5.3.1
1.5.3.2
1.5.3.3
1.5.3.4
1.6
1.6.1
1.6.2
1.6.3
1.6.4
1.6.5
1.7
Introduction
Brief History of Growth of Microwaves
Applications of Microwave Engineering
Maxwell’s Equations
Some Physical Parameters & their uses in Electromagnetics
Phasor Concept
Lumped Element
Decibel Units
Decibel below or above one Watt (dBw)
Decibel below or above one milliwatt (dBm)
Neper
Distributed Element
RF/Microwaves versus DC or Low AC Signals
Presence of Stray Capacitance
Presence of Stray Inductance
Skin Effect
Radiation
Quality Factor Q
Modes in Microwave Transmission Lines
References
CHAPTER 2
TRANSMISSION LINES
2.1
2.2
2.3
2.4
2.5
2.6
2.6.1
2.6.2
2.6.2.1
2.6.2.2
2.7
2.7.1
2.7.2
Introduction
Circuit Model of a Transmission Line
Wave Propagation Constant
Characteristic Impedance
Physical Significance of Propagation Constant Equations
Propagation Factor & Characteristic Impedance of Transmission Line
Ideal or Lossless Line
Line with Small Losses
Attenuation in Transmission Line with Low Losses
Characteristic Impedance of Transmission Lines with Low Losses
Waveform Distortions
Frequency Distortion
Delay Distortion
1
4
7
10
12
12
12
12
13
13
13
14
14
14
14
14
14
14
15
17
17
17
22
22
25
26
27
27
29
29
30
30
31
(vi)
2.8.1
2.8.2
2.8.2.1
2.8.2.2
2.9
2.10
2.10.1
2.10.2
2.11
2.11.1
2.12
2.12.1
2.12.2
2.12.3
2.13
2.13.1
2.13.2
2.13.2.1
2.13.2.2
2.13.3
2.13.4
2.13.5
2.13.5.1
2.13.5.2
2.13.6
2.13.7
2.13.7.1
2.13.8
2.13.8.1
2.13.9
2.13.10
2.13.11
2.14
2.14.1
2.14.2
2.14.3
2.14.3.1
2.14.3.2
2.14.4
2.14.5
2.14.5.1
2.14.6
2.14.7
2.14.7.1
2.14.7.2
2.14.8
2.14.9
The Open Two-wire Line
Coaxial Line
A Typical Coaxial Line
Analysis of Coaxial Line
Transmission Line at High Frequencies
Impedance & Admittance of Short-circuited & Open-circuited Lines
Input Impedance of Open Circuited Line
Quality Factor (Q) of Resonant Lines
Quarter Wave Line
Impedance Matching by Stubbing
Impedance Measurement using Transmission Lines
Position of Minimum when Impedance is Resistive
Position of Minimum when the Load is Purely Inductive
Position of Minimum when the Load is Purely Capacitive
Microwave Waveguides
Maxwell’s & Helmholtz Relations
Boundary Conditions
Helmholtz Equations
Wave Equations in Rectangular Coordinates
Non Propagation of TEM Mode in a Rectangular Waveguide
TE Modes in Rectangular Waveguide
Dispersion Relation & TE Modes
Modes in Rectangular Waveguide
Dominant TE10 Mode
TM Modes in Rectangular Waveguide
Excitation of Modes in Rectangular Waveguides
Field Patterns for some Modes
Circular Waveguide
TE Modes in Circular Waveguide
TM Modes in Circular Waveguide
Fields in Circular Waveguide
Excitations of Modes in Circular Waveguide
Some other Microwave Transmission Lines
Dielectric Waveguide
The Strip Line
Microstrip Line
Empirical Formula for Effective Dielectric Constant
Attenuation Factors
The Coplanar Waveguide
The Slot Line
Slot Wavelength
Suspended Microstrip Line
Fin Lines
Galerkin’s Method in Spectral Domain
Design Considerations
Ridge Waveguide
Mono Strip Lines & Integrated Fin Lines
31
33
33
35
41
41
43
43
45
47
51
52
54
54
56
56
56
57
57
60
61
65
66
67
67
69
70
72
75
78
80
80
80
80
83
84
87
88
90
93
94
96
97
98
99
100
100
(vii)
2.14.10
2.15
Transition between two Transmission Lines
Concluding Remarks
References
101
109
CHAPTER 3
SCATTERING MATRIX
115
3.4.2.1
3.4.2.2
3.4.2.3
3.5
3.6
3.7
3.7.1
3.8
3.9
3.9.1
3.9.2
3.10
133
136
139
147
148
150
153
156
159
160
161
162
163
3.1
3.2
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
3.3
3.4
3.4.1
3.4.2
Introduction
Network Representation
Impedance Representation
Admittance Representation
Hybrid Representation
G Parameter Representation
ABCD Parameter Representation
Inverse Chain Parameters
Parameter Conversion
Scattering Parameters
Conversion of S-parameter to other Network Parameters
General Properties of Scattering Matrices of Linear Lossless
Microwave Devices
Application of Scattering Matrix Concepts to E- and H-plane Tees
Magic or Hybrid T
Another Microstrip Realisation of Hybrid Junction (Magic Tee)
Translation of Reference Planes
Flow Graphs or Two Port Devices
Signal Flow Graph for Three and Four Port Devices
Four Port Devices
Crossing
Some Aspects of a Two Port Junction Scattering Matrix
Shunt Susceptance
Series Reactance
Scattering Transfer Parameters
References
CHAPTER 4
MICROWAVE TRANSISTORS
4.1
4.2
4.2.1
4.2.2
4.3
4.3.1
4.4
4.4.1
4.4.2
4.5
4.5.1
Introduction
Microwave Bipolar Transistor
Silicon Bipolar Junction Transistor & its Small Signal Model
Scattering Parameters of Transistors
Microwave FET
Brief Description of Noise Performance of MESFET
DC Biasing
Temperature Stability
Bias Decoupling
Microwave Transistor
Microwave Transistor Amplifier
115
115
116
117
118
119
120
121
121
124
129
167
167
168
168
176
184
192
197
199
199
200
201
(viii)
4.5.1.1
4.5.1.2
4.6
4.7
4.7.1
4.7.2
4.8
4.8.1
4.8.2
4.9
Gain & Stability
Stability
Transistor Oscillator
Dielectric Resonator Oscillator
Dielectric Resonator Oscillator using Parallel Feedback
Configuration using Series Feedback
Other Microwave Sources
Microwave Tubes
Solid State Sources
Conclusion
References
201
203
206
207
208
208
208
209
210
211
CHAPTER 5
FREQUENCY SYNTHESIZER
215
CHAPTER 6
SMITH CHART & IMPEDANCE MATCHING
233
5.1
5.1.1
5.2
5.2.1
5.2.2
5.3
5.3.1
5.3.2
5.4
5.5
6.1
6.1.1
6.1.2
6.2
6.3
6.3.1
6.4
6.4.1
6.4.2
6.4.3
6.4.3.1
6.4.3.2
6.4.3.3
6.4.4
6.5
6.6
6.7
6.8
6.8.1
6.8.2
6.9
Introduction
Principle & Types of Synthesizer
Synthesizer Architectures
Direct Analogue Synthesis
Indirect Synthesis
PLL Synthesizers
Single Reference PLL Synthesizer
Double Loop Radar Synthesizer
Piezo Electric Synthesizer
Conclusion
Introduction
Decibels & Nepers
Derivation of Reflection Coefficient based on Simple Transmission
The Smith Transmission Line Chart
Application of Smith Chart
Determination of Unknown Impedance
Impedance Matching
Quarter Wave Transformer
Quarter Wave Transformers with Extended Bandwidth
Stub Matching using Smith Chart
Single Stub Matching
Double Stub Matching
Forbidden Regions in Double Stub Matching
Matching with Three Stubs
Compressed Smith Chart
The Normalised Impedance & Admittance Smith Chart
The Normalised Z-Y Smith Chart
Impedance Matching Using Lumped Elements
Impedance Matching Networks
Microstrip Matching Network
Conclusion
References
215
215
220
220
223
229
229
229
229
232
233
234
235
236
241
244
248
250
254
256
256
261
267
267
268
271
271
272
276
281
291
(ix)
CHAPTER 7
DESIGN OF MICROWAVE NETWORK,
MODELLING & COMPUTER-AIDED DESIGN
7.1
7.2
7.2.1
7.2.2
7.3
7.3.1
7.3.2
7.3.3
7.3.4
7.4
7.4.1
7.4.2
7.4.3
7.5
7.5.1
7.5.2
7.6
7.6.1
7.6.1.1
7.6.1.2
7.6.2
7.6.3
7.6.4
7.6.5
7.7
7.7.1
7.7.2
7.7.3
7.8
7.8.1
7.8.1.1
7.8.1.2
7.8.2
7.8.2.1
7.8.3
7.8.3.1
7.8.3.2
7.8.3.3
7.8.4
7.8.4.1
7.8.4.2
Introduction
Frequency Domain Analysis of Microwave Networks
Connection Scattering Matrix in Circuit Analysis
Formation of Connection Matrix
Computation of Network Response Function
Input Port Reflection Coefficient
Insertion Loss
Group Delay
Voltage Transfer Gain
Solution of Systems of Circuit Equations using Sparse Matrix
Techniques
Bifactorization Techniques
Algorithm for Bifactorisation
Computation of the Solution Vector
Microwave Network Sensitivity Analysis
Transposed Matrix Method for Sensitivity Analysis
Sensitivity Computation of the Overall Scattering Parameters
Microwave Network Optimisation
Philosophy & Methods of Optimisation
Objective Functions & Constraints
Minimax Approximation
Direct Search Optimisation Method
Razor Search Method
The Simplex Method
Gradient Methods for Optimisation
Microwave CAD Programmes
Subroutines
An Overview of Available CAD Programmes
Some CAD Programes
Electromagnetic Techniques in Computer Aided Design
of Microwave Components & Circuits
Finite Difference Methods
Explicit Method
Implicit Method
Finite-Difference Time Domain Technique
Yee’s Finite Difference Algorithm
Moments Methods
Green’s Function
Transformation to Matrix Equation
Evaluation of Matrix Elements
Finite Element Method
Finite Element Discretisation
Elements Governing Equations
293
293
294
294
297
300
300
303
304
304
305
306
307
308
308
310
311
312
319
319
321
321
322
322
323
324
324
325
326
329
329
332
333
334
335
338
339
341
343
344
345
346
(x)
7.8.4.3
7.8.4.4
7.8.5
7.8.6
7.9
7.10
Assembling all Elements
Solving the Resulting Equation
Iterative Method
Band Matrix Method
Solution to Some Problems
Conclusion
References
CHAPTER 8
ELECTRONIC WARFARE
8.1
8.2
8.2.1
8.2.1.1
8.2.2
8.2.3
8.2.4
8.2.5
8.2.6
8.2.7
8.2.8
8.2.8.1
8.2.9
8.3
8.3.1
8.3.2
8.3.3
8.4
8.4.1
8.5
8.5.1
8.5.1.1
8.5.2
8.5.2.1
8.5.2.2
8.5.3
8.6
8.7
8.8
8.8.1
8.8.2
8.8.2.1
8.8.3
8.8.3.1
8.8.4
8.8.4.1
8.8.4.2
Introduction
Electronic Support Measures
Noise, Probability & Information Recovery
Elementary Probability Theory & Statistics
Recovery of Signal from Noise
Detection & Correlation
Characteristics of Microwave Receivers
Tangential Sensitivity
False Alarm Rate & Probability of Detection
Introduction to Dynamic Range
Direct Detection Receiver & Superheterodyne Receiver
Superhetrodyne Detection Receivers
Spectrum Analysers
Direction Finding Techniques
Direction finding by Amplitude Measurement
Direction finding by Phase Measurement
Directions finding by Time of Arrival Measurement
Location Measurements
Evolution of the System
Electronic Countermeasures (ECM)
Pulse Radars
Pulse Radar Range
CW Radars
Doppler Filter Bandwidth Limitations
CW Power Range
Pulse Doppler Radars
General Radar Concepts
Effect of ECM on Radars
Some Jamming Techniques
Communications in Comparison with Radar Jamming
Cover & Deceptive Jamming
Power Management
Deceptive Jamming Techniques
Range gate pull-off Techniques
Inverse Gain Jamming Techniques
AGC Jamming
Velocity Gate Pull-off
349
351
352
353
354
374
379
379
380
380
381
383
384
387
388
390
392
393
393
394
394
394
395
395
395
396
397
397
401
403
404
406
407
412
414
418
419
419
421
421
422
422
423
424
(xi)
8.8.5
8.8.5.1
8.8.5.2
8.8.6
8.8.6.1
8.8.6.2
8.8.6.3
8.9
8.9.1
8.9.2
8.9.3
8.9.4
8.9.5
8.9.6
8.9.7
8.9.8
8.9.9
8.9.10
8.10
Appendix A
Appendix B
Index
Deceptive Jamming Techniques used against Monopulse Radars
Radar Resolution Cell
Blinking Jamming
Decoys: Applications & Strategies
Saturation Decoys
Detection Decoys
Seduction Decoys
Radar Counter Countermeasure Techniques
Surveillance Radars – ECCM Considerations
Tracking Radar ECCM Consideration
Radar Range In Presence of Jamming
ECCM with Antenna
Integrated SLC
Side Lobe Blanking
ECCM with Transmitter
Pulse Compression
ECCM with Receiver
ECCM with Signal Processing
Concluding Remarks
References
424
425
426
426
427
427
427
427
428
430
432
434
435
436
436
437
437
438
439
441
443
447
PREFACE
In this book, an attempt has been made to give the fundamental concepts
a fresh perspective. It lays emphasis on transmission lines, various types of
transmission lines generally used in the microwave engineering. Emphasis has also
been made on microwave network analysis and scattering matrix. Microwave
semiconductor active devices have also been discussed. Smith chart and its
application in microwave circuit design and matching has also been discussed.
Attempt has also been made to describe the essential features of computer-aided
design of microwave circuits, although the subject is so vast that it is not possible
to describe all features and take up some practical designs. Some packages have
been mentioned to obtain useful results. A microwave engineer who can apply
these concepts towards actual design objective is most likely to be successful.
The last chapter describes elements of electronic warfare. This is an area
where lot of literature is available. While going through literature, I have found
that these papers conceal much more than they reveal. However, I have made an
attempt to describe them systematically and have included many useful circuits
and their principles of operation. Most of it deals with radar applications but other
circuits have also been described.
It should be accepted that I have consulted all good literature available
for writing this book. Wherever I have found that some points have been described
better in standard books, I have not hesitated to use them.
The author gratefully acknowledges the authors and publishers of these
books and also the authors of research papers as it is not possible to acknowledge
them individually.
Dayalbagh
Agra -282 005
December 2004
G P SRIVASTAVA
CHAPTER 1
INTRODUCTORY BACKGROUND OF
MICROWAVE ENGINEERING
1.1
INTRODUCTION
The term microwaves refer to that portion of electromagnetic waves
whose frequency lies between 300 MHz and 300 GHz, i.e., somewhere between
3x108 Hz to 3x1011 Hz. Why these are known as microwaves? The wavelength
corresponding to the frequencies mentioned above are 1 m and 1 mm. Sometimes
signals with wavelength of the order of millimetres are called millimeter waves.
These wavelengths are much smaller than RF waves, say, the wavelength of RF
waves at 1 MHz is 300 m. Thus, the microwave wavelengths are much smaller and
therefore they are known as microwaves.
Figure 1.1 shows the location of microwave frequency band in the
electromagnetic spectrum. The various bands are also shown in Table 1.1(a).
Table 1.1(b) shows telecommunication designations. Table 1.1(c) shows the
terrestrial links. Table 1.2 shows the 12 GHz satellite broadcasting bands in Europe
and Table 1.3 shows the band designations both old and new.
In microwave engineering, one must begin with Maxwell's equations.
However, one uses parameters like power, impedance, voltage and current which
can be used with concept of circuit theory.
FREQUENCY (Hz)
103
102
10
3 1010
3 1011
Microwaves
10-1
3 1012
3 1013
10-2
10-3
10-4
10-5
3 1014
Visiblelight
3 109
Infrared
3 108
Far Infrared
VHF TV
Shortwave radio
3 107
FBI Broadcast radio
3 106
AM Broadcast radio
Longwave radio
3 105
10-6
WAVELENGTH (m)
Figure 1.1. The electromagnetic spectrum as a function of wavelength/frequency
Microwave Circuits & Components
Microwave components are often distributed elements where phase of a
voltage or current changes significantly over the physical extent of the device. It
must be noted that unlike RF frequency range or visible frequency range, the
dimensions of components are of the same order as wavelength of microwaves. In
rf region the wavelength is much larger than components. In the optical frequency
region the components have much larger dimensions than the wavelength.The
lumped circuit approximations are not valid at microwave frequencies. Of course,
now with improvement in the technology, lumped elements have been fabricated
and can be used up to 10 GHz.
Table 1.1(a). The various bands of electromagnetic spectrum
Typical frequencies
AM broadcast band
Approximate band designations
L-Band
1–2 GHz
MHz
S-Band
2–4 GHz
FM broadcast band
88–108 MHz
C-Band
4–8 GHz
VHF TV (2–4)
54–72
MHz
X-Band
8–12 GHz
VHF TV (5–6)
76–88
MHz
Ku-Band
12–18 GHz
UHF TV (7–13)
174–216 MHz
K-Band
18–26 GHz
UHF TV (14–83)
470–890 MHz
Ka-Band
26–40 GHz
U-Band
40–60 GHz
Shortwave radio
Microwave ovens
535–1605 kHz
3–30
2.45 GHz
Table 1.1(b). Telecommunication frequency designations (satellite)
Frequency Range
(GHz)
Designation
Comments
2.5–2.69
Broadcast
Arabsat 2.6/6 GHz
3.4–4.2
Major telecoms
Intelsat IV, V (down)
Telecom I & II
4–17 GHz (up/down)
4.4–4.9
European satellite
Symponie (down)
5.7–8.4
European satellite
Symponie (up/down)
5.925–6.4
Major telecoms
Intelsat IV, V (up)
10.7–11.7
Telecoms
OTS (MK1) & ECS
11.49–11.7 GHz (down)
ECS 11.45–11.78 GHz (down)
Intelsat IV, V (down)
Contd...
2
Introductory Background of Microwave Engineering
Frequency range
(GHz)
Designation
Comments
11.7–12.7
Broadcast
TV-Sat 11.7–12.7 GHz (down)
14.0–15.35
Telecoms
OTS 14–14.5 GHz (up)
Intelsat IV, V (up)
17
Broadcast
17.3–19.7
19.0–21.4
22.5–23.6
27.0–27.5
39.5–40.5
40.5–42.5
TV-Sat 17.3–18.1 GHz (up)
New Satcoms
Broadcast
Broadcast
European Telecommunication Satellites
ECS
European Communication Satellite (Launched 1983)
MARCES
Maritime Telecommunication Satellite
OTS
Orbital Test Satellite
DBS
Direct Broadcast Satellite
TV- Sat, DBS
Direct Broadcast Satellite (Launched 1983)
Telecom I & II
French Commercial Digital Telecom Satellite
USA Intelsat IV, V
Major Telecommunication Satellites over Atlantic and Indian
Oceans for Worldwide Network
Table 1.1(c). Telecommunication frequency designations (terrestrial)
Frequency
range(GHz)
Designation
Comments
1.40–1.53
1.70–2.7
–
2 GHz band
ITU Allocation Region 1
60-90 ch FDM system (BT)
3.40–4.2
4 GHz band (lower)
4.40–5.0
4 GHz band (upper)
1800 ch FDM system (BT)
Future–Digital
5.85–6.4
6 GHz band (lower)
High capacity 2700 ch FDM systems (BT)
Future – Digital 140 MBIT/S (BT)
6.40–7.0
6 GHz band (upper)
New TV/4 Voice ch FDM system (BT)
7.25–7.75
7 GHz band
Low capacity 300 ch FDM System
8.20–8.5
8 GHz band
900 ch FDM System
11 GHz band
New digital140 MBIT/S 4 phase system (BT)
9.30–10.68
10.7–11.7
Contd...
3
Microwave Circuits & Components
Frequency
range(GHz)
Designation
11.70–12.5
14 GHz band
Radio relays (FDM) mainly TV
and studio transmission over short range
17 GHz band
New digital radio relay systems
12.75–13.25
14.40–14.5
14.50–15.5
17.70–21.2
21.20–29.5
31.00–31.3
36.00–40.5
42.50–43.5
47.20–50.2
Comments
Short range systems
Table 1.2. European satellite bands (12 GHz broadcasting bands)
Channel
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Country
F
D
LUX
GB
F
D
LUX
GB
F
D
LUX
GB
F
D
LUX
GB
F
D
LUX
GB
Frequency (MHz)
Channel
11727.48
11746.66
11765.84
11785.02
11804.20
11823.38
11842.56
11861.74
11880.92
11900.10
11919.28
11938.46
11957.64
11976.82
11996.00
12015.18
12034.36
12053.54
12072.72
12091.91
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Country
B
CH
NED
I
B
CH
NED
I
B
CH
NED
I
B
CH
NED
I
B
CH
NED
I
Frequency (MHz)
12111.08
12130.26
12149.44
12168.62
12187.80
12206.98
12226.16
12245.34
12264.52
12283.70
12302.88
12322.06
12341.24
12360.42
12379.60
12398.78
12417.96
12437.14
12456.32
12475.50
Note: Even number channels are left circular polarized and odd channels are right circular polarized.
1.2
BRIEF HISTORY OF GROWTH OF MICROWAVES
The field of microwave engineering is often considered a fairly mature
discipline because the fundamental concepts of electromagnetics were developed
more than hundred years ago. The foundations of modern electromagnetic theory
was formulated in 1873 by Maxwell1, who hypothesized from mathematical
considerations, the four famous relations of electromagnetic theory, now known as
Maxwell's equations. He established light as a form of electromagnetic waves.
4
Introductory Background of Microwave Engineering
Maxwell's formulations of electromagnetic theory were given in modern form by
Oliner Heaviside during 1885 and 1887. Heinrich Hertz, a German Professor of
Physics was a gifted experimentalist who carried out set of experiments during
1887-1889 that theory of Maxwell. Professor JC Bose 2 is considered as the
founding father of Microwave Technology3. JC Bose succeeded in generating
electromagnetic waves of 5 mm wavelength. He also gave the term millimeter
waves. His radiator comprised two hollow hemispheres interposed with a small
sphere and generated radio waves through electric discharge. These
electromagnetic waves were radiated through a device now known as Horn
Antenna. He used a Galena crystal to filter the millimeter waves. He succeeded in
demonstrating reflection, refraction, total internal reflection, double refraction and
polarization by using the crystals of Tourmaline and Nemalite of these
electromagnetic waves and also proved beyond doubt their identity with light
waves. In 1895 he gave a public demonstration in Calcutta.
Because of the lack of reliable microwave source and other components,
nothing worthwhile happened in research up to activity in early twentieth century.
It was only in thirties that two microwave sources were developed. Magnetron 4
was developed by Hull and Klystron5 was developed by Varian Brothers. In the
same decade quantum and wave mechanics were developed. The first experimental
verification of wave mechanics was sought in the inversion spectra of ammonia. If
the coordinates of all nuclei of any non-planar molecule are inverted at the centre
of mass the resulting configuration is also an equilibrium structure of the molecule.
Consequently, every non planar molecule has identical energy minima separated
by potential barriers and corresponding to equally stable molecular structures.
These structures, moreover, cannot be obtained by simply reflecting at the plane of
symmetry. If the barrier is sufficiently high, the two forms are stable isomers, often
separable and often with different optical property. If barrier is low and narrow,
however, the molecules resonates between the two possible structures. The
solutions to quantum mechanical wave equation are linear contributions of the
wave functions corresponding to these two structures. One of the solution is
symmetric and another anti-symmetric with respect to inversion of coordinates.
The emergence of these two solutions differ by a small amount, Einv which is
largest for light molecules with low, narrow potential barriers. The molecule may
be treated as an oscillator of reduced mass moving in a double minimum potential
well. Dennison and Uhlembeck6 relates Einv. Their result is
Einv
1
Ev
A2
A exp
(1.1)
x1
2 v E
where,
2
h
0
12
u
dx
v = the separations of vibrational levels
X = inversion coordinates (dimensionless)
5
(1.2)
Microwave Circuits & Components
Table 1.3. Band designations (old & new)
Band designations
New
Old
HF
3–30 MHz
Frequency
(GHz)
A
0–250 MHz
VHF
0.1
0.15
0.2
30–300 MHz
UHF
300–1000 MHz
L
1–2 GHz
100 cm
C
0.5
60 cm
500–1000 MHz
0.75
40 cm
D
1.0
30 cm
1.5
20 cm
2
15 cm
1–2 GHz
F
3–4 GHz
G
4–6 GHz
C
4–8 GHz
H
6–8 GHz
X
8–12 GHz
I
8–10 GHz
J
O
7.5 cm
5
6 cm
6
5 cm
8
3.75 cm
10–20 GHz
15
2 cm
K
20
1.5 cm
20–40 GHz
30
10 mm
L
40
7.5 mm
40–60 GHz
50
6 mm
M
60
5 mm
40–60 GHz
E
10 cm
4
3 cm
KA
27–40 GHz
3
10
J
Ku
12–18 GHz
K
18–27 GHz
1.5 m
0.3
2–3 GHz
2–4GHz
3m
2m
B
250–500 MHz
E
S
Q
Wavelength
60–90 GHz
75
60–100 GHz
100
6
4 mm
3 mm
Introductory Background of Microwave Engineering
X = 0 at maximum of potential hill
X = x1at V = Ev
Ammonia is unique among molecules investigated for microwaves.
Cleeton and Williams7 claimed to have observed transition in 1934. This was the
birth of microwave spectroscopy. This experiment was performed in 1934, when
microwave technology had not been developed. They used Klystron for generation
of microwaves but the other components were classical as the technology
developed with advent of radar. Cleeton and Williams believed that they have seen
inversion transition in Ammonia. However, no one believed in these observations.
During Second World War microwave technology got a thrust as radars had to be
developed for detection of aeroplanes. A Radiation laboratory was established at
Massachusetts Institute of Technology (MIT) in USA. This laboratory had to work
on microwave theory and technology. A number of scientists including
N Marcuritz, I I Raber, J Schwinger, H A Bethe, E M Percell, C G Montgomery
and R K Dicke among others, were gathered for what turned out to be a very
intensive period of development in microwave field. Their work included the
theoretical and experimental treatment of waveguide components, microwave
antenna, small scale coupler and the microwave network theory. Many of these
researchers were physicists who went back to physics research after the war (many
later received Nobel prizes) but their microwave work is summarised in the
classical 28 volume Radiation Laboratories series that still finds application today.
After the development of radar, the area of microwave communication
was developed. It has certain advantages like wide bandwidth and line of sight
propagation. These have proved to be important from the point of view of
terrestrial and satellite communication. Terrestrial and satellite communication
bands are listed in Tables 1.2 and 1.3. Shiner has described these in detail 8. The
microwave spectroscopy and laser are also extension of developments in
microwave theory.
1.3
APPLICATIONS OF MICROWAVE ENGINEERING
Some of the applications of microwave are mentioned below.
(a)
Microwave communications systems: Microwave communications links
are an important practical application of microwave techonology and are
used to carry voice, data and television signals. Microwave communication
systems can be classified into two types (a) Guided-wave systems; where
the signal is transmitted through low-loss cable or guide (b) Radio links;
where signal is transmitted through space. Microwave radio propagation
is essentially line-of-sight and repeater stations communications satellite
are required to connect two widely separated earth stations.
Radar systems: Radar or radio detection and ranging is probably the most
prevalent application of microwave technology. In basic operation, a
transmitter sends out a signal which is partly reflected by a distant target
and then detected by sensitive receiver. Some of the typical application of
radar are: (a) Civilian applications: which include airport surveillance,
(b)
7
Microwave Circuits & Components
marine navigations, weather radar, altimeter, aircraft landing, burglar
alarms, speed measurements (police radar) and mapping, and (b) Military
applications: which include air and marine navigation, detection and
tracking of aircraft, missiles, spacecrafts (including unmanned ones),
missile guidance, weapon fuses. The scientific applications are astronomy,
mapping and imaging. The applications also include study of basic
properties of solids, liquids, gases and plasma. The effective masses of
carriers are determined by cyclotron resonance experiments. Even the
researches on Ammonia Maser lead to development of lasers which has
many applications. Remote sensing is one of the natural applications.
Radiometry is a passive technique which gathers information about a target
solely from the microwave portion of black body radiation (noise) that it
emits directly or reflects from surrounding bodies.
Microwave has already entered kitchen in the form of microwave ovens
besides industrial and medical applications for microwave heating. During the
microwave heating the inside body of material gets heated first. The process
through which it occurs involves resonance of water molecules and conduction
loss in material with large loss tangent. The efficiency of microwave oven is nearly
50 per cent which is significantly high compared to conventional heating.
Microwaves is being used in microwave hyperthermia for treatment of cancer.
Military radar and communication systems can be limited or prevented
from performing their intended function by deliberate means such as interference,
jamming and other counter measures. This is generally called as electronic warfare.
Electronic warfare includes Electronic Support Measures (ESM). This involves
use of threat warning receivers to detect presence of wide band jamming signals.
Wide open microwave intercept receivers are used in radar warning and electronic
intelligence (RWR, ESM, ELINT) applications of Electronic Warfare (EW) to
detect, measure, analyse and identify radar signals and platforms. Optimising
receiver detector sensitivity is dependent on
(a)
(b)
(c)
(d)
RF circuit flatness across the instantaneous reception band
Detection threshold false alarm rate
Acceptable parameter measurements inaccuracy due to system noise
The efficiency of demodulators.
These aspects are analysed by studying receiver sensitivity9.
The Electronic Countermeasures (ECM) includes both active and passive
techniques to either confuse or deceive a radar or communication system. ECM
applications utilise the serrodyning technique to deceive the velocity technique, or
velocity detection tracking or other velocity related functions of a solid-state victim
radar missile system. These are solid-state, phase shifting serrodyne techniques
and are high performance, light weight, high reliability and low-cost alternative to
medium power Travelling-Wave Tube Amplifier (TWTA) generated deception
jamming used in conjunction with Digital Signal Processing (DSP) techniques.
8
Introductory Background of Microwave Engineering
This approach provides the next generation of intelligently controlled deception
waveforms for EW applications. The Electronic Counter Countermeasures
(ECCM) counters the effect of ECM.
Many modern analysis technique are also being used in electronic
warfare. These can be used by wavelet transformation to distinguish Quadrature
Amplitude Modulated signal from Phase Shifting Key signal and the Frequency
Shift Keying signal. The approach is to use wavelet transform to extract transient
characteristics in a digital modulations signal and apply distinct pattern in wavelet
transform domain for simple identification. The relevant statistics for optimum
threshold situation are devised under the condition that input noise is additive white
gaussian. The percentage of correct identification is 97 per cent.
Fuzzy logic is also used in countermeasures. In the present day electronic
warfare, aircraft protection is done by very fast data acquisition and almost instant
decision. A good fuzzy logic and fuzzy processor can help in decreasing the time
required for decision, process and improving the quality of decision.
Defensive radio frequency receivers require wide bandwidth coverage
for high probability of intercept operation. Most defensive receiver systems are
known to be susceptible to saturation from high duty cycle continuous wave (CW)
and RF agile emitters. High duty cycle CW emitters can usually be filtered out by
slow tuning (millisecond) Yittrium Iron Garnets (YIG) microwave filters. High
temperature superconducting (HTS) filters can be applied in receiver processing
chain as close to the front end of the receiver as defensive receiver architecture
permits. The HTS filters effectively free up the limited signal and data processing
capability of the defensive receiver to find other signals that may be asked by the
agile RF emitters. Much progress has been made in developing high quality HTS
microwave devices that possess much lower insertion loss than conventional
devices. Switchable and limitable band reject filter technology has been developed.
The unique capabilities of millimetre (MM) waves has been used for radar
and communications. For instance, the atmospheric windows at 35 GHz and
95 GHz are used for communication and radar while heavy atmospheric
attenuation at 60GHz makes possible for secure or correct communication . In
general, MM waves can provide a narrow and directed antenna beam. Compared
to infrared and optical waves, the MM wave has better capacity in penetrating the
odd weather and dust. These capabilities have been applied to certain civilian
applications like automobile collision avoidance, traffic monitoring and control.
The MM waves have many military applications such as seeker for smart
weapons, the adverse weather alternative to IR guided system, etc. Millimetre wave
radar has some of the following advantages:
(a)
(b)
(c)
(d)
(e)
High gain with small aperture
High track and guidance accuracy
Reduced ECM vulnerability
Operation at low elevation angles without significant multipath ground
clutter interface.
Multipath target discrimination
9
Microwave Circuits & Components
The MM wave radar would be specially effective against low altitude
airborne targets and ground targets as it is necessary to reduce clutter return. The
field of mobile communication is also moving rapidly. Cross border application in
Europe have paved the way for a Group Special Mobile (GSM) communication.
Mobile telephones communicate at 0.9 or 1.8 GHz within cell sizes of 1 to 5 mm in
diameter.
Since the entire microwave circuitry is developed using the Maxwell's
equation it would be appropriate to mention the Maxwell's equation in the
introductory chapter. The differential form is given first then integral form and
finally boundary conditions to be applied.
1.4
MAXWELL’S EQUATIONS
The Maxwell’s equation in the differential form can be written as
E B M
t
(1.3a)
H D J
t
(1.3b)
D ρ
(1.3c)
B 0
(1.3d)
where,
E
= Electric field intensity in volt/m
H
= Magnetic field intensity, in ampere/m
D
= Electric flux density, in coulomb/m2
B
= Magnetic flux density, in weber/m2
M = Fictitious magnetic current density in volt/m2
J
= Electric current density in ampere/m2
= Electric charge density in coulomb/m2
In these expressions m is metre. M is the magnetic current which is
fictitious. It is only mathematical convenience. The real source of magnetic current
is always a loop of electric current. Magnetic monopole charge is not known to
exist, only magnetic dipole exists.
seen that
The differential equations can be expressed as integral ones. It can be
D . ds ρ dv Q
(1.4)
B. ds 0
(1.5)
v
10
Introductory Background of Microwave Engineering
where, Q represents total charge contained in closed volume v
E .dl
B .ds M .ds
c
t
(1.6)
which, without M, is usual form of Faraday’s equation
H .dl
D .ds J .ds
D .ds I
t
t
(1.7)
where,
s
s
I J . ds
s
s
s
s
is total electric current flow through surface s. These equations
when applied to a propagation problem have to follow certain boundary conditions,
none of it is a perfect electric conductor. The tangential components of the electric
and magnetic field are continuous as expressed by
n E1 E 2 0
volt m
n H1 H 2 0
(1.8)
ampere m
where, n is the unit vector to the interface directed from medium 2 towards
medium 1. Indices 1 and 2 specify the medium within which the field is defined.
When, 0 , the boundary conditions for normal components of phasor-vectors
are automatically satisfied when Eqns. 1.7 and 1.8 are met.
On the surface of a perfect electric conductor (pec, ), the electric
phaser vector must meet the condition
n E 0
(1.9)
volt m
This condition is nearly satisfied on metallic surface (short-circuit)
On the surface of perfect magnetic conductor (pmc, ) without
surface current, the magnetic phasor-vector must meet the condition
n H 0
(1.10)
ampere m
At microwave frequencies magnetic materials do not have sufficiently
large permeability to satisfy the above condition. It can however be used to take
into account geometrically symmetry (open-circuit plane) or to nearly represent
the interface between a very high permittivity dielectric and one having much
smaller permittivity.
Boundary conditions on normal components of electric field yield
n r1 E1 r2 E2
s
0
volt m
(1.11)
Surface charges s can only be found on the interface between two
materials when at least one of the two has a non-zero conductivity .
Similarly, the magnetic field must satisfy
n r1 H1 r2 H 2 0
ampere m
11
(1.12)
Microwave Circuits & Components
1.5
SOME PHYSICAL PARAMETERS & THEIR USES IN
ELECTROMAGNETICS
1.5.1
Phasor Concept
The following conditions should be present before the phasor concept is
used effectively:
•
•
•
the circuit is linear
sources are sinusoidal, and
steady state response is desired.
Consider a sinusoidal waveform of voltage current or electromagnetic
wave given by
x t Am cos Wt
(1.13a)
using Euler's identity this can be written as
x t Re Am e j it
(1.13b)
The coefficient, Am e of exponential term jt is a complex number that
carries the amplitude and phase angle of given sinusoidal function. This complex
number is defined as phasor representation of the given sinusoidal wave form
A Am .e j
Sometimes it is also referred as phasor-vector.
1.5.2
Lumped Element
This is defined as a self-contained element that offers one particular
electrical property throughout the frequency range of interest. These are resistor,
capacitor and inductor.
1.5.3
Decibel Units
The decibel is defined as the ratio of two powers or intensities or the ratio
of a power to a reference power. It is one tenth of the international unit known as
Bel. It was used to measure attenuation in telephone cables. Bel is defined as
logarithm to base 10 of power ratio
P
Bel log10 2
P1
(1.14a)
To convert from dB to power ratio
P2
10 N dB 10
P1
(1.14b)
12
Introductory Background of Microwave Engineering
1.5.3.1 Decibel below or above 1 Watt (dBw)
In this case the reference one watt then
P
N dBm 10log10 2
1W
(1.15a)
1.5.3.2 Decibel below or above 1 milliwatt (dBm)
Reference in this case is 1 milliwatt
P
N dBm 10log10 2
1mW
(1.15b)
Decibel above or below 1 microwatt
P
N dBW 10 log10
1W
(1.15c)
1.5.3.3 Neper
Neper is unit of attenuation used for expressing the ratio of two currents,
voltages or fields by taking natural logarithm (logarithm to base e) of this ratio. If
the voltage V1 is attenuated down (always a number) to V2 then,
V2
eN
V1
(1.16)
Then N is attenuation in Neper and is defined as
V
N Np log e 2
V1
1
N
l n 2
N1
1
(1.17)
The unit Neper is named after John Napier, a Scottish scientist and
inventor of natural logarithms.
In the circuit matched in impedance, following conversion between Neper
and dB can be derived
1 Np ln
V2
V
1
2
V1
V1 e
(1.18)
Therefore
1
1Np 20 log10
20 log10 e 8.686 dB
V2 V1
1 Np is a larger unit than dB by a factor of 8.86. Conversely
1 dB = 0.115 Np
13
(1.19)
(1.20)
Microwave Circuits & Components
1.5.3.4 Distributed element
An element whose property is spread out over an electrically significant
length or area of a circuit instead of being concentrated at one location or within a
specific component.
1.6
RF/MICROWAVES VERSUS DC OR LOW AC SIGNALS
There are several major differences between signals at higher radio
frequencies or microwaves (MW) and their counterparts at low AC frequency or
DC. These differences which greatly influence electronic circuits and their
operations become increasingly important as frequency is raised. The following
four effects provide a brief description of MW and AC circuits.
1.6.1
Presence of Stray Capacitance
The capacitance which influence the behaviour of MW signals are:
•
•
•
Capacitance between conductors of the circuit
Capacitance between conductors or components and ground
Capacitance between components
1.6.2
Presence of Stray Inductance
The following inductance will influence the behaviour of circuit at MW
frequency.
•
•
The inductance of conductors that connect components
The parasitic inductance of components themselves
1.6.3
Skin Effect
It is well known that AC signals penetrate a metal partially and flow in
narrow band near the outside surface of each conductance. For RF signals the
current density falls off exponentially from the surface of the conductor towards
the centre. At a critical depth called skin depth or depth of penetration, signal
amplitude 1/e or 36 per cent of its surface amplitude. Skin depth is given by
1
(1.21)
f
1.6.4
Radiation
This is caused by leakage or escape of signal into air. This means that the
signal is lost in atmosphere and therefore add to the losses. The radiation factor
causes coupling effects to occur as follows:
•
•
•
Coupling between elements of circuits
Coupling between the circuit and its environment.
Coupling from environment to circuit Electromagnetic Interference (EMI)
also called radio frequency interference or RF Noise
Quality Factor Q
This is the ability of an element or circuit to store energy. The general
definition of quality factor Q is
1.6.5
14
Introductory Background of Microwave Engineering
Q 2
Energy stored
Energy dissipated per cycle
(1.22)
Q is also a figure of merit of reactive for element and it can be shown that
for inductor
Q
X L L
R
R
(1.23)
And for capacitor it is
Q
XC
1
R
wCR
(1.24)
1.7
MODES IN MICROWAVE TRANSMISSION LINES
The presence or absence of longitudinal field components affects the
propagation behaviour (Table 1.4).
Table 1.4. Modes in transmission lines
(a)
Ez
Hz
Name
Acronym
Other
denomination
=0
=0
Transverse
electromagnetic
TEM
—
0
Transverse electric
TE
H
(c)
0
=0
Transverse
TM
E
(d)
0
0
Hybrid
—
EH or HE
(b) = 0
Most of the topics have been introduced in this chapter and discussed in
detail in other chapters. The appendix A shows the graphical symbols utilised in
microwaves. Appendix B shows the glossary of some physical, mathematical and
parametric symbols used in the text. Appendix C shows symbols and abbreviations
used in the text.
REFERENCES
1. Maxwell, J.C. A treatise on electricity and magnetism. Dover M.Y., 1954.
2. Bose, J.C. Collected physical papers. Longman, Green & Co., New York,
N.Y. 1927.
3. Ramsey, J.F. Proc. I.R.E, 1958, 46, 405.
4. Collins. Microwave magnetron (6th Volume) In M.I.T. Radiation Lab Series,
McGraw Hill Book Co., New York, N.Y. 1948-50.
5. Hamilton, K & Kupper. Klystron and triodes (7th Volume) In M.I.T. Radiation
Lab Series. McGraw Hill Book Co., New York, N.Y. 1948-50.
6. Dennison, D.M. & Uhlenbeck,G.E. Physics Rev,1932, 41, 313.
7. Cleeton, C.E. & Williams, N.H. Physics Rev, 1934, 45, 234.
15
Microwave Circuits & Components
Shiner, A.A. Historical perspectives on microwave field theory. IEEE Trans.
Microwave Theory and Techniques, 1984, MTT-32, 1022-45.
9. East, P.W. Microwave intercept receiver sensitivity estimation. IEE Proc.
Radar, Sonar Navig, 1997, 144, 186.
10. Van Brunt Leroy, B. Applied ECM. EW Engineering Inc.1978, p. 1.
11. Nyiri, E.J. & Madani A.M. Low cost high resolution, serrodynable, solid state
amplifier system for ECM applications. Proc. Military Microwave
Conference,1982, pp. 250-55.
8.
16
CHAPTER 2
TRANSMISSION LINES
2.1
INTRODUCTION
For circuit analysis it is assumed that circuits consist of lumped elements.
It is also assumed that current at every point of a series circuit has the same value
and the voltages between all pairs of opposite points on connecting wires are
identical. These assumptions hold good only at low frequencies.
Circuit analysis assumes that the physical dimensions of network are
much smaller than the electrical wavelength. However, in the frequency region
where this condition does not hold, the transmission line is considered to have
distributed parameters where current and voltage can vary both in magnitude and
phase. Hence, it can be viewed that the transmission line bridges the gap between
field analysis and basic circuit theory. The phenomenon can be approached using
the circuit theory as well as using Maxwell’s equations.
2.2
CIRCUIT MODEL OF A TRANSMISSION LINE
A transmission line is schematically shown as a two-wire line. Some of
the results obtained can be used by all types of transmission lines.
Figure 2.1(a) shows a transmission line of small length z . It can be
considered to consist of a series resistance Rz, a series inductance Lz, shunt
conductance Gz and shunt capacitance Cz , where, R, L, G and C are series
resistance per unit length, series inductance per unit length, shunt conductance per
unit length and capacitance per unit length respectively. R is due to finite
conductivity of conductor, L represents the total self-inductance of two conductors.
C appears due to close proximity of the two conductors, and G is due to leakage of
current in the dielectric between two conductors. These are known as distributed
parameters.
If we assume I as the current in the line at any point, V the voltage between
the conductors at any point and l is the total length of the line, the elemental
section z can be assumed to carry current I. The potential drop in the length dz is
dV IZ dz
where, Z R j L , so that
(2.1)
Microwave Circuits & Components
I(z,t)
V(z,t)
z
(a)
I(z,t)
R
I(z+z,t)
z
L z
V(z,t)
C
G
z
V(z+ z, t)
z
(b)
z
Figure 2.1.(a) Transmission line, and (b) Equivalent circuit
dV
IZ
dz
(2.2)
Similarly, the shunt admittance per unit length is Y mhos (Siemens) which
is equal to G j C , so that the change in current over length z is
(2.3)
I VY dz
which leads to relation
dI
VY
dz
(2.4)
The Eqns. 2.3 and 2.4 may be differentiated with respect to z , so that
d 2V
dz
2
Z
dI
dz
(2.5)
Y
dV
dz
(2.6)
and
d 2I
dz
2
Using Eqns. 2.2 and 2.4, this leads to differential equations
d 2V
dz 2
ZYV
(2.7)
18
Transmission lines
and
d 2I
dz 2
ZYI
(2.8)
These are the differential equations of the lines, fundamental to the
circuits of distributed elements. If we assume V V0 e j t and I I 0 e j t , they
may be the forms of a wave equation.
Solution to Eqns. 2.7 and 2.8 follow direct non-conventional methods.
Eqution 2.7 becomes
2
ZY V 0
so that
ZY
Equations 2.7 and 2.8 represents second order differential equations.
Here,
is the propagation constant. The results indicate two solutions; one for
positive sign and other for negative sign before the radical. The solutions of the
differential equations then are
V Ae
ZY dz
Be
ZY dz
(2.9)
I Ce
ZY dz
De
ZY dz
(2.10)
and
A,B,C and D are the arbitrary constants of integration.
Since, the distance is measured from the receiving end of the line, it is
possible to assign boundary conditions of the type, when
z 0 , I I L ,V V L
so that
VL A B
(2.11)
(2.12)
IL C D
Since, a second set of boundary conditions are not available, the same set
can be used again as a new set of equations formed by differentiating Eqns. 2.9
and 2.10
dV
Ae ZY dz Be ZY dz
dz
from the Eqn. 2.2 this becomes
or
IZ A ZY e
IA
Y
e
Z
ZY z
ZY z
B
B ZY e
Y
e
Z
ZYz Z
ZY z
(2.13)
19
Microwave Circuits & Components
In a similar manner
dI
C ZY e
dz
ZY z
ZY z
D ZY e
or,
YV C ZY e
V C
Z
e
Y
ZY z
ZY z
D ZY e
D
Z
e
Y
(2.14)
ZY z
ZY z
At load end z 0 , Eqns. 2.13 and 2.14 becomes
IL A
Y
Y
B
Z
Z
(2.15)
VL C
Z
Z
D
Y
Y
(2.16)
Simultaneous solution of Eqns. 2.11, 2.12, 2.15 and 2.16 along with the
Z
fact that V L I L Z L and factor
has dimensions of impedance, let us denote
Y
it as Z 0. This leads to solutions of constants A, B, C and D as
A
VL I L
2
2
Z0
Z VL
1
Y
2
ZL
B
VL I L
2
2
Z0
Z VL
1
Y
2
ZL
C
I L VL
2
2
I
Z
Y
L 1 L
Z
2
Z0
D
I L VL
2
2
I
Z
Y
L 1 L
Z
2
Z0
The solutions of the differential equations of the transmission lines may
be written as
V
VL
2
Z0
1
e
Z
L
ZY z
Z
1 0 e
Z
L
20
ZY z
(2.17a)
Transmission lines
Then
V VL
Z L Z0
2Z L
e
ZY z
Z L Z0
e
Z L Z0
ZY z
(2.17b)
and
I
IL
2
ZL
1
e
Z0
ZY z
ZY z
Z
1 L e
Z0
(2.18a)
Then
I IL
Z L Zo
2Z0
ZY z
e
Z L Z0
e
Z L Z0
ZY z
(2.18b)
Z L Z0
It is quite obvious from the above relations that Z Z
L
0
expression for reflection coefficient at the load end L .
L
Z L Z0
Z L Z0
is the
(2.19)
It can be seen from this expression that if Z L Z 0 , the reflection
coefficient L 0. Above relations then can be written as
V VL
and
I IL
Z L Z 0
2Z L
Z L Z 0
2Z 0
e
e
ZY z
ZY z
e
e
ZY z
(2.20)
ZY z
(2.21)
It can be concluded from this expression that wherever voltage maxima
occurs, current minima also occurs. Equations 2.17a and 2.17b may also be
rearranged as
e
V VL
e
I IL
ZY z
ZY z
e
2
e
2
ZY z
ZY z
I Z e
L
0
V
L
Z
0
e
ZY z
ZY z
e
2
e
2
ZY z
ZY z
and this can be expressed as
V z VL cos h ZY z I L Z 0 sin h ZY z
21
(2.22)
Microwave Circuits & Components
I z I L cos h ZY z
2.3
VL
sin h ZY z
Z0
(2.23)
WAVE PROPAGATION CONSTANT
Equation 2.23 may be written for the sending end current I S of line of
length l as
Z
I S I L cos h ZY l L sin h ZY l
Z
0
If the line is terminated in Z L Z 0 , then
I S I L cos h ZY l sin h ZY l
from which
IS
e ZY l
IL
(2.24)
This can be recognised as propagation equation of line and ZY can be
recognised as the propagation constant, . The result of Eqn. 2.24 is simply a
restatement for the line of basic relations between input and output currents. The
propagation constant, is defined per unit length of line. Since, Z and Y are
complex quantities, the propagation constant is also complex, i.e.,
j
(2.25)
where, is known as attenuation constant per unit length and is phase constant
per unit length.
2.4
CHARACTERISTIC IMPEDANCE, Z 0
For an asymmetrical T or network it has been shown1 that the interactive
impedance of a network is that when connected to one pair of terminals produces
like impedance at other pair of terminals. The asymmetrical T network is shown in
Fig. 2.2.
when Z R Z it
Z in Z1
Z 3 Z 2 Z it
Z it
Z 2 Z 3 Z it
This yields
Z it
Z1 Z 2
2
Z1 Z 2 2 Z
1 Z2
4
22
Z3
(2.26)
Transmission lines
Z1
Z2
Z3
ZIN
ZR
Figure 2.2. A symmetrical T-network
Similarly, the image impedances of a network are the impedances which
will simultaneously terminate the input and output in such a way that the
impedances in both directions at both terminals are equal.
For the network shown in Fig. 2.3 the image impedance Z I1 at input
terminal 1
Z I 1 Z1
Z2 Z I Z3
2
Z2 Z3 Z I
2
and the image impedance at the terminal Z I 2
Z1 Z I 1 Z 3
ZI 2 Z2
Z1 Z 3 Z I 1
Solution of these two equations yields
ZI 1
Z1 Z 3
Z 1 Z 2 Z 2 Z 3 Z 3 Z 1
Z 2 Z3
(2.27)
ZI 2
Z 2 Z3
Z1Z 2 Z 2 Z 3 Z 3 Z1
Z1 Z 3
(2.28)
Z2
Z1
Z3
Z11
Z12
Figure 2.3. Image impedance in a symmetrical network
23
Microwave Circuits & Components
For a symmetrical T section, two series arms are equal, i.e., Z 1 Z 2 .
Under this condition
Z it Z 1 2 2 Z 1 Z 3
and
Z I 1 Z I 2 Z1 2 2 Z1 Z 3
Therefore the image and iterative impedance combine into single value
which is given a special name, the characteristic impedance, Z 0 . For network
shown in Fig. 2.4, the characteristic impedance
Z 0 Z1 Z 2
Z12
4
Corresponding to Z1 of lumped iterative structure one has Zz for the
1
transmission line and corresponding to Z 2 , it is
. Thus
Yz
Z 2
Z
Z 0 Z1 Z 2 1 Z1 Z 2 1 1
4
4Z 2
Substituting the value of Z1 and Z2 for transmission line
Z ZY z 2
1
4
z 0 Y
Z 0 lim
Z
Y
R j l
G j C
(2.29)
Sometimes, the characteristic impedance is defined as input impedance of
line of infinite length when the effect of reflected wave is absent. This is not a good
definition since infinite line does not exist in practice. Better definition perhaps is
that the input characteristic impedance is the impedance of a line terminated in its
characteristic impedance or in other words: The characteristic impedance of a line
is that impedance on termination, produces the same impedance at the input.
Z1/2
Z0
Z1/2
Z2
Z0
Figure 2.4. Symmetrical T-network terminated in characteristic impedance Z 0
24
Transmission lines
The importance of characteristic impedance Z0 will be discussed later
whether it is complex quantity or real. If it has to be real quantity, how can it be
achieved? This will also be discussed later.
2.5
PHYSICAL SIGNIFICANCE OF PROPAGATION CONSTANT
EQUATIONS
Equations 2.20 and 2.21 can be written as
V z V L
Z L Z 0 e z e z
(2.30)
I z I L
Z L Z 0 e z e z
(2.31)
and
2Z L
2Z 0
Similarly, Eqns. 2.22 and 2.23 can be written as
V z VL cos h z I L Z 0 sin h z
I z I L cos h z
VL
sin h z
Z0
(2.32)
(2.33)
These two sets of equations lead to
Z z
V z
Z0
I z
Z z
e z e z
(2.34)
e z e z
V z
Z cos h z Z 0 sin h z
Z0 L
I z
Z 0 cos h z Z L sin h z
(2.35)
For a lossless line 0 and j . Therefore for a lossless line
Z z
V z
Z cos z jZ 0 sin z
Z0 L
I z
Z 0 cos z jZ L sin z
(2.36)
As a particular case it is of interest to find the value of Z z , the sending
end input impedance when the line is terminated in its characteristic impedance,
i.e., when Z L Z 0 this conforms to the definition of characteristic impedance for
the lumped network and establishes the validity of operation in the definition
Z 0 Z Y . Thus, the description of circuit performance through Z 0 and is
same for both circuits with lumped elements and distributed elements.
Thus, a line of finite length terminated in a load equivalent to its
characteristic impedance appears to the sending end generator as a line of input
impedance Z 0 . For a line of finite length terminated in its characteristic impedance
i.e., when Z L Z 0 .
25
Microwave Circuits & Components
V z V L e
z
(2.37)
I z I L e z
(2.38)
where, V L and I L are load end voltage and current. Thus, voltage and current
values change with distance in accordance with the relation e z. The relation may
also be written with respect to the sending end.
V z V S e z
(2.39)
I z I S e z
(2.40)
where, V S and I S are sending end voltage and current and are also functions of
time j so that
V z VS0 e j t e z e j z
Thus, V z is function of both time and distance. This is the property of
any solution to the wave equation. It is seen that voltage and current become
progressively smaller because of the factor e z . This is the logic of calling as
attenuation constant. Also voltage and current lag progressively more and more as
z increases because of the increasing angle inherent in e j z which is equivalent to
z .
Thus, there exists a voltage and current wave travelling down the line
from the generator. It should be remembered that although analysis is given in
terms of voltage and current, it is actually the physical quantity that varies as the
wave is propagating energy, and its transfer is taking place in terms of electric and
magnetic field to the surrounding region. In a simple two-wave transmission line,
voltage and current are convenient means of observing the field.
2.6
PROPAGATION FACTOR & CHARACTERISTIC IMPEDANCE
OF TRANSMISSION LINE
We have seen that
Z R j L ,
and Y G j C
as
The propagation constant and the characteristic impedance can be written
ZY
Z0
Z
Y
R
j L G j C
R j L
G j C
26
Transmission lines
The transmission lines are considered to be of three types as far as
attenuation characteristics are concerned. These are:
(a) Ideal or lossless line
(b) Line with low losses
(c) Line with high losses
Of these three only line of interest is line with low losses. However, the
concepts involved with the first case, i.e., an ideal line with no losses is also
important. Therefore, in the present list only two, i.e., (a) lossless line, and (b) line
with low losses will be discussed in detail.
2.6.1
Ideal or Lossless Line
For this case both R and G are zero, therefore
j LC
(2.40a)
Z0 L C
(2.40b)
The real and imaginary parts of are
(2.41a)
0
(2.41b)
LC
It is to be noted that the phase velocity
1
vp
LC
We know that L and C are distributed line parameters, i.e., inductance
per unit length and capacitance per unit length. It should be further noted that the
characteristic impedance is purely resistive and is real. It also means that the
current and voltage in this case for a traveling wave are in phase, i.e.,
V z Ae z , z IZ 0
(2.42a)
and
I z
A z V
e
Z0
Z0
(2.42b)
It should be further noted that since the voltage and current are varying
sinusoidally the power growing is the simple time average of product VI which is
P
2.6.2
1
1V2 1 2
VI
I Z0
2
2 Z0 2
(2.43)
Line with Small Losses
In problems involving long lines or lines with appreciable attenuation the
above simplifying assumptions are no longer valid. In many cases, it will be true
27
Microwave Circuits & Components
that R L and G C which are valid assumptions for the practical lines.
For such cases the expressions for and Z 0 can be simplified to
12
R
j LC 1
j L
12
G
1
j C
(2.44)
This can be expanded by the binomial theorem, the solution becomes
R
R2
G
G2
j LC 1
1
2 j L 8 j 2 2 L2
2 j C 8 j 2 2 C 2
Neglecting the terms higher than the second order,
R
G
2 jC
2 jL
j j LC 1
R2
RG
G2
2 2
4 2 LC
8 2 C 2
8 L
2
R
G 1 R
G
j LC 1
2 j L 2 j C 2 2 j L 2 j C
If the primed symbols indicate quantities for lines with no losses, then
' LC
and
2
'
(2.45)
2
1 C d
2 ' '
' 1
(2.46)
where, C and d are the conductor attenuation constant and dielectric attenuation
constant. They are discussed in the next section. The wavelength relation is
2
1 C d
2
' 1
(2.47)
It may be noted that when C d , and have same values as for
the ideal case, i.e., lines with no losses, this statement is equivalent to specifying
R
G
R G
L C or, L C
28
Transmission lines
The greatest change in obviously occurs when one type of attenuation
is large compared to other.
2.6.2.1
Attenuation in transmission line with low losses
Writing the real part of Eqn. 2.44, one obtains
R
2
C G
L 2
L
C
neper/m
Using the ideal line characteristic impedance
Z 0
L
C
and the ideal line characteristic admittance defined as
Y0
C
L
1
Z 0
Equation for may be written as
R
G
2 Z 0 2Y0
neper/m
(2.48)
It is easily recognised that the first term on the right hand is due to
conduction loss and the second is due to dielectric loss. The Eqn. 2.48 may be
expressed as
C d
where
2.6.2.2
(2.49a)
C
R
2Z 0
(2.49b)
d
G
2Y0
(2.49c)
Characteristic impedance of transmission lines with low losses
As shown earlier the characteristic impedance is
12
Z0
L
R
1
C
j L
Z0
L
C
G
1
j C
1 2
R
R2
2 2 2
1
2
L
8j L
29
G
3G 2
2 2 2
1
2
j
C
8j C
Microwave Circuits & Components
If terms of order higher than the second are dropped from above
multiplication these become
Z0
L 1 R 2
RG
3G 2
1
C 2 4 2 L2 2 2 LC 4 2 C
R
G
j
2 C 2 C
Simplifying in the same manner as in Eqn. 2.49(a) finally one obtains
1
3
Z 0 Z 0 1 C d C d j C d
2
(2.50)
It can be seen from the above equation that if C can be made equal to
d , the characteristic impedance is equal to its value for an ideal line with zero
losses which is real and purely resistive. This is a very important result since a line
terminated in its characteristic impedance gives real or resistive impedance. When
connected to a generator having purely resistive internal impedance and having the
value equal to the characteristic impedance of the line, will transfer maximum
energy to the transmission line. The transmission line having primarily reactive
elements, will, not absorb or dissipate energy and will transmit practically all
energy (or power) to the terminating impedance. Therefore, transmission line with
low losses can be made to have real and resistive characteristic impedance Z 0 .
Thus, we have seen that the dielectric and conductor losses add in case of
, but bothand Z 0 remain equal to their ideal line values ' and Z 0 so long as
the conductor losses are equal to dielectric losses. It may be noted that in all cases,
the attenuation is in nepers per radian of the line length and it enters rather in
simple way into perturbation of each quantity from its ideal line value. In and in
real part of Z 0, the attenuation is squared and is therefore unimportant. In
imaginary part of Z 0 it enters to the first power and is therefore appreciable.
2.7
WAVEFORM DISTORTIONS
Under this section only two types of distortions are considered. One is
frequency distortion and the other is delay distortion.
2.7.1
Frequency Distortion
Very often the attenuation of line is dependent on frequency. The
resistance varies with frequency if the skin effect (which is frequency dependent) is
effective. Even the dielectric properties to some extent depend upon frequency.
Therefore, both losses are functions of frequencies. If the length of line is
sufficiently long the attenuation becomes frequency dependent, thus, the amplitude
of different frequency components of a signal is not the same at the load end of the
line as it is at the input end, which distorts the signal. This is known as frequency
distortion. It becomes important if the distance of transmitter from studio is large
as in the case of audio transmission for broadcast. Broadcast centres can be many
30
Transmission lines
kilometers away from the transmitter end. To correct it, the signal at the transmitter
end is passed through an equaliser before it is amplified and used for modulating
the characteristic of the line rf signal. The equaliser should have opposite frequency
characteristics with respect to frequency, i.e., whenever output is large the equaliser
should reduce the signal strength and whenever signal is small the reduction in
signal strength should be small. This is to obtain the signal as it originated at the
studio. If the line length is more than one tenth of wavelength this distortion
becomes effective.
For video transmission, frequency distortion is not so important as eye is
not sensitive to this distortion. Even if it is effective, it can be adjusted at the
receiver end. For audio transmission this is very important since ear is very
sensitive to this distortion. The quality of music programme will be greatly affected
by this distortion.
2.7.2
Delay Distortion
Since the phase velocity of propagation is
vp
It is apparent that if both and are not frequency dependent in the same
manner, then the velocity of propagation will in general, be some function of
frequency. All the frequencies applied to the transmission line will not be
transmitted at the same time. Some frequencies would be delayed more than others.
For an applied voice voltage wave the received wave form will not be identical
with the input wave form at the sending end since some components will be delayed
with respect to other component. This phenomenon is known as delay or phase
distortion.
Delay distortion is of relatively minor importance to voice and music
transmission because of the characteristic of the ear. However, it is important for
circuits used for picture transmission. Application of coaxial cable has been made
to overcome this difficulty. In such cables the internal inductance is low at high
frequency because of skin effect, the resistance is small because of large
conductance, capacitance and leakage are small because of the use of a dielectric
with minimum spaces. If propagation velocity is raised, it becomes nearly equal for
all frequencies.
2.8.1
The Open Two-wire Line
Two-wire lines are used upto frequency of 500 MHz. TV twin-lead line
which is used to connect antenna to television set is an example of this type of line.
It is not used at higher frequencies because the loss due to radiation becomes too
large. A two-wire line is shown in Fig. 2.5 along with electric and magnetic lines of
force. For propagation of waves on the open two-wire line both electric and
magnetic field are perpendicular to the direction of propagation 2. This mode of
transmission is generally referred as TEM mode. However, it must be noted that
other modes of propagation are also possible in which electric and magnetic fields
have longitudinal components as well.
31
Microwave Circuits & Components
r0
r0
d0
d0
Figure 2.5. Schematic diagram of parallel two wire transmission line
The electric and magnetic line of force for a parallel two-wire
transmission line is shown in Fig. 2.6.
For TEM mode, the expression of inductance and capacitance per unit
length for no distortion is given as
L
C
0 R
S
n
a
(2.51a)
0 R
S
n
a
(2.51b)
The symbols have their usual meanings.
12
It may be noted that 0 4 107 H/m and 0 8.854 10 F/m .
These are the permittivity and permeability of free space and R and R are
relative permittivity/dielectric constant and relative permeability respectively. For
air, their values are unity.
Figure 2.6. Field patterns in parallel wire transmission line
32
Transmission lines
The characteristic impedance Z0 is given by the expression
Z 0 120
R S
S
l n 276 R log
R a
R
a
(2 52)
and the velocity of propagation is
v
1
0 0 R R
c
R R
(2.53)
Approximate expressions for shunt conductance and series resistance
have also been calculated. These are
(a)
Shunt conductance per unit length
G
(b)
0 R
ln
S
a
tan
(2.54)
Series resistance per unit length
1
R
a S
(2.55)
where, s is the conductivity of conducting material and S is the skin depth.
Since proximity effect is neglected which means that these expressions are valid
for >>4a.
2.8.2
Coaxial Line
The coaxial line consists of two connective conductors; one is hollow
metallic conductor (outside) and the other is solid conductor (inside). The
schematic diagram is shown in the Figs. 2.7 and 2.8 (Rizzi 3 ).
, , are dielectric constant, conductivity and permeability of region
between two conductors; 1, are same constants for the conducting region.
The two conductors act as two lines of the transmission line structure. It
can be seen that in this case the field remains confined between the two conductors,
therefore the radiation losses will be minimum. The TEM mode for a coaxial line is
shown in Fig. 2.9. Higher order modes can also propagate. One of these modes
TE11 is shown in Fig. 2.10.
2.8.2.1
A typical coaxial line
The coaxial line can be rigid as well as flexible. The dielectric inside such
line makes it flexible. Their characteristic impedances are either 50, 75 or 90 W.
Note that it is real and purely resistive. It has been earlier explained why it should
be so. Rigid coaxial lines use beads for holding the central conductor. To avoid
discontinuity in characteristic impedance it is of different radius at the point where
beads are situated as shown in the Fig. 2.11.
33
Microwave Circuits & Components
SOLID
CONDUCTOR
METALLIC CONDUCTOR
Figure 2.7. A typical coaxial line
1,
1,
2,
METALLIC PART
Figure 2.8. View of coaxial transmission line
ELECTRIC LINE OF FORCE
MAGNETIC LINE OF FORCE
Figure 2.9. TEM mode for coaxial line
(b)
(a)
OUTER
INNER
CONDUCTOR
CONDUCTOR
Figure 2.10. The field pattern for the TE 11 mode in coaxial line
34
Transmission lines
b
2ad
2a
Figure 2.11. Support for inner conductor using dielectric beads
In practice, however, the radius as shown in figure does not result in
perfect match because of discontinuity in shape of the inner conductor which
produces fringing fields. It is found that if inner radius is decreased by another
10 percent, this result in good match over a large frequency bandwidth of the order
of many octaves.
2.8.2.2
Analysis of coaxial line
It would be analysed under two sections; first for ideal case then for
coaxial line with small losses.
Ideal coaxial line
The appropriate relations for ideal or lossless coaxial line are
=0
LC
Z0
L
C
The basic assumption for low losses leads to assumption that the
conductor has infinite conductivity. This suggests that because of skin effect the
current flows entirely on the surface of the conductor. Referring to the Fig. 2.8, the
inductance and capacitance for unit length of ideal coaxial line may be written as
L
1
lnb a
2
(2.56)
C
21
lnb a
(2.57)
The proof of this can be found in any standard text. As is evident 1 and
1 apply to dielectric medium between conductors. Thus,
35
Microwave Circuits & Components
11
(2.57)
0 0 R1 R1
c
R1 R1
Therefore, the phase velocity v p
vp
c
(2.58)
R1 R1
and the wavelength
vp
where, is the frequency of operation. The expression for can be modified to
c
R1 R1
0
(2.59)
R1 R1
Usually for non-magnetic material R1 has the value of unity,,
consequently
2
0
2
c
vp
R1
R1
0
R1
It will be noted that the phase velocity is independent of frequency; that is
an ideal coaxial line is a non-dispersive transmission line. Consequently, group and
signal velocities are equal to the phase velocity. Similarly
Z0
1 1
1 0
2 1 lnb a 2 0
R1
R1 lnb a
Let us restrict the consideration to dielectrics for which
Inserting the numerical value 376.7 for the quantity
36
R1 1 .
0
, the so called
0
Transmission lines
impedance of free space, we obtain
Z0
60.0
R1
lnb a
thus
Z0
138.0
b
log10
a
R1
(2.60)
The power in the propagating wave may be written as
V 2 R1
1V2
P
2 Z 0 120 lnb a
(2.61)
If the electric field intensity at the centre conductor is denoted by E a , the
voltage may be shown as
b
V
a
Ea a
dr E a a lnb a
r
E a 2 a 2 R1
1
2 2
2
P
E a a lnb a
lnb a
2Z 0
120
(2.62)
Actual or low loss line
The line characteristics require specific evaluation of parameters C , d,
and Z 0' (for ideal line primed symbols will be used).
The value of attenuation due to conductors is
R
C '
2Z 0
Here, Z 0' is the characteristic impedance neglecting losses and is given
by the expression
1 0 R1
Z 0'
ln( b / a )
(2.63)
2 0 R1
Since, the current flows near the surface of the conductors, a calculation
of effective resistance requires consideration of skin effect. The current density
37
Microwave Circuits & Components
has its maximum value at the surface of the conductor and falls off exponentially to
1 e of the maximum value at a distance
1
(2.64)
2 2
where, is skin depth. The losses are the same as if total current of uniform
distribution flowing in the walls of the tubular conductor of wall thickness . The
effective resistance per unit length of centre conductor is then
Ra
1
1
2a 2 2a
2
2
A similar expression may be obtained for the outer conductor replacing
a by b. The total resistance is therefore
Ra Rb R
C
1
2
1
2
2 1 1
2 a b
2 1 1 1
. .2 0
0
2 a b 2
R1
R1
1
lnb a
2 R1 1 1 1
1
2 R1 a b lnb a 376.7
1
2
2.63 10 5
lnb a
R R
1
2
2 R
1
12
1 1
a b
neper/m
(2.65)
The attenuation due to dielectric is given by
G
d
2Y
0
gives
A simple calculation for a dielectric whose effective conductivity is 1 ,
G
21
lnb a
d
1
2
1 21 1 1
2 lnb a 2 1 lnb a
1
1
38
(2.66)
Transmission lines
It may be noted here that the losses are independent of the dimensions of
the line. The effective conductivity may be true conductivity which could be
measured with DC ohmmeter. Conductivity may be due wholly or part by
hysteresis to which occurs in molecules of dielectric as they are subjected to
polarisation by high frequency fields. Dielectric constant 1 and effective
conductivity 1 specify the dielectric constant and conductivity of the material.
The current density in the dielectric medium is
J 1 E 1
E
t
By Ohm’s law 1 E is the conduction current including both true
conduction current and current supplying hysteresis losses. The term 1 E is the
t
displacement current. For a harmonic voltage
E E0 j t
J 1 j1 E
The conduction current is in phase with the electric field and therefore
represents power loss. The displacement current is out of phase and does not
represent any losses. Since the conduction current is usually small compared to
displacement current.
J j1 1 j 1 E
1
It is convenient to define a complex dielectric constant
C1 1 1 j
1
J j C1 E
(2.67)
The conductivity no longer appears explicitly but is contained in the
complex dielectric constant. If C1 is substituted for 1 in the equation derived on
the basis of an ideal dielectric with a simple dielectric constant 1, the resulting
equation will take into account the non-ideal character of the dielectric. The
complex dielectric constant is usually expressed as
C1 1' j1"
where, 1' 1 , 1"
1
(imaginary part of dielectric constant )
39
Microwave Circuits & Components
Thus the results are presented diagramatically in the Fig. 2.12. The power
loss per unit volume is
P1 JE cos
JE sin
Therefore, the power factor
p cos sin
the ratio
1"
tan
'
(2.68)
is called the loss tangent and for small angles, it is almost identical for the power
factor. Therefore, the dielectric attenuation is
d
1
2
1
1
11 neper/m
1
1
d C1 R1 0 0 tan
c
R R tan
1
tan
1
neper/m
neper/m
neper/m
(2.69)
1 is the line wavelength
The dielectric attenuation factor d is therefore,
d tan
1
tan
2
nepers/line wavelength
neper/radian
(2.70)
Displacement
Current
J
j ' E
'' E
Figure 2.12. Tan : The need and imaginary electric pack of dielectric constant
(concept of displacement circuit).
40
Transmission lines
2.9
TRANSMISSION LINE AT HIGH FREQUENCIES
At frequencies above 100 MHz the physical length of lines are normally
small compared to wavelength. This is because at 100 MHz the wavelength is
3.0 m and complete standing wave pattern can be seen over a length of 1.5 m.
Entire impedance variation from maximum to minimum occurs over a length of
1.5 m. Of course at high frequencies the losses would increase, but since the
physical length is small it can be neglected in most of the cases and the lines may be
derived from the following equation
j 2 z
V z V1 e j t e j z 1 e
I z
V1
Z0
j 2 z
e j t e j z 1
(2.71a)
(2.71b)
j t j z
and remember that the reflection coefficient
If we write V1 V1 e e
Z L Z0
. Also remember that e j
Z L Z0
V z V1e j z 1 e j 2 z
I z
2.10
V1 j z
e 1 e j 2 z
Z0
(2.72a)
(2.72b)
IMPEDANCE & ADMITTANCE OF SHORT-CIRCUITED &
OPEN-CIRCUITED LINES
Equation 2.37 can be written for a lossless line when the length of line is l
and the terminating impedance Z L .
Z cos l jZ 0 sin l
Z in Z 0 L
Z 0 cos l jZ L sin l
(2.73)
The input impedance of short circuited line is found by setting Z L 0
the input impedance is therefore
Z SC jZ 0 tan l
(2.74)
A graph of input impedance as a function of l is shown in Fig. 2.13. It
can be seen that input impedance assumes all possible reactive values ranging from
positive infinity to negative infinity as l varies from 0 to. The length varies
from 0 to one-half of a wavelength. The properties of short circuited quarter-wave
line resembles those of anti resonant circuit. This property recurs when l is odd
41
Microwave Circuits & Components
multiple of 2 . The property of line with length corresponding to even multiple
of 2 resembles that of a resonant circuit with zero impedance. The results are
valid for all frequencies. Short circuited lines are known as stubs. The short
circuited stubs are used as variable reactances. Open circuited lines are rarely used
because it is difficult to vary their lengths and because of radiation losses at ends.
When the losses are taken into account the hyperbolic functions may be
used for the impedance. The input impedance for short circuited line with losses is
sinh l
Z SC Z 0 tanh l Z 0
cosh l
(2.75)
sin h l cos l jcos h l sin
Z0
cos h l cos l jsin h l sin
For an anti resonant stubs, l
values
2
l
l
(2.76)
, cos l 0, sin l 1 . Using these
cosh l
Z SC Z 0
sinh l
(2.77a)
Under the assumption that l 1 , i.e., cos h l 1 and sin h l l,
it follows that
Z SC
Z SC
Z0
l
(2.77b)
Figure 2.13. Input impedance of a short circuited line
42
l
Transmission lines
1
C
R
R
2
L 2Z 0
where, Z 0 is the characteristic impedance of a line with zero attenuation. The
short circuit input impedance
If G 0 ,
Z SC
2Z 0 2
R
(2.78)
where, R in this expression is different from earlier expression, where it was
resistance per unit length. In this expression R is the total ohmic resistance of stub
and the radiation losses are neglected.
2.10.1
Input Impedance of Open Circuited Line
The input impedance of the open circuited line of length l can be obtained
by writing the expression of input impedance in the following form
cos l j Z 0 sin l Z L
Z in Z 0
Z 0 cos l Z L j sin l
(2.79)
Taking the limit of right hand side as Z L , gives the following results
cos l j Z 0sin l Z L
Z 0C limZ L Z 0
Z 0cos l Z L jsin l
(2.80)
Z 0cos l
jsin l
Z 0C jZ 0 cot l
(2.81)
The graph of Z 0C as a function of l is shown in Fig. 2.14
2.10.2
Quality Factor (Q) of resonant lines
The Q of a resonant line can be defined as
Q
W
Maximum energy stored
Energy dissipated per second
P
(2.82)
where, W is the maximum stored energy and P is the average dissipated power.
It is shown earlier, the instantaneous voltage and current distributions
along a line are
v z,t 2 VL e j t Z Lcos z jZ 0sin z
(2.83a)
2 VL
i z,t
Z 0
(2.83b)
e j t Z 0cos z jZ Lsin z
43
Microwave Circuits & Components
Z OC
l
3/2
Figure 2.14. The values of ZOC for different values of l
It can be shown that voltage and current are 90° out of phase. The total
energy stored remains constant in a resonant or an antiresonant line. It is, therefore,
possible to calculate W from either the magnetic field or electric field when either
voltage or current remains zero at every point in the line. If the excitation is cosine
wave current when voltage is zero everywhere, the current distribution is given by
i z 2 I Lcos z
(2.84)
the voltage zero everywhere. Therefore,
W
2
1 l0
L 2 I L cos z dz
2 0
(2.85)
If the length of line is 4 , the stored energy is
W
L IL
2
(2.86)
4
The average power PG lost in shunt conductance is found to be
PG
G
2
2
0
2
I L GZ 0
2
2
2 I L Z 0 sin z dz
2
2
0
(2.87a)
2
2 I L Z 0 sin z dz
I L GZ 0 2
4
(2.87b)
(2.88)
The average power dissipated in series resistance
44
Transmission lines
PR
1
R
2
2
0
2
2 I L cos z dz
(2.89a)
2
I L R2
4
Therefore,
W
Q
P P
G
R
(2.89b)
2
L IL
LC
2
2
RC LG
I L R Z0 G
(2.90)
It may noted that if G 0
Q
L
R
which is a standard relation.
2.11
QUARTER WAVE LINE
A quarter wave line can be used for impedance matching. This depends
on impedance transformation by quarter wave line. It can be seen from Eqn. 2.73
that the input impedance of a line for which l
Z in
2
Z0
ZL
2
is
(2.91)
As shown in Fig. 2.15, quarter wave line transforms an impedance in a
manner that is analogous to the way a transformer with unity coupling in lumped
circuits transform impedance Z 2 in the secondary to M 2 2 Z 2 in the primary
where, M is the mutual inductance. A quarter wavelength line can match two
impedances (resistive) Z1 and Z 2 if the characteristic impedance of the quarter
wavelength line is Z 0 given by the following relation
Z 0 Z 1 Z 2 , since
Z1
Z02
Z2
(2.92)
t
4
Z1
Z0
Z2
Z1
Figure 2.15. Quarter wave line transforms impedance analogous to transformer
45
Microwave Circuits & Components
The two disadvantages of this matching circuit are (a) It will match only
resistive impedance and (b) Such a matching arrangement is a narrow band device.
In case, it is desired to match a line of resistive characteristic impedance
Z 0 to a complex load Z L R L jX L, it has been shown that there are two series
of points at which the impedances are purely resistive. These are the voltage
maxima and voltage minima. The impedances are
1
Z Vmax
Z 0 SZ 0
1
(2.93)
and
1
Z0
(2.94)
Z Vmin
Z 0
S
1
where, Z Vmax is the input impedance at voltage maxima and Z Vmin is the
impedance at voltage minima. It can be seen that these are purely resistances.
Therefore, matching at these points can be obtained as shown in Fig. 2.16.
Z1
Z0
Z2
Z1
2
Z0
Z2
Figure 2.16. Shows the relation of the characteristic impedance Z 0 with the
impedances Z1 and Z 2 .
In above expression Sis the Voltage Standing Wave Ratio (VSWR) which
is defined as
S
Vmax 1
Vmin 1
where, is the voltage reflection coefficient.
Figure 2.17(a) shows that a line of characteristic impedance
Z0
S
can be
used to match a complex impedance Z L to a line of characteristic impedance Z 0 .
The figure shows the line of characteristic impedance Z 0 Z 0 can
match a complex impedance Z L with a line of characteristic impedance Z 0 .
46
Transmission lines
t / 4
CHARACTERISTIC
IMPEDANCE QUARTER
S
WAVELINE Z0=Z0
MAXIMUM
Z'0
Z0
Z0
VOLTAGE
Z 0 R1 jX
IMPEDANCE = Z0S
Figure 2.17(a). Matching at voltage maximum where impedance is product of
Zo and the voltage standing wave ratio S.
c
h
a
r
a
c
t
e
r
i
s
t
i
c
i
m
p
e
d
a
n
c
e
These behaviours are shown in Figs. 2.17(a) and 2.17(b). Of course it is true that
this technique is very inconvenient and it is rarely used.
It can be seen that similar matching can be carried out with voltage
minimum where the impedance as
characteristic impedance
this impedance.
CHARACTERISTIC
IMPEDANCE QUARTER
WAVELINE
Z0
S
Z0
. At this point if a quarter wave line of
S
is connected it will match the rest of the line with
Z'0=Z0/
VOLTAGE MINIMUM
S
t / 4
Z0
Z'0
Z0
Z1=R1
IMPEDANCE = Z0/S
Figure 2.17(b). Matching at voltage minimum where impedance is product of
1
characteristic impedance Zo and S where S is the voltage standing
wave ratio.
2.11.1
Impedance Matching by Stubbing
Suppose a line of characteristic impedance Z0 is to be matched to a
complex impedance ZL. The principle of matching of impedance ZL is by
transforming impedance ZL to Z0 + jB by the impedance transforming property of
47
Microwave Circuits & Components
d
zL
z0
I
Figure 2.18. Matching a line of characteristic impedance Z 0 with a complex
impedance ZL by a single stub.
a line and cancel impedance by using a stub. Since stub is connected in parallel it is
advisable to use admittance instead of impedance (Fig 2.18).
The input admittance of line at a distance of Z from the terminating end
(where ZL is connected) can be calculated by using following transmission line
equation for lossless line
Y z
I z
Z 0 cos z jZ L sin z
V z Z 0 Z L cos z jZ 0 sin z
(2.95)
If the distance of voltage minimum from the stubbing point is d, the
1
stubbing point is selected where admittance is
jB, i.e.,
Z0
Z
Z 0cos d j 0 sin d
1
S
jBS
(2.96)
Z
Z0
Z 0 0 cos d jZ 0sin d
S
This expression can be simplified to
1 S j tan d 1
jBS
Z 0 1 jS tan d Z 0
(2.97)
where, S is the voltage standing wave ratio (VSWR)
Rationalising the left hand side of the above equation yields
S S tan 2 d
Z 0 (1 S 2 tan d )
tan d S 2 tan d 1
j
jBS
2
Z 0 (1 S tan d ) Z 0
Equating the real and imaginary part results in
48
(2.98)
Transmission lines
S S tan 2 d
1
1 S 2 tan 2 d
and
(2.99)
(1 S 2 ) tan d
BS
Z 0 (1 S 2 tan 2 d )
(2.100)
where, B S is the reactive component of the admittance at the stubbing point.
The solution of first equation yields
S S tan 2 d 1 S 2 tan 2 d
(2.101)
S 1 S S 1 tan 2 d
(2.102)
so that
tan d
1
S
(2.103)
Above equation gives the distance of stubbing position from the voltage
minimum in terms of voltage standing wave ratio.
Equation 2.74 suggests that the shunt susceptance introduced by a shorted
line of characteristic impedance Z 0 and the length l is
jB
1
j cot l
jZ 0 tan l
Z 0
(2.104a)
The purpose of introducing short-circuited stub is to neutralise the effect
of susceptance introduced by terminating impedance at a distance of d S .
Therefore
B BS
(2.104b)
Equation 2.100 for susceptance B S can be simplified to
1 S 1 S 1 S 1 S B
Z 0 1 S
S
Z0
(2.105)
Therefore, using the Eqns. 2.104(a) and (b)
1 S 1 S
Z0
1
cot l
Z 0
(2.106)
So
tan l
Z0 S
Z 0 1 S
(2.107)
49
Microwave Circuits & Components
Z 0 Z 0 ,
If
tan l
S
1 S
(2.108)
Both these equations define position of stubbing and the length of the
stub. It can be said that a single stub can be used for matching a complex
impedance. Two variables are needed for matching the real and imaginary parts of
impedance. In the present case these are
Position of stubbing d from the voltage minimum. Normally the first
voltage minimum from the load is used though it is possible to use any
voltage minima.
The length of stub which shunts the impedance at the point of stubbing
(a)
(b)
which in the present case is l . The expression for d and l are
d
1
1
tan 1
2
S
(2.109)
S
1
tan 1
2
1 S
(2.110)
and
l
These relations obtained in the present form are rarely used. Smith chart
provides much more convenient method (described in the subsequent chapter).
Transmission lines of any shape, structure or construction can make use of these
general relations. It could be waveguide, stripline, microstrip line, H-guide,
dielectric guide or any form of transmission line. This technique may be of
convenient form for wave guides but is very inconvenient for a coaxial line. It is
more convenient to use two stubs instead of single stub as shown in the Fig. 2.19.
The expressions for stubs are not developed for impedance since they are
quite involved. Smith chart is much better to use for finding out lengths l1 and l 2
d
z0
I2
z0
s1
s2
I1
Figure 2.19. Double stub impedance matching
50
zL
Transmission lines
for impedance matching remembering that impedance is to be matched at S 2
where the line impedance is Z 0 . This form of impedance matching is very
convenient for rigid coaxial line. One such arrangement is shown in Fig. 2.20.
Figure 2.20 shows double stub impedance matching for a coaxial line
system. The length l1 and l 2 are varied to get best possible matching. The
3
,
and so on.
separation between two stubs can have any value, typically
8 8
2.12
IMPEDANCE MEASUREMENT USING TRANSMISSION LINES
Here we describe the principle of measurement of unknown impedance
and not exactly how can it be used for measurement as it requires the description of
components including microwave generators, detector and other components.
From the principle point of view, the input impedance of line of
characteristic impedance Z 0 and terminated in load impedance Z L is given by
relation
Z L jZ 0 tan z
Z 0 jZ L tan z
Z in Z 0
(2.111)
In above relation it is assumed that the line is lossless. If the first minimum
i
s
s
i
t
u
a
t
e
d
a
t
a
d
i
s
t
a
ZVmin Z 0
n
c
e
o
f
d from the load end
Z L jZ 0 tan d
Z 0 jZ L tan d
(2.112)
Z0
, where S is
Remembering that the impedance at voltage minima is
S
the voltage standing wave ratio. Substituting it in the above relation
ZVmin
Z0
Z jZ 0 tan d
Z0 L
S
Z 0 jZ L tan d
(2.113)
I1
I2
Figure 2.20. Double stub matching for a coaxial line
51
Microwave Circuits & Components
It gives
Z 0 jZ L tan d S Z L jZ 0 tan d
(2.114)
i.e.,
Z L S j tan d Z 0 1 jS tan d
Therefore,
Z L Z0
1 jS tan d
S j tan d
(2.115)
All the quantities in this equation can be experimentally determined. This
relation can be used for determining the unknown terminating impedance. The
terminating impedance may be complex. Therefore, two relations are needed. As a
matter of fact the above relation will serve this purpose when the real and imaginary
parts are separated. However, this method takes lot of time to solve. It will be seen
later that use of Smith chart reduces this time to a few minutes.
The voltage standing wave ratio (S), can be fairly well defined and
determined. However, it is difficult to define d as it is not a quantity which is
absolutely defined, it is the distance of some voltage minimum to some point where
load is considered to be located. At low frequencies there is seldom doubt as to the
location of the terminals of the load. At higher frequencies close to or equal to
microwave frequency the definition of the terminals of the load, i.e., the reference
point is very often arbitrary.
Normally what is done is that the line is terminated in a short, the position
of minimum is taken as the reference point. Therefore, d is defined as separation
between this reference point and the minimum point when the line is terminated by
the impedance whose value is to be determined.
2.12.1
Position of Minimum when Impedance is Resistive
It is often convenient to recognise the nature of the terminating
impedance. Looking at the general expression for the impedance at voltage minima
Z0
Z jZ 0 tan d
Z0 L
S
Z 0 jZ L tan d
gives
(2.116)
Assuming ZL to be purely resistive and rationalisation of right hand side
Z
Z
tan d 0 L
2
Z0
1 tan d
Z L Z0
Z0
j
Z0 Z L
Z0 Z L
S
2
tan 2 d
Z Z tan d
Z
Z
0
L
0
L
(2.117)
The left hand side of this equation is purely resistive. This is only possible
if the imaginary part, i.e.,
52
Transmission lines
Z
Z
tan d 0 L
Z L Z0 0
Z0 Z L
tan 2 d
Z L Z0
(2.118)
This is possible under three conditions:
(a) Z L 0 ; there are no standing waves; the line is matched to the terminating
impedance.
t
(b) tan d 0 ; This would mean that d 0 ,
, t and so on. Under this
2
condition
ZL
Z0
,
S
therefore
Z L Z0
t 3 t
(c) tan d , this would mean that d
,
and so on. Under this
4
4
condition
Z L Z0 .
If the load impedance is purely resistive, minima occurs at two set of
points. These are
d 0,
t
, t ....... or
2
d
t 3 t
,
........
4
4
This is clarified in the Fig. 2.21. The figure shows the shape of standing
waves and the position of minima: (a) When Z L Z 0 (b) When Z L Z 0
2t
2t
3t
2
7 t
4
3t
2
5t
2
t
t
t
t
t
2
4
0
2
3t
4
Figure 2.21. Shape of standing waves and the position of minima
53
Microwave Circuits & Components
2.12.2
Position of Minimum when the Load is Purely Inductive
If the load inductance is purely inductive, i.e., when . The position
of minima can be derived as before. Let Z L jX L , then
Z0
jX L jZ 0 tan d
Z0
S
Z 0 X L tan d
(2.119)
The reactive load is situated at d 0 . Since, there is no imaginary term
on the left hand side
X L Z 0 tan d 0
(2.120)
or, if
tan d
XL
Z0
XL
. In this
Z0
respect it is different from the previous case when the line was terminated in a pure
resistance.
In this case the position of minima depends upon the ratio
However, if X L 0 , d tends to
If X L , d tend to
If
X L Z 0, d
t
3 t
,
,...
2
2
t 3 t
,
,...
4
4
t
3 t
,
,...
8
8
Therefore, in this case minima will be between
t
t
and
,
4
2
3 t
and t and so on.
4
The magnitude of inductance can be computed from the relation
X L Z 0 tan d
2.12.3
Position of Minimum when the Load is Purely Capacitive
If the load is purely capacitive, i.e., Z L jX C , then
Z 0 jX C jZ 0 tan d
S
Z 0 X C tan d
(2.121)
54
Transmission lines
The left hand side is zero in this case, therefore
tan d
XC
Z0
t
It is evident from the figure that if X C 0 , d 0 ,
,.....
4
if
XC , d
t 3 t
,
,...
4
4
Therefore, d is located between (a) 0 and
(c) t and
relation
t
t
3 t
, (b)
and
,
4
4
2
5 t
, and so on. The value of X C in this case be computed using the
4
X C Z 0 tan d
This is shown in Fig. 2.22.
Figure 2.22. Position of standing wave pattern for different types of load
55
Microwave Circuits & Components
2.13
MICROWAVE WAVEGUIDES
Some transmission line other than parallel and coaxial lines are described
here. Prof Jagdish Chandra Bose, in his experiments (1895) developed the concept
of microwave propagation in waveguides and used Microwave Horn for radiation
of millimeter waves. Lord Rayleigh visited his laboratory and saw the experiments
and then developed vigorous theory of propagation of electromagnetic waves in
rectangular waveguide. Therefore, in 1930s when radar was developed the
transmission line used for guiding microwaves were rectangular waveguide and
appeared to be different from the two wire transmission line. However, the
principle is similar and many of the relations and concepts that were developed for
transmission line can be applied here. The theory of propagation of microwave in
waveguides is based on Maxwell's equations and Helmholtz equation. These are
reproduced below though excellent texts are available on these topics.
2.13.1
Maxwell's & Helmholtz Relations
Without going into derivation of Maxwell's equation it is produced below.
The Maxwell's equations are in differential form (Liao 5 )
E j B j H
(2.122a)
H J j D ( j E )E A/m2
(2.122b)
(2.122c)
.H 0
(2.122d)
. H
where,
E
electric phasor vector
H
magnetic phasor vector
B
inductor phasor vector
D
displacement phasor vector
J
current density phasor vector
ρ
σ
change density phasor
conductivity
2.13.2
Boundary Conditions
On the interface separating two different materials (none of which is a
perfect conductor) the tangential components of electric and magnetic field are
continuous as expressed by
n ( E1 E 2 ) 0 V/m
2.123a)
n ( H1 H 2 ) 0 A/m
(2.123b)
56
Transmission lines
On the surface perfect electric conductor ( )
n E 0 V/m
(2.124a)
For perfect magnetic conductor ( )
n H 0 A/m
(2.124b)
Boundary conditions on normal components
(2.125a)
(2.125b)
ρS
V/m
ε0
Similarly, the magnetic field must satisfy
n E r1 E1 E r2 E r2
n r1 H1 r3 H 2 0 A/m
2.13.2.1 Helmholtz equation
Taking curl of Eqn. 2.122a
E jωμo ( H ) jωμ jω σ E k 2 E
(2.126)
where
k 2 jωμ( jωε σ )
Remembering that
E (.E ) 2 E V/m3
(2.127)
As there are no free charges within the medium, the divergence of electric
field vanishes
2 E k 2 E
(2.128a)
which is the Helmholtz equation for the electric field. Similarly, the Helmholtz
relation for the magnetic field is
2 H k 2 H
(2.128b)
2.13.2.2 Wave equations in rectangular coordinates
In most of the treatment –k2 is written as , therefore the vector wave
equations are
2E 2E
2H 2H
where,
(2.129)
jμ(σ jε) α jβ
57
Microwave Circuits & Components
Rectangular waveguide and the rectangular coordinates are shown in
Fig. 2.23. They are usually a right hand system. The rectangular components of E
and H satisfy the complex scalar wave equation or the Helmholtz equations where
it is
2ψ γ 2ψ
where, ψ can be either electric or magnetic vector. The Helmholtz equation in
rectangular coordinates is
2 2 2
2
x 2 y 2 z 2
(2.130)
This is a linear inhomogeneous partial differential equations in three
directions. The solution can be assumed in the following form
(x1 y1 z ) X (x )Y (y )Z (z )
1 d 2 X 1 d 2Y 1 d 2 Z
2
X dx 2 Y dy 2 Z dz 2
Since the sum of three terms on the left hand side is a constant and each
term is equal to a constant
d2X
dx
2
k x2 X
(1.131a)
z
a
x
a
Figure 2.23. Rectangular waveguide and rectangular coordinates
58
Transmission lines
d 2Y
dy
2
d 2Z
dz 2
k y2Y
(2.131b)
k z2 Z
(2.131c)
In their relations, k x2 , k y2 , k z2 are the three terms. Then
k x2 k y2 k z2 k 2 2
Evidently, the solutions are
X(x) = A sin(kxx) + B cos(kxx)
(2.132a)
Y(y) = C sin(kyy) + D cos(kyy)
(2.132b)
Z(z) = E sin(kzz)+ F cos(kzz)
(2.132c)
Thus, the solution of Helmholtz equation can be written as
ψ = [A sin(kxx) + B cos(kyy)][C sin(kyy)
+ D cos(kyy)][E sin(kzz) +F cos(kzz)]
(2.133)
Conventionally, propagation is considered to be in z direction. The
propagation constant g in the guide is different from free space propagation
constant . Thus
(2.134)
g 2 2 k x2 k y2 k c2
where, k c k x2 k y2 is called the cut-off wave number. For lossless dielectric
=0
2 2
Then
(2.135)
g 2 kc2
It can be seen that there are three different values of g for different
conditions.
Case-I:- There is no wave propagation if c 2 k c2 it can be seen that g 0 .
This is critical cut-off propagation. The cut-off frequency can be expressed
as
fc
1
2
k x2 k y2
(2.136)
59
Microwave Circuits & Components
Case-II:- The wave will propagate in the guide if 2 k c2 , thus
f2
g j g j 1 c
f2
(2.137)
Thus, the operating frequency must be greater than the cut-off frequency
Case-III:- The wave will be attenuated if 2 k c2 and
f
2
g I g c 1
f
Thus, if the frequency is below the cut-off frequency, the wave will decay
exponentially. The solution of Helmholtz equation is thus
[ Asin( k x x ) B cos( k x y )][Csin( k y y ) D cos( k y y )] e
j g z
(2.138)
In this expression it is assumed that wave is propagating in z direction
with propagation constant j g . Apparently, for the wave to propagate (Case-II)
g 0 .
The modes of propagation of electromagnetic waves are tabulated in
Table 1.4. The four propagating modes are reproduced in the Table 2.1.
Table 2.1. Classification of modes of propagation
S.No.
1.
The value
Ezz
Hz
=0
=0
2.
=0
3.
4.
Name of the mode
Acronym
Other names
Transverse
electromagnetic waves
TEM
–
0
Transverse electric
TE
H
0
=0
Transverse magnetic
TM
E
0
0
Hybrid mode
–
EH or HE mode
In a rectangular waveguide it can be seen that TEM mode cannot
propagate.
2.13.3
Non Propagation of TEM Mode in a Rectangular Waveguide
It can be easily shown that the expressions for Ex, Ey, Hx and Hz can be
obtained from the Maxwell's equation. According to Maxwell's equation
.E
ρ V/m2
ε
60
Transmission lines
and .H 0 A/m2
It can be seen that
t Ez
t Ex
z
Ey
x
y
Thus, if 0
t .Et
E z
0
z
and
H z
0
z
For a TEM mode Ez and Hz = 0. Thus,
t .H t
t Et 0
t Et 0
t H t 0
t H t 0
Thus, if both E z H z 0 and lead to the trivial solution E t H t 0 ,
the boundary condition imposes a constant potential on the metal tube. Therefore,
an empty hollow waveguide cannot propagate a TEM mode. Thus, the modes that
can propagate are TE and TM modes. These are now discussed in two subsections.
2.13.4
TE Modes in Rectangular Waveguide
The TE modes in a waveguide are characterized by Ez = 0. Following
three relations will hold good in this case
(2.139a)
E jωμ H
(2.139b)
E jωμ E
and
(2.139c)
2 H z 2 H z
When the boundary condition is imposed the solution is of the form
m x
m x
n y
j z
n y
H z Am sin
Cn sin
Bm cos
e g
Dn cos
a
a
b
b
(2.140)
61
Microwave Circuits & Components
It can be seen that k x m , k y n . The six components of
a
b
Eqns. 2.139a and 2.139b are
E z E y
j H x
y
z
(2.141a)
E x E z
j H y
z
x
(2.141b)
E y
x
E x
y
(2.141c)
j H z
H z H y
j Ex
y
z
(2.141d)
H x H z
j E y
z
x
(2.141e)
H y H x
j E z
x
y
(2.141f)
Substituting these relations j g and E z = 0
z
g E y H x
(2.142a)
g Ex H y
(2.142b)
E y Ez
j H z
y
(2.142c)
H z
j g H y j E x
y
(2.142d)
jg H x
H z
j E y
x
(2.142e)
H y H z
0
x
y
(2.142f)
Differentiating the expressions for H z with respect to x and y and
substituting in above relation gives
62
Transmission lines
Ex
j H z
kc2 y
(2.143a)
Ey
j H z
kc2 x
(2.143b)
(2.143c)
Ez 0
Hx
j g H z
.
x
kc2
(2.143d)
Hy
j g H z
kc2 y
(2.143e)
n y
n y j g z
m x
m x
Bmcos
Dncos
e
H z Amsin
Cnsin
a
a
b
b
(2.143f)
The boundary conditions are again applied to these field equations.
Remembering that
tangent E field = 0/ at the surface
normal H field = 0/ at the surface
Since Ex = 0, then H z 0 at y = 0, b. Hence, Cn = 0. Since, Ey = 0, then
y
H z
0 . Hence, Am = 0.
n
It is generally concluded that normal derivative of Hz must vanish at the
conducting surface, i.e., H z 0 at the guide walls. Thus
n
m x
n y j g z
H z H oz cos
cos b e
a
(2.144)
Thus the six field equations are
m x n y j g z
E x Eox cos
sin
e
a b
(2.145a)
n y g z
m x
E y Eoysin
cos b e
a
(2.145b)
Ez 0
(2.145c)
63
Microwave Circuits & Components
m x
n y g z
H x H ox sin
cos
e
a
b
(2.145d)
m x n y j g z
H y H oy cos
sin
e
a b
(2.145e)
m x
n y j gZ
H z H oz cos
cos
e
a
b
(2.145f)
It can further be seen that
2
2
m
n
kc
c
a
b
fc
2
m
n
b
2 a
1
(2.146)
2
(2.147)
The propagation constant g can be expressed as
fc
f
2
(2.148)
g 1
g
g
b
g
f
1 c
f
2
(2.149)
f
1 c
f
2
(2.150)
The waveguide impedance that
Zg
E y
Ex
Hy
Hy
g
0
0
where
f
1 c
f
64
2
(2.151)
Transmission lines
2.13.5
Dispersion Relation & TE Modes
The constant Rc is known as cut-off wave number associated with cut-off
frequency c and cut-off wavelength
c
2
2
k c c
(2.152)
The cut-off frequency and cut-off wavelength are geometrical parameters
depending upon waveguide cross-section
by
The propagation constant of frequencies greater than the cut-off is given
2 2 k c2 2 c2
or 2 2 2 c 2
(2.153)
Therefore,
2 2 2 c 2
(2.154)
This may be considered as dispersion relation which shows variation of
with . It can be used for finding the phase group velocities and waveguides
1 c 2
p
2 z z
1
2
c2
c 2
R R
1
2
where, c is the phase velocity corresponding to cut-off frequency. Thus, the
phase velocity
p c
c
R R c
1
2
(2.155)
Differentiating Eqn. 2.154
.2d 2 2 d 2
Thus, the group velocity
g
d
2
d 2
(2.156)
Thus
2
f
2 1 c 1 c
f
65
2
Microwave Circuits & Components
p
1
2
2
f
1 c
f
1
(2.157)
2.13.5.1 Modes in rectangular waveguide
In order to determine which mode may propagate at a given frequency in
a waveguide, all modes having cut-off frequencies below this value have to be
determined. For rectangular waveguide, one makes use of
fmn
2
C m
n
2 a
b
2
2
c
c
m2 n2
a
2
2b
2
Suppose it is desired to find which modes can propagate at 15 GHz in a
rectangular waveguide of 3 1.5 cm cross-section. The two quantities appearing
under square root is given by
C 3 1010 cm s
5.109 s-1 56 Hz
2a
2 3cm
C
3 1010
10 GHz
2b 2 1.5cm
One must then have
fmn 2 m 2 .25 n 2 .100 225 15 2 f 2
Solutions obtained are
n=1
m 1, f 10 5 GHz
m 2, f 20 10 GHz
m 3, f 30 15 GHz
n=1
m 0 , f 01 10 GHz
m 1, f 11 11.18 GHz
m 2 , f 21 14.142 GHz
Phase and group velocities for these modes are formed by using relations
p
2
2
c p
c
pc
1
2
ms
and
2
2
1
c p
2
d
bc
g
c
1
ms
2 c2
d
66
Transmission lines
2
p
It is to be noted that m and n represent half-cycle variation along the major and
minor axis.
2.13.5.2 Dominant TE10 mode
where cut off wavelength is c
To avoid multimode propagation, the use of waveguide is restricted to the
part of the frequency range over which a single mode propagates and where
dispersion is acceptable. Evidently the cut-off wavelength is 2a and the guide
wavelength is
10
2
10
1
2a
2
(2.158)
It is to be noted that the dominant mode has lowest cut-off frequency of
all possible modes in a rectangular guide.
The field components for this mode are
Et E y
2
x
sin
Aexp j 10 z B exp j 10 z V/m
ab
a
(2.159a)
Ht Ex
2
x
sin
Aexp j 10 z B exp j 10 z A/m
ab
a
(2.159b)
Hz
1
j a
x
2
Aexp j 10 z B exp j 10 z A/m
cos
ab
a
(2.159c)
The three impedences are
ZVI
b
(2.160a)
2b
a
(2.160b)
2a
Z PV
2 b
(2.160c)
4 2a
where, V, I and P are potential, current and power respectively and is the free
space wave impedance
Z PI
2.13.6
TM Modes in Rectangular Waveguide
The TMm,n modes in a rectangular waveguides are characterized by Hz = 0.
Therefore, the Helmholtz equation to be used in this case is 2 E z 2 E z which
gives
67
Microwave Circuits & Components
m x
x
Bm cos
E z Amsin
a
a
n y
n y j g Z
Cn sin
Dn cos
e
a
b
(2.161)
The boundary condition of Ez requires that the field vanish at the
waveguide since tangential component of the electric field Ez = 0 in the conducting
surface. This requires that
Ez = 0 | x = 0, a then Bm = 0
Ez = 0 | y = 0, b then Dn = 0
Then the solution is of the form
m x n y j g z
E z Eoz sin
sin
e
a b
Following the same method as for TE mode, the TM m,n mode field
equations in rectangular waveguide are
m x n y j g z
E x Eox cos
sin
e
a b
(2.162a)
m x
n y j g z
E y Eoy sin
cos
e
a
b
(2.162b)
m x n y j g z
E z Eoz sin
sin
e
a b
(2.162c)
m x
n y j g z
H x H ox sin
cos b e
a
(2.162d)
m x n y j g z
H y H oy cos
sin
e
a b
(2.162e)
Hz 0
(2.162f)
The other characteristic equations for TM mode are
fc
1
m2
2
a2
n2
(2.163a)
b2
68
Transmission lines
fc
f
2
(2.163b)
g 1
g
g
f
1 c
f
p
f
1 c
f
2
(2.163c)
2
g
(2.163d)
f
Zg
1 c
f
2.13.7
2
(2.163e)
Excitation of Modes in Rectangular Waveguides
The desired modes in waveguide can be established by means of a probe
or loop coupling. A probe can be located at a point where it excites the desired
mode. The coupling loop generate the magnetic field intensity for the desired
mode. If more then one probe or coupling loop are used, proper phase relationship
has to be ensured. In order to excite TE 10 mode in one direction the two exciting
devices are used to enforce propagation in one direction but cancel the other. Some
of the excitation modes are shown in Fig. 2.24.
COAXIAL
CABLE
A N TE N N A
PROBE
4
TE20 MODE
TE10 MODE
SHORT CIRCUITED
END
A N TE N N A
PROBE
TE11 MODE
TE21 MODE
Figure 2.24(a). Rectangular waveguide: Methods of exciting narrow modes
69
Microwave Circuits & Components
1
OUT PHASE
2
p
1
1
2
IN PHASE
2
4
WAV EGUI DE
ANTENNA
WAV EGUI DE
ANTENNA
PROBE
PROBE
RF INPUT
Figure 2.24(b). Unidirectional TE 10 mode
2.13.7.1 Field patterns for some modes
T
The field pattern for TE10 mode at t = 0 and = ; periodic time is T. The
4
field pattern is shown in Fig. 2.25. The conduction current is shown in Fig. 2.26
and field pattern of higher modes is shown in Fig.2.27a and 2.27b.
E LINES
H LINES
a
OUTWARD DIRECTED LINES
INWARD DIRECTED LINES
g
3
g
4
g
g
2
4
x
y
y
a
b
x
(a) FIELD PATTERN at t = 0
g
3
g
4
g
g
2
4
x
z
a
x
y
y
a
b
x
(b) FIELD PATTERN at t = T/4
Figure 2.25. TE10 mode pattern
70
z
x
Transmission lines
y
z
x
g
2
g
2
b
a
Figure 2.26. Conduction current in rectangular guide
b
TE01 MODE
a
c 2b
TE02 MODE
c a
(a)
TE30 MODE
c
2
a
3
TE11 MODE
c
2ab
a 2 b2
E LINES
OUTWARD DIRECTED LINES
H LINES
INWARD DIRECTED LINES
y
y
3
4
4 w
(b)
3
4
x
a
2
w
3
4
(c)
a
2
a
a
Figure 2.27(a). Higher order modes
71
Microwave Circuits & Components
z
g
3
g
4
g
g
2
4
E LINES
H LINES
a
OUTWARD DIRECTED LINES
INWARD DIRECTED LINES
g
x
y
y
2ab
a 2 b2
a
b
z
Figure 2.27(b). Field pattern and cut-off wavelength for TM 11 mode
2.13.8
Circular Waveguide
There are certain applications that require dual polarization capability.
For example, a waveguide connected to a circularly polarized antenna must be able
to efficiently propagate both vertically and horizontally polarized waves. Circular
waveguide is the most common form of a dual polarization transmission line. Many
modes can be excited in a circular waveguide. Figure 2.28 shows some modes.
TE11 MODE
TE21 MODE
c 1029
. D
c 1706
. D
E LINES
H LINES
OUTWARD DIRECTED LINES
TE01 MODE
c 1306
. D
TE01 MODE
INWARD DIRECTED LINES c 082
. D
D = INNER GUIDE DIAMETER
Figure 2.28. Field patterns for some common circular waveguide
72
Transmission lines
The scalar Helmholtz equations in cylindrical coordinates is given by
1 1 2 2
2
r
r r r r 2 2
z 2
(2.164)
Using the method of separation of variables the solution is assumed to be
R(r ) () Z ( z )
So that
2
to γ g
1
d dR
1 d 2 1 d 2 Z
2
2
rR dr dr r d 2
z dz 2
Each of the three terms must be constant. The third term may be set equal
d 2z
2
2g Z
dz
The solution to this is the form
Z Ae gz Be gz
2
where, g is the guide propagation constant γ g in the third terms of above
equation one obtains
1 d 2
2 g2 r 0
r
R dr dr d 2
The second term in the Eqn. 2.165 yields
d dR
(2.165)
d 2
n 2
d 2
The solution is of the form
An sin n Bn cosn
Therefore
r
2
d dR
2
r
kc r n R 0
dr dr
This is the characteristic equation of Bessels function. For the lossless
guide the characteristic equation reduces to
g 2 Rc 2
73
Microwave Circuits & Components
1.0
J0
VALUE OF Jn (kcr)
0.8
J1
0.6
J2
J3
0.4
0.2
0
-0.2
-0.4
-0.6
2
0
8
6
4
10
12
16
14
ARGUMENT OF Jn (kc,r)
Figure 2.29. Bessel function of the first kind
The solution is Bessel function of first kind representing a standing wave
of cos(kcr) for r < a.
Nn (kcr) is the nth order Bessel function of second kind representing
standing wave of sin(kcr) for r > a. Therefore, the total solution of Helmholtz
equation is
[C n J n (k c ) Dn N n (k c r )[ An sin(n ) Bn cos(n )]I e j g z
(2.166)
Jn (kcr) and Nn(kcr) are shown in Eqn. 2.166 and Figs. 2.29 and 2.30.
0.6
0.4
N1
N0
N2
N3
VALUE OF Nn (kcr)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
0
2
4
6
8
10
12
ARGUMENT OF Nn (kc,r)
Figure 2.30. Bessel function of the second kind
74
14
16
Transmission lines
At r = 0, however, kcr = 0 then function Nn approaches infinity, so Dn= 0.
This means that r = 0, on the Z-axis, the field must be finite. The two sinusoidal
functions can be condensed into one. Thus the solution of Helmholtz equation is
0 J n ( kc r ) cos( n )e
(2.167)
j g z
2.13.8.1 TE modes in circular waveguide
For TE mode
2H z 2H z
The solution will be of the type
H z H oz J n (kc r )cos(nQ )e
j g z
Corresponding to Maxwell's equation
E jH
H jE
Er
E
j i H z
kc2 r d
(2.168a)
jw H z
kc2 r
(2.168b)
Ez 0
Hr
(2.168c)
j g H z
kc 2 r
H j
g
kc2
r
H z
H z H oz jn (kc r ) cos(n )e
2
kc
(2.168d)
(2.168e)
(2.168f)
i g z
(2.169)
2 - g2
It can be seen that
E 0 at r=a,
H z
ra0
r
75
Microwave Circuits & Components
H z
H r 0 at r = a,
ra0
r
H z
j z
H oz J n ' ( kc a ) cos( n )e g
r
Thus, J n ' (k c a ) 0
where, J n ' indicates derivatives of j n
'
Since, j n are oscillatory functions j n (kca) are also oscillatory functions.
a
An infinite sequence of k c satisfies above equations. Tables are available for pth
zeros of J n ' (k c a) for TEnp modes.
The permissible values of k c can be written as
kc
X ' np
r
a
This yields complete solutions
jg z
X ' np
Er Eor J n
r sin( n )e
a
(2.169a)
i g z
X ' npr
E Eo J 'n
cos( n )e
a
(2.169b)
Ez 0
(2.169c)
Hr
H
Eo
g z
X ' npr
Jn
cos( n )e
Zg
a
Eor X ' npr
j g z
Jn
sin(n )e
Zg
a
g z
X ' npr
H z H oz J n
cos( n )e
a
0
(2.169d)
(2.169e)
(2.169f)
The first subscript n represents the number of full cycles of field variation
one revolution through 2 radians of . The second subscript p indicates the
X ' npr
number of zeros of E , i.e., J n '
along the radius of guide, but zero on
a
the axis is excluded if it exists.
76
Transmission lines
Therefore,
X ' np
a
g 2
kc
fc
2
(2.170)
X ' np
c
a
(2.171)
X ' np
(2.172)
2a
and the phase velocity for the TE in the circular guide is then given by
g
g
f
1 c
f
2
(2.173)
g
Zg
p
f
1 c
f
g
2
(2.174)
f
1 c
f
(2.175)
2
where, p = wavelength of unbounded directive
f
u0
0
intrinsic impedence in an unbounded dielectric
The pth zeros of J n ' (k c a) for TE mode is shown in Table 2.2.
np
Table 2.2. pth zeros of J n ' (k c a ) for TEnp modes
p
n=0
1
3.832
2
7.016
3
4
1
2
3
4
5
1.841
3.054
4.201
5.317
6.416
5.331
6.706
8.015
9.282
10.50
10.173
8.536
9.969
11.346
12.682
13.987
13.324
11.706
13.170
77
Microwave Circuits & Components
2.13.9
TM Modes in Circular Waveguide
As mentioned earlier the TMnp modes in a circular guide are characterized
by Hz = 0, which means that E z 0. The Helmholtz equation for Ez in circular
waveguide is
2Ez 2Ez
(2.176)
Its solution is of the type
E z Eoz J n (kc r ) cos(n )e
(2.177)
j g z
The boundary conditions require that the tangential component of electric
field Ez at r = a is zero. Consequently
(2.178)
J n (k c a) 0
Jn(kcr) is oscillatory function shown in Fig. 2.30. There are infinite
numbers of roots of Jn(kcr). Some of the values are tabulated in Table 2.3.
Table 2.3. pth zeros of J n ' (k c a ) for TM np modes
p
n=0
1
2
3
4
5
1
2.405
2
3.520
3.832
5.136
6.380
7.588
8.771
7.106
8.417
9.761
11.065
12.339
3
4
8.645
10.173
11.620
13.015
14.372
11.792
13.324
14.796
Following the conventional method the field relations are given by
Er
j g E z
k c2 r
(2.179a)
E
j g E z
k c2
(2.179b)
E z Eoz J n (kc r ) cos(n )e
j g z
(2.179c)
Hr
j 1 E z
kc2 r
(2.179d)
H
j E z
kc2 r
(2.179e)
(2.179f)
Hz 0
78
Transmission lines
where, kc2 2 g2 has been replaced. Finally
jg z
Xnpr
Er Eor J n '
cos( n )e
a
(2.180a)
jg z
Xnpr
E Eo J n
sin( n )e
a
(2.180b)
j g z
Xnpr
E z Eoz J n
cos( n )e
a
(2.180c)
Hr
H
Eo
zg
j g z
Xnpr
Jn
sin( n )e
a
Eor
jg z
Xnpr
Jn '
cos(n )e
zg
a
(2.180e)
(2.180f)
Hz 0
Zg
(2.180d)
g
E
Er
and k c Xnp
H
Hr
( )
a
Some of the TM mode characteristic equations are
Xnp
a
g 2
kc
fc
g
2
(2.181a)
Xnp
c
a
(2.181b)
Xnp
(2.181c)
2a
g
p
f
1 c
f
2
(2.181d)
79
Microwave Circuits & Components
g
Zg
f
1 c
f
2
g
f
1 c
f
(2.181e)
2
(2.181f)
2.13.10 Fields in Circular Waveguide
The electric and magnetic field of some of the modes are shown in Fig. 2.28.
2.13.11 Excitations of Modes in Circular Waveguide
If a device is inserted in an circular guide in such a way that it excites only
a z component of electric intensity the wave propagating through the guide will be
the TM mode. If a device is placed in a circular waveguide in such a manner that
only 2 components of magnetic intensity exists, the travelling wave will be the TE
mode. These are shown in Fig. 2.31.
COAXIAL LINE
SHORT CIRCUITED END
A NT E NN A
PROBE
TM01 MODE
TM11 MODE
Figure 2.31. Excitation of the TM01 and TE11 mode in circular waveguide
2.14
SOME OTHER MICROWAVE TRANSMISSION LINES
Some of other transmission lines used in microwave region are shown in
Figs. 2.32, 2.33 and 2.34.
2.14.1
Dielectric Waveguide
The solid rectangular dielectric waveguide is perhaps one of the simplest
waveguide. This is particularly true if a standard metallic waveguide operated in
TE10 mode is used as feeder. Figure 2.35 shows a possible configuration for
launching fields in dielectric waveguide where the external dielectric waveguide
dimensions are the same as the internal dimensions of the metallic waveguide.
Unlike metallic walls, boundary conditions at dielectric do not constrain the
tangential components of electric field to vanish and, for low to moderate values of
the relative dielectric permittivity affect normal and tangential field components in
80
Transmission lines
Homogeneous
Inhomogeneous
two conductor line
insulated two
conductor line
one conductor line
Goubau line
microstrip
strip-line
slot line
coplanar line
coaxial line
loaded coaxial line
loaded metallic wave
guides
metallic
waveguides
dielectric
waveguides
optical fibres
Figure 2.32. Cross-section of main microwave transmission lines (includes wave
guides also).
Figure 2.33. Microwave transmission lines
81
Microwave Circuits & Components
Figure 2.34. Some specialised microwave transmission lines
FIELD
AMPLITUDE
z
x
k1
(a)
k2
y
2a
Ex
Ex
k
k
Hz
Hz
Hy
Hy
TE MODE
(b)
k
k
Hz
Hz
Hy
Hy
Ex
Hx
Ey
Hx
Ez
k
k
Hz
Ey
Ez
Hx
TM
MODE
(c)
k
k
Ez
Ez
Ey
Ey
Hx
Figure 2.35. Dielectric slab waveguide (a) Attenuated field outside the slab
(b) Field for TE mode (c) Field for TM mode.
82
Transmission lines
7
6
5
x
z
1
9
4
y
8
3
3
Figure 2.36. Nine different regions of dielectric waveguide
a similar manner. Morita has discussed matter of extending the boundary
conditions for analyzing guided modes of dielectric waveguide and permittivity of
1 and permeability 0 immersed in free space with constants 0 and 0 . Such
a system in which a dielectric waveguide cross-section and number of regions are
shown in Fig. 2.36. In the dielectric waveguide two basic set of modes exists; one
with line only polarized transverse electric field and the other with transverse
magnetic field. In each case, these modes may be either symmetric (even) or
aymmetric (odd) with respect to coordinate axes X and Y. It must however be
noted that no simple and exact solution to the field equation is a dielectric
b
a
waveguide. The boundary between region (1) and (2) where y
and x ,
2
2
we must have
0 E y2 1 E y1
On the other hand boundary between regions 1 and 4
E y4 E y
On the corner where regions 1, 2, and 4 meet, both the equations must
hold simultaneously, it means that electric field cannot be continuous at this point.
Preferred continuous solutions for electric field and thus does not impose the
continuity of normal component of electric displacement at the dielectric
boundaries. He was mainly interested in the fields inside the waveguide and on free
space boundaries and so did not consider regions from 6 to 9. Due to space
constraint, this topic is not dealt in further depth.
2.14.2
The Strip Line
The schematic diagram of stripline transmission line is shown in Fig. 2.37.
The magnetic field at the position of the configuration of figure, due to current I in
the central conductor is
83
Microwave Circuits & Components
GROUND
PLANE
CENTRAL
H
CONDUCTOR
t
w
(a)
GROUND PLANE
MAGNETIC FIELD
CENTRAL CONDUCTOR
H
y t
y
w
(b)
Figure 2.37. (a) Strip transmission line (b) Magnetic field in strip line
I
(2.182)
2W 2t 4 y
The contribution from ground plane is cancelled since the magnetic field
from uniform infinite current sheet does not depend upon distance from it. The
average field is then given by
h
h
1 H
Idy
H 0 2W 2t 4 y
(2.183)
Integrating this quantity
h
1
W t 2H
ln
4H W t
(2.184)
and approximate relation for voltage can be estimated by
(2.185)
V Edy EH
These two equations can be combined to give the characteristic
impedence as
W t 2H
Z 0 30 ln
W t
2.14.3
(2.186)
Microstrip Line
There are many shapes of transmission line like, stripline, H-guide, fin
lines, dielectric guide, coaxed line-coplanar and few others; of these some are
planar. Amongst all planar transmission lines, microstripline has gained much
84
Transmission lines
ELECTRIC FIELD
MAGNETIC FIELD
(a)
220
t/b=0.25
0.20
0.15
0.10
0.05
0
200
OHMS
180
b
r
w
160
140
t/b=0
0.05
0.10
0.15
0.20
0.25
120
100
80
0.1
0.2
0.3
0.7 1.0
w/b
0.5
2.0
3.0 4.0
(b)
Figure 2.38. Strip lines (a) Field pattern, and (b) Characteristic impedance
popularity and importance in planar circuit technology. A microstrip line is a
transmission line consisting of a strip of conductor of thickness t, width w and
ground plane separated by a dielectric medium of thickness h as shown in Fig. 2.39.
Because of its open conduit for wave transmission not all of the electric or magnetic
STRIP
CONDUCTOR
t
DIELECTRIC
w
h
GROUND
PLANE
H–FIELD
E–FIELD
Figure 2.39. Microstrip line and E and H-fields inside such a line
85
Microwave Circuits & Components
fields will be confined to the structure (Pozer6). This fact, along with the existence
of a small axial E-field, leads to a purely TEM wave propagation but to QuasiTEM mode of propagation. These are extensively used in microwave planar circuit
design and microwave integrated circuit (MIC) technology. Microstrip line is very
popular and superior to other types of planner transmission lines. The dielectric
used in fabrication of the microstrip line are characterised by a dielectric constant
defined by
r
, 0 8.854 1012 F m
0
(2.187)
where, and 0 are dielectrics and vacuum primitively respectively. The most
popular dielectrics are mentioned in Table 2.4.
Table 2.4. The microwave substrate dielectrics
S No.
Name
Dielectric Constant er
1
2
3
4
5
6
7
8
RT Duroid 5880
RT Duroid 6006
RT Duroid 6010.5
Alumina 85%
Alumina 96%
Quartz
Silicon
Epsilam-10
2.23
6.00
10.50
8.00
8.90
3.70
11.70
10.00
The EM-wave propagation in microstrip line is approximately nondispersive below the cut-off frequency ( f0 ) which is given by
f 0 GH Z 0.3
Zo
(2.188)
h r 1
where, h is in centimetres.
Z0 in ohms (characteristic impedance). The phase velocity of quasi-TEM
wave is given by
Vp
c
eff
(2.189)
where, c is the speed of light and eff is the effective relative dielectric constant.
Because the field lines are not constant in a structure and some exist in the air and
the effective dielectric constant satisfies the relation.
1 eff r
(2.190)
86
Transmission lines
In general the effective dielectric constant is a function not only the
substrate material, i.e., but also of dielectric thickness h and conductor width w.
The characteristic impedance Z0 is given by
Z0
1
V p C0
(2.191)
where, C0 is the capacitance per unit length.
The wavelength , of propagating wave in the microstrip line is given by
0
VP
12
f
eff
(2.192)
c
is the wavelength in the free space. Figure 2.32 shows the crosswhere, 0
f
section of different transmission lines which can be used at microwave frequencies.
2.14.3.1 Empirical formula for effective dielectric constant
The essential empirical formulas for microstrip line are following:
(a)
The effective dielectric constant is given by, assuming that the dimensions
of microstrip line (w, h) are known.
w
1
For
n
eff
For
1
r 1 r 1
1 12 wh 2 0.04 1
2
2
w
h
2
(2.193)
w
1
n
eff
1
r 1 r 1
1 12 wh 2
2
2
(2.194)
The effective dielectric constant can be thought of as a dielectric constant
of a homogeneous medium that will fill the entire space, replacing air and dielectric
region.
(b)
Assuming that the dimensions of microstrip (w, h) are known, the
characteristic impedance is given by,
0
8h w
ln
eff w 4h
60
0 eff
(2.195)
wh 1.393 0.667 ln wh 1.444
87
(2.196)
Microwave Circuits & Components
(c)
Assuming eff and Z0 are given, then microstrip dimensions (w, h) can be
found as follows
For
w
2
h
w
8 eA
2
h e A 2
For
(2.197)
w
2
h
1
w 2
0.61
B 1 ln 2 B 1 r
l n B 1 0.39
h
r
2 r
(2.198)
where
A
0
60
r r 1
0.11
0.23
2 r 1
r
(2.199)
and
B
(d)
377
(2.200)
2 0 r
The wavelength in the microstrip line is given by
w
0.6
For
h
u
12
r
r 1 0.6 1
r
For
w 0.0297
h
(2.201)
w
0.6
h
u
12
r
r 1 0.63 1
r
w 0.1255
h
2.14.3.2 Attenuation factors
The total attenuation factor is composed of two components 7
d c
88
(2.202)
Transmission lines
where
d = dielectric loss factor
c = conductor loss factor
The two loss factors are
Attenuation due to dielectric loss
For low-loss dielectric
d 27.3
1
tan r eff
0 r 1 eff
dB
cm
where, tan is the loss tangent is given by tan
(2.203)
For high-loss dielectric
12
1
0 1 eff
0 r 1 eff
d 4.34
dB
cm
(2.204)
where, is the conductivity of dielectric, 0 4 107 4 m is the permittivity
of free space.
Attenuation due to conductor loss
Attenuation due to the conductor identified by conductor-loss factor c .
w
Using quasi-TEM mode of propagation is for given approximately by
h
w
h
c
Rs
Rs
N m
Z0w
f 0
cc d
lines.
The last topic to be introduced in this chapter is modes in transmission
89
Microwave Circuits & Components
1000
100
r
1
2
4
6
8
10
16 12
10
5
0.1
1.0 W/h
10
Figure 2.40. Characteristic impedance of microstrip line vs. W/h
TEM
1.3
0
r
free-space wavelength
0
1.25
1.20
1.15
12
10
8
6
4
2
r
1.10
1.05
1.0
0.1
1.0
W/h
10
Figure 2.41. Normalised wavelength of microstrip line vs. W/h
2.14.4
The Coplanar Waveguide
The coplanar waveguide consists of central metallic ground strips. The
Fig. 2.42 shows the structure of coplanar line. For approximate estimation of
characteristic impedance, the quasi-static electric and magnetic field are shown in
the Fig. 2.43. As mentioned in the beginning the series and shunt elements can be
90
Transmission lines
h
W
S
W
r
Figure 2.42. Coplanar waveguide
integrated in this case; the magnetic field is elliptically polarized for calculating
characteristic impedance. In this case, it may be easier to estimate the quasi-static
capacitance. Mapping can be used in this case. Using complex variables
W u jv
in the W-plane corresponding to one in the Z-plane
Z x jy
W
S
-b1 -a1
W
a1
b1
z1 - PLANE
-a
a
0
ELECTRIC FIELD
MAGNETIC FIELD
-a-jb
(a)
a-jb
f,e
z - PLANE
(b)
W=2q
r
r
2
H
h qsin h 0
0
0
1
(c)
Figure 2.43. (a) Electric and magnetic fields in a coplanar guide (b) Conformal
transformation between coplanar waveguide and equivalent parallel
plate capacitor (c) Elliptic diagrams of shielded slot lines and coplanar
guides.
91
Microwave Circuits & Components
for a given function
W f (W )
Using this method the dielectric half-plane Z in Figs. 2.42 and 2.43 may
be transformed into the interior of a rectangle in the W-plane, the capacitance of
which can be calculated easily if we know the dimension. The transformation may
be written as
dW
dZ
Z
A
2
a1
Z
2 12
2
b12
(2.205)
12
The ratio of u/v is deduced by integrating
b
1
W u jv
0
Z
udZ
2
2
1
a
Z
12
2
b12
12
(2.206)
Assuming
u K( k )
v K' ( k )
a
where, k 1
b1
(2.207)
The identity which may be used to evaluate K’(k) is
K' ( k ) K ( k' )
where, K' 1 k 2
12
The capacitance of the dielectric half-space is thus
C r 0 r
K( k )
F/m
K' ( k )
and that of free space is
C0 0
K( k )
F/m
K' ( k )
(2.209)
The effective dielectric constant is estimated by taking averages
eff
r 1
2
and the phase velocity is therefore described in terms of free space by
C
p
m/s
eff
92
Transmission lines
Therefore, characteristic impedance is given by
Z0
0
K( k )
eff K' ( k )
Thus the elliptic functions have to be estimated in this case.
2.14.5
The Slot Line
The slot line is another planar structure which is commonly used. It
consists of a dielectric substrate in which a slot is etched in the metal of the
substrate. The other surface is without any metallisation. The series and parallel
elements can be fabricated without much difficulty. The structure of slot line, the
electric and magnetic fields of wave propagating in the slot is shown in the
Fig. 2.44. The field pattern is quasi-TE one. It is very similar to a semi-elliptical
dielectric-loaded waveguides. The figure shows one to one equivalence between
the two. The field inside the dielectric is unaffected by the details of the line.
Slot line planar transmission structure was proposed in 1968 and are
analysed by following methods
(a) Transverse resonance
(b) Galerkins’ method in Fourier transform domain
(c) Finite-difference time domain technique
The approximate analysis is briefly discussed here.
The slot line field contains six field components; three electric field
components and three magnetic field components. The longitudinal component of
electric field is very weak since the energy propagates between two conductors.
Normally, the slot width w is much smaller than free space wavelength 0 . Under
this assumption the electric field across the slot may be represented by an
equivalent line source of magnetic current. Then the far field only contains three
components: Hx, Hr, EQ . These may be written as
H x AH 0 (1) (k c r )
Hr
EQ
x H x
kc
2
r
(2.210)
A
1 s
0
2
H1(1) (k c r )
j H x
H r s
2 r
kc
0
(2.211)
(2.212)
where, x is the propagation constant along the x-direction, which is direction of
propagation and kc the cut-off wave number and is related to wavelength s by
equation
93
Microwave Circuits & Components
H
r
W
(a)
ELECTRIC FIELD
MAGNETIC FIELD
(b)
Figure 2.44. Slot lines (a) Structure and (b) Field pattern in slot lines
kc j
2
0
0
s
2
1
(2.213)
2.14.5.1 Slot wavelength
Slot line field components are not confined to the substrate only but
extend into region above the slot and below the substrate also. Thus, the energy is
distributed between the substrate and air regions. The effective dielectric constant
is re . Therefore the slot line is less than the substrate permittivity r . For
infinitely large thick substrate the average dielectric constant of the two media is
re
and therefore,
r 1
2
s
2
0
r 1
(2.214)
94
Transmission lines
h
The cut-off thickness
0
h
0
is given by
c
0.25 r 1
c
(2.215)
Though physical picture becomes quite clear using approximate theory,
the picture does not provide any expression for the characteristic impedance. A
few expressions of characteristic impedance have been obtained. These are
Z 0g
for
Z0 s
for 0.02
v t
v g
0
s
v
v g i
W
0.2
h
h
s
W
W
0.923 0.448 log r 0.2 0.29 0.047 log 102
h
h
0
0
W
W
0.02 01
.
h
h
Z0 72.62 3519
. log r 50
W
h
W
log 102 44.28 19.58 log r
h
W
0.32 log r 011
.
107
. log r 144
.
h
(2.216)
h
114
102
. 6.07 log r
0
Another planar practice transmission line is suspended stripline. It is
described in the next subsection.
95
Microwave Circuits & Components
2.14.6
Suspended Microstrip Line
It consists of a dielectric sheet metallised with a circuit element on one
side and mounted on a ground metallic plate. Figure 2.45 shows the schematic
diagram of suspended microstrip line. Then in this case characteristic impedance is
found by finding the effective dielectric constant and is given by the relation
Z
Z0
(2.217)
r 0
where
2
f u
2
1
Z 0 60l n
u
u
u
W
ab
and
30.666 0.7528
f ( u ) 6 ( 2 6 ) exp
u
Semi-empirical equation obtained for such a structure is
W
r ( 0 ) 1 a1 b1l n
b
1
1
r
1
The coefficient a1 and b1 are specified by
a
a1 0.155 0.505
b
t
(2.218a)
DIELECTRIC SUBSTRATE
W
STRIP
r
a
b
GROUND
PLANE
Figure 2.45. Suspended microstrip line
96
Transmission lines
a
a
b1 0.023 0.1863 0.194
b
b
for 0.2 a 0.6
b
2
(2.218b)
and
a
a1 0.307 0.293
b
for
(2.218c)
a
b1 0.0727 0.0136
b
a
0.6 1
b
2.14.7
(2.218d)
Fin Lines
Fin lines are not exactly planar structure. However, we may call it quasiplanar. It was first proposed in 1972 by Meir. The main characteristics of fin lines
are large bandwidth, compatibility with planar circuit technology and absence of
radiation. The structure of commonly used fin lines is shown in Fig. 2.46. A fin line
can be considered as a shielded slot line. The slot line is mounted in the E-plane of
waveguide. The fin line dimension should be such that it is commensurate with the
dimensions of waveguide. The other structures are unilateral, bilateral and
antipodal fin lines as it can be seen from the Fig. 2.47. The dominant mode of
propagation is the HE mode of propagation. The cut-off frequency is lowered in
this structure thus, increasing the bandwidth. Large power densities can be used
resulting in better matching. However, the concentration of field will result in larger
conductor and dielectric losses. The attenuation is of the order of
0.1d / wavelength.
The unilateral fin lines are most suitable for fabrication of the components
whereas bilateral produces lower losses. Antipodal fin lines can offer impedance
levels of the order of 10 and are suitable for transition from microstrip line to
waveguides. The typical transformation ratio is to the order of 20:1. The presence
of sharp edges and inhomogenous dielectric loading complicates the analysis of fin
line structures. The modes that can propagate in such structures are of hybrid types,
both HE and EH. These modes have dominant E2 and H2 fields. Near cut-off it
reduces to TE and TM modes. There are various methods of analysis. Some of
these techniques are (a) transverse resonance method, (b) transmission line matrix
method (c) space and spectral-domain techniques (d) the Ritz-Galerkin method,
and (e) the mode matching method.
The most widely used method is Galerkin's method of spectral domain.
The main features and results are described in the following section.
97
Microwave Circuits & Components
x
b
r
(a)
a
(3)
(2)
h1
h
W
y
(b)
r
(d)
r
(1)
h2
a
(c)
r
Figure 2.46. Cross-section of several fin lines (a) Unilateral (b) Insulated
(c) Bilateral, and (d) Antipodal.
2.14.7.1 Galerkin’s method in spectral domain
An accurate analysis of fin lines can be carried out using Galerkin’s
method in Fourier transform domain. The method used for slot lines can be applied
here with modification of boundary conditions.
The basic functions are given as [P]
UNILATERAL
BILATERAL FINLINE
ANTIPODAL
Figure 2.47. Commonly used fin lines
98
Transmission lines
p
Ex am Ex
m 1
E xm
m
2x
cos m 1
1
W
x W
0
2
2
2x
1
elsewhere
W
(2.219)
Q
E z bm E z m
m 1
E zm
2x
sin m
1
W
0
2
2x
1
W
W
2
elsewhere
x
(2.220)
Out of three possible definitions of characteristic impedance of non-TEM
mode, the most frequently used definition is based on the slot voltage and time
averaged power flow. These quantities are
V E x n 0
and
P
1 h1 d
~
1
RC
E x n , y H y n , y E y n , y H x n , y dy
2
b n h2
(2.221)
The integration is carried out analytically and evaluated numerically in
three regions each.
2.14.7.2 Design considerations
The treatment given above involves too much computer time. Simple
design can be carried out by modelling fin lines as ridged waveguide uniformly
filled with dielectric of relative permittivity Ke. Based on this model the guide
wavelength can be given by relation
f
0
(2.222a)
re f
where, re f is frequency dependent effective dielectric constant and defined as
99
Microwave Circuits & Components
re f Ke 0
a
2
(2.222b)
ca is the cut-off wavelength. The cut-off of both unilateral and bilateral
fin lines
b
a
0.245Wb 0.173 for
Wb
1
1
Wb
16
4
W
h
a
, ha , and a h
p
a
h
(2.222c)
(2.222d)
The accuracy of other relation is about ±1 per cent
The expression for equivalent dielectric constant at cut-off is
2
cd
KC
1 F1 ha ,W0 r 1
2a
(2.222e)
F1 is the connection factor.
2.14.8
Ridge Waveguide
Ridge waveguide is essentially a capacitively loaded guide. A single ridge
version is shown in the Fig. 2.48. In this case the reduced height portion of ridge
guide represents the low impedance line.
2.14.9
Mono Strip Lines & Integrated Fin Lines
As is well known microstrip is widely used in design and fabrication of
microwave integrated circuits. At millimeter wavelengths however it is difficult
fabricate microchip circuits with high degree of precision required in modern
systems. To overcome this and other difficulties (radiation, higher modes, etc.)
integrated fin lines are often utilized as a medium for construction of millimeter
wave circuits. In practice the metallic fins are incorporated into dielectric slab
using printed circuit technique. Complex components or circuits can be processed
on a single dielectric substrate which is then inserted into rectangular guide. For
this slabs with low values offer the fin line structure that is essentially the same as
narrow width ridge guide.
The usable bandwidth of fin lines is an octave or greater since it is
basically a capacitively loaded rectangular guide. In general its attenuation is
slightly greater than that of main strip. Due to high field concentration in the edge
of the fins, the fin line configuration is restricted to low and medium power
application.
100
Transmission lines
r 10
.
r 10
.
H'
W
r
r
h
a'
(a)
2S
(b)
h
a'
Figure 2.48. A ridge waveguide has been used as model for microstrip line
2.14.10 Transition between two Transmission Lines
A microwave transition is an interface between two microwave
transmission lines to launch microwave power from one to another with the
minimum possible reflective and dissipative losses. The characteristics of a
transition are:
(a)
(b)
(c)
(d)
Low transmission and reflection losses over the operating bandwidth
Easily connectable and disconnectable
Inline design and simple fabrication.
Adaptability to operating conditions
The mechanical design meant for transition takes care of the electrical
and magnetic field distributions between the two media and is as close as possible
to keep the discontinuity reactance as small as possible. The electrical design
matches the impedances. Some of these transitions are discussed below:
(a)
Coaxial to Rectangular Waveguide Transition
Transition between the TEM coaxial mode and the TE 10 mode in
rectangular guide are frequently used in microwave system. One such configuration
is shown in Fig. 2.49(a). The coaxial line is connected to broad wall of the
waveguide with its outer conductor terminating on the wall. The central conductor
protrudes into rectangular waveguide. As TEM mode wave enters the waveguide
section the electric field lines follow along the conductor walls. The equivalent
circuit is shown in Fig. 2.49(b). The function of quarter-wave shorted stub is to
provide an open circuit in shunt with the Z0 and Z01 times.
(b)
Coaxial-to-Microstrip Transition
A coaxial-to-microstrip transition is simple. It is broadband as both
transmission lines support the TEM mode. Figure 2.50 shows typical in-plane and
right-angle coaxial-to-microstrip transitions. The centre conductor pin in these
connectors is generally soldered to the microstrip. The coaxial-to-microstrip
transitions can be represented by a simple equivalent circuit as shown in Fig. 2.51.
The VSWR of transitions should be lower than 1.2.
101
Microwave Circuits & Components
REDUCED-HEIGHT
WAVEGUIDE PORT
E LINES
COAXIAL PORT
SHORT CIRCUIT
bs
b
b
bs
a
ls
z0
SIDE VIEW
END VIEW
(a)
Zo
WAVEGUIDE
COAXIAL PORT
PORT
Z01
Z0s
SHORT
ls
EQUIVALENT CIRCUIT
(b)
Figure 2.49. Transmission lines (a) Coaxial to waveguide transition, and
(b) Equivalent circuit.
TEFLON
LAUNCH REGION
FL AN GE
50 Ohms SMA-TYPE COAX. CONNECTOR
MICROSTRIP
h
MICROSTRIP
SUBSTRATE
FIXTURE
Figure 2.50. Coaxial to waveguide transmission
(c)
Rectangular Waveguide to Microstrip Transitions
Such transition is shown in the Fig. 2.52. The ridged waveguide shown in
the figure in used for impedance matching.
102
Transmission lines
COAXIAL
MICRO-STRIP
Ls
Cs
Figure 2.51. Coaxial to waveguide transmission circuit
RIDGED WAVEGUIDE
STRIP
CONDUCTOR
DIELECTRIC SUBSTRATE
GROUND
BONDED
PLANE
Figure 2.52. Waveguide to microstrip transition
(d)
Coaxial to CPW Transition
Figure 2.53 shows two such transitions. The first one is for non-hermetic
case. The second one is for hermetic seal connector. In both transitions, the pin of
the coaxial connector rests on the strip part of the CPW and the ground planes of
the CPW and coaxial connector make electrical contact with each other. A VSWR
of <1.35 (below 18 GHz) has been obtained.
01.8 mm
2.2 mm
2.2 mm
(a)
2.2 mm
(b)
Figure 2.53. Two Coaxial to CPW transition
103
Microwave Circuits & Components
(e)
Microstrip to CPW Transition
Figure 2.54 shows the two type of Microstrip to CPW transitions. Figure
2.54(a) is non-planar. It has CPW and microstrip lines on different substrates.
Figure 2. 54(b) shows the case when the ground planes of the CPW and microstrip
are made common by placing the microstrip line on the top of the CPW ground
planes. Strips of two transmission lines are joined to each other through a metal
W=2.2 mm
h=0.635 mm
h1=0.635 mm
W
h1
h
W
(a)
(b)
h
Figure 2.54. Microstrip to CPW transition
ribbon. The open-end capacitance of the microstrip line, ribbon inductance and
open end capacitance of CPW constitute a low-pass filter whose cut off frequencies
can be varied by varying the substrate thickness of microstrip line. VSWR less than
1.4 has been obtained between 2 and 18 GHz frequencies. The planar transition
between a microstrip line and a CPW uses three symmetrical three-coupled
microstrip lines as intermediate transition.
(f)
Slot line to CPW Transition
Various types of transitions proposed are shown in Fig. 2.55. The
transition in Fig. 2.55(a) is based on CPW slot line T-junction. Similar is the case
Rs
(a)
(c)
(b)
(d)
Figure 2.55. Slot line to CPW transition
of Fig. 2.55(b). This junction works like power divider. One of the port of power
divider is terminated in open circuit, the result is a CPW to slot line transition of
Fig. 2.55(a). In the next figure quarter-wave sorted stub replaces the open circuit.
In Fig. 2.55(d) the slot is terminated in 90° radial slot line stub. Ho and
104
Transmission lines
associates8-10 have developed many other transitions as shown in the figure with
different properties and characteristics.
(g)
CPW to CPS Transition
Figure 2.56 shows a transition between a CPW and CPS. It has a CPW to
slot transition of Fig. 2.55(d) as an intermediate transitions. The widths of the
conductors of slot line is reduced to get a transition from slot line to CPS. An
insertion loss of <1 dB is achieved over a frequency band of 1.6 GHz to 7.0 GHz.
Figure 2.56. CPW to CPS transition
(h)
CPS to Slot Line Transition
Figure 2.57 shows a uniplanar CPS to slot line transition. Simon11 in 1994
developed a biplanar transition between CPS and slot line. It is shown in Fig. 2.57.
CPS is first transformed into coupled microstrip lines by using the conductor of
slot line as ground plane for coupled microstrip lines. The spacing between the
coupled lines is then slowly flared so that they get uncoupled and cross the slot line
at right angles in a way exactly similar to microstrip slot line transition.
Figure 2.57. CPS to slot line transition
(i)
Microstrip to Slot Line Cross-junction Transition
The fabrication of microstrip to slot line can easily be included in the
MIC fabrication routine. One such transition is shown in Fig. 2.58. The slot line
which is etched on one side of the substrate is crossed at right angles by a microstrip
conductor on the opposite side. The microstrip extends about one quarter of
105
Microwave Circuits & Components
STRIP ABOVE SUBSTRATE
s / 4
m
4
SLOT BELOW SUBSTRATE
Figure 2.58. Microstrip to strip line transition
wavelength beyond the slot. The transition can be fabricated using photo etching
process. Coupling between the slot line and microstrip line is achieved by means of
magnetic field.
(j)
Coaxial to Slot Line Transition
The slot line coaxial line transition consists of miniature coaxial line
placed perpendicular to and at the end of an open-circuited slot line. This is shown
in Fig. 2.59. The outer conductor of the cable is electrically connected to the
metallisation in left half of slot plane. The inner conductor is extended over the slot
over the slot and connected to the metallisation on the opposite line of a slot.
COAXIAL
SLOT
MOVABLE
SHORT
Figure2.59. Coaxial to slot line transition
(k)
Microstrip to Slot Line Cross Junction Transition
Microstrip to slot line transition can be easily fabricated using the MIC
technique. Such a transition (Figs. 2.60 and 2.61) shows the mechanism of
production of 180 phase difference. The slot line, which is etched on one side of
substrate, is crossed at a right angle by a microstrip conductor on the opposite side.
The microstrip extends about one quarter of wavelength beyond the slot. This
transition can be fabricated using the photo etching process and is easily
reproducible.
106
Transmission lines
MICRO STRIP
SLOTLINE
MICRO STRIP
Figure 2.60. Microstrip to slot line junction transition
y
E
x
b
INPUT
a
E
MICRO-STRIP
SLOTLINE
x
Z
c
E
d
y
E
E
x
Figure 2.61. Mechanism of 180° phase difference in transition
(l)
Fin Line to Waveguide Taper Transition
Figure 2.62 shows such a transition. This employs a tapered fin line in
which the slot width of fin line is gradually increased to full height of the
waveguide. At the waveguide end of taper transition the characteristic impedance
is different from the waveguide impedance Z0 due to dielectric loading by the
substrate. This effect is minimized by the using quarter-wave transformer.
Z(0)=Z2
Z3
Z(L)=Z1
W
r
Z=0
Z=Zl
Zin(Z)
Figure 2.62. Transition (a) Fin line and waveguide transition (b) Tapered
transmission line with a load.
107
Microwave Circuits & Components
(m)
Fin Line to Microstrip Taper Transition
This transition uses all the three types of transmission line, viz, fin line, to
waveguide and microstrip, i.e., it is a fin line to waveguide to microstrip transition.
It also uses antipodal fin line. Figure 2.63 shows the fins of the antipodal fin line,
h
GROUNDPLANE
MIC
ANTIPODAL TO
MICROSTRIP
TRANSITION
ROS
TRI
P
LIN
SHORTED
END
E
lo
b
FRONT METALLIZATION
BACK METALLIZATION
a
SUBSTRATE
b
W
R
S
l
0 r
W
h
c
c
a
Figure 2.63. Fin line to microstrip taper transition
108
Transmission lines
on opposite sides of substrate, and tapered to form a nearly circular arc over
transition length L. Beyond this arc, one of fins forms the ground plane of the
microstrip and other forms the top strip. The portion of waveguide below the
microstrip ground cannot propagate the signal as the cut-off frequency is well
above the waveguide band. It can be terminated short without affecting the
performance of transition.
(n)
Slot line to CPW Transition
This transition has been studied most comprhensively. Various types of
transitions that has been proposed is based on the CPW slot line T-junction of
Fig. 2.64a. This junction works like a power divider. If one ot the output ports of
power divider is terminated in an open circuits, we obtain the CPW T-junction of
Fig. 2.64a. This junction works like a power divider. If one of the output ports of
power divider is terminated in an open circuits, we obtain the CPW-to-slot line
transition of Fig. 2.64a. In this transition there is complete transfer of currents from
the CPW to the slot line. While current on the strip of CPW flows into one of the
1
Z om
1
SLOTLINE
Z om
n
2
I
Zos
3
(b)
(a)
Figure 2.64(a). Schematic and equivalent circuit of a microstrip slot line T-junctions
Zos
Z os
Z om
Z
os
Z om
Figure 2.64(b). Schematic and equivalent circuit of a microstrip slot line parallel
T-junctions.
conductors of the slot line, the backward current on ground plane of the CPW
flows from the other conductor plane of the slot line through a bending wire.
2.15
CONCLUDING REMARKS
There are many applications and these type of transmission lines for
which excellent texts are available. A comprehensive study of their properties is
109
Microwave Circuits & Components
beyond the scope of this monograph. Some of the common types have been
described in detail but specialized type are described in brief, so that anyone who is
really interested may study them elsewahere. Some of the properties developed in
the present text are used for all types of transmission kind. The powerhandling
capacity of various types of lines are different as they mostly depend upon the
connectors used. The waveguide has the maximum power capacity but is very
bulky. Size can be reduced as frequency is increased and can be easily fabricated
upto 100 GHz and therefore used for physical experiments upto that frequency.
Four types of planer transmission line suitable for microwave integrated
circuits discussed in the text are (i) microstrip line (ii) slot line (iii) coplanar wave
guide and (iv) coplanar strips. The range of characteristic impedances that can be
practically realized by any transmission line are limited by two factors.
Technological processes such as photo etching limit the minimum strip width and
the spacing between the two strips. The minimum dimension for the present test
has been taken as 5 mm. The other limitation comes from possibility of execution
of higher order modes. To avoid higher order, the substrate thickness and lateral
dimensions should be kept below a quality wavelength. Table 2.2 compares the
impedance limits of four types of line.
Table 2.2. Comparison of Z0 limits ( r = 13, h =100} at frequency of 30 GHz
Transmission line
Lower limit of Z0(O)
Upper limit ofZ0(Q)
Microstrip
11(m)
110(d)
Slot line
35(d)
250(m)
Coplanar waveguide
20(m,d)
250(m,d)
Coplanar strips
20(m,d)
250(m,d)
m = limit due to modes
d = limit due to small dimension
3
b 50 m
2.5
CPW
2
b 100m
1.5
1
b 200m
0.5
MICRO STRIP
00
0
20
40
60
80
100
120
140
CHARACTERISTIC IMPEDANCE Ohm
Figure 2.65. Conductor loss for microstrip and CPW lines
substrate width 100 m, er = 13, frequency = 20 GHz.
110
on
GaAs
Transmission lines
S=0.02 mm
2.0
CPW
SLOT LINE
CPW
1.5
S=0.1mm
1.0
S=0.4mm
.5
0
20
40
80
60
100
120
140
Figure 2.66. Conductor losses for slot line CPW GaAs substrate width 100 m,
er = 12.8, frequency = 20 GHz.
Attenuation constant is yet another characteristic to be taken into
consideration. Figures 2.65 and 2.66 compare these values. It can be seen that
high-impedance lines are less lossy if realised in slot line configuration and a CPW
configuration with wider strips should be chosen for low impedance levels.
Fabrication constraints are also less stringent if under slots and strips are
used. Impedance variation caused by tolerances are expressed in terms of
maximum value of VSWR presented to an ideal line connected at the input. These
are compared in the next Table 2.3.
Table 2.3.
Comparisons on effect of tolerances on various lines (G = 13
10.1; h =100I5 m, =s = 1 m and Z0= 50 .
Transmission line
Microstrip
Max.VSWR
Max
re
re
1.033
0.013
1.015
0.0036
1.07
0.044
(W/h = 0.731)
Slot line
(W/h = 0.1, h/= 0.01
Coplanar guider
(h/b = 8, a/b = 0.4)
The fabrication accuracy of strip width and gap width has been assumed
to be 1m. The assumed tolerances in h, r are 5 m and 0.1 m respectively. It
can been be seen that slot impedance is less sensitive to variations in parameters as
compared to other lines.
111
Microwave Circuits & Components
Several other parameters of four types of lines are compared qualitatively
in Table 2.4.
Table 2.4. Comparison of various MIC lines
Characteristics
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Effective dilative
constant (Er = 13
and h = 10 m)
Power handling
capability
Radiation loss
Unloaded Q
Dispersion
Mounting of
components in
shunt configuration
In series
configuration
Technological
difficulties
Elliptically
polarized
magnetic field
configuration
Enclosure
similar ions
Microstrip
Slotline
8.6
5.07
High
Medium
Medium
Low
Medium
Medium
Low
Medium
Medium
Small
Difficult
Large
Easy
Small
Easy
Easy
Difficult
Easy
Easy
—
—
~7
Ceramic
—
hole edge plating
Not
Available
available
Small
Coplanar
waveguide
Large
Coplanar
strip
~7
Medium
Medium
Low
(lower impedance);
High
(higher impedance)
Small
Easy
Available
Available
Large
Large
The best feature of coplanar lines is ease in mounting components in
series and shunt configurations, whereas microstrip lines are communications for
series mounting and slot lines can accommodate only shunted-mounted
components. A coplanar waveguide has the advantage of easier fabrication with no
via holes and good grounding for active devices. Disadvantages are larger size,
parasitic odd modes and poor heat transfer for active devices.
There are two more aspects to any consideration of power handling in
microstrip continuously working (CW) and power operation. Under CW condition
the major problems and limitations are thermal whereas under pulse conditions the
principal limitation is dielectric break down. An expression for temperature rise is
T
d
0.2303h c
c/w
K
Weff 2Weff ( f )
where, c and d are conductor and electric losses respectively in dB/m, Weff and
Weff (f) are effective microstrip widths, K is thermal conductivity of substrate. This
gives following values for substrate:
Silicon
0. 04 c/w
Gallium arsenide (GaAs)
0.06 c/w
112
Transmission lines
Quartz
Polystyrene
0.2 c/w
1.0 c/w
If we consider temperature rise of 75°C above ambience, a 50
microstrip line on alumina substrate could be
Pma (maximum average power) =
=
Therefore
T
75
0.02
Pmc = 3.75 W
Fin line is a quasi planar transmission lines. This can be used for circuit
applications at millimeter wavelengths. The main characteristics of fin lines are
large bandwidth, compatibility with planar circuit technology and absence of
radiation. Although waveguides are available for long time and have relatively less
losses but they cannot be fabricated by integrated circuit technology. Microstrip,
slot line and coplanar lines have been used extensively in integrated circuits. These
lines, however have problems of tolerance requirement with the very narrow strip
widths required at millemeter wavelength. Multimoding and radiations are also of
concern.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Everitt, W. E. & Anner, G.E. Communication engineering (3 rd Ed).
McGraw Hill Book Co. Inc. 1956.
Halliday, D. & Resnick, R. Physics Part II. Wiley Eastern Pvt. Ltd., 1970.
Rizzi, Peter A. Microwave engineering passive circuits. Prentice Hall,
1988. Chapter 3.
Potter , H. L. & Fich, S. J. Theory of networks and lines. Prentice Hall of
India, 1965. Chapter 8.
Liao, S. Y. Microwave devices and circuits. Prentice Hall, 1990. Chapter 3.
Pozer, David M. Microwave engineering, (2 nd Ed.). John Wiley & Sons,
1988. Chapter 3.
Collin, Robert E. Foundation for microwave engineering. McGraw Hill
International Edition, 1992. Chapter 3.
Ho, T.Q. & Hart ,S. M. A broadband coplanar wave guide to slot transitions
IEEE Microwave Guided Wave Litt. 1992, 2, 415.
Ho, T.Q.; et al. Broadband unipolar hybrid ring couples. Electronic Litt.
1993, 29, 4.
Ho, T.Q.; et al. Broadband uniplanar hybrid ring and branch line couples.
IEEE Trans, 1993, MTT-41, 2116.
Simon, R.N. Novel coplanar stripline to slot line transition on high
restriction to silicon. Elec. Litt. 1994, 309, 654.
113
CHAPTER 3
SCATTERING MATRIX
3.1
INTRODUCTION
Measurement of commonly accepted design parameters such as
impedance Z (series-series), admittance (parallel-parallel) Y or hybrid parameters
(series-parallel) H becomes increasingly difficult to measure at frequencies beyond
100 MHz. If these parameters is to be measured for frequencies above 1 GHz in
open circuit or short circuit conditions, the design of the measurement system
becomes very complex. The parameters which can be easily measured at
microwave frequencies on transmission lines are the standing wave ratio, the
reflection coefficient and the power. Another directly measured parameter is the
transmission coefficient through a circuit or junction. In other words, the amplitude
and phase angle of waves reflected, scattered or transmitted through a junction
relative to the incident wave amplitude and phase can be measured. The scattered
wave amplitude are linearly related to the incident wave amplitude. The concept
of travelling waves along a transmission line and the scattering matrix of a junction
of transmission lines are well known and they play important roles in theories of
microwave circuits. However, the travelling wave concept is more closely related
to the voltage or current along the line. But it is not possible to measure voltage and
current at high frequency. It is convenient to measure power which is related to
voltage and current. Therefore, it is easy to extend these concepts in development
of Network Analysers which are widely used for measurement of ‘S’ parameters
and other parameters or responses.
In the design of microwave transistor amplifiers two port scattering
parameters1,2 of transistors are widely used. However, it should not be thought that
these scattering parameters are widely different from other parameters. In this
chapter first, the concept of scattering parameters is introduced, proper definitions
are given, an introduction of other parameters are given and their relationship with
scattering parameters is derived. Secondly, at the end of the chapter some
applications of scattering parameters are described.
3.2
NETWORK REPRESENTATION
This section provides a summary of network characterisation techniques
based on linear time-invariant networks. The circuit elements when connected
together forms a network. Figure 3.1 shows a simple T-network. The voltage and
current parameters are also shown. The pair of external terminals used to connect a
Microwave Circuits & Components
stimulus or termination is known as a port, and a network with n-ports is known as
n-port network. Two currents I1 and I1 flow through the source-end termination
and must be same. The voltage at the source end is V1 and at load-end is V2 where
I 2 I 2 . Therefore, there are two parameters per port. Any two of the four
parameters of a two-port network can be taken as independent variables and the
other two as dependent variable. There are six combinations of voltage and current;
taking two at a time, we have six different representations of a network.
I'1
I1
1
z1
2
z2
v1
v2
z3
1'
I'2
I2
2'
Figure 3.1. Simple T-network
3.2.1
Impedance Representation
If we select current I1 and I 2 as independent variable and voltage V1
and V2 as dependent variable, the relation between them can be written as
V1 Z11 I 1 Z12 I 2
V2 Z 21 I 1 Z 22 I 2
(3.1a)
or it can be written in matrix notation as
V1 Z 11
V Z
2 21
Z 12 I 1
Z 22 I 2
(3.1b)
V Z I
The four parameters Z11 , Z12 , Z 21 and Z 22 can be defined as
Z11
V1
I1
Z12
V1
I2
I 2 0
I1 0
Z 21
V2
I1
I 2 0
Z 22
V2
I2
I1 0
116
(3.1c)
(3.1d)
Scattering Matrix
These coefficients are known as impedance parameters, as these
parameters have dimensions of impedance. These are generally known as
Z-parameter. When I1 or I 2 is zero, the parameters are open-circuit parameters.
Very often normalised parameters are convenient to use. The normalised
Z n matrix is given by
Z11
Z n ZZ 0
21
Z0
Z12
Z 0
Z 22
Z 0
The network has following characteristics:
(a)
When Z11 Z 22 , the network is unchanged when one or two ports are
interchanged, the network is said to be symmetrical.
(b)
When Z12 Z 21 , the response is identical when the excitation is applied
at port 1 or port 2, the network is said to be reciprocal.
3.2.2
Admittance Representation
Selecting V1 and V2 as independent variables and I1 and I 2 as
dependent variable, it can be written as
I 1 Y11V1 Y12V 2
(3.2a)
I 2 Y21V1 Y22V1
and the parameters are known as admittance parameter and the matrix
representation of above network is
I1 Y11 Y12 V1
I Y
2 21 Y22 V2
(3.2b)
I Y V
These coefficients can be defined as
Y11
I1
V1
Y12
I1
V2
V2 0
V1 0
Y21
I2
V1
V2 0
Y22
I2
V2
V1 0
117
(3.2c)
(3.2d)
Microwave Circuits & Components
If V1 V2 0 it will be closed circuit parameter whereas Y-parameter is
short circuit parameter. The normalised admittance matrix Y n is given by
Y Z
Y n 11 0
Y21Z 0
Y11 Y12
Y12 Z 0 Y0 Y0
Y22 Z 0 Y21 Y22
Y
0 Y0
Y0 is reference admittance and Z 0 is reference impedance. The
characteristics of this network representation is
(a)
If Y-parameters Y11 Y22 the network is said to be symmetrical.
(b)
If Y-parameters Y12 Y21 the network is said to reciprocal.
3.2.3
Hybrid Representation
If the current at the input port and voltage at the output port are
independent variables and voltage at the input port and current at the output port
are dependent variables, then
V1 h11 I1 h12V2
I 2 h21 I1 h22V2
(3.3a)
or the matrix representation is
V1 h11
I h
2 21
h12 I1
h22 V2 ;
(3.3b)
The coefficients h11 , h12 , h21 and h22 are defined as
h11
V1
I1
h12
V1
V2
V2 0
I1 0
h21
I2
I1
V2 0
h22
I2
V2
I1 0
(3.3c)
(3.3d)
h11 has units of impedance and h22 has units of admittance. The two
transfer coefficients h12 and h21 are voltage and current ratios which are
dimensionless. These coefficients are known as h or hybrid parameters.
The normalised h-parameters define the normalised H-matrix H n
h11
H Z 0
n
h21
h12
h22 Z 0
118
Scattering Matrix
It may be noted that
(a)
The network is symmetrical if
h11h22 h12 h21 1
which is normally written as n 1
(b)
The network is reciprocal if
h21 h12
3.2.4
G Parameter Representation
Reversing the dependent and independent variables of h-parameter, the
relation obtained between these parameters are
I 1 g11V1 g12 I 2
V2 g 21V1 g 22 I 2
(3.4a)
or
I1 g11
V g
2 21
g11
g 21
g12 V1
g 22 I 2 ;
g12
g 22
G
(3.4b)
These four coefficients are defined as
V
I
g11 1
g 21 2
V1 I 0
V1 I 0
2
and
g12
I1
I2
g 22
V1 0
V2
I2
V1 0
The normalised g-parameter matrix is
G n
(a)
g11 Z 0
g 21
g12
g 22
Z 0
The network is symmetrical if
g11 g 22 g12 g 21 1
or, g 1
(b)
(3.4c)
2
The network is reciprocal if
g 21 g12
119
(3.4d)
Microwave Circuits & Components
3.2.5
ABCD Parameter Representation
Remember that z, y, h and g-parameters are defined by using independent
and dependent variables. Another useful representation is obtained by taking
current and voltage at the input port as dependent variable and the voltage and
current at the output port as independent variables. These are used particularly for
cascading networks. If V2 and I 2 are voltage and current respectively at the output
port and are independent variables and V1 and I1 are voltage and current
respectively at input port are the dependent variables, the relation between them
can be expressed as
V1 AV2 BI 2
I1 CV 2 DI 2
(3.5a)
or in the matrix form as
V1 A B V2
I
1 C D I 2
(3.5b)
A , B , C and D coefficients are defined as
A
V1
V2
B
C
I 2 0
V1
I2
D
V2 0
I1
V2
I 2 0
I1
I2
V2 0
(3.5c)
(3.5d)
A and D are dimensionless coefficients, B has dimension of impedance
and C has dimensions of admittance. These coefficients are known as
ABCD parameters or chain parameters. The normalised chain parameters are
Ch n A
CZ 0
B
Z0
D
The characteristics of this representation are following:
(a)
The network is symmetrical if
AD
(b)
The network is reciprocal if
AD BC 1
or
1
120
Scattering Matrix
3.2.6
Inverse Chain Parameters
The inverse chain parameters are rarely used. They are almost similar to
chain parameters. In this representation V2 and I2 are used as dependent variables.
V2 EV1 FI1
I1 GV1 HI1
(3.6a)
V2 E
I
2 G
(3.6b)
or
F V1
H I1
The four coefficients are defined as
V
I
E 2
G 2
V1 I 0
V1 I 0
1
1
F
V2
I1
H
V1 0
I2
I1
V1 0
In this case E and H are dimensionless coefficients. F has dimension of
impedance and G has dimension of admittance. These parameters are known as
inverse chain parameters. The normalised matrix of these coefficients are
I n Chn
E
GZ 0
F
Z0
D
The characteristics of this representation are following:
(a)
The network is symmetrical if
EH
(b)
The network is reciprocal if
EH FG 1
or, 1
3.3
PARAMETER CONVERSION
A network can be represented by using any of six parameter sets. In many
applications it is necessary to convert from one parameter set to another. It can be
easily noticed that
Y Z 1
G H 1
I n Ch Ch1
The conversion between other parameter is not quite as simple. For
121
Microwave Circuits & Components
example, to convert from Z- to h-parameter, for V2 0 condition the following
equation may be compared.
Comparing Eqns. 3.3c and 3.2c
h11
V1
I1
V2 0
This gives the relation h11
gives Y11
Thus,
Y11
I1
V1
V2 0
(3.7)
1
. Inverting the Z matrix to form Y matrix
Y11
Z 22
where Z Z11 Z 22 Z12 Z 21
Z
h11
Z
Z 22
(3.8)
When the input current is zero it is possible to compare Eqns. 3.3d and
3.1d to yield
h22
I2
V2
which gives h22
I1 0
Z 22
V2
I2
1
Z 22
I1 0
(3.9)
(3.10)
For I1 0 , we find that
h12
V1
V2
(3.11a)
I1 0
then from Eqn. 3.1d for V1 and V2
V1 Z12 I 2 and V2 Z 22 I 2
(3.11b)
and by substituting in Eqn. 3.11b, we get
h12
Z12
Z 22
Similarly, we can write for h21 from Eqn. 3.3c and solve for I 1 and I 2
from Eqn. 3.2c, then we find that
122
Scattering Matrix
h21
Y21
Y11
(3.12)
V2 0
Thus,
h21
Z 21
Z 22
(3.13)
V2 0
Table 3.1 shows the conversions between the five parameter sets. There
is no need for the sixth conversion, as it is very rarely used.
Table 3.1. Interconversion parameter set
To
Z
Z
z11 z12
Z0 Z 0
z21 z22
Z0 Z 0
Y
z22 z12
z
z
z z
21 11
z z
G
1
z
12
z11 z11
z21 z
z11 z11
ABCD
H
Y
y22
y
y21
y
y
12
y
y11
y
y11 Z 0 y12 Z0
y12 Z 0 y22 Z0
y
y22
y21
y22
z z12
z22 z22
z
21 1
z22 z22
z11
z
z21 z21
1 z
22
z21 z21
G
1
y11
y21
y11
y22
y21
y
y21
y12
y22
1
y22
y
12
y11
y
y11
1
y21
y
21
y21
H
1
g
12
g11 g11
g21 g
g11 g11
g12
g
22
g22
1
g
21 g
g22 22
g
g11Z0
g21
g12
g22
Z0
g22 g
12
g
g
g21 g
11
g
h
h22
h21
h22
1
h11
h21
h11
ABCD
h12
h22
1
h22
h12
h11
h
h11
h22
h
12
h
h
h21 h11
h h
h11
Z0
h12
g
h21 h22 Z0
1 g22
g21 g21
g11 g
g21 g21
h
h
11
h21 h21
h
1
22
h21 h21
123
A
C
1
C
C
D
C
D
B
B
1 A
B B
C
A
A
1 B
A A
B
D
1
D
D
C
D
A
B
Z0
CZ 0 D
Microwave Circuits & Components
3.4
SCATTERING PARAMETERS
The difficulty in measuring voltages and currents for non TEM lines (such
as waveguides or microstrip line) has already been mentioned. It is difficult to
measure magnitude and phase of voltage and current at microwave frequencies.
The equivalent voltages, currents, impedance and admittance parameters at
microwave frequency becomes complex. The ideas of incident, reflected and
transmitted waves are exploited by defining scattering matrix. The direct
measurement of scattering coefficients is possible.
The question now is that how to define scattering coefficients and
scattering matrix. Evidently this must be related with the six representations
mentioned earlier in this chapter. Figure 3.2 shows an N-port device.
The classic method of solving circuits using concept of voltage and
current, impedance and admittance is replaced by electric fields, magnetic fields,
incident, reflected and transmitted waves. In the Fig. 3.2 the question is how to
define a k and bk ? This however, will be dealt in a later section. In establishing the
scattering coefficients description of a microwave device, the incident and
reflecting travelling waves of voltage at the device are separately dealt with. The
voltage wave is defined in terms of power but normalised by conventional usage, in
a different way. The complex voltages at the terminals are defined in terms of a
and b , representing the incident and reflecting waves, respectively. In this
representation voltage at terminal say k is given by
V k a k bk
For the present these voltages are normalised so that power carried by a k
is Wk a k a k . In terms of these scattering coefficients are defined such that
voltage wave bi leaves the i th port due to a voltage wave a j incident upon the j th
port, when no waves enter in any other ports.
bi S i j a j
(3.14)
Generally, when a voltage is incident upon all of n ports of the device,
each incident wave will make contribution to total resultant wave departing from
ak
bk
R
a2
b2
1
N
2
bn an
b1
a1
Figure 3.2. N-port device
124
Scattering Matrix
i th port. Therefore,
bi S i 1a1 S i 2 a 2 S i 3 a3 S i n a n
(3.15)
The scattered wave departing from each of the port can be written by a set
of scattering equations
b1 S11a1 S12 a 2 S13 a3 S1n a n
b2 S 21a1 S 22 a 2 S 23 a3 S 2n a n
b3 S 31a1 S 32 a 2 S 33 a3 S 3n a n
.................................................................
................................................................
bn S n1a1 S n 2 a 2 S n3 a3 S nn a n
(3.16)
It may be noted that these coefficients are complex and the above
equations can be written in the form
b1 S11
b S
2 21
b3 S 31
b S
n n1
S12
S 22
S 32
S13
S 23
S 33
S14
S 24
S 34
S n2
S n3
S n4
S1n a1
S 2n a 2
S 3n a 3
S nn a n
(3.17a)
This can also be written as
b S a
(3.17b)
b and a are column matrices while S is the scattering matrix.
Therefore, the scattering matrix is
S11
S
21
S 31
S
S
n1
S12
S 22
S 32
S n2
S13
S 23
S 33
S n3
S14
S 24
S 34
S n4
125
S1n
S 2n
S 3n
S nn
(3.17c)
Microwave Circuits & Components
The diagonal element S jj of the scattering matrix is the reflection
coefficient at the j th port and represents the reflected voltage wave which would
be observed at this port with an incident voltage wave of unit magnitude and zero
phase, when all other ports are terminated in matched impedances (and hence no
waves are reflected back).
The power waves and generalised scattering matrix were defined very
elegantly in a paper by Kurokawa 3. These parameters were introduced previously
by Penfield4, 5 and also by Youla 6 for positive real reference impedances. The term
scattering comes from the similarity between the behaviour of light in the field of
optics and microwave energy. These parameters can be measured using any
convenient termination. The equipment which is more conveniently used is
Network analyser. This instrument can be used to measure the scattering
parameters; both the magnitude and the phase for one or two-port network from
0.5 GHz to 100 GHz. Built in microprocessors provide error correction, a high
degree of accuracy and also a wide choice of display formats. Figure 3.3 shows a
two-port network driven at port 1 by Z1 voltage source and terminated at the
port 2 by a Z 2 impedance.
a1
b2
z1
+
Vs
-
I1
I2
V1
V2
b1
Z2
a2
Figure 3.3. A two-port network
The scattering parameters use either a1 and b1 or a2 and b2 as
dependent variables and the two networks can be cascaded with proper use of
S-matrix. Assuming the reference impedance Z 0 , we can define the parameters
ai and bi as
ai
Vi Z i I i
2 ReZ i
i 1,2,3, , n
(3.18a)
and
bi
Vi Z i I i
(3.18b)
2 ReZ i
where, Z i is the complex conjugate of Z i . ReZ i is the real component of the
reference impedance. In terms of scattering parameters following equation can be
126
Scattering Matrix
written as
b1 S11a1 S12 a 2
(3.19a)
b2 S 21a1 S 22 a 2
(3.19b)
In matrix notation it can be written as
b S a
(3.20)
for any n-port network. The physical concept of scattering parameters in terms of
physically measurable quantities are explained now. S ii is explained in terms of
Return loss at port i and S ji is explained in terms of power gain (transducer) of the
circuit. Setting a 2 0. It is important to know how can it be set equal to zero. It
is very simple if we put V2 I 2 Z 2 . Then from Eqn. 3.18(a)
a2
V2 Z 2 I 2
2
Re Z 2
I2 Z2 Z2 I2
2
Re Z 2
0
From transmission line theory
Vi Vi I Vi R
(3.21a)
Vi I Vi R
Zi
Zi
(3.21b)
Ii
I and R denote incident and reflected components. Assuming Z i is real
ai
Vi Z 0 I i
2
ReZ i
Vi I
V
V
Vi R Z i i I i R
Z
Zi
i
2
ReZ i
Vi I
ReZ i
(3.22a)
and
bi
Vi Z i I i
2
ReZ i
Vi I
V
V
Vi R Z i i I i R
Zi
Zi
2
ReZ i
Vi R
ReZ i
(3.22b)
127
Microwave Circuits & Components
Zi Zi
which indicates that ai is a function of incident voltage and bi is the function of
reflected voltage and can be simplified as
ai
2
bi
2
Vi I
2
Vi R
2
(3.23a)
ReZ i
(3.23b)
ReZ i
2
Thus ai is incident power and bi
seen from the figures that
V V1
V1 Z1 S
Z1
2
a1
2 ReZ1
2
is the reflected power. It can be
2
VS
2
4 ReZ1
(3.24)
2
a1 is the power available from the source. If the reflected power by the
network is subtracted from the power available from the source we have
2
ai
bi
2
ai ai bi bi
(3.25)
ai and bi are the complex conjugate of ai and bi
Thus
2
ai
bi
2
V1 Z1I1 V1 Z1 I1 V1 Z1 I1 V1 Z1 I1
4 ReZ1
4 ReZ1
2 Z1 V1 I1 V1 I1
Z1
Re V1 I1
4 ReZ1
ReZ1
(3.26)
Z i Z i
which is the power delivered to the network. If the source is only connected to
port 1, a 2
2
can be taken as zero and
128
Scattering Matrix
2
b2
2
V2 Z 2 I 2
Re Z 2
2
Re Z 2 I 2
2
(3.27a)
which is the power delivered to the load.
2
The S i j coefficient connects power incident at port j with power
reflected out (reflected power) of port i. At port i we have
Si i
bi Vi Z i I i Z in I i Z i I i Z in Z i
ai Vi Z i I i
Z in I i Z i I i
Z in Z i
(3.27b)
where, Z in is the impedance looking into port i. Thus
Si i in Reflection coefficient at port i
and
Si i
2
bi
2
ai
2
Power reflected from the input port
Power available from the generator
(3.28)
Return loss at the port i
At any port j, where, i j
Sji
2
bj
ai
2
2
Power delivered to the load
Power available from the generator
(3.29)
Transducer power gain
These parameters are related with other parameters discussed earlier. A
relation between them can be found out. These conversions are discussed in the
next section.
3.4.1
get
Conversion of S-parameter to other Network Parameters
Assuming that the reference impedance is Z 0 , if ai is added to bi we
2
ReZ 0 a i bi Vi Z 0 I i Vi Z 0 I i 2Vi I i Z 0 Z 0
(3.30)
Subtracting bi from ai
2
ReZ 0 ai bi Vi Z 0 I i Vi Z 0 I i I i Z 0 Z 0
129
Microwave Circuits & Components
(3.31)
Solving this equation for I i
Ii
2
ReZ 0 ai bi
(3.32)
( Z 0 Z 0 )
and substituting this value in the Eqn. 3.30 for obtaining relation for Vi
Vi
2
Re( Z 0 ) ai Z 0 bi Z 0
( Z 0 Z 0 )
(3.33)
when Z 0 is real we can write relation for I i as
Ii
Z0
Z0
ai bi
(3.34)
and
Vi
Z 0 ai bi
(3.35)
If Z 0 is positive and real we can write the voltage and current relationship
for n-port in terms of matrices
V
[ Z 0 ] [ a ] [b ]
(3.36)
all the terms in this relation are matrices
I
1
[Z 0 ]
[a] [b]
(3.37)
The Z-parameters are
V Z I
(3.38)
Substituting the values for V and I in Eqn. 3.38 from Eqns. 3.36 and
3.37, we obtain
Z 0 [a ] [b] Z
1
Z0
[a] [b]
(3.39)
Dividing by the square root of the reference impedance and rearranging
130
Scattering Matrix
we obtain
1
Z b 1 Z a a
Z0
Z0
(3.40a)
1
1
Z Z U b Z Z U a
0
0
(3.40b)
b
or,
where, U is the unit matrix which has elements equal to one on main diagonal
and zero elsewhere.
Multiplying both sides by
1
Z Z U
0
1
we get
b 1 Z U
Z0
(3.41)
1
1
Z Z U a
0
(3.42)
Thus
S b 1 Z U
a Z 0
1
1
Z Z U
0
(3.43)
which gives a relation for finding out the scattering matrix from the impedance
matrix. Impedance matrix can be found from the scattering matrix using the relation
1
Z U S 1 U S
Z0
(3.44)
Similar relations can be found between Y and S remembering
Y Z 1
It can be seen that for converting Y to S following relation can be
used
S U Z 0 Y
U Z
1
0
Y
(3.45a)
Similarly, S can be converted to Y by using the relation
Z 0 Y U S
U S
1
131
(3.45b)
Microwave Circuits & Components
The parameters Z and Y are used by taking current or voltage as the
independent variable whereas other parameters are defined by using voltage and
current together as independent variable. For an n-port network the relations
become complicated and are difficult to obtain. However, it is possible to derive
relation for two-port network. Leaving ABCD-parameters, there is some
commonality between other conversions and it is possible to write a simple
algorithm. Let us define a parameter as
2
1 S11 1 S 22 S12 S 21
then S to Y conversion can be written as
1 S 22 1
.S12
.S 21
1 S11 1
Y
(3.46)
By making minor changes and appropriate substitutions, relations for
other conversion can be written as
(c)
For converting Y to S use S to Y conversion.
For converting S to Z replace S with – S , then use S to Y
conversion.
S12 S to Y conversion and then replace
For conversion from Z to S use
(d)
For conversion of S to H parameters replace S11 with – S11 and S12
(a)
(b)
`
(e)
(f)
(g)
Y with – Y .
with S12 , then use S to Y conversion.
For conversion from H to S use S to Y conversion and then replace
S11 with – S11 and S12 with – S12 ,
S to G replace S 22 with – S 22 and S 21 with
– S 21 , then use S to Y conversion.
For conversion from G to S use S to Y conversion, then replace
For conversion of
S 22 with – S 22 and S 21 with – S 21 .
We must also remember
1
Y Z
1
Z Y
1
H G
1
G H
for obtaining proper conversion
132
Scattering Matrix
Table 3.2 gives two-port conversion from S-parameters as well as
conversion to S-parameters.
3.4.2
General Properties of Scattering Matrices of Linear Lossless
Microwave Devices
(a)
The first property makes application of conservation of energy as applied to
transmission through a lossless junction. Let it be assumed that a wave of unit
voltage enters in one of the arms say 1 of an n-port device and no voltage wave
enters in any of the other ports. Hence, the power entering the device is
Win a1a1 . The power leaving the j th port is equal to b j b j . The law of
conservation of energy gives
a1a1 b1b1 b2 b2 b3b3 bn bn
(3.47)
Remembering that b j S j i ai . The scattering coefficient S ji connects
port j to port i , i.e., wave coming out of port j is b j and b j is S ji ai where ai
is the wave entering into the port i . Therefore
a1a1 S11a1 S11
a1 S21a1 S21
a1 Sn1a1 Sn1a1
(3.48)
and hence
1 S11 S11
S 21 S 21
S n1 S n1
(3.49)
Therefore
S i j S i j S i j
i j
2
1
(3.50)
ii
(b)
If the device is reciprocal S i j S j i . Most of microwave devices are
reciprocal, but not all. Devices which make use of ferrite material (magnetic
material) are non-reciprocal when put in the magnetic field. The reciprocal
property is quite evident; a reciprocal device means that the fraction of wave
coming out of port j if the wave is entering in the port i is equal to wave coming
out of port i if same voltage wave is entering the port j . This would mean that
S j i Si j
(3.51)
For reciprocal devices it further means that
Si j
j
2
S ji
2
1
i
(c)
Another important property of scattering matrix can be derived by taking
the case in which voltage waves enter in all the ports. If the device is lossless at all
133
134
H
G
Y
Z
From S Parameters
To S Parameters
Microwave Circuits & Components
Scattering Matrix
junctions and in between junctions from conservation of energy, we get the relation
that total incident power must be equal to the total output power, i.e.,
Win Wout
i.e.,
a j a j bk bk
j
(3.52a)
k
It is known from scattering matrix coefficients that
bk
S k n an
(3.52b)
k
If we substitute this value of bk in the right hand side of Eqn. 3.52(a)
j
a j a j
/
S j k S j k a j a j
j k
n
m
S
S
a
a
jm jn m n
j
/
S
S
a
a
jm jn m n
n j
m
(3.53)
where, primes on the summation over m means that the term m n is not included
in this sum as it has been already included in the earlier term. Remembering that
S j k S j k 1
k
The first term will reduce to
a j a j . Therefore, this will cancel with
j
the term on the left hand side. Therefore,
/
n
m
/
S
S
a
a
jm jn m n +
n
j
m
S
S
a
a
0
jm jn m n
j
Also recalling that
F F 2 ReF
Therefore,
/
2 Re
n
m
S
S
a
a
0
j
m
j
n
m
n
j
135
(3.54)
Microwave Circuits & Components
This expression though derived in general should hold good for any two
values of m and n (note that m n ).
If the wave is transmitted at ports 1 and 2 with all other a j set equal to
zero. Then Eqn. 3.54 becomes
2 Re S11 S12
S 21 S 22
S 31 S 32
S n1 S n2 a1a 2 0
(3.55)
Since the phases of waves a1 and a2 are arbitrary, the quantity in the
parentheses of Eqn. 3.55 must be equal to zero, i.e.,
S
11 S12
S 21 S 22
S 31 S 32
S n1 S n2 0
(3.56)
Equation 3.56 consists of sum of members of the first column of scattering
matrix, multiplied by complex conjugates of the second column of the matrix,
which is equal to zero. This also applies to sum of similar multiplication of a term
with complex conjugate of second row of matrix.
Thus remember that for symmetrical devices the scattering matrix terms
are same, i.e., S12 S 21 , S 31 S13 ..... and so on, i.e., the scattering matrix is
symmetrical about the diagonal terms for symmetrical devices.
The three properties of scattering coefficients have been derived by a
very simple method. However, they have been obtained in much more rigorous
way in books like those of Montgomery7 and have been reproduced by Altman8.
3.4.2.1 Application of scattering matrix concepts to E- and H-plane T's
Figure 3.4(a) and (b) show E-plane and H-plane T's. In one case the side
branch is in E-plane and in the other case it is in H-plane. These are poorly
matched devices since if two arms have matched impedances the third one does not
present a matched impedance to source. In terms of scattering coefficients it means
that if for terminals 2 and 3 S 22 S 33 , S11 is not equal to zero. It can be seen from
the figure that with the arms of the T , a wave entering the side arm of E-plane T
yields outgoing waves at arms 1 and 2 which are relatively reversed in phase at
equal electrical distances from the centre of the T .
2
3
1
Figure 3.4a. E-plane T
136
Scattering Matrix
This relative phase reversal of the waves leaving main guide arms does
not occur in H-plane T. The structural symmetry of the T’s simplifies the scattering
matrix. From the symmetry, it is clear that S13 S 23 and S11 S 22 in either
T. Hence, the general matrix for three port device is given as
S11
S S 21
S 31
S12
S13
S 23
S 33
S 22
S 32
it reduces to
S11
S12
S13
S13
S S12
S13
S13
S 33
S11
(3.57)
Here the positive sign refers to the H-plane T, and the negative sign to
E-plane T. It can be shown that measurement of two reflection coefficients, S11
and S 33 are sufficient to completely determine the scattering matrix of a
symmetrical T junction. These can be conveniently determined by simple
conventional method at each port mentioned above by slotted line method with
the remaining two ports terminated in matched loads. Applying Eqn. 3.50 to
H-plane T, we obtain from Eqn. 3.57
S13
2
S13
2
S 33
2
(3.58)
1
(third column) of Eqn. 3.57, and from the first column
S11
2
S12
2
S13
2
(3.59)
1
Equation 3.58 gives
S13
2
1 S 33
2
2
(3.60)
Since, the magnitudes of S 33 has already been measured the magnitude
of S13 can be found by using Eqn. 3.60. Using Eqn. 3.59 it can be seen that
S12
2
2
1 S11 S13
2
1 S11
2
1 S 33
2
(3.61)
2
This relation can be used to determine the magnitude of S12. For
determining phase a relation in Eqn. 3.57 is used so columns first and second of
137
Microwave Circuits & Components
2
3
1
Figure 3.4b. H-plane T
scattering matrix gives
S11 S12
S12 S11
S13
which gives
2
0
S11 S12 e j 11 12 e j 11 12 S13
2
0
it can also be written as
cos
11 12
S13
2
2 S11 S12
(3.62)
which further reduces to
12
11
cos
S13
1
2 S11
2
S12
(3.63)
This relation can be used to determine 1 2 and for determining 1 3 ,
column first and conjugate of column three can be used as
S12 S13
S11 S13
S13 S 33
0
this can be reduced to
S12 S11 S13 e j 13
which reduces to
e 2 j 13
S13 e j 13 S 33
S12 S11
S 33
Therefore
138
(3.64)
Scattering Matrix
13
2
S S11
1
ln 12
2j
S 33
(3.65)
This application shows the use of scattering matrix. We will discuss
another application i.e., to Magic T.
3.4.2.2 Magic or hybrid T
Figure 3.5 shows a Magic T. The symmetrical four-port junction of an Hplane arm and an E-plane arm with a section of waveguide is a microwave circuit
device. This is known as Hybrid T since it is a junction of two different T. Why is
it known as Magic T? This would be explained a little later with the help of
scattering matrix of this device. Since, it is a four-port device the general scattering
matrix of this device would be
S11
S
21
S 31
S 41
S12
S 22
S 32
S 42
S13
S 23
S 33
S 43
S14
S 24
S 34
S 44
This matrix would be simplified using some evident properties. A wave
entering arm 4 (H-plane) in Fig. 3.5 excites equal waves of like phase in arms 1
and 2 and a wave entering the E-plane arm (arm 3) excites equal waves of opposite
phase in arms 1 and 2. Due to geometrical symmetry a wave in arm 4 excites no
dominant mode in arm 3. There is no direct transmission from arm 4 to 3. It should
also be remembered that S i j S j i , thus symmetry also leads to S13 S 23 ,
S11 S 22 and also S14 S 24 . It should be noted that since ports 3 and 4 are all
independent of each other
S 34 S 43 0
In view of these properties the scattering matrix is reduced to
3
2
4
1
Figure 3.5. Magic T
139
Microwave Circuits & Components
S11
S
S 12
S13
S
14
S12
S 22
S13
S14
S13
S13
S 33
0
S14
S14
0
S 44
(3.66)
We can make use of properties
S kj
2
1
(3.67)
j
S k j S k i 0
(3.68)
k
It can be further noted that if matching devices are added to the E and H
arms of a hybrid junction such as to make the reflection coefficients at these arms
vanish, i.e., S 44 0 and S 33 0 . If the two arms are matched it can be shown by
using the scattering matrix that all the four arms are matched. This property makes
Magic T an important component for many applications 7.
Consider now the application of normalisation conditions to rows three
and four of Eqn. 3.66.
2 S13
2
S 33
2
1
2 S14
2
S 44
2
1
Since the ports (3) and (4) are matched
1
S13 S14
2
(3.69)
rows one and four gives
S11
2
2 S14
S12
2
2
S 44
S13
2
2
S14
2
1
(3.70a)
(3.70b)
1
These two equations are further reduced to
S11
2
S12
S 44
2
0
2
(3.71a)
0
(3.71b)
This equation can only be satisfied if
S11 0
S12 0
140
Scattering Matrix
as both the terms are positive. Thus, if the E and H plane arms of the junction are
matched, the other two arms are also matched. There is no transfer of power
between the arms 1 and 2 of matched junction. Because of these two properties,
i.e., (a) if the arms 3 and 4 are matched, ports 1 and 2 are also matched, (b) there
is no transmission of waves between ports 1 and 2 if the junction is matched this
hybrid junction is called Magic T. The scattering matrix of Magic T is thus
0 0 1
0 0 1
S 1
2 1 1 0
0
1 1
1
1
0
(3.72a)
0
in which input terminal planes have been chosen so that it yields the above
scattering matrix. The length of each arm from the junction is
arms are altered to
8
4
. If the length of
, then
j
j
0 0
0 0 j j
S 1
2 j j 0 0
0 0
j
j
(3.72b)
Magic T can be used for many applications which include an impedance
bridge, and discriminator. A simple application is discussed in the next section to
highlight the concepts so far developed for making scattering matrix more clear.
Consider the case of four matched generators, synchronised to the same
frequency, feeding the Magic T. If a1 , a 2 , a3 and a 4 are the normalised incident
waves respectively, at ports 1, 2, 3 and 4 the output waves are obtained by using the
relation
b S a
The expansion of this relation is
1
b1
0 0
b
0 0 1
2 1
b3
2 1 1 0
0
1 1
b4
1 a1
a3 a 4
1 a 2
1 a3 a 4
0 a 3
2 a1 a 2
0 a 4
a1 a 2
(3.73)
so that the scattered waves and the corresponding power out of the ports are
b1
1
2
a 3 a 4 ,
P1
1 2 1
b1 a3 a 4
2
4
141
2
Microwave Circuits & Components
b2
b3
b4
derived
1
2
1
2
1
2
a3 a 4 ,
P2
1
b2
2
2
1
a3 a 4
4
a1 a2 ,
P3
1
b3
2
2
1
a1 a 2
4
2
a1 a 2 ,
P4
1
b4
2
2
1
a1 a 2
4
2
2
For these general equations many special cases may immediately be
(a) If a1 a 2 a 4 0 then
1
1
2
b1
a3 , P1 a3
4
2
b2
1
2
a3 ,
b3 b4 = 0 ,
P2
1
a3
4
2
P3 P4 0
Thus, the input to port 3 is split up equally and in phase at port 1 and port
2 with no power back to port 3 or out of port 4.
(b) If a1 a 2 a3 0 , then
b1
b2
1
2
1
2
P1
1
a4
4
2
a 4 , P2
1
a4
4
2
a4 ,
b3 b4 = 0 ,
P3 P4 0
The conclusions are similar to previous case
(c) If a3 a 4 and a1 a 2 0 , then
b1 2 a3 2 a 4 , P1 a3
b2 b3 b4 0 ,
2
a4
2
P2 P3 P4 0
Thus, the two equal signals combine at port 1 and cancel out at port 2. No
power is reflected back to port 3 and to port 4.
(d) If a3 a 4 and a1 a 2 0
b2 2 a3 2 a 4 , P2 a3
2
142
a4
2
Scattering Matrix
b1 b 3 b4 0 ,
P1 P3 P4 0
Similar conclusion can be made in this case also
(e) If a 2 a 3 a 4 0
b1 b2 0 ,
b3
b4
1
2
1
2
P1 P2 0
a1 ,
P3
1
2
a1
4
a1 ,
P4
1
2
a1
4
(f) If a1 a3 a 4 0
b1 b2 0 ,
1
b3
b4
2
1
2
P1 P2 0
a2 ,
a2 ,
P3
P4
1
a2
4
1
a2
4
2
2
(g) If a1 a 2 and a3 a 4 0
b1 b2 0 ,
P1 P2 0
b3 2 a1 2 a 2 , P3 a1
b4 0 ,
2
a2
2
2
a2
2
P4 0
(h) If a 2 a1 and a3 a 4 0
b1 b2 0 ,
P1 P2 0
b3 0 ,
P3 0
b4 2 a1 2 a 2 ,
P4 a1
The conclusions in these cases are more or less similar.
143
Microwave Circuits & Components
1
0 0
0 0 1
S MT 1
2 1 1 0
0
1 1
1
1
0
0
(3.74)
When all reference planes are moved
electrical radians ( radians
4
4
away from the junction) the scattering matrix becomes
1
0 0
0 0 1
S 1
2 1 1 0
0
1 1
1
1
0
0
(3.75)
The unitary property of S is stated as
S S U
(3.76)
This property can be derived from the first property of scattering
coefficient. From Eqn. 3.64
S S
(3.77)
Hence, when Eqn. 3.77 is substituted in Eqn. 3.76
S 2 U
(3.78)
The admittance matrix
Y can now be introduced through the equation
Y U S U S 1
By the associative property of matrix operations, it becomes
Y U S U S U S 1 U S 1
U 2S S 2 1 S 2
1
(3.79)
so that
Y S
(3.80)
since
S 2 U
(3.81)
and the admittance matrix is
144
Scattering Matrix
I2
I1
Y0 A
Y0
V1
V2
Y0
/4
Figure 3.6. Quarter wave line of characteristic admittance Y0
1
0 0
0 0 1
Y j
2 1 1 0
0
1 1
1
1
0
0
(3.82)
This admittance can be obtained by means of quarter-wave lines for
4
3
positive coefficients and three quarter of wavelength for negative
4
coefficients of admittance.
Consider two ports of a quarter-wave line (Fig. 3.6) of characteristic
admittance Y0A inserted in transmission line of Y0 . The system operates in TEM
mode of lines which includes microstrip line. It can be shown that admittance
matrix of a section of line of length z is
Y
1
j Y0 cos z
cos z
1
sin z
(3.83)
where, is the propagation constant of the line. Consequently, the admittance
matrix of a quarter-wave length line of characteristic admittance Y0A normalised on
a Y0 basis
0 j
Y0 j 0
Y Y0A
(3.84)
If the line is three-quarter wavelength long Eqn. 3.84 becomes
0 j
Y0 j 0
Y Y0A
(3.85a)
These equations shows that any device whose admittance matrix is such
that diagonal elements are zero and off-diagonal elements are pure imaginary
145
Microwave Circuits & Components
Y0
2
Y0 A = Y0/
Y0
2
Y0 A
Y0 A
3
Y0 A
4
1
Figure 3.7. Two ports of a quarter-wave line
numbers, may be synthesised by means of 4 and 3 4 TEM elements of the
correct characteristic admittance.
Thus referring to Eqn. 3.82 a device with this admittance is fabricated by
means of 4 and 3 4 lines of admittance
Y0A
1
(3.85b)
Y0
2
The
self
admittances
Y11 , Y22 , Y33 and Y44
are
zero.
Since
Y12 Y34 0 there is no direct element between ports 1 and 2 and between ports
3 and 3. But between ports 1 and 3 there is a 4 element of normalized
1
characteristic admittance
2 and this holds true between ports 1 and 3.
1
Between ports 2 and 4, a 3 4 line of normalized characteristic admittance
2
j
must be inserted since Y24
. The complete synthesis is shown in Fig. 3.7.
2
4
2
/4
/4
1
3
/4
Figure 3.8. Rat-race hybrid structure
146
Scattering Matrix
g/4
b b/ 2
g/4
g/4
g/4
Figure 3.9. Microstrip hybrid T structure
The fabrication of such a device is on a microstrip line. Figure 3.8 shows the ratrace hybrid which can be realised using microstrip or strip line structure. The actual
design is shown in Fig. 3.9.
3.4.2.3 Another microstrip realisation of hybrid junction (magic T)
By moving the reference planes of Magic T by respective electrical length
corresponding to
1 3
2
2 4
The scattering matrix then becomes
0 0 1 j
0 0 j 1
S 1
2 1 j 0 0
j 1 0 0
(3.86)
The corresponding admittance matrix obtained by usual method
Y j
0
1
0
2
1
0
2
0
0
2
0
1
2
0
1
0
(3.87)
which again can be realised using 4 elements of proper characteristic
admittances. The realisation is shown in Fig. 3.10 and for coaxial line in Fig. 3.11.
The same ideas can be extended to a microstrip and strip lines.
147
Microwave Circuits & Components
/4
/4
2
3
Y0
Y0
/4
/4
4
1
Figure 3.10. Second synthesis of magic T
g/4
3
2
/4
4
1
Figure 3.11. Second coaxial synthesis of magic T
3.5
TRANSLATION OF REFERENCE PLANES
In the earlier section the property of shifting the reference plane has
already been discussed to some extent. The reciprocal devices possess several
plane of symmetry. In addition if the reference planes are located symmetrically
the terms of scattering matrix related to the symmetrically located ports are either
equal or of opposite sign depending upon the orientation of the electric field. In the
Fig. 3.12 the origin of Z i coordinate on line i is defined arbitrarily. The reference
plane may be shifted along the transmission line by a distance Z i . The displaced
coordinates in the new system Z id Z i Z i . The input voltage wave ai and
backward wave bi maintain their respective amplitude, but their phases are shifted
148
Scattering Matrix
i
Zi
Zi
Figure 3.12. Translation of the reference plane at port i
in proportion to the distance Z i .
ai ai exp j i
bi bi exp j i
W1 2
(3.88a)
W1 2
(3.88b)
remembering that i i Z i
Shifts may be carried out at the same time within all n-ports of the device.
The column vector of incoming signal a and outgoing signal b becomes
a diag exp j a
W1 2
(3.89a)
b diag exp j b
W1 2
(3.89b)
where, W stands for power..
The diagonal matrix is defined as
0
exp j1
0
exp j 2
diag exp j
0
0
S
0 0
0 0
0
0
0 0 exp j n
(3.90)
The scattering matrix within system of the displaced reference planes
is obtained by using the relations already developed.
It can therefore be seen that
b diag exp j S
diag exp j a
149
Microwave Circuits & Components
S a
W1 2
we know that b S a
the equation for each component of matrix then becomes
S i j S i j exp j i j
A shift of the reference planes produces a change in argument of terms of
scattering matrix. The matrix is not modified.
3.6
FLOW GRAPHS OF TWO PORT DEVICES
Chow and Cassignol9 in 1965 gave a physical meaning to scattering
matrix which has been found to be useful in many cases. Every port is represented
by two modes, one at which input wave ai enters and the other at which output
wave bi departs. Every one of S ij terms are indicated in the diagram by a thick
arrow showing the direction from input to output at two different ports as well as at
the same port. Such a signal flow graph is shown in Fig. 3.13.
S 21
b2
a1
S11
S 22
b1
S 12
a2
Figure 3.13. Flow graph of two port device
It can be seen that it is just pictorial representation of two-port scattering
coefficient equations:
b1 S11a1 S12 a 2
W1 2
b2 S 21a1 S 22 a 2
W1 2
The full arrow indicates an incoming or outgoing signal. The contribution
of bigger white arrows are the product of the transfer function S ij associated to
the arrow multiplied by the value of signal wave at node j .
Signals flow graphs are of interest in inter connection analysis of several
devices using the simple deduction rules. They are
150
Scattering Matrix
(a) Multiplication: Two arrows showing signal flow represents multiplication of
two transfer function. This is shown in Fig. 3.14.
S2
S1
a
b
s
Figure 3.14. Flow graph multiplication
It can be easily shown that the resulting transfer function S is the product
of S1 and S 2
S S1 S 2
(3.91)
(b) Addition: If the two arrows are in shunt they can be replaced by sum of their
transfer functions, i.e.,
(3.92)
S S1 S 2
as shown in the Fig. 3.15
S1
S
a
a
b
S2
b
Figure 3.15. Flow graph addition
(c) Feedback loop: The schematic is as shown in Fig. 3.16. It is similar to above
case except one of the arrow is pointing in the reverse direction. It can be seen
from the figure that
S
S1
1 S1S 2
It can be obtained in the following way. It can be seen that
b S1 a S 2 b
W1 2
(3.93)
or in other terms
b1 S1 S 2 S1a
and therefore
151
Microwave Circuits & Components
S1
a
a
b
S
b
S2
Figure 3.16. Feedback loop flow graph
S
S1
b
a 1 S1 S 2
(3.94)
Example 3.1. Find the equivalent scattering coefficient of arrangement shown in
Fig. 3.17.
Solution: The useful equations in this case are
S 22
V1
S 21
S 32
V1
V3
Figure 3.17. Flow graph of a simple network (self-loop)
V 2 S 21V1 S 22V 2
(3.95)
V3 S 32V2
(3.96)
We see from Eqn. 3.95
V2 1 S 22 S 21V1
Substituting the value of V2 from Eqn. 3.96
V3
1 S 22 S 21V1
S 32
V3 S 21 S 32
V1 1 S 22
which gives for equivalent transfer function
S
S 21 S 32
1 S 22
This is known as self loop rule
152
Scattering Matrix
3.7
SIGNAL FLOW GRAPH FOR THREE & FOUR PORT
DEVICES
The scattering matrix of a three port devices, as written earlier is
reproduced
S11 S12 S13
S S 21 S 22 S 23
(3.97)
S 31 S 32 S 33
Fig. 3.18
The signal flow graph representing this scattering matrix is shown in
As mentioned earlier that the three ports cannot be matched at the same
time at all three ports. It has been shown earlier but it can be established in a more
general way. Let us assume that Sii 0 , so that the port i is matched and if the
device is reciprocal Sij S ji . Using the properties of scattering coefficients
S12
2
S13
2
1
(3.98a)
S12
2
S 23
2
1
(3.98b)
2
(3.99)
2
S13 S 23 1
and
S13
S 23 0
(3.100a)
S12
S 23 0
(3.100b)
S12
S13 0
(3.100c)
a1
b2
s21
s11
s22
s12
b1
a2
s31
s23
s32
s13
s33
b3
a3
Figure 3.18. Flow graph of a three port device
153
Microwave Circuits & Components
Let us assume that one term is different from other, for instance S13 0 .
From Eqn. 3.101(a), one must they have S 23 0 and from Eqn. 3.101(c) and
Eqn.3.101(b) S12=0.
2
2
This will lead to S12 S 23 0 which relates to Eqn. 3.100(b). This
indicates the absurdity as it is not possible to construct a three port device which is
reciprocal as well as matched at all three ports.
Therefore, if the device is matched it has to be non-reciprocal [an example
of this is circulator (Fig. 3.19)]. The circulator can be analysed by using Fig. 3.20
S 21
2
S 31
S12
2
S 32
2
2
1
(3.101a)
1
(3.101b)
The energy conversion expression
s21=1
a1
b1
b2
s13=1
s32=1
a2
a3
Figure 3.19. Flow graph of a circulator
a1
s21
b2
s12
a2
b1
s33
a3
Figure 3.20.
a1
b3
Lossless three port device at two ports. It appears to consist of two
physically independant devices.
154
Scattering Matrix
n
Sij* Sik S jk
i 1
yields in this case
S13
2
S 23
2
(3.101c)
1
S12
S13 0
(3.102a)
S 21
S 23 0
(3.102b)
S 31
S 32 0
(3.102c)
Assuming that S13 0 , one obtains the following sequence
S13 0 S12 0 S 32 1 S 31 0 S 21 1 S 23 0 S13 1
The reference planes can be placed at such places that the three non-zero
terms are positive real
0 0 1
S 1 0 0
0 1 0
(3.103)
The device as mentioned earlier is a circulator. (Fig. 3.21)
However, Fig. 3.20 shows a lossless three port device. As it can be shown
from the figure it appears as two independent devices.
2
1
3
Figure 3.21. Symbol of circulator
155
Microwave Circuits & Components
s21
a1
s11
b2
s22
s12
a2
b1
s23
s31
s13
s41
s24
s14
s42
s32
a4
s43
a3
s33
s44
b3
s34
b4
Figure 3.22. Flow graph of four port device
3.7.1
Four Port Devices
The scattering matrix of a four port devices is shown below
S11
S
S 21
S 31
S 41
S12
S 22
S 32
S 42
S13
S 23
S 33
S 43
S14
S 24
S 34
S 44
(3.104)
This is shown in Fig. 3.22. One of the four port device is Magic T or
hybrid junction which is discussed earlier. Let us consider another four-port device
which is reciprocal Sij S ji , matched Sii 0 and lossless.
of them
Energy conservation condition provides 16 relations. If we consider two
S13
S14 S 23
S 24 0
(3.105a)
S13
S 23 S14
S 24 0
(3.105b)
and Eqn. 3.105(b) by
and
Multiplying Eqn. 3.105(a) by S14
S 23
subtracting the second resulting equation from the first one
2
2
S13
S14 S 23 0
(3.106)
There are two ways to satisfy this equation. First, one may let S13 0 .
The terms S 23 and S 24 are different from zero. To satisfy Eqn. 3.106 S24 can be
set equal to zero. The resulting device is known as directional coupler. The
scattering matrix satisfying the conditions mentioned earlier is
156
Scattering Matrix
S12
0
S 23
0
0
S
S 12
0
S14
S14
0
S 34
0
0
S 23
0
S 34
(3.107)
A second possible way is to set
S14 S 23
terms
(3.108)
Reference planes may then be selected to have two purely imaginary
S14 S 23 j
Then for energy conservation
S12
2
S13
2
S14
2
1
(3.109a)
S13
2
S 23
2
S 34
2
1
3.109b)
selecting the reference planes in such a way that
S 12 S 34
A conservation expression yields
S12
S 24 S13
S 34 0 S 24 S13
(3.110)
(3.111)
Thus
S 24 0
S13
(3.112)
There are two possible solutions
S13 S 24 0
(3.113)
The second solution is
0
which has decoupled four ports. It can be concluded that a lossless matched
reciprocal four-port device must be a directional coupler. The scattering matrix of
the perfect directional coupler is
0
S
0
0
0
0
0
0
0
(3.114)
157
Microwave Circuits & Components
The two parameters of directional coupler are coupling C and directivity
D. These are defined as
C 20 log
in dB
(3.115a)
in dB
(3.115b)
and
D 20 log
where, is S13 S 31 S 24 S 42
which should be zero for perfect directional coupler. From the points of view of
flow graph two other cases are discussed here briefly. One is the case of
symmetrical coupler for which the scattering matrix is
0 j
0
0 j
0
S
(3.116)
0 j 0
0
0
j
The corresponding signal flow-graph is shown in Fig. 3.23. In the general
scattering matrix expressed in Eqn. 3.116, a directional coupler would have
scattering matrix given below
1
0 j
0
1
j 0
0
S
0 j 0
1
1
0
j 0
(3.117)
a1
b2
a2
b1
j
a4
a3
b3
b4
Figure 3.23. Signal flow graph of a symmetrical coupler
158
Scattering Matrix
a1
b2
a2
b1
a3
a4
b4
b3
Figure 3.24. Signal flow graph of an anti-symmetrical coupler
The last case which will be briefly described here that of anti-symmetrical
directional coupler. In this case scattering matrix is of the form
0
S
0
0
0
0
0
0
0
(3.118)
It is quite clear why such a directional coupler is anti-symmetric.This is
shown in Fig. 3.24. The term coupler is generally associated with a device made of
two transmission lines, most often of same kind. The power transfer takes place
either at certain discrete locations or in a distributed fashion all along the structure.
The performances of an ideal directional coupler is realisable partially in practice
because the real devices have some losses. In addition, it is not possible to perfectly
match every port of the device, particularly over a broad frequency ranges. As a
result S i i 0 , a transfer of signal takes place towards the fourth port, which is
perfectly isolated in an ideal coupler. This means that
S13 0
S 24 0
Thus, in definition of directivity we have introduced a term which is
equal to S13 S 24 .
3.8
CROSSING
Symmetrical crossing is shown in Fig. 3.25. One transmission line is
connected to three other transmission lines. If all lines are identical, the
159
Microwave Circuits & Components
Figure 3.25. Crossing of two microstrip
approximate scattering matrix neglecting reactive elements within the junction is
1
1
1 1
1 1 1
1
S 1
1 1 1
2 1
1
1
1 1
(3.119)
Menzel and Wolff10 analysed the waveguide model with different
impedances. Gopinath, et al11 and Wu, et al12 have used the integral method.
3.9
SOME ASPECTS OF A TWO PORT JUNCTION SCATTERING
MATRIX
We have already obtained a relation for V1 V1 for a two port device if
it is terminated in impedance Z L . Refer to Eqn. 3.98
V1
S S
S11 12 21 L
S 22 L 1
V1
(3.120)
It shows how the input reflection coefficient is modified when the output
guide is not terminated in matched load. For a reciprocal junction S12 S 21 . If
the junction is lossless, the scattering matrix gives following relations
S11 S11
S12 S12
1
(3.121a)
S 22 S 22
S12 S12
1
(3.121b)
S11 S12
S12 S 22
0
(3.121c)
The coefficient of first two equations show that
S11 S 22
Hence the reflection coefficients on the input and output side is the same.
One relation is obtained, i.e.,
160
Scattering Matrix
2
S12 1 S11
(3.122)
This gives, using Eqn. 3.122c
1
2
S11 1 S11 2 e j 1 e j 2 0
where
S11 S11 e j 1
S22 S11 e j 2
and
S12 1 S11
1
2 2
ej
or equivalently
e j 1 2 e 2 j
or
1 2 2 2n
and
1 2
2
2
n
(3.123)
The relations at Eqns. 3.123 and 3.124 completely specify the
transmission coefficients S12 in terms of reflection coefficients S11 and S22 . The
scattering parameters S11 and S22 can be easily measured and a knowledge of
these suffices for complete description of lossless junctions. Thus, the scattering
coefficients are convenient way for describing a lossless microwave two port
circuits.
To evaluate scattering matrix parameters two simple cases are discussed.
These are
(a)
Shunt susceptance j
(b)
Series reactance jX
3.9.1
Shunt Susceptance
Figure 3.26 shows a shunt susceptance j which is connected across a
transmission line with characteristic admittance Y0 . To find S11 assuming that
output line is matched so that V 2 0 . The reflection coefficients on the output
side is
161
Microwave Circuits & Components
Z0
j
Z0
Figure 3.26. Shunt element on transmission line
S11
Y0 Yin Y0 Y0 j
j
Y0 Yin
2Y0 j
2Y0 j
(3.124)
S 22 S11 , S 22 can be evaluated using Eqns. 3.123 and 3.124 by finding
the transmitted voltage V2 with the output line matched.
For a pure shunt element
V1 V1 V2 V1 1 S11
Since
(3.125)
V 2 S 21V1
S 21 1 S11 S12
2Y0
2Y0 j
(3.126)
3.9.2
Series Reactance
Figure 3.27 shows series reactance jX system. In this example the
characteristic impedances of two lines are different, so we must choose normalised
voltages. Power flow for a single propagating wave is given by
2
2
1
1
Y1 V1
and
Y2 V2
2
2
There are
1 2
V1
and
2
1 2
V2
2
where, V 1 and V 2 are the normalised voltages
If output line is matched
V1
V1
V1
V2
S11
Z in Z1 Z 2 Z1 jX
Z in Z1 Z1 Z 2 jX
when input line is matched
V1 V 1
Z Z 2 jX
S 22 1
Z
V1
2 Z1 jX
V2
162
(3.127)
Scattering Matrix
I2
I1
jX
Z0=Z1
Z0=Z2
Figure 3.27. Series element on transmission line
we have.
To find, S 21 again consider the output line is matched on the input line
V1 V1 V1 V1 1 S11 and
I1 Y1 V1 V1 Y1V1 1 S11
The current is continuous through a series element, hence
I 2 I 2 I1 Y1V1 1 S11
But,
I 2 Y2V2 ; so Y2V2 Y1V1 1 S11
S 21 S12
Z
2
Z1
V2
12
V1
Y
2
Y1
12
V2
V1
Z
2
Z1
12
1 S11
(3.128)
2 Z1 Z L
2 Z1
Z1 Z 2 jX Z1 Z 2 jX
3.10
SCATTERING TRANSFER PARAMETERS
This parameter has been mentioned in the present chapter many times.
However, the scattering transfer parameters can be formally defined. These are
also called the chain scattering parameters or simply T-parameters. For a two port
network these parameters are defined by
a1 T11b2 T12 a 2
(3.129a)
b1 T21b2 T22 a 2
(3.129b)
163
Microwave Circuits & Components
a1
a2
T
a2'
a1'
T'
b1
b2
b1'
b2'
Figure 3.28. Cascading of two port networks
As it can be seen that the T-parameters give us relationship between wave
variables at port 1 to the variables at port 2, which are naturally independent
variables. Relationships between T-parameters and S-parameters can be easily
found.
1
T11
(3.130a)
S 21
T12
S 22
S 21
(3.130b)
T21
S11
S 21
(3.130c)
T22 S12
S11 S 22
S 21
(3.130d)
The T-parameters are useful in analysing cascaded networks (Fig. 3.28).
It is shown in the figure that two two-port network are connected together. If the
two ports are described by scattering parameter T and T' , then we can write
T12
b2
a1 T11 T12 T11
b T
T22
a 2
1 21 T22 T21
(3.131)
The scattering transfer matrix for overall network can be found by matrix
multiplication. This can be extended to any number of two ports connected in
cascade. Suppose there are n two port networks cascaded together then
T T1 T2 T3 Tn
(3.132)
REFERENCES
1.
2.
3.
Leed, D. & Kummer, O. A loss and phase set for measuring transistor parameters
and two port networks between 5 and 250 MHz. Bell Syst. Tech. J., 1961, 84184.
Frchner, W.H. Quick amplifier design with scattering parameters. Electronics,
1967, 41, 100-09.
Kurokawa, K. Power waves and the scattering matrix. IEEE Trans. MTT, 1965,
194.
164
Scattering Matrix
Penfield, P. (Jr). Noise in negative resistance amplifier, IRE Trans-CT, 1960,
CT-7, 166-70.
5. Penfield, P. (Jr). A classification of lossless three ports. IRE Trans-CT,1962,
CT-9, 215-23.
6. Youla, D.C. Scattering matrices normalised to complex port numbers. Proc.
IRE, 1961, 49, 122.
7. Montgomery, C.G. (Ed.). Technique of microwave measurements. McGraw
Hill Book Company Inc., New York, 1948.
8. Altman, J.L. Microwave circuits. D.Van Strand Co, Inc., New York, 1964.
9. Chow, Y. & Cassignol, E. Theorie et application des graphes de transfert. Paris,
Dunod, 1965.
10. Menzel, W. & Wolff, I. A method for calculating the frequency dependent
properties of microstrip discontinuities. IEEE Trans. Microwave Theory &
Technique, 1977, 25, 107-12.
11. Gopinath, A.; Thomson, A.F. & Stephenson, I.M. Equivalent circuit parameter
of microstrip step change in width and cross-sections. IEEE Trans. 1976, MTT24, 142-44.
12. Wu. S.C.; Yang, H.Y.; Alexopoulos, N. & Wolff, I. A rigorous dispersive
characterisation of microstrip cross and T-junctions. IEEE Trans. MTT, 1990,
38, 1837-990.
4.
165
CHAPTER 4
MICROWAVE TRANSISTORS
4.1
INTRODUCTION
Microwave transistors can be classified into four groups: Silicon bipolar
transistors, gallium arsenide field effect transistors, modulation doped gallium
arsenide field effect transistors (FET) and hetrojunction gallium arsenide bipolar
transistors. Two transistors that have not made any impact to date are gallium
arsenide bipolar and silicon FET emitter, because of no planar technology, poor
efficiency, high base resistance, and due to poor microwave performance.
The two microwave transistors, which are commonly used, are
(a)
(b)
Silicon bipolar transistor
GaAs MESFET
GaAs MESFET gives higher output because of higher critical field and
higher saturated drift velocity. The approximate power-efficiency squared limit is
given by1
2
where,
EV 1
pf 2 c S
2 X c
(4.1)
EC = effective electric field before avalanche breakdown.
VS = drift velocity of carriers (electrons)
XC = device impedance level
Since, EC and VS are higher for GaAs, the GaAs MESFET is intrinsically
a higher-power device.
Table 4.1(a). Comparision of microwave transistors
Parameters
GaAs MESFET
Silicon bipolar transistor
4 GHz
8GHz
12 GHz
18 GHz
Gam(dB)
20
16
12
8
Fmm
0.5
0.7
1.0
1.2
4 GHz
8 GHz
12 GHz
15
9
6
2.5
4.5
8
contd...
Microwave Circuits & Components
Parameters
GaAs MESFET
Silicon bipolar transistor
4 GHz 8GHz 12 GHz 18 GHz
Power
output W
25
15
8
2
4 GHz
8 GHz
12 GHz
6
2
0.25
Oscillator
noise1/f
corner frequency
---------------- 30 MHz -------------
--------------- 10 KHz------------
1981
Pf 2 (W/S2)
--------------- 1021 ----------------
-------------- 1.5 1020----------
--------------- 1.8 1021------------
-------------- 1.5 1020-----------
1988
Pf 2 (W/S2)
Theoretical
Pf 2 (W/S2)
------------------- 5 1021------------
----------------- 5 1020-----------
Table 4.1(a) gives the comparison of performance of silicon bipolar
transistor and GaAs MESFET. The advantages of silicon are (i) lower cost (ii)
higher thermal conductivity and (iii) lower thermal power.
4.2
MICROWAVE BIPOLAR TRANSISTOR
4.2.1
Silicon Bipolar Junction Transistor & its Small Signal Model
Figure 4.1(a) shows a bipolar transistor. As is well known, bipolar
transistor is a current controlled device. The base current modulates the collector
current of the transistor. The small-signal equivalent circuit of bipolar transistor is
given in Fig. 4.1(b). For this device structure, the distributed T-equivalent circuit
BASE
EMITTER
n+
p+
BASE
p+
p BASE
n– COLLECTOR
dx~1.5 m
n+ COLLECTOR
SUBSTRATE
B
~200 m
B
B
C
Figure 4.1(a). Bipolar transistor cross-section
168
Microwave Transistors
EMITTER PITCH
EMITTER
BASE
ie
p+
p+
Cpkg
Cpkg
p BASE
n
n
C OL L E CT O R
Figure 4.1(b). Bipolar transistor circuit
of Fig. 4.2 has been found to be an effective small signal model at fixed bias
condition. Another bipolar transistor equivalent circuit is shown in Fig. 4.3.
Microwave power transistor technology has advanced significantly
during in the last four decades. The microwave is non linear and its principle of
C OL L E CT O R
Re
C bp
C3
C2
C1
BASE
R bc
R3
R2
R1
C ep
Cte
Re
ie
R ec
EMITTER
Figure 4.2. Small signal equivalent circuit of microwave bipolar transistor
169
Microwave Circuits & Components
ie
Cie
Cc
re
E
ie
C
b
Z0
Lb
B
Figure 4.3. Simplified T-equivalent circuit
operation is similar to that of a low frequency device, but requirement for
dimensions, process control, heat sinking and packaging are more severe. For
microwave applications, the silicon (Si) bipolar transistor dominate up to frequency
of 10 GHz. The range can be extended up to 22 GHz. The silicon bipolar transistors
are cheap, durable, integrative and offer much higher gain than available with
GaAs MESFET. The noise characteristics are about 10-20 dB superior to GaAs
MESFET. These are often selected for lower microwave frequencies and are used
as local oscillators.
All microwave transistors are now planar in the form and almost all are of
silicon n-p-n transistors. The geometry can be characterized by the structures which
are (a) Inter Digital (b) Overlay (c) Matrix or mesh (Fig. 4.4). Figure 4.5 shows
the various surface geometries.
Table 4.1(b). Microwave transistor characteristics
Parameters
MODFET (Al GaAs/In GaAs)2,3
12 GHz
18 GHz 36 GHz 60 GHz
HBT (AlGaAs/GaAs)4,5
12 GHz 18 GHz 36 GHz 60 GHz
Gain(dB)
22
16
12
8
20
16
10
7
Fmin(dB)
0.5
0.9
1.7
2.6
4
--
--
--
Power
output
--
--
0.15
0.10
0.4
--
--
--
Oscillator
noise1/f
corner frequency
------------- 30 MHz-----------
----------------- 10 MHz----------Contd...
170
Microwave Transistors
Parameters
MODFET (Al GaAs/In GaAs)2,3
12 GHz
Lg() or
emitter
width
18 GHz 36 GHz 60 GHz
HBT (AlGaAs/GaAs)4,5
12 GHz 18 GHz 36 GHz 60 GHz
0.5
0.5
0.25
0.25
--------------- 1.2 --------------
Power
density
(W/mm)
--
--
0.9
0.5
4
--
1.5
--
Power added
efficiency
--
--
38
28
4
8
--
28
In Tables 4.1(a) and 4.1(b) the definition of various parameters can be
obtained from references 2-5. The effective values for GaAs are
X C 1
EC
VS
Emax
105 V/cm
4
Vsat
4 106 cm/s
5
Pf 2 5 1021 W/S2
gave
CW performance of 10W GaAs FET and 10 GHz (achieved in 1980)
pf 2 10 1010 1010 1021 W S2
In 1988, class-A performance of 8 W at 15 GHz of microwave transistor 6
2
21
2
0.5W at 10 GHz and pf 1.8 10 W S with pf 2 for the case7 is
pf 2 1.5 1010 and pf 2 1.5 1020 respectively. As regards configuration in
which transistors can be used are only those which are used at lower frequencies,
i.e.,
(a) Common-base configuration
(b) Common-emitter configuration
(c) Common-collector configuration
For small-signal operation, the non-linear or AC parameters of a hybrid-
equivalent mode which can be expressed as
Vce hie ib hre Vce
(4.2a)
i c h fe ib hoe Vce
(4.2b)
171
172
BASE METALISATION
p+ BASE DIFFUSION
EMITTER METALISATION
(b) OVERLAY
Figure 4.4. The geometry of n-p-n microwave transistor (a) Inter digital (b) Overlay (c) Matrix
(c) MATRIX
(a) INTEGRATED
BASE METALISATION
EMITTERS
EMITTER METALISATION
Microwave Circuits & Components
Microwave Transistors
OVER LAY
M
l+p
EP
BA
2(l w)
( w s)(l p)
EMITTER
w
w+s
w
INTEGRATED
2(l w)
l ( w s)
w+s
w+s
l+p
MATRIX
2(l w)
( w s)(l p)
w
EMITTER
BASE p +
Figure 4.5. Figure of merit (M) of various surface geometries
where
hie
Vbe
V = constant
ip ce
(4.3)
hre
Vbe
ib = constant
ib
(4.4)
h fe
Ve
Vce = constant
ib
(4.5)
hoe
ic
ib = constant
Vce
(4.6)
173
Microwave Circuits & Components
All the symbols are standard and are well known; therefore, they are not
explained here.
An incremental change of emitter voltage Vbe at input terminal will
induce an incremental change in collector current I c at the output terminal. This
defines an initial conductance or transconductance of small signal transistor as
gm
I c
Vie
Vb' c
(4.7)
Thermal equivalent equilibrium carrier density at the function is equal to
minority density times the forward bias voltage factor. That is
n p 0 n po e
v f vt
(4.8)
and
Ic
q A Dn p 0
(4.9)
Ln
Substitution of Eqn. 4.8 into Eqn. 4.9 and differentiation yields
gm
ic
(4.10)
VT
where, Vt = 26 mV (at 300 k) is voltage equivalent at temperature.
As the width of base region is very narrow, Cb >> Cc , the diffusion
capacitance in the base region is
C be
d Qb
dVbe
(4.11)
The diffusion capacitance is then derived using relation
Qb
q np 0Wb A
2
Then diffusion capacitance is expressed by
C be
q AWb n p 0
2VT
gm
Wb2
2Dn
(4.12)
(4.13)
All symbols have conventional meanings. The voltage across the diffusion
capacitance can be written as
174
Microwave Transistors
Vbe
ib
jc
(4.14)
The small signal input conductance of emitter function looking at the
input of base is defined as
gb
Ic
g
I
1
B
m
R p VT hFE VT hFE
(4.15)
where, all the symbols are standard.
Figure 4.6(a) shows circuit of common emitter amplifier circuit while the
Fig. 4.6(b) shows the hybrid- model of common emitter configuration. The
Fig.4.6(c) shows n-p-n transistor and the two hybrid- equivalent model of
transistor.
However, it must be remembered that when the dimensions of a bipolar
junction transistor becomes very small their ZY hybrid-model parameters cannot
ie
C
ib
B
Vce
Vout
Ri
V be
Vin
Vce
E
E
(a)
Figure 4.6(a). Common emitter transistor amplifier
B
Ri
Vc
b'e
Rb
Vb'e
Vb'e
E
C
Ro
E
Figure 4.6(b). Hybrid -model of transistor
175
Microwave Circuits & Components
b
Ri
B
b'
Vb'e
C9
gmVb'e
Co
Ro
Figure 4.6(c). Simplified hybrid model used for analysis of transistor
be measured because the input and output terminals cannot be openly and shortly
realized. Therefore, S-parameters are measured in design of transistor parameters.
Figure 4.7 (a,b) show a schematic diagram of a bipolar function transistor
describe n-p-n transistor and for integrated chip type respectively.
INSULATED LAYER SiO2
EMITTER CONTACT
BASE CONTACT
BASE MATTER JUNCTION
n BASE LAYER
COLLECTOR
CONTACT
n EMITTER LAYER
n COLLECTOR LAYER
Figure 4.7(a). Schematic diagram of a bipolar junction transistor for discrete
n-p-n planar.
4.2.2
Scattering Parameters of Transistors
The scattering parameters in genaral form has been defined in earlier
chapter and their use in design of circuit has been explained. However, in case of
transistor it needs better explanation. The S-parameters of microwave transistors
in chip and packaged form are provided by the manufacturer(Table 4.2). The
transistors in the chip form are used for best performance, low noise high gain and
176
Microwave Transistors
EMITTER CONTACT
INSULATED LAYER SiO2
COLLECTOR
CONTACT
n
n
n
SUBSTRATE
p
p
BASE CONTACT
Figure 4.7(b). Schematic diagram of a bipolar junction transistor for integrated
chip type n-p-n BJT.
bandwidth. Of course packaging introduces parasitic elements, which decides the
performance of transistors. Manufacturers usually provide the common-emitter or
common-source S-parameters of transistors as function of frequency at a given DC
bias. The minimum noise figure, output power and maximum gain requires
Table 4.2.
VCE
Volts
50
LC
(mh)
10
25
50
50
10
25
50
S-parameter data for Motorola MR1-962 transistor (from motorala
data manual).
f frequency
MHz
S11
S21
S12
S22
1000
1500
1000
1500
1000
1500
0.78
0.79
0.79
0.81
0.81
0.82
2.26
1.51
2.72
1.82
2.89
1.96
0.078
0.098
0.067
0.086
0.061
0.082
0.24
0.31
0.32
0.34
0.30
0.40
1000
1500
1000
1500
1000
1500
0.76
0.77
0.77
0.78
0.77
0.79
2.58
1.78
3.19
2.13
3.42
2.30
0.071
0.085
0.062
0.080
0.069
0.078
0.24
0.31
0.20
0.25
0.23
0.27
different biases. Therefore, the manufacturer typically provides the parameters at
three or four biasing parameters (Table 4.3).
177
Microwave Circuits & Components
Table 4.3. Parameter values for 60 m emitter periphery n-p-n and p-n-p HBT
npn
Parameter
Ft
fmax
pnp
22 GHz
40 GHz
0.93
2 ps
65 GHz
0.06 pf
0.01 pf
0.4 pf
1.0 106
10
17
27.5
0
fb
C1
C2
C3
R1
R2
RB1
RB2
Parameter
19 GHz
25 GHz
0.96
4 ps
35 GHz
0.04 pf
0.1 pf
0.3 pf
1.0 106
6.8
3.0
4.4
npn
C5
Rc1
Rc2
RE
CBC
CBE
CCE1
CCE2
CE
LB
LE
LC
pnp
1.34 pf
1
4
8.5
0.012 pf
0.022 pf
0.012 pf
0.06 pf
0.022 pf
0.165 nH
0.032 nH
0.06 nH
0
7.4
3.3
7.0
0.012 pf
0.022 pf
0.012 pf
0.08 pf
0.03 pf
0.26 nH
0.04 nH
0.134 nH
It is possible to convert one S-parameter like common-emitter to
common-base S-parameters from each other. Tables are available in standard texts,
text conversion. For converting common-emitter S-parameters is first converted
into common-emitter RF-parameter then convert common emitter-Y parameter
into common base Y-parameter and finally convert the common-base of parameter
into common base S - parameters.
EMITTER
BASE
N
Ip
C OL L E CT O R
P
In
E
nn= 1.7 x 101
nn= 1.7 x 1011
np= 1.7 x 1014
19
15
12
pn= 3.7 x 10
Rs
nn= 3.7 x 10
C
np= 3.7 x 10
B
vvb
v
Figure 4.8. Carrier density of n-p-n transistor
Before we continue with discussion on S-parameters, a typical carrier
density is shown in Fig. 4.8. As an example of three configurations one
configuration namely common base configuration is shown in Fig. 4.9. These
figures to indicate that S-parameters, though related are different. The p-n-p
junction transistor is a complementing structure of the n-p-n BJT by inter178
Microwave Transistors
n-p-n TRANSISTOR
p-n-p TRANSISTOR
p
n
B
V EB
p
n
V EB
V CB
E
B
B
n
V CB
E
C
V EB
p
C
V EB
V CB
B
V CB
CB CONFIGURATION
CB CONFIGURATION
Figure 4.9. Common base configuration for both p-n-p and n-p-n transistors
changing p for n and n for p. The p-n-p BJT is basically fabricated by first
forming an n-type layer in the p-type substrate j then P +-type region is developed
for n-layer. Finally, metallic contacts to P + and p-layer through windows opens
the oxide layer.
For recalling the definition of scattering matrix it may be written that the
generalised scattering matrix of n-port network is defined as
[b] = [s] [a]
(4.16)
The definition Eqn. 4.16 of scattering matrix shows that different
impedance Zoi, produce different values of normalized impedances Zoi and different
values of scattering matrix. Therefore, the generalised scattering parameters are
defined in terms of specific normalized impedances.
From Eqn. 4.16 the elements of [S] are given by
Sii
and
bi
ai
(4.17a)
bR
a R where, R i R 1,..........n
(4.17b)
ai
Sii can be recognized as the input reflection at port I with all ports matched
observing that
S Ri
Vi zi I i
The reflection coefficient can be written as
Sii
Zi Z0,i
(4.18)
Zi Z0,i
179
Microwave Circuits & Components
In Chapter 3, the conversion relations for converting S-parameter into Z
or Y and other parameters are written. The table is already given.
The frequency characteristics of a network can be represented as a
continuous impedence or reflection coefficient in the Smith chart. A typical plot of
S11 for a transistor in the common emitter configuration is shown in Fig. 4.10. The
plot of S11 is given for the transistor in chip and packaged form. The biased
2.0
3.0
5.0
1.0
Z SMITH CHART
0.5
10.0
10
VCE = 10V
0.2
PACKAGED
10
02
CHIP
0.2
IC = 25 mA
100
30 50
05 10 20
f in GHz
10.0
t
5.0
t=1
3.0
0.5
1.0
2.0
Figure 4.10. S11 of a transistor (BJT ) in chip and packaged form
conditions are also stated. It can be seen that S11 for this transistor in the chip form
follows a constant resistance circle with capacitative reactance at lower frequencies
and inductive reactance at higher frequencies. The equivalent circuit is shown in
Fig. 4.11. The resistance R represents base to emitter resistance to which contact
resistance is added. The capacitance C is due to junction capacitance from base-toemitter. The inductance L is due to the reflection properties of a transistor where
the emitter resistance h fe (w) is complex, produces an inductive reactance across
the base-to-emitter terminals.
L
R
L
R'
C
C'
Cpkg
S11
S11
(b)
(a)
Figure 4.11. Input equivalent circuit for a transistor for (a) chip form, and (b)
packaged form.
180
Microwave Transistors
2.0
3.0
1.0
5.0
0.5
10
CHIP
0.2
PACKAGED
f=1
10
02 05
10.0
10
t
20
30 50 100
0.2
10.0
0.5
1.0
5.0
2.0
3.0
Figure 4.12(a). S 22 of a transistor circuit in common emitter configuration of a
chip form and packaged form.
The equivalent circuit for transistor in the packaged form is shown in
Fig. 4.11(b). In the packaged inductance LPRg and the packaged capacitance CPRg
contribute to reflection coefficient variations at higher frequencies. A typical S22
plot for a chip and packaged transistor in common emitter configuration is shown
in Fig. 4.12(a). The forward and reverse transmission coefficients are S21 and S12.
These are given in polar plots (Fig. 4.12). The parameter S21 is constant for
frequencies below the beta cutoff frequency, i.e., f and the decays at 6dB/ octave.
The transducer cutoff frequency (fs ) is the frequency where S21 –1. The parameter
S21 increases at approximately 6 dB/octave, levels off around fs and decays at
90
1GHz
1GHz
CHIP
PACKAGED DEVICE
0
180
VCB = 15
IVC = 15
mA
270
Figure 4.12(b). S 21 of a transistor circuit in common emitter configuration of a chip
form and packaged form.
181
Microwave Circuits & Components
90
P AC KA GE D
DEVICE
180 0.4
3
1GHz
0.3 0.2 0.1
4
2
5
0
10
CHIP
VCB = 15
IVC = 25
mA
270
Figure 4.12(c). S 12 of a transistor circuit in common emitter configuration of
chip and packaged form.
higher frequencies. A typical Bode plot8 of S21 , S12 and the product S12S21 are
shown in Fig. 4.13.
The common-emitter S-parameter and its frequency variation of transistor
are shown in Figs. 4.14 and 4.15. These illustrate some of the information provided
by manufacturers. A transistor can be considered to be a three-port device as shown
in Fig. 4.14. In this case, the scattering of matrix is defined by relation
b1 S11 S12
b S
2 21 S22
b3 S31 S32
S13 a1
S23 a2
S33 a3
(4.19)
107
30
S 21
20
10
dB
0.1
FREQUENCY
S12 S 21
-10
-20
S12
-30
Figure 4.13. Frequency response of (a) S21 (b) S12 and (c) S12 S21
Source: Reproduced from Hewlett Packard Design 8
182
Microwave Transistors
100
50
150
25
250
10
15
10
100
05
0
10
02
t
05
10
S1101
500
02
S 2201
25
250
50
150
100
Figure 4.14. Input/output reflection coefficients versus frequency V CE=10V, Vc
50 mA.
The S11 is
S11
b1
a2 0 ,a3 0
a1
(4.20)
That is, to measure S11, reference resistance of 50 are used at ports 2
and 3. In a two-port common-emitter, S11 is measured with the emitter grounded.
Therefore, the value of S11 in Fig. 4.10 will be different from the value of S11 in a
two- port common-emitter configuration.
90
S21
120
60
01
S12
03
04
05
06
07
10
150
180
20
16
12
8
15
10
07
05
30
01
0.1
0.05
4
0
S12
S21
150
30
60
120
90
Figure 4.15. Forward and reverse transmission coefficient versus frequency,
VCE=10V, ic 50 mA .
183
Microwave Circuits & Components
4.3
MICROWAVE FET
The GaAs MESFET is more commonly used in microwave integrated
circuit designs because of higher gain, higher output power, and lower noise in
amplifier. The higher gain is due to higher mobility of electrons (compared to
silicon). The improvement in output power is due to higher electric field and higher
saturated drift velocity of electrons given by relation
2
E V 1
pf 2 c s
(4.21a)
2 X e
The lower noise figure is partially due to higher mobility of electrons
carriers. Fewer noise sources are present compared to bipolar junction transistor.
The disadvantage of GaAs MESFET is higher if noise compared to silicon bipolar
transistors. The cross section of GaAs MESFET 9 is shown Fig. 4.16. The
S-parameter and its variations10 is shown in Figs. 4.17 (a to d). The reverse bias of
Schottky barrier gate allows the width of channel to be modulated. There are
several interesting of contrasts between FET and bipolar transistors (Table.4.4).
As can be seen from Table 4.4, the frequency limitation of the FET is due
to gate length which should be as small as possible. The short-current gain is
I
g V
h21 out mo c
(4.21b)
I in
I in
BASE
BASE
EMITTER
0.2 m
n
AlGaAs
GaAs
p+
p
IMPLANTATION
DAMAGE AREA
GaAs
n-
GaAs
COLLECTOR
n+
0.1 m
GaAs
p+
1.5 m
AlGaAs ( 0.25 mm)
(a)
BASE
E
B
C
n+
2 x 1019 cm-3 p
5 x 1019 cm-3 n
19
-3
5 x 10 cm
SEMI-INSULATING
GaAs SUBSTRATE
+
n
0.06 mm
0.07 mm
0.06 mm
1.5 m
B
IMPLANTATION
DAMAGE
100 m
(b)
Figure 4.16. Hetero junction bipolar transistor structure (HBT)
(a) Single structure
(b) GaAs monolithic circuit structure (HBT).
Source:
Reproduced from Asbeck et. al.9
184
100 m
Microwave Transistors
1
2
0.5
5
0.2
26.5 GHz
0.5 GHz
-1
0.5
0
-0.5
1
Figure 4.17(a). S 11 parameter of a HBT upto 26.5 GHz
1
2
0.5
5
0.2
0.5 GHz
20 15
10
5
26.6 GHz
-1
-0.5
0
0.5
1
Figure 4.17(b). S 22 parameter of a HBT up to 26.5 GHz
Table 4.4. Characteristics of bipolar junction transistor as well as for MESFET
Property
Geometry
Modulation
Control Signal
Frequency limitation
Low-frequency
transconductance
Common-emitter bipolar
Vertical
Base current
Current
Base length
High
Commmon-source
MESFET
Horizontal
Gate voltage
Voltage
Gate length
Low
185
Microwave Circuits & Components
5
10
15
20
5 GHz
26.5 GHz
-0.5
-1
1
0.5
0
Figure 4.17(c). S 21 parameter of a HBT
0.4
5
10
-5 GHz
0.6
-0.3
0.8
-0.15
15
26.5 GHz
1
0
1.2
1.5
0.15
0.3
Figure 4.17(d). S 12 parameter of a HBT up to frequency of 26.5 GHz
I in
Vin
RC 1 5wC gs
(4.22)
As low frequencies
I in Vin jwl gs Vc jwc gs
h21
(4.23)
g mo
jwc gs
(4.24)
186
Microwave Transistors
f1
g
1
mo
f
2C gs f
h21
(4.25)
The frequency at which short-circuit current gain be compted is
g
ft mo
(4.26)
2C gs
This is an important parameter of transistor. The equivalent circuits of
GaAs MESFET equivalent circuit is shown in Figs 4.18(a & b) can be estimated
using the figure. They are
1
Rc 1 j Cgs
y11
(4.27)
(4.28)
y21 g mo
y 22
1
j C o
R0
(4.29)
(4.30)
y 21 0
Lq
Cgd
Rg
G
Ld
D
C gs
gmnt
R ds
Cds
Rc
Rs
Ls
(a)
G
S
Rc
D
Vc
gmt
Cgs
S
Ro
Co
S
(b)
Figure 4.18. (a) Complete model of GaAs MESFET (b) Simplified -model of GaAs
187
Microwave Circuits & Components
These Y-parameters can be converted into S-parameters which can also
be measured. In terms of L which is complex and can be written as
(4.31)
L U jV
For two-port device or network its can be proved that U >1. U can be
expressed in terms of Z or X parameters. The Y-parameters
U
Z 21 Z12
2
4 Re (Z11 ) Re (Z 22 ) 4 Re (Z12 ) Re (Z 21 )
(4.32)
For Y- parameters
U
y21
2
(4.33)
4 Re (y11 )Re (y22 )
1 1 g mo
U
4 f 2 2 C gs
2
f
max
f
2
R
o
Rc
(4.34)
Thus, fmax is given by
f1 R0
(4.35)
2 Re
A high-gain FET requires a high ft , a high output resistance, a low input
resistance. Under normal bias conditions the device is biased for maximum drift
velocity of electron carriers which gives
f max
ft
g mo
V
1
s
2C gs 2C 2L s
(4.36)
This equation shows the importance of short gate length Lg. Another
important parameter is gmo
Lg Z
A
d
d
Vs C gs Vs z
Lg
d
C gs
(4.37)
g mo
(4.38)
g mo V s
z
d
(4.39)
188
Microwave Transistors
GATE
SOURCE
DRAIN
Id
EPITAXIAL
LAYER
d
f
BUTTER
LAYER
SUBSTRATE
(kV/cm)
I
20
10
03
v
(cm/s)
ELECTRON VELOCITY
X
I
E
2
107
1
107
vs
v(E)
X
Figure 4.19. GaAs MESFET at high electric field
Source:
Reproduced from lwichti, C. A 11
Figure 4.19 shows the GaAs FET11 and field and electron velocity (cm/s)
as a function of x, the length and distance is FET. The Fig. 4.20 shows the electron
drift velocity versus electric field. Referring to Fig. 4.19 it can be seen that channel
can be considered to be two regions; a low-field region with a constant number of
carriers and a high field region with a constant velocity which is important. The
current continuity is required
I DS
qn x v x
(4.40)
A
where, n(x)max = ND
V(x)max = VSat = V s
The number of carriers must increase above ND in region II. This causes
an electron accumulation at drain edge of the channel followed by electron
depletion. Thus, a change dipole occurs a drain edge of channel, which has a very
small capacitative effect in the model ( 0.05 pf in Fig. 4.18).
189
Microwave Circuits & Components
DRIFT VELOCITY OF ELECTRON
(cm/s)
v
2 x 107
GaAs
1 x 107
Si
E
(kV/cm)
Figure 4.20. Electric velocity vs electric field
In GaAs the electron carriers will slow down at electric field greater than
3 kv/cm. The electron move from a high mobility state to a low mobility state in
about 1 ps and thus velocity of carrier reaches a peak and slows down in the middle
of the channel. This is amply clear from the Fig. 4.20.
An estimate of the drain current IDSS can be made using the relation 4.40.
For estimate for drain current IDSS in GaAs MESFET for which
ND = 2 x 10 17cm-3
VSat = 2 x 10 7 cm/s
A = z(t-d) = z(0.03) (10 -4) cm2
I DSS
q N D VSat 1.6 1019 2 1017 2 107
A
6.4 105 A cm 2
I DSS
0.3 6.4 A cm 200 mA mm
A
The measurements also shows the same values. The high-frequency GaAs
MESFET is maximized by achieving the minimum gate length without introducing
excessive device parasities. The 0.3 m gate GaAs MESFET s are probably within
a factor of 3 of highest fmax that can be achieved from the present device structure.
A typical values for ft and fmax are
ft = 20 GHz
fmax= 72 GHz
These are the values for transistor, AT – 825 for VDS = 5V, Z = 500 m,
IDS = 50 mA and LG = 0.3 m.
190
Microwave Transistors
AuGeNiAu
SOURCE
Ti / Pt / Au
DRAIN
n+ GaAs
200 Å
n+ AlGaAs
250–500 Å
GATE
100 Å
200 Å
20–60 Å UNDOPED AlGaAs
ELECTRON
GAS
1 m UNDOPED GaAs
100 m
SEMI-INSULATING
GaAs
SUBSTRATE
Figure 4.21. MODFET and HEMT structure
By using hetero junction semiconductor material AlGaAs interfacing with
GaAs a new field effect microwave semiconductor devices are fabricated with
superior microwave performance. This device is the MODFET (modulation-doped
field effect transistor), which is also called a HEMT (High electron mobility
transistor), a SDHT (selectively doped hetero structure transistor), or a TEGFET
(two-dimensional electron gas FET). The cross-section of this transistor is given6,7
in Fig. 4.21.
The basic properties of below junctions can be understood from
difference in energy gap between the two materials, which cause band bending.
This results in electron gas, which has high mobility (electron) due to n provided
by the donors in AlGaAs. The band bending results in quantum well. This has a
large population of electrons and therefore forms a 2-dimensional gas. This can be
easily modulated by the gate voltage. This, thus becomes a n-channel MOSFET
where the number of conduction electrons in channel is controlled by the gate
voltage. The band bending forces the electron to be resident in undoped GaAs
layer. Thus the electrons have very high mobility and high Vs even at room
temperature. This explains the superior performance of microwave devices.
Figure 4.21 shows the structure of MODFET. The region beneath the gate
metal and GaAs buffer layer can be analysed to explain the behaviour. If the width
of the n-AlGaAs donor layer is very thin ( 250 Å) , then the depletion layer below
the Schottky gate extends into the undoped GaAs electron gas and interrupts the
electron gas. This makes the structure an enhancement mode FET. Since no
channel flows if VGS = 0, for the flow of electron, a positive VGS is needed. If the
n-AlGaAs donor layer is thicker ( 500 Å) the depletion region only reaches the
undoped AlGaAs layer. This would also be depleted. Thus, it provides the
depletion mode FET. The voltage VGS will modulate the population of electrons in
quantum well and therefore the IDS of FET. Since the electrons travel in undoped
GaAs region with a few ionized donors, the mobility of Us is more for
N D 1017 cm 3 .
191
Microwave Circuits & Components
OHMIC
SOURCE
SCHOTTKY GATE
OHMIC DRAIN
n EPITAXIAL LAYER
ND 1017 cm-3
0.3 m
3 m
OPTIONAL BUTTER LAYER
HIGH-FREQUENCY SUBSTRATE
ND 1015. 1016 cm-3
100 m
Figure 4.22. GaAs MESFET cross-section
4.3.1
Brief Description of Noise Performance of MESFET
The cross-section of GaAs MESFET (metal-semiconductor field-effect
transistor) is shown in Fig. 4.22. The name MESFET has been adopted because of
similarly to MOSFET. The noise model of MESFET is based on the performance
of junction-field-effect transistor (JFET) originally proposed by Schottkey12.
To summarize the noise performance of gallium arsenide, MESFET and
silicon bipolar transistors, the thermal noise of JFET by Bruneke and Vander Ziel 13
NOISE POWER dBm/Hz
-130
-140
AT 22000 10 V, 10 mA
BIPOLAR[3.21]
-150
-160
AT 10500 3 V, 40 mA
MESFET[3.21]
-170
1 KHz
2
5
10 20
500 100 200
500
2
5
FREQUENCY
20 KHz
2MHz
Figure 4.23. 1/f noise of microwave transistor
192
10 MHz
Microwave Transistors
20 dB/decade
1/f
(dB) fmin
10 dB/decade
1/f
10 dB/decade
1/f
10 dB/decade
1/f
3
2
Si BIPOLAR
1
GaAs MESFET
1KHz
Figure 4.24.
1MHz
1GHz
FREQUENCY
1THz
fmin as a function of frequency for (a) silicon bipolar transistor and (b)
GaAs FET.
can be represented by equivalent an input resistor Rn = 0.8/gm. The high frequency
induced gate noise was added by Vander Ziel14, 15, and others16–18. The comparison
of performance of these two-microwave transistors is given in Figs. 4.23 and 4.24.
A detailed paper has been published by Su, et al19 on this subject.
Band gap and band structure engineering can also help in developing
materials which can be used for generating microwaves. It is possible to
superimpose some additional periodicity over a larger dimension, using the
repetitions of a combination of layers, having their own individual periodicity and
constituent elements. Such a modified lattice is known as super lattice, and has
some additional energy band structure properties over the regular band structure of
constituent layers. The second possibility is to replace appropriate number of atoms
in a unit cell itself. This can be done by mixing different species in proper
proportions for forming single alloy structures. Such combinations are ternary and
quaternary compound-semiconductors from III rd and Vth group elements. Even
elemental semiconductors like germanium and silicon could be used in mixed type
of band such as MODFET structure based on GeSi alloy and 2DHG concept as
shown in Fig. 4.25.
Aluminum and gallium arsenide can mix together in all proportions
without any precipitate or mixibility gap. Both materials have high melting point
and are mechanically strong. For all the compositions the resultant material, i.e.,
Alx Ga1-xAs, is a direct band type semiconductor having almost perfect lattice
193
Microwave Circuits & Components
SOURCE
SiO2
G AT E
DRAIN
Ti
Al
Al
MODULATION BF 2
DOPED Si IMPLAN T
SiO2
BF
2
30 mm IMPLAN
T
50 mm
2 DHQ
Ge0.2Si0.8
p-Si SUBSTRATE
Figure 4.25. MODFET based GeSi alloy using two-dimensional concept
DRAIN
SOURCE
G AT E
n-GaAs
n-GaAs
n-AlxGa1-xAs
Undoped GaAs
CHANNEL
(100) GaAs Si SUBSTRATE
Figure 4.26. AlGaAs MODFET – typical structure for device fabrication
10nm GaAs
50nm n-AlGaAs
INTERFACE-3
30nm GaAs
10nm AlGaAs
INTERFACE-2
50nm n-AlGaAs
0.2m AlGaAs
INTERFACE-1
0.1m GaAs
GaAs Si SUBSTRATE
Figure 4.27. MODFET with 2-dimensional channel for improving device
characteristics.
194
Microwave Transistors
match (0.14 per cent) to GaAs at room temperature 21. It is an ideal material for
ternary heterostructure family. AlxGa1-x As has almost similar thermal coefficient
as GaAs and therefore structure stability is quite good. Due to close match between
so many properties it is possible to have variations in band gap over relatively large
span by varying mole fraction of Al. Compositional dependence of energy gap in
III-V ternary compound semiconductors is shown in Table 4.5. GaAs
heterostructure in the form of a single quantum well and GaAs-AlAg superlattice is
shown in Figs. 4.26, 4.27 and 4.28. Very interesting transconductance property 22
of HEMT is experimentally observed in case of single quantum well structure made
out of AlGaAs-Gaps heterostructure having well dimensions of 300 Å.
Table 4.5.
Alloy
AlxIn1-x
AlxGa1-xAs
AlxIn1-xAs
AlxGa1-xSb
AlxIn1-xSb
GaxIn1-xP
GaxIn1-xAs
GaxIn1-xSb
Ga Px As1-x
Ga AsSb1-x
In Px As1-x
In Asx Sb1-x
Energy gap III-V ternary alloys semiconductor – composition
dependence.
Direct Energy Gap(eV)
1.34 + 2.23x
1.424 +1.247x (x<0.45)
1.424 +1.087x +0.430x2 (x>0.45)
0.36 +2.35x+0.24x2
0.73+ 1.10x+0.47x2
0.172 + 1.621x+0.43x2
1.342 + o.511x+0.6043x2(0.49<x<0.55)
0.356+ 0.7x+0.4x2
0.172+ 0.165x+0.413x2
1.424 + 1.172x+0.1863x2
0.73 – 0.5x+1.2x2
0.356 + 0.675x +0.32x2
0.18 – 0.41x+0.586x2
`
Hiroyoki Sakaki23 in 1990 proposed Quantum Effect Ultrafast
Semiconductor Devices (QEUSD). Remarkable developments in semiconductor
technology have now allowed the preparation of a variety of layer microstructures
(LMS) with thickness controllability precision of nearly one atomic layer. The
AlAs
n-GaAs
GaAs
EC
EF
2 DEG
LOWEST ENERGY BAND OF
SUPPERLATTICE
EV
Figure 4.28. Al As and GaAs superlattice – a method for realising 2-dimensional
device.
195
Microwave Circuits & Components
electrons traversing through or confined in such structures exhibit unique and
unprecedented properties since quantum wave nature of electrons manifest
themselves particularly in their motion along the stacking direction.
Extending the physical phenomenon of HEMT, Sakaki 23 proposed
Resonant Transmitting Diode (RTD) that provide excellent negative resistance
characteristics. RTD is the electron wave analog of Fabry Perot etalon. The
ultimate response time of RTD is determined by the energy DE (or the bandwidth)
h
of transmission filter characteristics through the uncertainity principle
.
AE
This device can be used upto a maximum frequency of 700 GHz. There is
possibility of controlling the resonant tunneling process and other electron
interferences by electrical signal to create novel Quantum Interference Transistor;
one of them is BRINT (Bragg Reflection and Interference Transistor) in which a
periodic potential is introduced in FET channels so that electron wave of specified
wavelengths undergo Bragg Reflection.
50
ICE(mA)
IFE = ICE / IBE = 100
IB = 0.50 mA
0.375 mA
0.25 mA
0.125 mA
25
10
5
VCE (V)
(i)
IDSS( VGS= 0)
gm =IDS/ VGS= 0.05 mS
= 50 ms
Ips(mA)
100
50
VGS = -0.5 V
= -1.0 V
= -1.5 V
2.5
5.0
7.5
VDS (V)
(ii)
Figure 4.29(a). DC characteristics of (i) Silicon bipolar transistor AT-41400
(ii) GaAs MESFETAT-8251.
196
Microwave Transistors
bc
CLASS B
OUTPUT VOLTAGE
A
AB
C
0
B
2
wt
3
CLASS A
OUTPUT VOLTAGE
(NO DISTORTION)
CLASS B
OUTPUT VOLTAGE
Figure 4.29(b). Transistor characteristics and class A, class B and class C
operations.
4.4
DC BIASING
The common-emitter or common-source biasing configuration for DC
biasing is discussed here. The DC biasing is completely independent of the RF
two-port configuration. A transistor may operate as an amplifier in the commongate mode, but DC biasing circuit is in common-source mode. The DC collector
and drain characteristics of a typical bipolar and FET is given in Fig. 4.29 (a).
The output current for both the type of transistors is controlled by one
biasing voltage. For the bipolar transistor the voltage across the reverse biased gate
source junction determines the channel width and therefore the drain output
current. The definitions of class A, AB, B and C are given graphically in
Fig. 4.29(b). The safe operating regions for the bipolar region is determined by
(a)
(b)
(c)
(d)
Maximum collector current
Maximum collector-emitter voltages
Secondary break down24
Maximum power dissipation (common junction transistor temperature
= 200 oC for silicon)
197
Microwave Circuits & Components
+Vcc
+V Cc
Rc
R1
VD
Rc
V CE
R2
Vcc
V EE
R1/R2
RE
VT
RE
- VEE
Figure 4.30. Bipolar transistor bias circuits
For GaAs MESFET, the safe operating region is determined by
(a) Maximum drain current
(b) Maximum drain-source voltage
(c) Maximum power dissipation (maximum junction temperature for GaAs
presently)
(d) Maximum input power to gate
For GaAs the junction temperature is limited for avoiding a possible
chemical reaction on metal degradation25,26 to 175 oC for reliability. Because of lower
thermal conductivity of this material, excellent heat sinking must be provided. Some
simple biasing circuits are shown in Fig. 4.30. For silicon bipolar the emitter-base is
forward biased at about 0.7 V and the collector-base is reverse biased at a voltage
dependent on device breakdown and desired point is typically 8 to12 V for small
signal microwave device.
-VDD
RD
RD
V DD
V GG
IG =0
2
RS
V DD
-VSS
L or R
IG =0
-VSS
1
RD
IG =0
RD
IG=0
L or R
L or R
RS
RS
-VSS
(a)
(b)
(c)
Figure 4.31. MESFET bias circuit (a) Two-supply bias (b) Single-supply bias and
(c) Dual gate bias.
198
Microwave Transistors
4.4.1
Temperature Stability
One of the most important considerations in circuit design is the
temperature stability of the transistor bias points. The parameters that must be
controlled VBE are for bipolars and –VGS for FET, which can be set with either a
passive bias circuit or an active bias circuit.
For silicon bipolar transistors, the collector current will increase with
temperature as hFE increases due to the property of forward biased emitter base
junction. A complete discussion of temperature current effects on hFE is given by
Sze24. Although the junction breakdown voltages increase with temperature 24,
normal selection of the DC bias point will never approach junction breakdown.
Since, hFE increases with temperature, the S-parameters will change, leading to
variations in RF performance. Collector current rise results from increased hFE, the
junction temperature will also rise, lowering device reliability or possible damage
to the transistor. A passive bias circuit using two power supplies is shown in
Fig. 4.31 for either a bipolar or FET. In this circuit some of the resistor values may
be zero. The temperature stabilizing resistors are R1 (collector to drain feedback)
and R5 (either or source feedback). If the collector current starts to increase, the
increased voltage drop across the resistors will reduce the collector drain current.
V1( 0)
V1( 0)
R1
R2
R1
VC
2N2907
R2
R4
MICROWAVE
DEVICE
R3
R3
R5
V2( 0)
R6
V3( 0)
V2( 0)
(a)
(b)
Figure 4.32. Biasing circuits (a) Passive bias circuit using two power supplies
(b) Active bias circuit with pnp transistor.
A complete analysis of the problem is straight forward using SPICE or other nodal
analysis programme. Passive and active biasing circuits are shown in Figs. 4.32.
4.4.2
Bias Decoupling
The bias decoupling networks prevent the DC bias circuit from
releasing unwanted impedances to the microwave device. Improper impedance
loading can reduce the gain and output power capability and often results in
oscillations. Table 4.6 shows some of the offset frequencies.
199
Microwave Circuits & Components
Table 4.6. Offset frequencies at different frequency levels
S. No.
Frequency
1
2
3
100kHz
3MHz
15MHz
Offset frequency
–110 dBc/Hz
–135 dBc/Hz
–142 dBc/Hz
The RF chokes may be inductors, quarter wave, high impedance shorted
stubs, or shorted stubs that are part of the matching circuits. Most shorted stubs
used as chokes are shorted through a bypass path to ground. If the resistors in the
bias network are large (e.g., 10) compared to the RF impedance at the point of
attachment RF chokes may not be necessary. Extra decoupling is sometimes
necessary when PNP transistor is used as active bias circuits. Any or all terminals
of the PNP may need to be AC grounded through bypass capacitors to ensure that
the RF effects or oscillators due to the gain present in the biasing circuits
(Fig. 4.33).
Impedance matching of the networks are not discussed here. However,
Gonzalez27 or Liao28 have discussed these in detail.
+ V CC
CB
RFC j L 500 MIN
RC
R1
CB
RFC 2
1
jC
1 MAX
CB
RFC 1
RF OUT
RF IN
RFC 2
RE
CB
R2
Figure 4.33. Decoupling dc bias circuit from the rf part of the circuit
4.5
MICROWAVE TRANSISTOR
Having described the various microwave transistors, it is desirable to
describe the application of amplifiers and oscillators in brief. Some problems on
design of microwave transistor amplifier is given at the end of Chapter 6 on
Smith Chart. However, some of the points are described briefly here.
200
Microwave Transistors
Zs
+
V 1+
Vs
V1
–
S
[S]
V 2+
ZL
–
(Zs)
IN
-–
2
V
OUT
L
Figure 4.34. A two-port network with source and load impedance
4.5.1
Microwave Transistor Amplifier
Broadly, microwave transistor amplifier consists of input matching of
network, transistor, and output matching network. The impedance transforming
properties of transmission lines can be used in the design of matching networks.
A microstrip line can be used as a series transmission line as an open-circuited
stub or as a short-circuit stub. A series microstrip line together with a shortcircuit or open-circuited shunt stub can transform a 50 resistor into any value
of impedance. The eight lumped element circuit can also be used for impedance
matching.
4.5.1.1 Gain & stability
Consider an arbitrary two-port network S connected to source and load
impedances ZS and ZL respectively as shown in Fig. 4.34. Three types of power
gain in terms of S-parameters of two-port network and the reflection coefficients
S and L of source and load.
(a)
(b)
(c)
PL
is the ratio of power dissipated in the load ZL to the
Pin
power delivered to the input of two-port network. It may be noted that this
gain is independent of ZS but in practice the power gain may strongly depend
on ZS.
Power gain = GP =
P
Available gain GA= avn is the ratio of power available from two-port
Pavs
network to the power available from the source. This assumes conjugate
matching for both the source and the load. It is dependent on ZS but not on ZL.
PL
is ratio of power available from the
Pavs
source. This depends on both ZS and ZL.
Transducer power gain GT =
It may be noted that if the input and the output are both conjugately
matched to the two-port, then the gain is maximised and GP = GA= GT . Further, if
the reflection coefficients
Z Z0
L L
(4.41)
Z L Z0
201
Microwave Circuits & Components
and
S
Z S Z0
Z S Z0
(4.42)
where, Z0 is the characteristic impedance, then
GP
1
1 in
S21
2
2
1 L
2
(4.43)
2
1 S22 L
S11,S12,S21are S22 the S-parameters of two port network and
in S11
GA
S12 S21 L
1 S22 L
1 S
2
1 S11S
2
S21
(4.44)
1
2
(4.45)
2
1 out
where,
L
S S
S22 12 21 S
1 S11S
out
(4.46)
and
GT
1 S
2
1 in S
1 S
2
2
1 S11S
2
S21
1 L
S21
2
2
1 L
2
1 S22 L
(4.47)
2
2
1 out L
(4.48)
2
A single-stage microwave transistor amplifier can be modelled by circuit
Z0
INPUT
OUTPUT
MATCHING
CIRCUIT
GL
TRANSISTOR
(S)
GO
MATCHING
CIRCUIT
GS
S
OUT
IN
L
Figure 4.35. Transistor for amplifier circuit
202
Z0
Microwave Transistors
of Fig. 4.35, where a matching network is used on both sides of the transistor to
transform the input and output impedance Z0 to source and load impedances ZS and
ZL. It is possible to define separate effective gain factors for input (source)
matching network, the transistor itself and the output (load) matching of network
as follows:
GS
1 S
1 in S
G0 S21
GL
2
(4.49)
2
2
(4.50)
1 L
2
1 S22 L
(4.51)
2
4.5.1.2 Stability
The amplifier considered will start oscillating if either the input or output
port impedance has negative real part which means that in 1 or out 1 .
in and out depend on the source and load matching networks. The stability of
amplifier depends on S or L as presented by the matching networks. There are
two types of stability:
(a)
Unconditional stability: The network is unconditionally stable if
in 1 and out 1 when S 1 and L 1
(b)
Conditional stability: If the above mentioned conditions are satisfied for a
certain range of source or load impedance the amplifier is said to be
conditionally stable for the ranges defined by the circuit.
Thus, the stability is frequency dependent. The amplifier may be stable
for the design frequency but may be unstable for other frequencies. The condition
for stability is therefore,
in S11
S12 S21 L
1
1 S22 L
out S22
(4.52)
S12 S21S
1
1 S11S
(4.53)
If the device is unilateral these conditions reduce to S11 1 and S22 1 .
Finding this range for S and L can be facilitated by using Smith chart and
plotting input and output stability circles. What are these stability circles? The
stability circles are defined by making use of boundaries between stable and
potentially unstable regions of S and L . S and L must be on Smith chart
for which S 1 and L 1 .
203
Microwave Circuits & Components
It can be shown that in the complex plane, an equation of the form
c R is obtained which represents a circle with centre at C (a complex
number) and a radius R (a real number). The output stability circuit thus obtained
has centre at CL and has radius RL which are:
CL
S
22
*
S11
2
2
S22
RL
S12 S21
2
*
2
S22
(centre)
(4.54)
(radius)
(4.55)
where, is determinant of scattering matrix and is
S11S22 S12 S21
(4.56)
Similarly, for input stability circle, centre is at CS and the radius of circle is
CS
S
RL
11
*
*
S22
2
S11
S12 S21
2
S11
2
2
(centre)
(4.57)
(radius)
(4.58)
If the S-parameter of transistor is known, the input and output stability
circles can be plotted to define where in 1 and out 1 , on one side of the
stability we have out 1 , while on the other side out 1 . Similar results can
be obtained for the output stability circle.
Figure 4.36 shows the output stability circle plotted in L plane for
S11 1 and S11 1 . If ZL is matched to Z0 i.e., ZL = Z0 , L = 0, in S11 . So
if S11 1 , in 1 . This thus means that the centre of the Smith chart ( L 0 ) is
in stable region; but if S11 1 , in 1 for L = 0, the stable region does not
enclose the circle. In the figure stable region is shown by shaded region.
Similar results will be obtained for S22 . The sufficient condition for
unconditional stability can also be written in the form
204
Microwave Transistors
CL
RL
in 1
(STABLE)
(a)
CL
RL
in 1
(STABLE)
(b)
Figure 4.36. Output stability circuit (a) S11 1 (b) S11 1
205
Microwave Circuits & Components
2
k
1 S11 S22
2
2 S12 S21
2
1
(4.59)
and
1
4.6
TRANSISTOR OSCILLATOR
Like any other oscillator the oscillation in a transistor of microwave oscillator
is triggered by transients/noise. The device must have negative resistance. In a transistor
oscillator, a one port negative resistance is effectively created by a potentially unstable
transistor with an impedance designed to drive the device in the unstable region. The
circuit model is shown in Fig. 4.37. Common source or common gate FET
configurations are used (common emitter or common base for bipolar devices), often
with positive feedback to enhance the instability of device. The output stability circuit
can be drawn in T plane and T is selected to produce large value of negative
resistance at the input of transistor. The load impedance ZL is choosen to match Zin
because such a design uses the small-signal S-parameters. A value of
RL
Rm
3
(4.60)
is used so that R L Rm 0 . The reactive value of ZL is choosen to resonate the
circuit
X L X in
(4.61)
For steady-state oscillation at input port, we must have L m 1 . Thus
S S
S T
1
in S11 12 21 T 11
L
1 S22 T
1 S22 T
(4.62)
which results in
LOAD
NETWORK
(TUNING)
TRANSISTOR
TERMINATING
NETWORK
(S)
L
IN
(ZL)
(ZIN)
OUT T
(ZOUT)
(ZT)
Figure 4.37. Circuit model for a transistor oscillator
206
Microwave Transistors
0.319
0.238
S
50
D
0.346
G
50
5 nH
ZL
50
LOAD
ZIN
T
ZT
Figure 4.38. An oscillator circuit
T
1 S11 L
S22 L
(4.63)
The out is given by expression
out S22
S12 S21 L S22 L
1 S11 L
1 S11 L
(4.64)
which shows T out 1 and therefore ZT = - Zout
Figure 4.38 shows a oscillator at 4 GHz which uses a GaAs FET in
common gate configuration with 5 nH inductor in series with the gate to increase
the instability.
4.7
DIELECTRIC RESONATOR OSCILLATOR
Oscillator stability would be increased if a high Q tuning network is used.
A dielectric cavity resonator has Q of the order of a few thousands. It is compact
and can be easily integrated with planar circuit. It is made of ceramic material and
has excellent temperature stability. The transistor dielectric resonator oscillators
(DROs) are now being widely used as microwave source. Two types of oscillator
configuration are commonly used.
DR
LOAD
TERMINATING
NETWORK
Z0
Figure 4.39. Parallel feedback dielectric resonator oscillator
207
Microwave Circuits & Components
Figure 4.40. Series feedback dielectric resonator oscillator
4.7.1
Dielectric Resonator Oscillator using Parallel Feedback
Figure 4.39 shows this circuit. The configuration uses a resonator coupled
to two microstrip lines functioning as a high Q-band pass filter. This circuit couples
a portion of transistor output to input. The amount of coupling can be controlled by
controlling the spacing between the resonator and the lines. The phase is controlled
by length of lines.
4.7.2
Configuration using Series Feedback
It uses only a single microstrip feedline. It does not have tuning range as it
can be obtained by parallel feedback. Design of oscillator with parallel feedback is
most easily done using a microwave CAD package.
A typical dielectric resonator transistor circuits is shown in Fig. 4.41. In
this circuit the dielectric resonator is placed from the open end of the microstrip
l1
DIELECTRIC
RESONATOR
l2
l
r
Z0
4
OUT
L'
L
T
IN
Figure 4.41. A typical dielectric resonator oscillator circuit
line, the line length lr can be adjusted to match the phase of required value of L .
The output load impedance is part of the terminating network.
4.8
OTHER MICROWAVE SOURCES
Microwave power source is essential for any microwave system.
Communication and radar use relatively high power for transmitter, low power
source for receiver. Radar transmitters generally operate in pulse mode. Electronic
warfare system uses sources like radar with additional requirement of tunability.
These requirements can be met with solid-state as well as tube sources. In general,
208
Microwave Transistors
tubes are used for high power and solid state sources for low power. Solid state
sources have advantage of small size and ruggedness. They are cheaper and can be
integrated easily. Very high power applications are dominated by microwave tubes.
4.8.1
Microwave Tubes
Magnetron was the first microwave tube developed in 1930s by Hull.
This was used in radar during the World War II. Since then variety of tubes have
been developed. They are still essential for high power and high frequency i.e., in
millimeter tube region. Power required may be somewhere between 10 kW to
10 MW for frequencies greater than 100 GHz. Tubes in general involve interaction
between electron beam and electromagnetic field inside a glass or metal vacuum
envelope. Cathodes are generally fabricated from barium oxide coated metal
surface sometimes impregnated tungsten surface is used. The electron stream is
then focussed by focussing grids specially designed for this purpose. Some times
magnetic field of a solenoid magnet is used. The electron beam interacts with RF
field in such a way that energy is transferred from the electron beam to RF field.
Microwave tubes can be grouped into two categories; first one is linear
beam or ‘O’ type tubes. The electron beam traverses the tube and the electric
field is parallel to the electron beam. The second one is of crossfield of m type
in which focussing is perpendicular to the accelerating electric field.
The klystrons are linear beam tubes and are widely used as amplifier or
oscillator. It is a multicavity tube in which the RF field modulates the electron
beam. Using four cavity klystron gain of about 90 dB. The reflex klystron is a
single cavity klystron tube which operates as an oscillator by using reflector
electrode. The klystrons are narrow band devices. Travelling wave tube (TWT)
is a wide band devices as slow-wave structures is used instead of cavity. The
slow wave structure generally used is a helix structure. The RF signal enters at
one end of helix and interacts with the electron beam in the helix space. There are
four modes of interacting; in one of them there is transfer of energy from electron
beam to RF It gets amplified and output is taken at the other end of the helix. The
TWT has highest bandwidth; this makes it very useful for electronic warfare
systems. However, it cannot be used as microwave source. A modified version of
TWT is backward oscillator (BWO) in which feedback takes place through
backward mode (one of the four modes). A very useful feature of the BWO is that
frequency of oscillation is tuned by varying the DC voltage between cathode and
helix.
There are many crossed field devices like magnetron in which interaction
takes place in a region where both electric and magnetic field exist. Crossed field
amplifier (CFA) is another such device. Amplifier tubes have also been used.
One of the important tube recently developed is gyration. It is relativistic
tube in the sense that the variation of mass with velocity is used for bunching. This
tube also employs magnetic field. The field forces the electrons to travel in tight
spirals down the length of the tube. The electron velocity is high enough so that
relativistic effects occurs. Bunching occurs and the energy from the transverse
209
Microwave Circuits & Components
10MW
KLYS TRON
1MW
POWER
COUPLED CAVITY
TWT
AVERAGE
100kW
GYROTRON
10kW
HELIX TWT
1k W
GRIDDED
100W
0.3
1
3
10
30
100
300
FREQUENCY(GHz)
Figure 4.42. Power Vs frequency of some microwave tubes
component of electron velocity is transferred to the RF field. The significant feature
of gyration is that frequency of operation is determined by the bias field strength and
electron velocity and not on the dimension of the tube. This makes the gyration useful
for millimeter waves. The power output is high. It has high efficiency in the millimeter
wave region; can give power output as high as 100 kW. The variations of this tube are
gyro klystron, gyro-TWT, ubitron and perhaps one can classify even periotron versus
which is not a relativistic tube.
Figure 4.42 summarizes the power versus frequency performance of
microwave tube amplifiers and oscillators.
4.8.2
Solid State Sources
Two most commonly used solid state microwave diodes are gunn diode
and IMPATT(IMpact ionization Avalanche Transit Time) diode. Both of them
directly convert DC power into RF power and can operate in frequency region
between 2 GHz to 100 GHz and more. The Gunn diodes are transferred electron
devices. It used bulk semiconductor like GaAs and InP. The electrons are
transferred from main conduction valley to satellite valley where the effective mass
is more. This results is negative differential mobility which can be used for
generation of microwaves.
The IMPATT diode uses a reverse biased p-n junction to generate
microwave power. The material is usually silicon, however, GaAs, can also be
used. The diode is operated at relatively high (70–100 V) to achieve avalanche
breakdown. When coupled with high Q resonator and biased at appropriate
operating point, a negative resistance effect is observed because of the time taken
210
Microwave Transistors
for movement in the intrinsic region. It has essentially a p-i-n type structure. High
power output can be obtained, however, noise is also high. IMPATT diodes can
also be used as negative resistance amplifiers.
4.9
CONCLUSION
In this chapter while transistor oscillator design has been discussed briefly
the device fabrication which is more important, has been discussed in detail.
Transistor sources generally have lower frequency and power capabilities when
compared to Gunn or Impatt sources but do offer several advantages. First, the
oscillators using GaAs FETs are readily compatible with MIC/MMIC circuitry. Easy
integration with FET amplifiers and mixers are possible. The transistor oscillator
circuit is much more flexible compared to Gunn or Impatt sources. This is due to the
fact that negative resistance oscillation mechanism of a diode is determined and
limited by physical characteristics of diode itself whereas the operating
characteristics of transistor source can be adjusted to a greater degree by the oscillator
circuit. Thus, transistor oscillators allow more control of frequency of oscillation
temperature stability and the output noise than do diode sources. Transistor oscillator
circuits can be easily frequency tuned, phase or injection locked and also be easily
modulated. Transistor oscillators are relatively efficient but cannot deliver high
power.
For local oscillator application, transistor dielectric resonator oscillator is
quite suitable. A very good temperature stability can be obtained and it is compatible
with MIC design. Tunable sources are needed in many types of electronic
warfare systems,frequency hopping radar and communication system.
Transistor oscillators can be made tunable by using adjustable elements like
varactor diode or a magnetically based YIG sphere. When reverse biased, the
function capacitance of a varactor diode can be controlled with DC bias
voltage. Thus, a voltage controlled oscillator (VCO) can be made by using
reverse biased varactor diode in the tank circuit of the transistor oscillator. In
YIG tuned oscillator (YTO) a single crystal YIG sphere is used to control the
inductance of a coil in the tank circuit of the oscillator.
The power capability of solid state device is being constantly
enhanced by power combining techniques. However, even now there is large
gap between power generated by tubes and solid state devices.
Various types of microwave oscillators are discussed in this chapter.
However, one type which is not mentioned here is a Frequency Synthesiser. This is
being widely used for precise time, frequency and phases in communication
transmitters as well as in receivers. They are also being used in radar pulse
compression. Digital implementions such as direct digital synthesis (DDS) are
frequently selected because of their inherent advantages. Any desired frequency
resolution can be obtained merely by increasing number of bits, fast switching and
setling times and very wide tuning range is available.
Because of many applications it is necessary to discuss these synthesisers
in little more detail and therefore the next chapter describes the broad principle and
types of these devices.
211
Microwave Circuits & Components
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Johnson, E.O. Physical limitations on frequency and power parameters
of transistors. RCA Review, 1965, 26, 163-77.
Smith, P.M.; Chao, P.C.; Duh, K.H.G.; Lester, L.F.; Lee, B.R. &
Ballingall, J.B. Advances in HEMT technology and applications. IEEE
MTT-S International Microwave Symposium Digest, 1987, II, 749-52.
Sovero, E.; Gupta, A.K.; Higgins, J.A. & Hill, W.A. 35 GHz performance
of single and quadruple power hetro junction HEMT’s. IEEE Trans. on
Electron Devices, 1986, 33, 1434-438.
Sheng, N.H.; Chang, M.F.; Asbeek, P.M.; Wang, K.C.; Sullivan, G.J.;
Miller, D.L.; Higgins, J.A.; Sovero, E. & Basit, H. High power AlGaAs/
GaAs HBTs for microwave applications. Tech. Digest of IEDM, 1987,
619.
Higgins, J.A. Report. Rockwell Science Centre, 1988.
Drummond, T.; Masselink, W.T. & Morkoc, H. Modulation doped GaAs/
AlGaAs hetro junction field effect transistors MODFET’s. Proc-IEEE,
1986, 74, 773-822, (correction in December 1986, 1803).
Mimura, T.; Joshin, K. & Kuroda, S. Device modelling of HEMTs. Fuji
Scientific and Technical Journal, 1983, 243-77.
S-Parameter Design. Hewlett Packard application note 154, April, 1972.
Asbeek, P.M.; Chang, M.F.; Wang, K.C.; Miller, D.L.; Sullivan, G.J.;
Sheng, N.R.; Sovero, E.A.; & Higgins, J.A. Hetero junction bipolar
transistors for microwave and multi-meter wave integrated circuits. IEEE
Microwave and Millimeter-wave Integrated Circuits Symposium Digest,
1987, 1-4.
Bayraktaroglus, B. & Camilleri, N. Microwave performances of npn and
pnp AlGaAs/GaAs hetro junction bipolar transistors, IEEE Trans of MTT,
1988, 12, 1869-873.
Liechti, C.A. Microwave field effect transistors. IEEE Trans on
Microwave Theory and Techniques. MMTC, 1976, 279-300.
Schottky, W. Proc. IRE, 1952, 50, 1365-376.
Bruncke, W.C. & Vander, Ziel, A. Thermal noise in junction-gate field
effect transistors. IEEE Trans. Electron Devices, 1966, ED-13, 323-29.
Vander Ziel, A. Thermal noise in field-effect transistor. Proc. IRE, 1962,
50, 1808-812.
Vander Ziel, A. Gate noise in field-effect transistors at moderately high
frequencies. Proc. IRE, 1963, 51, 461-67.
Baechfold, W. Noise behaviour gates field effect transistors with short
gate lengths. 1972, ED-19, 674-80.
Pereel, R.A.; Hans, H.A. & Hstatz. Signal and noise properties of gallium
arsenide field effect transistors. In Advances in Electronics and Physics,
Academic Press, 1975, pp. 145-264.
Fukui, H. Addendum to design of microwave GaAs MESFET’s for broadbroad low noise amplifiers. IEEE Trans., 1981, MTT- 29, 1119.
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Microwave Transistors
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
Su, C.; Rohdin, H. & Stollte, C. 1/f noise in GaAs MESFET’s. Tech.
Digest of IEDM, 1983, 601-04.
Kasper, E. et. al. ESSDERC 1986, Conference Proc. P.105 Cambridge,
Sept. 1986.
Inoue , K. et.al. Japan. J. Applied Physics, 1984, 23, 161.
Weisbuch, C. Fundamental properties III-V semiconductor, Two
dimensional quartered structures; basic for optical and electronic device
applications, In Semiconductors and Semi metals, Vol. 24, Academic
Press Inc.
Sakaki, H. Quantum effect ultrafast semiconductor devices. Proceedings
of YAG1 – the Symposium on advanced technology binding the gap
between light and microwaves. 1990, Sendai.
Sze, S.M. Physics of semiconductor devices. Zerets, Wiley, New York,
1981, pp. 19, 809, 170-74.
Irwin, J.C. & Loya, A. Failure mechanism and reliability of low-noise
GaAs FETs. Bell System Technical Journal, 1978, 57, 2823-46.
Irwin, J.C. The reliability of GaAs FETs. In GaAs FET principles and
technology, Edited by De Lorenzo, J.V. & Khandelwal, D.K. Artec
House, Norwood, Mass, 1982. pp.349-402.
Gonzalez, G. Microwave transistor amplifiers analysis and design.
Prentice Hall Inc., Englewood Cliffs, 1984. pp. 55-80.
Liao, Samuel Y. Microwave circuit analysis and amplifiers design.
Prentice Hall Inc., 1987. pp. 8-63.
213
CHAPTER 5
FREQUENCY SYNTHESIZER
5.1
INTRODUCTION
A synthesizer is a device which takes an input or source frequency and
from it produces an output frequency which is either directly or indirectly related
to it. Figure 5.1 shows schemes for direct and indirect synthesizer.
N
RF OUT
(A)
CONTROL
FREQUENCY
RF OUT
N
M
(B)
CONTROL
FREQUENCY
RF OUT
N
M
(C)
Figure 5.1. Synthesizer types (a) Direct (b) Indirect, and (c) Indirect with mixer
5.1.1
Principle & Types of Synthesizer
The direct synthesizer (Fig. 5.1a) produces an output which is directly
proportional to the input. In this arrangement fOUT is given by
fOUT N . f IN
(5.1)
Microwave Circuits & Components
where, fIN is input frequency and N is a multiplication factor. The value of N can be
fractional (i.e., 1/3); also, it can be an integer value. The combinations of fractional
and integer multipliers between the synthesis frequency source and output can
produce an output which has strange multipliers between the input and output. One
such example is 61/8. The Fig. 5.2 shows how X–3 multiplication may be done. In
both direct or indirect synthesizer, it is necessary to have some form of filtering.
This is to ensure that harmonics and other related spurious frequencies are deleted.
Figure 5.1 (b) and (c) operate by locking the output of a frequency source
likeVCO to that of other cleaner source known as reference frequency. The
reference frequency source usually have better phase noise and is more stable both
for mechanical and temperature fluctuations. The mixing stages allow generation
of high frequencies but requires a fine tuning range.
Figure 5.3 shows the basic block of a phase lock loop. The VCO is locked
to the reference frequency, fref by dividing the reference, fref , by some integer R,
the VCO output, fOUT by some integer N and then comparing the phase of the two
signals, generating an error signal. The error signal is then amplified and filtered to
remove phase comparison frequency components and modify the phase response
of the loop to provide closed loop stability. The output frequency is then given by
N
f ref
(5.2)
R
The phase detector in a phase lock loop (PLL) can take many forms, such
as XOR gate (type 1), a mixer (type 1) and dual D types (type 2) amongst many.
The last one is more common.
fOUT
2
fIN
fOUT
Figure 5.2. X-3 multiplier
fREF
R
PHASE
DETECTOR
K PHA SE
LO O P
FILTER
Z(S)
N
Figure 5.3. A phase locked loop
216
VCO
K PHA SE
fOUT
Frequency synthesizer
In a loop where input frequency exceeds either the maximum rf or
reference input frequency of a synthesizer it may be desirable to use a fixed divide
by M prior to N or R functions. M has been used to highlight the difference
between M and N. N is to be referred to as the division performed by the synthesizer
IC used. The output frequency will now be
f OUT
M .N
. f ref
R
(5.3)
Dual modulus pre-scalers are a simple way of implementing a high
frequency N within PLL. The division ratio commonly available are 8/9, 16/17,
32/33, 64/65. Figure 5.4 shows a general implementation of a dual modulus
pre-scaler.
HIGH
HIGH
FREQUENCY
RF IN
FREQUENCY
DIVISION
LOW
FREQUENCY
CONTROL
CONTROL BLOCK
(i.e.,SYNTHESIZER
CHIP)
A/A+1
TO VCO
CONTROL
Figure 5.4. Implementation of dual modulus pre-scalar
Mixers can be used to aid a division in feedback loop as shown in
Fig. 5.1(c). The mixer essentially provides a down conversion in the loop which
lessens the amount of actual division required to synthesize required output.For a
reference frequency fref , the division ratio of N and R, the frequency fmix applied to
mixer down converter, the resulting output frequency will be
N
f OUT
f
R ref
f mix
(5.4)
This results in improved noise performance.
Figures 5.5 and 5.6 show a standard third order filter used in most of the
synthesizers. The elements of filter can be calculated using relations
C2
KVCO . I cp
(5.5)
2
.N
BW
R1 2 . p .
N
K VCO .I cp .C 2
(5.6)
217
Microwave Circuits & Components
2nd ORDER LOOP FILTER
EXTRA POLE
R2
V OUT
R1
C2
C3
C1
Figure 5.5. A third order loop filter
C1
C2
R1
R2
_
Vcccp
2
+
Figure 5.6. An active third order loop filter
C1
C2
12
(5.7)
C3
R1 .C2
20 R2
(5.8)
R2 3.R1
(5.9)
where, p is the damping factor and
BW 2 f BW
(5.10)
Active filters of the type shown in Fig. 5.6 can be used
F (S )
1 SR2 (C1 C2 ) 1
.
SC1 R1
SC2 R2 1
(5.11)
Figures 5.7(a) and (b) show Bode plots for open loop gain and phase. The
gain margin is defined as magnitude of the reciprocal of open-loop transfer
function at the frequency where the phase angle is –180° and is relative measure
by stability. It can be seen from the phase plot that is 180° phase shift at 100 KHz.
The gain margin is the value at this offset. It is desirable that this value should be
larger than 15 dB. In the present case it is 25 dB.
218
GAIN
Frequency synthesizer
FREQUENCY SCALE 180°°
PHASE ANGLE OUTPUT
Figure 5.7(a). Bode plot for open loop gain
FREQUENCY SCALE 180°
Figure 5.7(b). Bode plot for phase angle
The phase margin is defined as 180° phase plus the phase angle of open
loop transfer function at 0 dB or unit gain. This is also a measure of relative
stability. In the present case it is larger than 130°; usually 30° is sufficient. A plot
of synthesizer output is shown in Fig. 5.8.
The main application of frequency synthesizers are in: (a) communication
(b) radar (c) navigational aids, and (d) measurement, where direct and indirect
synthesis and combination of both can be used. In direct synthesis, the frequencies
are summed, divided or multiplied to generate output signal. In indirect synthesis,
a voltage controlled oscillator is phase locked on one or several reference signals.
The design depends also on physical parameter differences like size or weight, the
electrical performance may also be different for different applications. The radar
and navigational aids require low and medium bandwidths, say about 20 per cent
and frequency accuracy of say about 105 and sometimes 106. They require medium
219
Microwave Circuits & Components
10
10 dB
40 dB
2.32999939 GHz
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
Centre 2.329999365 GHz
5 kHz
Span 50 kHz
Figure 5.8. Synthesised 2.33 GHz signal
frequency resolution of the order of either MHz or in terms of 10 4 – 106 MHz
frequencies may be used in communication systems of medium to wide relative
bandwidths upto several octaves. The corresponding resolution generally extend
from hertz to Kilohertz. Also spectral purity may be different, phase noise of about
–125 dBc/Hz in the band region. Short or very short switching time is required (a
few microseconds). Synthesizers for measurement purpose must have large
bandwidth like 100 MHz – 25 GHz with resolution between 1 – 100 Hz.
5.2
are:
(a)
(b)
5.2.1
SYNTHESIZER ARCHITECTURES
Microwave signal generation are of two major types of synthesis. These
Direct (analogue) synthesizer
Indirect synthesizer
These are discussed in following subsections.
Direct Analogue Synthesis
This consists of several reference sources associated with frequency
multipliers and dividers, mixers, amplifiers, switches and filters. These must be
accurate and stable sources with very high quality resonators. It consists of
generators of many frequencies making a combination of sources (S1,S2………., Sn)
with their output being selected by a switch. The sources can be stable microwave
oscillators using dielectric resonators or HF, VHF or UHF oscillators using
conceptual resonators. This type of device is convenient only when low number of
frequencies are requested.
Synthesizer architectures using several banks of sources successively
associated by means of mixers are used when high numbers of frequencies are
requested. If Na,Nb………..,Nn the total number of frequencies available at
synthesizer output is
220
Frequency synthesizer
Nt = NaxNbxNcx………Nn
(5.12)
or with equal number N of frequencies in each bank
Nt N n
(5.13)
Design and manufacture of synthesizer according to this principle is done
by using crystal oscillators which are switched to deliver the desired frequencies at
the synthesizer output.
A synthesizer used in a radar system is shown in Fig. 5.9. The basic
synthesis was done by three banks of five high spectral purity crystal oscillators.
Frequency selection is performed by very efficient diode switches properly
controlled to obtain the desired frequencies at the synthesizer output. Quartz crystal
frequencies lie in the region of 50 to 200 MHz. The first bank is designed to
produce the worse frequency steps whereas second is devoted to fine frequency
steps and the third one produces the intermediate frequency steps. The second
frequency mixing acts as a frequency subtracted, so as to maintain a low spurious
levels. Each of the mixers is followed by an amplifier, a band pass filter used to
reduce out of band spurious signals (in band spurious are minimised by proper
selection of frequencies and levels of the mixed signals).
Amplifiers are also used to derive mixers and particularly the frequency
multiplier chain. This chain depends on the frequency chain derived at the output
of the synthesizers so several modules were designed to produce signals at S, X
and Ku bands. These modules are (2 x 3) for S band, (2 x 3 x 3) for X band and (2 x
3 x 7) for Ku band. The multiplication, SSB signal to phase noise ratios of these
three versions of synthesizer is about –135 dBc/Hz at S band, –125 dBc/Hz at X
band and –118 dBc/Hz at Ku band for output frequency of about 100 KHz from the
carrier. White noise asymptotic curve is located approximately at 5 KHz from the
X.0 X.0 X.0 X.0 X.0
AMP
MIXER
B.P.F
MIXER
AMP
AMP
B.P.F
AMP
fff
fff
FREQUENCY
MULTIPLIER
FREQUENCY
CONTROL
X.0 X.0 X.0 X.0 X.0
X.0 X.0 X.0 X.0 X.0
BAND P AS S
FILTER
AMP
SYNTHESIZER
OUTPUT
Figure 5.9. Non coherent synthesiser used
221
Microwave Circuits & Components
carrier. The main advantages of direct incoherent synthesis are a relatively low cost
and a good flexibility as the sources utilized in the synthesizers are independent so
their number is chosen according to number of frequencies available. Nevertheless,
this type of synthesizer has its own inconveniencies such as low frequency stability
(contribution of several sources) and a limited number of available frequencies
(cost increases with number of sources used in the device). Sometimes, the noncoherence of all the sources of a synthesizer can also induce disturbances such as
signal beating in the receivers.
Next is the direct coherent synthesis. The most popular principle used in a
chain of mixers, filters and amplifiers with all signals that derive the mixers is
derived from a single source. Practically, this basic principle is often improved by
using the so called mix and divide system in which frequency divider is associated
to each mixer of the chain as shown in Fig. 5.10. The main advantage of this type of
synthesis is that very small frequency increments can be generated by cascading
the adequate number of stages since the basic increment is divided by Dn at the
output if D is the division factor and n the number of stages. This greatly simplifies
the problem of filtering that occurs when very narrow increments are needed. This
type of synthesizer operates through two different parts, first is divide and mix part
and second the reference generator section. The former is represented here as a
chain constituted with stable reference oscillator drawing a comb generator
followed by a bank of narrow band filters that feed a switching array. The reference
source is either a crystal or surface acoustic wave (SAW) generator. The interesting
way to make a comb generator is to use a step recovery diode as multiplier. A large
number of harmonics are generated. Multiple switches are used. These should be
high isolated from each other. Leakage is to be avoided. Quasi-identical stages
containing a frequency divider is followed by a mixer and band pass filter.
Amplifiers are not shown in Fig. 5.9. Frequency multipliers are to be used if the
ONE STAGE
fn
D
f4
fff
f3
REFERENCE
SOURCE
fr
D
fff
F OUT
D
fff
f2
f1
f
r
COMB
GENERATOR
SWITCHING
ARRAY
FILTER
BANK
FREQUENCY O
MULTIPLIER F
AND FILTER O U T
h.fr
FREQUENCY
CONTROL
Figure 5.10. Block diagram of a coherent synthesizer
222
Frequency synthesizer
synthesizer is to generate microwaves. The second input of each mixer is connected
to the switching matrix so as to feed the reference signal. For better mixer input,
frequencies are indexed f1 for the last (output) stage to fx for the first stage. Any
frequency f1, f2 ……, fm can be equal to fr or to one of its harmonics R. fr (R is an
integer) according to the selection made through the switching matrix.
Thus the total number of frequencies generated is
N t R n R m1
(5.14)
m being the total number of mixers used in the device and n = m+1, the total
number of reference signals applied to the mix and divide chain. The harmonics
cannot be increased beyond circuit limit because of filtering problem. The total
number of frequencies will be essentially determined by number of stages used to
make synthesizer. If D is the division factor of each stage, the expression of the
output frequency is
f out f1
f
f
f2 f3
.......... n 1 n
n
2
D D
D
D n 1
(5.15)
The output frequency may also be expressed as a function of reference
frequency fr (i.e., the crystal frequency) since fx=Rx.fr. So, Eqn. 5.15 may be
expressed as
R
R
R
R
f out R1 2 3 .......... n 1 n
n2
2
D
D
D
D n 1
fr
(5.16)
with R2 varying from 1 to Rx max. For regularly spaced frequency with the overall
bandwidth, the number of reference frequencies Rx max must be at least equal to
division factors D.
A coherent direct synthesizer is shown in Fig. 5.11. These four reference
frequencies are generated from a single oscillator frequency at 64 GHz. The
switching matrix consists essentially of four single pole 4 throw switches that
receive the four reference signals, each of the switches with output connected to
one of the four inputs of the mix and divide chain. The divide ratio is equal to the
number of reference frequencies (i.e., D = 4). The filters are located at the mixer
outputs. This synthesizer operates in the frequency range of 1360 – 1615 MHz
with frequency increments of 1 MHz. The total number of channels is 256. The
switching time is lower than 2 s to reach the phase steady state with an error
of 5o. The noise level is characterised by a PSD of less than –110 dBc/Hz at
100 KHz offset frequencies.
5.2.2
Indirect Synthesis
This is a very common type of frequency generation. An oscillator is used
for generating output at different frequencies and is controlled by a Phase Locked
Loop (PLL).
223
Microwave Circuits & Components
1280-1520 MHz
4
fff
f4
f3
1344-1596 MHz
4
fff
1360-1615 MHz
4
fff
fout
f2
f1
1024 MHz
fff
1088 MHz
fff
COMB
GENERATOR
1152 MHz
X.0.64
MHz
fff
1216 MHz
fff
SWITCHING MATRIX
FILTER BANK
Figure 5.11. Coherent direct synthesizer
Theory: This section concentrates on basic principle of PLL synthesizer1-3.
The block diagram of a typical microwave synthesizer is shown in Fig. 5.12. Two
reference signals at low and high frequencies (fRL and fRH) are generated by
frequency division and multiplication. The output signal at fOUT is obtained by a
Voltage Controlled Oscillator (VCO). A mixer in which the difference between fRH
and fOUT is obtained. The resulting frequency is divided by N in a variable ratio
divider. A phase comparator gives an error signal, which varies with the phase
difference between the signal at fRL and the signal at f OUT f RH N . The signal
is amplified and filtered.
Mathematical relations for the ratio fOUT / fRO can be obtained but the
relation for fOUT and fRH or fOUT and fRL is established to study the variation of
output versus each of the two reference signals. It is assumed that the signal is
small under linear conditions and the Laplace transform is used. The slope of VCO
is KVCO , i.e., the ratio of frequency generated and applied voltage. Phase detector
slope is KPD which is the ratio of voltage to phase. The other constant parameters
wrt Laplace variables are the multiplication factor M, producing the high frequency
reference fRH and fixed and variable division ratio P and N which are used to
generate the low frequency reference, i.e., fRL and the frequency variation of
synthesizer. As phase detector delivers a voltage and this resulting voltage is
applied to VCO input so as to control its frequency, transformation between phase
and frequency must be translated in Laplace notation. This corresponds to I/S
function introduced in the block diagram, which means that the frequency
difference is integrated to give phase difference. To close the loop, the loop filter is
added so as to filter out spurious signals and noise to amplify the phase error and
224
Frequency synthesizer
REFERENCE
OSCILLATOR
(X.0 OR
S A W O)
fRO
FRE QUEN CY
MULTIPLIER X M
fOUT
fRH
fOUT
V C O
PERFUME
fOUT– fRH
FRE QUEN CY
CONTROL
DIVIDER
P
FRE QUEN CY
CONTROL
VARIABLE RATIO
DIVIDER
N
LOOP FILTER
AND
AMPLIFIER
fRL
P HA S E
D ET EC TO R
FRE QUEN CY
DISCRIMINATOR
Figure 5.12. PLL microwave synthesizer with crystal of SAW source
ensure loop stability. The function is GLF(S) and it depends on frequency. The
frequency and phase differences are simply materialised by two subtractions in the
block diagram.
Assuming fOUT >M fRO , the closed loop transfer function is
F S
fOUT S
f RO S
N K vco GLF S K PD / N
M
P S T KVCO GLF S K PD N
OUT S
N
M H S
RO S
P
(5.17)
This may also be found out by means of feedback theory, i.e., by
application of general rule; transfer function from any point of the loop to the
output is equal to the ratio between the forward gain from that point to the output
(S )
(here 1 KVCO GLF
K PD N S from the phase detector inputs) and the sum
(S )
[1 + (open loop gain)], which is 1 KVCO GLF K PD N S .
Examination of Eqn. 5.6 where S = j , shows that the output signal at
frequency fOUT is controlled by the reference frequency jRO (or to fRL and fRH
(S)
together). At low offset frequencies KVCO GLF
K PD N S is generally large and
constant so j may be neglected and the spectrum of output signal is that of
reference signal multiplied by (M + M/P).
( S ) decreases and j becomes
At far offset frequencies, the loop gain G LF
larger and larger so that input and output frequencies are finally related by a
complex function and this means the output spectrum is no longer an image of
input spectrum. In other words, the phase-locked loop is a low pass filter that can
225
Microwave Circuits & Components
be used in frequency systems to copy the spectrum of one (or several) reference
signals. The efficiency of copy can be limited to lower part of reference spectrum
by selection of the loop cut-off frequency.
The basic mathematical representation of PLL microwave synthesizer is
shown in Fig. 5.13 and schematically in Fig. 5.14.
fOUT (S)
f RH (S )
or OUT (S)
M
K VCO
f RH (S ) fOUT (S )
MULTIPLIER
GL/F(S)
f RO (S )
1
N
or RO (S )
VARIABLE
RATIO
DIVIDER
K PD
f (S )
1
S
DIVIDER
1
P
VCO
f RL (S )
LOOP FILTER
PHASE
ERROR
PHASE
DETECTOR
PHASE
ERROR
[ RL (S )
(S )]
FREQUENCY
ERROR
[ f RL (S )
f (S )]
Figure 5.13. Basic PLL synthesizer
c
o
n
s
t
a
n
t
Let us consider the simplest PLL that can be fabricated. It is a loop with
(S )
GLF
K LF with respect to the offset frequency. The open loop
g
a
i
n
,
i
.
e
.
,
transfer function is then –jk/n. so that phase between input and output is 2 .
Then it is clear that whatever is the gain, the loop will be stable since at the
frequency where open loop gain is 1 (or 0) dB there is still a 2 phase margin that
prevents the loop from oscillating, as illustrated in Fig. 5.15. Also represented in
this figure is the asymptotic closed loop transfer function. In addition to the
theoretical Bode plot, influence of spurious elements has been represented by
means of dashed curves. It was supposed here that there were two spurious lowpass filters and for example they may be one of the loop amplifiers whose gain
decreases at high frequencies and one of the filters at VCO input (high input
impedance and parallel parasitic parallel capacitor). Their influence is not only to
lower the loop gain, but also to introduce a phase shift that quickly reduces and
cancels the phase margin. This is the case (Fig. 5.15) where it can be seen that the
closed loop is quite unstable since the loop gain is greater than unity when the
phase is equal to 180o. As a consequence, the designer needs to define a loop filter
to avoid loop instabilities. Numerous solutions can be used and the convenient
solution depends on certain characteristics of the synthesizer. As an example, the
case of the lag-lead filter (Fig. 5.16), which is partially suitable for wide acquisition
and hold in ranges, is briefly described.
226
Frequency synthesizer
OPEN LOOP GAIN
20 log
k
N
THEORETICAL
LOCUS
REAL LAWS ASYMPTOTIC
log
1/t1
OPEN LOOP
ASYMPTOTIC
0
1/t1
1/t2
SPHERICAL LOCUS
log
/2
REAL
LOCUS
ASYMPTOTIC REAL
LOCUS
3 / 2
CLOSED LOOP
N
GAIN
20 log M
P
K /N
1
K / N 2 2 2
N
20 log M
P
log
Figure 5.14. Bode plot for first order loop
The transfer function of the filter is
(S )
K LF
GLF
12S
1 (S / W2 )
K LF
1 1S
1 (S / W1 )
(5.18)
so the open loop gain is
(S )
GOL
KVCO K LF K PD 1 2 S K
NS
1 1S N
1 2S
1 1S
(5.19)
Figure 5.16 shows that phase shift increases when frequency is low then it
decreases as frequency increase. Thus, it is clear that the proper fixation of
cut-off frequencies is likely to ensure a stable operation of the loop. As a matter of
fact, the decreasing part of phase shift, i.e., say from to 2 is properly
selected, it counteracts the spurious phase shift, thus decreasing the phase margin
around the point loop which is equal to unity. The dc gain is also important for loop
stability. It has to be decided how it is affected by cut-off frequencies.
227
Microwave Circuits & Components
LOOP FILTER
OPEN LOOP GAIN (dB)
R1
K LF
R2
1
w1 =1/t1 =
(R1 +R2 )C
w1 = 1/t2 =
C
1
R2C
K
log w
N
0
w1= 1/t1
w2= 1/t2
K
t1N
t2
t1
K
N
OPEN LOOP GAIN ARGUEMENT
w1= 1/t1
0
w2= 1/t2
log w
/2
Figure 5.15. Bode plot for a second order loop with lag-lead filter
The transfer function of the loop when the loop filter is a lag lead circuit is
F S
fOUT S
S 1
N
N 2
M
2
P 1S 1 K N 2 S
f RU S
K
K
N
(5.20)
TO TRANSMITTER
VCO
962–1213 MHz
(252 FREQUENCIES)
LOOP
AMPLIFIER
AND FILTER
AMPLIFIER
AND POWER
SUPPLIER
FREQUENCY
CONTROL
TO RECIEVER
TO TEST
FIXED
DIVIDER
6
PROGRAMMABLE
DIVIDER
(
PHASE
DETECTOR
862–1213)
125 KHz
FIXED
DIVIDER
32
X.0
F0= MHz
Figure 5.16. An example of simple PLL synthesizer in a DME application
228
Frequency synthesizer
OUT S
RO S
5.3
PLL SYNTHESIZERS
Two such circuits are discussed here. They are
(a)
(b)
Single reference PLL synthesizer
Double loop radar synthesizer
5.3.1
Single Reference PLL Synthesizer
The single reference PLL synthesizer is used in Distance Measurement
Equipment (DME) as shown in Fig. 5.16. Its operation bandwidth is 962 to 1213
MHz with 1 MHz frequency increment. This has medium spectral purity and
narrow loop bandwidth VCO used in this circuit is a hybrid. Component variable
ratio divider working in L band is not available. An ECL fixed ratio divider was
used to lower the frequency at variable divider input, thus decreasing the phase
detector reference frequency to 125 KHz. So the frequency of stable frequency
crystal oscillator is divided by 32 to generate the step reference of the synythesizer.
The
main
characteristic
of
device
(temperature
range:
–10 to 55°C) are a long term frequency stability of I =10.10 -6, a PSD of –110 at
100 KHz, offset frequency and spurious elements is less than 60 dBc. The 251
frequencies (with 1 MHz step) are manually controlled.
5.3.2
Double Loop Radar Synthesizer
This type of synthesizer (Fig. 5.17) operates between 4560–5060 MHz
frequency increment. This device has low noise and relatively fast switching off
time. The double loop architecture was chosen in order to minimise the noise. The
first loop generates a stable signal whose frequency varies by 10 MHz fine step
increments while the second has its frequency changed by 100 MHz coarse steps.
To reduce noise, the division ratio of the second loop is only four or five,
the complimentary variation being generated by means of a divider (P = 3, 4, 6 or
12) located out of loop so there is no multiplication effect (p2) on the loop noise.
The low noise reference is delivered by a hybrid crystal oscillator whose frequency
is multiplied by 35(7X5) to provide the high frequency reference of the first loop.
A frequency control system converts the frequency and switching data into signals
that drive the programmable divider and the VCO pre timings. A similar
synthesizer has also been made for radars operating over 450 MHz at S band with
a 5 MHz step. The main characteristics of C band synthesizer are settling time of
less than 35 ms for Doppler filtering capability or less than 10 ms without phase
stability condition and a PSD of –115dBc at 100 Hz for the carrier (spurious
emission be at < 50 dBc).
5.4
PIEZO ELECTRIC SYNTHESIZER
Piezo electricity was discovered by the end of last century but it was used
for stabilization of transmitter frequency in transmitter by 1930.
229
Microwave Circuits & Components
250 MHz
REFERENCE
X.0
FREQUENCY
MULTIPLIER
X 35
5250
5060 TO 5160 MHz
M H z MIXER
MIXER
VCO 1
P R OG R A M M AB L E
FREQUENCY
DIVIDER
(9 TO 19)
FREQUENCY
DIVIDER TO MHz
FINE
15
STEP
R EF ER E NC E
VCO 2
PROGRA MMABLE
FREQUENCY
DIVIDER
(4TO 5)
TO
RECIEVER
5060 MHz
(51 FREQUENCIES)
TO TRANSMISSION
UPCONVERTER
FREQUENCY
CONTROL
AND
P RE T UN E
SYSTEM
P HA S E
COMPRATOR
SWITCHING F R E Q U E N C Y
DATA
DATA
FREQUENCY
MULTIPLIER
X2
300 MHz
PROGRAMMABLE
FREQUENCY
DIVIDER
(3,4,6 OR 12)
25
50 M H z
75
100
(COARSE STEP)
PHASE
COMPRATOR
Figure 5.17. Radar PLL synthesizer
Piezo electricity is a greek word meaning pressure. Electricity appears on
piezoelectric faces when pressure (mechanical) is applied to other pairs of faces.
This property is anisotropic and appears only in non-centro symmetric crystals.
Development towards high frequency has been made recently. Wide band devices
are mostly based on ion etched (or chemical etched resonators and on the
introduction of new piezo electric materials (Li Ta O 3, Al PO4) which offer stronger
piezo electric coefficients than quartz is the most widely used now. These crystals
are grown at high temperature, high-pressure autoclaves in aqueous solutions of
NaCO3 or NaOH. Resonators which cover the frequency range from 1 MHz to
1 GHz uses thickness shear waves of AT, BT or doubly rotated cuts shown in
Fig. 5.18. Acoustic waves in piezoelectric medium are solutions of a differential
equations of motion. The relations are
Tij = CijRlS Rl - emijEm
(5.21)
Dn = e mm E m + e nRl S Rl
(5.22)
where,Tij is the stress lessor, Em and Dn are the electric field and electric
displacement, CijRl SRl, emijEm are the elastic, piezo-electric and dielectric
parameters. SRl is the strain terms or described on first order of displacement by
S Rl
1
2
U
R1l
U ljR
(5.23)
Plane waves propagating along an S-direction are
S
U j a j exp jw t
V
(5.24)
230
Frequency synthesizer
Z
Z
Z
Z
SC
AT
X
X''
Figure 5.18. Thickness shear geometry in quartz peizoelectric resonator
a4 exp jw t VS
with S = nlxl and V=w/R
(5.25)
A resonator is created by a thin disc, metallised on both faces, the
direction of the normal to major faces is then propagating direction S. A stationary
wave is given by
r
r
r
U a exp jw 1 VS b exp jw t VS
(5.26)
h
2n 12 correspond to stationary waves
v r
whose amplitude goes to zero (anti-resonant frequencies). These are stationary
waves of maximum amplitude. These frequencies are related by
Frequencies given by
4 k r 2
rrn ran 1 2 2
N
(5.27)
where, N = Zn +1 is the overtone rank.
Environmental sensitivity affect the phase noise spectrum under
vibrations. This criterion is mostly sensitive in airborne radar or communication.
Figure 5.19 gives the performance of an up-to-date reference oscillator for airborne
radar.
Figure 5.20 gives typical phase locked configurations which can be used
either in radars, synthesizers or high frequency sources. In the figure a 1 GHz bulk
wave oscillator is locked on to a high stability 10 MHz radar synthesizer.
231
Microwave Circuits & Components
dBc/Hz
-60
-80
100
120
140
160
5
10
100 K
10 K
1K
100
FREQUENCY
Figure 5.19. Reference oscillator for airborne radar application
LO +
10 MHz
IF
1 GHz
-20 dB
HF
100
Figure 5.20. Phase locked loop oscillator for radar synthesizers of high frequency
source.
5.5
CONCLUSION
Frequency synthesizers are essential for precise time, frequency and
phase measurements which is important from the point of view of communication,
radar pulse compression and accurate measurement of frequency in network
analysers. PLL synthesizers are the most important elements in the design of
frequency synthesizers.
It may be noted that the topic of synthesizers is a vast one and it is very
difficult to deal with the design aspect of various frequency synthesizers are due to
space constrain. Every synthesizer design should be carefully assessed in its own
right. Careful calculation for any model should be done and critical areas on which
synthesizer characteristics depend should be made before they are used.
232
CHAPTER 6
SMITH CHART & IMPEDANCE MATCHING
6.1
INTRODUCTION
In 1984, Wheeler1 proposed a reflection chart for calculating various
parameters of a transmission line. However, it is not as popular or useful as another
chart proposed by Smith2,3 as early as 1939. It may appear that these days since
many scientific calculators and powerful computers are available, then why to use
a chart? However, the Smith chart is not just a graphical technique but displays
visual behaviour of many circuits with frequency and other parameters. A
microwave engineer can develop intuition for designing and impedance matching
in terms of Smith chart. It is essentially a polar plot of voltage or current reflection
coefficient . If one uses the relation e j then Smith chart can be seen as
a plot of as a function of θ , where θ is expressed in terms of distance from
either the load end or the source end. We know that θ l
2
l , therefore θ
can be expressed in terms of l. Using Smith chart, it is possible to convert reflection
coefficients to normalised impedances (or admittances) and vice-versa.
A few parameters connected with transmission line can be revised by
mentioning some of these. The voltage reflection coefficient is defined as
V Z Z0
0 L
V0 Z L Z 0
(6.1)
where, Z L and Z 0 are load and characteristic impedances, respectively..
When the load is mismatched, then all the available power from the
generator is not delivered to the load and the loss is called return loss (RL) and is
defined in terms of dB as
RL 20 log dB
(6.2)
It must be remembered that matched load has a return loss of dB ,
whereas a total reflection has a return loss of 0 dB
The standing wave ratio (SWR) is defined as
Microwave Circuits & Components
SWR
Vmax 1
Vmin 1
(6.3)
The voltage wave for z > 0 in the absence of reflections is outgoing only
and can be written as
V Z V0 Te j z
for
z0
Transmission coefficient T is
T 1 1
Z L Z0
2Z L
Z L Z0 Z L Z0
(6.4)
The transmission coefficient between two points in a circuit is expressed
in decibel (dB) as insertion loss IL
IL 20 log T
6.1.1
as
dB
Decibels & Nepers
The ratio of two power levels P1 and P1 can be expressed in decibels dB
10 log
P1
P2
dB
(6.5a)
It can also be expressed as
20 log
V1
V2
dB
(6.5b)
provided that the value of resistances R1 and R2 for V1 and V2 are same.
The ratio of voltages across equal load resistances can also be expressed
in terms of nepers as
ln
V1
Np
V2
(6.6a)
The corresponding expression in terms of powers are
1 P1
ln
Np
2 P2
(6.6b)
1 Np 10 loge 2 8.686 dB
It should be remembered that
P
10 log 1
is dBm
1 mW
Thus the power of 1mW = 0 dBm while the power of 1W is 30 dBm.
234
Smith chart & impedance matching
Example 6.1. Find the output of an amplifier whose gain is 40 dB and input is
1 mW. Express your result in dBm.
Solution: 1 mW is 0 dBm. Therefore, output is equal to 0 + 40 = 40 dBm
6.1.2
Derivation of Reflection Coefficient based on Simple Transmission
Figure 6.1 shows a simple transmission line. The source is connected at
the sending end and the load at the receiving end. The total length of line is l units.
z is the distance measured from the sending end and d is the distance from the
receiving end.
I+
I–
ZL
VS M
V+
Z0
ZL
V–
RECEVING
END
SENDING
END
L
Z
Figure 6.1.
D
Transmission line of characteristic impedance Z 0 of length L
terminated in impedance 2L.
Thus, for a line of length l terminated in a load impedance ZL, on solving
voltage and current at load L, one gets
VL V e l V e l
1
V e l V e l
I L
Z
0
(6.7a)
(6.7b)
But since,
V
V e l V e l
Z L L Z0
or,,
IL
V e l V e l
V e l
V e
l
Z L Z0
Z L Z0
(6.8a)
Therefore the reflection coefficient L at the load end is defined as
V _ e l
V e
l
V
I reflected
reflected
Vincident
I incident
Z L Z0
Z L Z0
(6.8b)
235
Microwave Circuits & Components
6.2
THE SMITH TRANSMISSION LINE CHART
The reflection coefficient in general is given by the relation
ZL
1
Z L Z0
Z0
Z 1
L
Z
Z L Z0
L 1 Z L 1
Z0
(6.9)
ZL
is the normalised impedance. Since ZL is complex, Z L is also
Z0
complex, therefore it can be assumed that
where, Z
Z
ZL
r jx
Z0
The above relation represents terminating impedance. However, the
impedance at any point d measured from the sending end is
Z d r d jx d
(6.10a)
This relation can be written as
Z r jx
Therefore the relation is
d
r jx 1
r jx 1
d
V e d
(6.10b)
where,
V e
d
V 2
e
V
d
e2 j
d
if the line is assumed to be lossless, solving the Eqn. 6.9 for Z, one gets
Z d
1 d
1 d
(6.11)
Therefore,
Z d r d jxd
j 2 d
1 d 1 e
1 d 1 e j 2 d
where it has been assumed that d j d , and d 0 , i.e., the line is
lossless. Writing for
d e j 2 d u jv
236
Smith chart & impedance matching
d is complex. Therefore, it can also be written as u jv . Hence
Z d r d jxd
1 u jv
1 u jv
Rationalisation on separating real and imaginary terms one gets,
r d
x d
1 u2 v2
(6.12a)
1 u 2 v 2
2v
(6.12b)
1 u 2 v 2
dropping d from Eqn. 6.12 for simplification gives
r u 2 v 2 2ru r 1 u 2 v 2
i.e., 1 r u 2 1 r v 2 2ru 1 r
This gives
u2
2 ru
1 r
v2
1 r
1 r
(6.13)
Therefore
u2
2 ru
r2
1 r
r2
v2
2
1 r 1 r
1 r 1 r 2
2
1
r
2
u
v
1
r
1 r 2
(6.14)
Equation 6.14 represents an equation of circle which has centre at
r
1
and radius
in u,v coordinates. This represents family of circles for
1 r
1 r
different values. The circles are shown in Fig. 6.2.
Table 6.1(a) and Table 6.1(b) give the values of centre and radii of circles
for different values of r. It can be seen that all the circles pass through the point
(1,0). These type of circles are seen in the Fig. 6.2. The circle with centre at (0,0)
and radius unity is known as the unit circle. The Smith chart is normally drawn
inside a unit circle.
237
Microwave Circuits & Components
A
r=0
r=1
1, 0
1, 0
0, 0
B
Figure 6.2. Constant resistance centre
Table 6.1(a). Variation of centre and radius of circles for same value of r for values
equal to 0 and >1.
S.No
Value of
r
Centre
Radius
r/1+ r, 0
1/1+ r
1
0
0, 0
1
2
1
1/2, 0
1/2
3
2
2/3, 0
1/3
4
3
3/4, 0
1/4
5
4
4/5, 0
1/5
Table 6.1(b). Variation of centre and radius of circles for same value of r for values
0< r < 1.
S.No
Value of
r
Centre
Radius
r/1+ r, 0
1/1+ r
1
1/2
1/3, 0
2/3
2
1/3
1/4, 0
3/4
3
1/4
1/5, 0
4/5
4
1/5
1/6, 0
5/6
5
1/6
1/7, 0
6/7
In the similar manner it follows from Eqn. 6.12b
x
2v
1 u 2 v 2
s x1 u 2 v 2 2v
238
Smith chart & impedance matching
2.0
1.0
A
1
x
0.
5
+jx
0
1
1
x
1.0
0
2.
0.
5
-jx
B
Figure 6.3. Constant-reactances circles
u 2 2u 1 v 2
u 12 v 1
2v 1
1
1
1 1
2
x x2
x
x2
2
1
x
x
2
(6.15)
1
Above equation represents a family of circles with centre at 1 , and
x
radius
1
. These family of circles (Fig. 6.3) are for fixed value of reactance x and
x
are called reactance circles.
Table 6.2(a) and Table 6.2(b) give the values of centre and radii of
circles for different values of x both positive and negative. Of course, as in the
case of r circles, more circles can be drawn in a similar manner for different
value of x as shown in Fig. 6.2. The values given in the table are only typical
but provide a clear picture on how circles of different values can be drawn.
These circles are drawn within the unit circle. The circles with positive values
of x have positive reactance and are called inductive circles when inductive
drawn in the upper semicircle. Similarly, the circles with negative values of x
have negative reactance, and are called capacitive circles when drawn in the
lower semicircle.
239
Microwave Circuits & Components
Table 6.2(a). Values of centre and radii of circles for positive values of x
S.No.
Value of x
Radius
Centre
1/x (1, 1/x)
1
1
1
(1, 1)
2
2
1/2
(1, 1/2)
3
3
1/3
(1, 1/3)
4
4
1/4
(1, 1/4)
5
5
1/5
(1, 1/5)
Table 6.2(b). Values of centre and radii of circles for negative values of x
S.No.
Value of x
Radius
Centre
1/x (1, 1/x)
1
-1
-1
(1, -1)
2
3
4
5
-2
-3
-4
-5
-1/2
-1/3
-1/4
-1/5
(1, -1/2)
(1, -1/3)
(1, -1/4)
(1, -1/5)
Thus Eqns. 6.14 and 6.15 show that loci of normalised input resistance
and reactance are the families of circles having centres and radii that are dependent
upon per unit parameters. Equation 6.14 shows that the circle for which r=1, passes
through the origin. This circle is important for impedance matching.
The resistance circles intersect the diameter of the unit circle drawn in the
figure, at right angles and reactance circles are tangential to the same diameter axes
at its right hand extremity.
In the transmission line chart wherever constant r circle cuts the diameter
passing through the centre. The diameter line is calibrated in terms of values of r.
Starting with 0 at the left end, its value increases up to 1.0 at the centre. On the right
side, values are increasing and at the extreme end, it becomes infinity. For Z-Smith
chart the upper circumference of circle is calibrated in terms of inductive reactance
component jx Z 0 . The circumference of lower semicircle is calibrated in terms
of capacitive reactance component jx Z 0 .
The resistance circle is also related to reflection coefficients on the right
side of scale below the Smith chart. It is calibrated in terms of reflection
coefficients which give the magnitude of reflection coefficients. The angle of
reflection coefficient is shown on the inner side of circumference of circle of Smith
chart. The angle can also be defined in terms of 2 d 2
240
2
d ; d can be either
Smith chart & impedance matching
towards the load or the generator. This is also shown above the angle on Smith
chart both towards the load and the generator in terms of d .
The other scales shown below the Smith chart are in terms of voltage
standing-wave ratio which can be calibrated in dB or kept as ratio. Wherever
needed these scales can be used.
In the Z-Smith chart one can find admittance rotating by 180° by joining
impedance point with the centre and extending by the same length on the other
side. Rotating by 180° means
Z
Y
1 e j
1 e j
1 e j
1 e j
(6.16)
1
g jb
Z
It is to be noted that in this expression is the voltage wave reflection
coefficient. g and jb are conductance per unit length, b is the susceptance per unit
length. Both are normalised wrt characteristic admittance. It may be recalled that
the reflection coefficient of the current is equal in magnitude to the voltage
reflection coefficient, but has a phase difference of . If represents the current
reflection coefficient, then Eqn. 6.16 can be written as
Y g jb
1 e j
(6.17)
1 e j
This equation is exactly similar to Eqn. 6.11. It therefore, follows that the
circles in Fig. 6.4 are loci of constant conductance and susceptance as well as
constant resistance and reactance.
Therefore, the Z-Smith chart can also be used as Y-Smith chart. The
circumference of upper semicircle for y-parameter represents positive susceptance
and the lower semicircle represents negative susceptance. The only difference is
that the positive susceptance refers to capacitance and the negative susceptance to
inductance.
6.3
APPLICATION OF SMITH CHART
The application of Smith chart can be demonstrated through the following
examples:
Example 6.2. Determine the length of a short-circuited stub having a characteristic
impedance of 200 and input reactance of -j100 .
j 100
j 0.5 . The unit
200
circle is the locus of pure reactance or susceptance. The origin of reactance of a
short-circuited stub is the left-hand intersection of the real-axis and the unit circle.
Solution: The per-unit or normalised input reactance
241
Microwave Circuits & Components
0.11
(+
jX
/Z
5
45
1.0
50
0.9
55
1.4
0.8
1.6
2.0
T
0.4
N
75
PO
NE
CE
CO
M
0.0
4
0.4
6
15
0
0.8
4.0
15
5.0
10
0.
8
ANG
0.6
TRA
L E OF
0.2
170
10
0.1
0.49
0.4
20
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
± 180
50
0.2
20
1.
0
NC
TA
2.0
dB
]
0.
0
-1 7
30
0.
43
0.4
2
0.0
8
0.4
1
0.4
0.39
0.38
20
1
1
0.0
1
0.9
0.8
0.9
0.1
3
15
2
0.7
4
0.6
0.8
0.2
10
3
4
0.5
0.4
0.7
0.3
5
0.4
2
8
6
0.3
0.6
2.5
7
8
0.2
0.5
0.5
1.8
6
5
9
10
0.1
0.4
0.6
0.3
0.7
1.6
1.4
4
3
12
14
0.05
0.2
0.8
1.2 1.1
2
15
30
0.01
0.1
1.1
0.1
0.9
TOWARD LOAD Ð>
10
7
5
1
20
0.99
CENTER
1
1.1
4
1.2
1.3 1.4
0.4
0.6
1.2
1.3
0.95
1.2
1.4
0.8
1.5
0.9
1.3
<Ð TOWARD GENERATOR
2
1
3
1.1
0.2
1.6
1
1.4
1.8
1.5
2
3
2
0.7
1.5
4
3
1.6 1.7 1.8 1.9 2
0.8
A
TT
EN
.[
40 30
0
5
0.6
1.6
5
4
0.5
1.7
10
5
2.5
3
0.4
6
0.3
1.8
20
10 15
4
0.2
1.9
5
I
0.7
-70
0
4
-1
(-
0.4
T
0
-65 .5
1.8
0.37
-12
0
0.0
9
0
EN
0.6
1.6
5
-4
0.36
0.12
CT
A
-60
1.4
1.2
1.0
0.9
0.13
RE
A
0.1
0.11
-100
-90
0.14
-11
0
-5
-4
0
0.15
0.35
0.8
6
0.1
4
0.3
ITI
VE
5
-5
-70
-3
5
AC
N
10
or
OR
CA P
PO
RADIALLY SCALED PARAMETERS
10040 20
TR
A
N
SM
.C
O
EF
F,
E
IN
-75
),
Zo
3
0.3
7
0.1
0.2
-60
-30
OM
06
0.
/
jX
2
-90
EP
C
US
ES
IV
CT
DU
0.3
NC
EC
4
0.4
0.
3
0.1 1
9
18
0.
0
-5
-25
-85
o)
jB/ Y
E (-
0.6
45
0.
5
0.0
0.8
-4
0
0 .3
-20
3.0
6
0.4
4
0.0
0
-15 -80
-1 5
4.0
V
WA
0
<Ð
-16
1.
0
8
0.2
9
0.2
1
-30
0.
2
4
0.
0.2
2
0.3
5.0
NG
ELE
8
0.
-10
0.2
0.2
7
0.6
-20
0.4
0.4
0.1
10
<
RD L OAD
TOW A
THS
-170
Ð
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
50
0.0
0.2
0.48
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLE CTIO N COE FFICIE N T IN DEG
L E OF
RE ES
AN G
N SM ISSION COE FFICIE N T IN D
E GR E
ES
IN D
U CT
I VE
0
1.
RE
1.0
2
AC
TA
N
80
Ð>
20
0
3.
0.6
0.3
TO
R
1.8
6
0.5 5
70
0.
06
0.
44
0
14
0.0
5
0 .4
7
25
0.4
0.2
0.4
0.
18
0.3
2
50
20
85
0.2
0.1
7
0.3
3
30
8
160
60
Yo)
jB/
0.2
90
0.3
4
35
30
TOW
ARD
GE N
E RA
0.48
(+
CE
AN
PT
0.1
6
70
1
0.2
9
0.2
N GTH S
0 .15
0.35
40
0.2
Ð > W A V EL E
0.14
0.36
80
0.3
0.49
P
CA
R
0.37
40
O
E
SC
0.13
0.38
19
0.
31
0.
,
o)
U
ES
IV
IT
AC
0.12
0.7
0.6 60
0
12
07
0.
43
0.
0
13
110
1
0.4
8
0.0
2
0.4
0.39
100
0.4
1.2
0.1
9
0.0
10
0.1
0
2
Figure 6.4. Z or Y Smith-transmission line chart
The short circuit, as represented by a point in the Smith chart (0 point), and the stub
length can be found by moving clockwise. Corresponding to motion along the line
from short-circuited termination towards the input value of –0.5, through the entire
upper-half of unit circle, the reactance is positive until the intersection of lower 0.5
reactance circle with unit circle. It is labelled A in Fig. 6.5. The displacement is
from 0 to 0.426 wavelength.
Therefore, the required length is 0.426 wavelength = 153.4°
Example 6.3. Determine the input impedance of a 200 line, the line at a length
of three-eighths of a wavelength is terminated by 100 load.
242
Smith chart & impedance matching
3/8
ZINPUT
Z0 = 200
P
100
Figure 6.5. Solution to the Example 6.3
Solution: The example is demonstrated in Fig. 6.6. The terminating impedance
ZL
Z L 100
0.5 j 0
Z 0 200
This is located as points B in the Fig. 6.6. This point is rotated clockwise
by
3
0.375 . The intersection of circle with line OC, which represents rotation by
8
S
d
1
S2
E
0
B
F
O
D
S1
S
A
C
2
l2
Figure 6.6. Solution to the Example 6.2, 6.3 and 6.4
243
Microwave Circuits & Components
0.375 , is the point D, which gives the normalised input impedance. This
impedance is found to be 0.8 j 0.6 .
Therefore, Z in 2000.8 j 0.6 160 j 120 .
6.3.1
Determination of Unknown Impedance
In Chapter 2, it has been stated that unknown impedance can be obtained
by finding out VSWR and the position of first minima from the termination. It can
be obtained by solving transmission line voltage and current equations, and then
the real and the imaginary parts are separated. But this procedure requires huge
calculations. Such examples can easily be solved using Smith chart as shown in
example 6.4. This reduces computation effort in terms of time and calculations.
Example 6.4. Determine the real and imaginary parts of an unknown impedance
terminating a line of characteristic impedance of 300 . The VSWR has been
determined as 4.48 and the first minimum is at 6 cm. from the termination when the
frequency is 200 MHz.
Solution: The frequency (f) is 200 MHz, which means that the line wavelength
3 1010
2 108
150 cm.
Hence, the first voltage minima occurs at
termination.
6
0.04
150
from the
As has been pointed out in the Chapter 1, the normalised impedance at the
first voltage minimum is purely resistive and its value can be obtained from the
following relation:
1
1
1
Z vol.min. Voltage standing wave ratio VSWR 4.48 0.22
In Fig. 6.6, this is represented by the point E. Therefore, the terminating
impedance is found by rotating this point E in the counter-clockwise direction by
0.04 wavelength to the point F. The coordinates of point F are
0.24 – j 0.24.
Therefore, the terminating impedance is
Z L 3000.24 j 0.24 72 j 72
Example 6.5. Find the terminating impedance when it is connected to a line of
characteristic impedance (Z0) of 50 . The voltage minimum is located at 2 cm
from the load-end. The VWSR has been found to be 5.2. The frequency of
operation is 3 GHz.
Solution: Frequency = 3 GHz
244
Smith chart & impedance matching
Line wavelength
3 1010
10 cm
3 10 9
The first voltage minimum occurs at 2 cm from the load-end, i.e., the
distance of first minimum from the load-end in terms of line wavelength
2
10 0.2
1
0.19
The normalised impedance at the voltage minima
5.2
First, draw the line from the centre at a distance towards the load 0.2
(Fig. 6.7). Locate the point 0.19 on r scale and then draw the circle and find the
point where it intersects the line. Read the impedance (r and x). It is 1.6 – j0.2.
0.11
(+
jX
/Z
5
45
1.0
50
0.9
55
1.4
1.6
2.0
T
0.4
N
75
PO
NE
CE
CO
M
0.0
4
0.4
6
15
0
0.8
7
15
5.0
10
0.
8
ANG
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLE CTIO N COE FFICIE N T IN DEG
L E OF
RE ES
AN G
N SM ISSION COE FFICIE N T IN D
E GR E
ES
0.2
IN D
U CT
I VE
0
1.
RE
0.4
4.0
1.0
2
AC
TA
N
80
0.3
Ð>
1.8
6
0.5 5
70
0.
06
0.
44
0
14
0.0
5
0 .4
TO
R
20
0
3.
0.6
0.2
85
25
0.4
20
0.6
90
0.
18
0.3
2
50
8
160
0.2
0.1
7
0.3
3
30
0.2
TRA
L E OF
170
10
0.1
0.49
0.4
TOW
ARD
GE N
E RA
0.48
60
Yo)
jB/
30
20
0.2
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
± 180
50
O
0.2
20
1.
0
NC
TA
2.0
dB
]
0.
0
-1 7
30
0.
43
0.4
1
0.4
0.39
0.38
0
1
1
0.9
0.8
5
2
0.7
4
3
15
0.6
10
3
4
0.5
0.4
5
6
0.3
2.5
2
1.8
1.6
8
6
5
4
3
9
10
12
14
7
8
0.2
0.1
0.05
1.4
1.2 1.1
2
15
30
0.01
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
TOWARD LOAD Ð>
10
7
5
1
20
0.1
1.1
0.99
CENTER
1
1.1
4
1.2
1.3 1.4
0.4
0.6
1.2
1.3
0.95
1.2
1.4
0.8
1.5
0.9
1.3
<Ð TOWARD GENERATOR
2
1
3
1.1
0.2
1.6
1
1.8
1.5
3
2
0.7
1.5
Figure 6.7. Solution to the Example 6.5
245
2
0.6
1.6
4
3
1.6 1.7 1.8 1.9 2
0.8
1.4
A
TT
EN
.[
20
5
4
0.5
1.7
10
5
2.5
3
0.4
1.8
6
0.3
20
10 15
4
0.2
1.9
5
I
0.7
-70
0
4
-1
0.4
0
-65 .5
1.8
0.37
(-
0.4
2
-12 0.0
8
0
0.0
9
0
T
0.6
1.6
5
-4
0.12
CT
A
-60
1.4
1.2
1.0
0.9
0.13
RE
A
0.1
0.11
-100
-90
0.36
-11
0
-5
0.14
-80
-4
0
0.15
0.35
0.8
6
0.1
4
0.3
ITI
VE
5
-5
-70
-3
5
AC
EN
10
40 30
or
OR
CA P
N
RADIALLY SCALED PARAMETERS
10040 20
TR
A
N
SM
.C
O
EF
F,
E
IN
-75
),
Zo
3
0.3
7
0.1
0.2
-60
-30
PO
06
0.
/
jX
2
-90
EP
C
US
ES
IV
CT
DU
0.3
NC
EC
OM
4
0.4
0.
3
0.1 1
9
18
0.
0
-5
-25
-85
o)
jB/ Y
E (-
0.6
45
0.
5
0.0
0.8
-4
0
0 .3
-20
3.0
6
0.4
4
0.0
0
-15 -80
1.
0
-1 5
4.0
V
WA
0
<Ð
-16
8
0.2
9
0.2
1
-30
0.
2
4
0.
0.2
B
2
0.3
5.0
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50
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0.8
9
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8
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100
0.4
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10
0.1
0
2
Microwave Circuits & Components
Therefore, the load impedance
= 50(1.6 – j10.2)
= (80 – j10)
This is the value of the load impedance.
Example 6.6. Determine the load impedance when connected to the line of
characteristic impedance Z0 = 50 produces a VWSR of 2.0. When the load is
shortened the minima shifts to 0.15 towards load-end.
Solution: Figure. 6.8 illustrates the example.
Sometimes, it is difficult to determine the point where the load is
LOAD-END
SHORT-END
0.15
ZS
S
Z0 = 50
VS
ZL
Figure 6.8. ZL connected to line of Z0= 50
connected. So the best way to determine the load position is by shortening the
load-end. It is easy to see that wherever minima occurs, the load is supposed to be
connected at that point, and therefore, the example 6.6 means that the first voltage
minima when the load is connected, occurs at 0.15 .
1
0.5. Locate this
VSWR
point on the Smith chart and move towards the load-end by 0.15 on the circle of
radius 0.6. As shown in Fig. 6.9 the point shifts to the point shown in the figure as
zl. Impedance at this point is found to be
Draw a circle for which VSWR = 2.0, Z
0.125
1.0
2.0
SWR CIRCLE
0.5
0
V MIN
0.5
0.15
B 1 2
0.5
0.25
Zl
1.0
0.375
2.0
Figure 6.9. Solution to the Example 6.6 using Smith chart
246
Smith chart & impedance matching
zl 1 j 0.65
Therefore, the exact impedance is
zl 1 j 0.65 50
50 j 32.5
Example 6.7. An unknown impedance is connected at the termination of a line of
characteristic impedance (Z0 ) 50 . The VSWR has been found to be 6.0. The
length of line is 5.3 cm and the frequency of operation is 3 GHz. The minima is
shifted to the left by 6.2 cm when the line is shortened. Find
(a) The unknown impedance connected at the termination.
(b) The input impedance at the input end.
(c) The variation in input impedance if the frequency is varied by 10% .
Solution: Frequency of operation = 3 GHz
Therefore, line wavelength
3 1010
3 10 9
10 cm
Hence, half the line width is 5 cm.
The minima shifts to the right by (5.2 – 5) = 0.2 cm. This means that
minima occurs when line is terminated and the unknown impedance is shifted by
0.2
0.02 , when the line is shortened. Thus, the normalised impedance at
10
1
voltage minima is 0.2
5
Draw a line from B to a point A at a distance of 0.02 from B'. Then with
point O as centre and BC corresponding to 0.2 as radius, draw a circle (Fig. 6.10).
This circle cuts the line at the point D. Read the values of r and x and find out the
normalised impedance. The normalised impedance is found to be
z l 0.21 j 0.13
Therefore, the load impedance is
50 0.21 j 0.13
10.5 j 6.5
5 .3
0.53 is the same as at 0.03 . If one moves
10
from the load-end by 0.03, one will reach point F' (towards the generator). The
normalised impedance at point F' is 0.21 j 0.007. The actual impedance (i.e.,
The impedance at
input impedance) is
247
O
P
CA
R
TA
1.2
1.0
50
0.9
55
0.8
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1.6
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0.2
10
20
0
1.
8
5.0
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0.25
0.26
0.24
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0.25
0.24
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0.8
0.7
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50
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RADIALLY SCALED PARAMETERS
10
0.2
0.2
0.4
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0.2
10040 20
0.23
AN G
L
AN G
10
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
4
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2
0.2
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0.0
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1
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8
0.0
2
0.4 120
45
Microwave Circuits & Components
10
0.1
0
2
Figure 6.10. Solution to the Example 6.7
Z in 500.21 j 0.007
10.5 j 0.35
If the frequency is varied by +10 per cent then it is 3.3 GHz. Therefore, line
wavelength
3 1010
3.3 10
9
30
9.09 cm 9.1 cm
3 .3
Rest of the calculations can be done in a standard manner.
6.4
IMPEDANCE MATCHING
The term matching has been used in two form and can be distinguished as
(a) In the first form, it means that a transmission line or waveguide terminated by
an impedance equal to its characteristic impedance (condition for no reflection
at the termination) is said to be matched.
248
Smith chart & impedance matching
(b) In second form, a load impedance is said to match to the associated generator
when power transferred is maximum.
Both definitions are more or less same because no reflection will occur if
all the power is transmitted to the load. In some of the books 4, there is some
confusion. According to some authors, the first definition means that two
impedances Z1 R1 jX 1 and Z 2 R2 jX 2 are matched when
R1 R2
and X1 X 2
i.e., R1 jX 1 R2 jX 2
Z1 Z 2
(6.18)
The second condition that must be satisfied with regard to impedance
represented by the generator and the load is that their resistive parts must be equal
and their reactive parts must be equal and opposite, i.e., if the impedance of
generator is Z g R g jX g and that of load is Z L R L jX L , then
RL Rg
X L X g
R g jX g R L jX L
Z g Z L
(6.19)
where asterisk (*) denotes complex conjugate of the value. Same can be said for
admittance, i.e.,
YL Yg
As a matter of fact, the second relation is correct and in the remaining
discussions, the same is being used. Many a times, the characteristic impedance is
purely resistive and obviously the two definitions of match are the same.
One method commonly used for impedance matching is using Quarter
wave line. However, for the design of Quarter wave line, Smith chart is rarely used.
It can be used in the cases where the terminating impedance is complex. Let these
be discussed separately. The first one is useful if the terminating impedance is
purely resistive, and the second is useful for complex impedance.
For example, if the terminating impedance is purely resistive then here it
may be presumed that the load to be matched is 100 . It is to be matched with a
line of characteristic impedance 50 . Then the characteristic impedance of
matching quarter-wave line is
Z 0 50 100 5000 70.8
249
Microwave Circuits & Components
The resulting device would be narrow band since it matches strictly only
at one frequency. However, bandwidth can be increased for a given tolerance 5.
Before this device is discussed in detail, it may be mentioned that during 1950–65,
this principle was used in slug-tuning or slug-matching.
6.4.1
Quarter Wave Transformer
Quarter wave transformers are used in connection of two lines with
different characteristic impedances. Another example is that of connection of an
empty waveguide to a waveguide partially or completely filled with dielectric. Two
guides or microstrip of different width, height, or both can be used for this purpose.
As shown in Fig. 6.11, if a transmission line of characteristic impedance Z2 and a
length of quarter wave is connected between main line and the load, the effective
load presented to the main line is
Z Z2
Z L j Z 2 tan 4
Z 2 jZ L tan 4
Z 22
ZL
(6.20)
If is the electrical angular length at frequency f so that
l
2 l
2 f l
c
is the phase constant. At any other frequency, the input impedance
presented to the main line is
Z in Z 2
Z L jZ 2 t
Z 2 jZ L t
(6.21)
where, t tan tan l . Consequently the reflection coefficient is
Z in Z1
Z in Z1
Z jZ 2 t
Z1
Z 2 L
Z 2 jZ L t
Z1
Z L jZ 2 t
Z1
Z 2
Z 2 jZ L t
Z2
ZL
t/4
Figure 6.11. Quarter wave transformer
250
Smith chart & impedance matching
Z jt Z
Z Z Z
Z 2 Z L Z1 jt Z 22 Z1Z L
Z2 Z L
1
2
2
1 L
Z L Z1
L Z1 2 jt Z1Z L
assuming Z 22 Z1Z L
Z L Z1
Z L
If
2
1
2
Z1 4t 2 Z 22 2
1
1
2
2 Z Z
2
1 L
1
sec
Z L Z1
(6.22)
, the above equation can be approximated to
Z L Z1
cos
2 Z1 Z L
(6.23)
If m is the maximum value of reflection coefficient that can be tolerated
the useful bandwidth is obtained by evaluating . The value of at the edge of
useful passband can be found from Eqn. 6.22 as shown in Fig. 6.12. The value is
m cos 1
2m Z1 Z L
Z L Z 1
(6.24)
1 m 2
ZL ZI
ZL ZI
Q
Pm
/2
3 / 2
Bl=Q
Figure 6.12. Bandwidth characteristics of a single section of quarter wave
transformer.
251
Microwave Circuits & Components
Bandwidth is given by
2f
f 2 f 0 f m 2 f 0 0 m
f 0 is the frequency6 for which
(6.25)
2
Example 6.8. An empty rectangular waveguide is to be matched with dielectric
filled rectangular waveguide by means of intermediate quarter wave transformer.
Find the length and dielectric constant of matching section. Use
f 1 0 G H z, a 2 . 5 cm , K 0 2 . 5 6 .
Note that approximate impedance to be matched is the wave impedance
of TE10 mode, K0 is the dielectric constant at the output guide.
Solution: It is known that
n m K02 Kc2n m
2
Kc
c
2
K0
1
2
c 2a
a
2 f
c
c velocity of light 3 1010 cm s
leads to value for
10
10
4 2 f 2
c2
2
(6.26)
a 2
2 2
2
A 4 f
2
2
c
a
9 10
4 2 1010
10 2
2
2
2.52
2.81 1.68
10
2 2
2
C 4 f
2
2
c
a K0
3.77 1.94
252
4 2
2
9
2.52 2.56
Smith chart & impedance matching
A
AIR DETECTIVE CONSTANT =1
B
C
K1
K2
Figure 6.13. Figure depicting the Example 6.8
The wave impedance (Fig. 6.13) is given by
K0
Z10
Z0
10
K0
Z0
2
2 377
Z
469.9 470
A 0
A
c
3 1.68
10
10
Z 10 A
Z10 C
given by
Z0
2
2 377
407
B
c 10
3 1.94
The impedance of the intermediate quarter wave transformer section is
Z10 B Z10 A Z10 C 407 470 437.37
10 B
K0Z0
2
377
1.81
B
3
437
.37
Z10
It can be seen
4 2 f
1.81
c2
2
2
a 2 K1
K1 = Dielectric constant of intermediate section
This leads to
K1
2
2.52
1
1.42
4 2
1.812
9
253
Microwave Circuits & Components
0.87 CM
A
Z = 470
Ki=1.42
K0= 2.56
Z0 407.37
g/4
Z0 = 437.37
Figure 6.14. Figure depicting Example 6.8
K1= 1.42
is
Since the quarter wave transformer is being used, the height of the section
L
g
4
3.47
0.87 cm
4
To match the free-space waveguide with the dielectric-filled guide one
requires the quarter wave transformer of length 0.87 cm filled with dielectric whose
dielectric constant ( r ) is 1.42 (Fig. 6.14).
6.4.2
Quarter Wave Transformers with Extended Bandwidth
Often it is necessary to transform one impedance level to another. One
reason perhaps could be that it is necessary to interface a component with entirely
different impedance level that can either be very large or small. In such cases, it is
necessary to use transformer levels which are either very large or very small. A
simple transformer, discussed in Section 5 may not be sufficient. To obtain wide
bandwidth, it is necessary to employ multi-section transformer. Transformers with
two or three sections can give bandwidth up to 150 per cent. Collin 7 and Arnold8,
have investigated transformers displaying Chebyshev equal ripple responses and
Butterworth, i.e., maximally flat responses. Two such cases have been illustrated in
Fig. 6.15. Figure 6.15(a) gives a two section transformer whereas 6.15(b) gives a
three-section transformer. Larger than n = 3 is rarely used. In Fig. 6.15(a) Z0 and
Z3 represent the resistances that require matching. In Fig. 6.15(b) Z0 and Z4 require
matching. The nominal length of each section in the transformer is generally taken
as one-quarter line, i.e., 4 is selected at central frequency of the transformer..
For stepped impedance transformer constructed from the dispersive sections, the
nominal lengths of each section is
l
gH gL
2 gH gL
(6.27)
254
Smith chart & impedance matching
Z0
Z2
Z1
Z3
(a)
Z0
Z2
Z1
Z3
Z4
(b)
Figure 6.15. Quarter wave stepped impedance transformer (a) two section (b) three
sections.
where, the L and H indicate the lowest and the highest frequencies at which the
transformer operates. In either case, at the midband frequency, the electrical length
of the line should be 90°, the fractional bandwidth B defined for the purposes of
discussion is
gH gL
B 2
gH gL
(6.28)
For three section transformers the relations are complex and computer
calculation is recommended. These can be found by using iterative process.
Example 6.9a. Design a two section stepped impedance transformer that will
operate over a 40 per cent bandwidth between unequal resistances of magnitude
50 and 100 . The transformer should display Chebyshev response.
Solution: Impedance ratio, R
100
2
50
Fractional bandwidth, B 40 per cent 0.4 . Therefore,
k sin 0.4 0.31
4
radians
Thus,
D
(2 1)(0.31) 2
0.0252
2{2 (0.31) 2 }
Therefore,
V12
1
2
0.0252 2 0.0252 1.44
2
255
Microwave Circuits & Components
V1 1.2
Denormalised to 50
Z1 501.2 60.0
502.0
83.3
1 .2
This is the complete design.
Z2
Example 6.9b. Extend example 6.9(a) to Butterworth design
Solution: In this example, D is small and the design becomes approximately
equivalent to Butterworth design, in which case the characteristic impedance
needed to complete the design becomes
Z1 59.5
Z 2 84.1
For large impedance ratio, R 1, the correspondence between
Chebyshev and Butterworth design becomes less marked.
6.4.3
Stub Matching using Smith Chart
Stub matching has already been described in Chapter 2. However, the use
of Smith chart for stub matching is described here. First, as mentioned in
Chapter 2, to match a load to the line, at least two variables are to be adjusted.
Since impedance is a complex quantity, the real and the imaginary parts are to be
matched separately. The condition of impedance matching has already been
mentioned in the beginning of this section. Even if the characteristic impedance of
the line is a real quantity, the reactive part is to be cancelled or its value reduced to
zero. Single and double stub matching and also both the types of stub, i.e., shortcircuited and open-circuited are discussed.
6.4.3.1 Single stub matching
In this case either short-circuited or open-circuited stub is to be used. Two
adjustable parameters are at a distance d from the load to the stub position and the
length of either open-circuited or short-circuited stub. For the shunt-short-circuited
case, the basic idea is to select d so that the normalised admittance looking into the
line at a distance d is equal to 1 jb then the normalised shunt admittance is
selected as jb , which results in the matched condition. For the series opencircuited case the distance d is selected so that normalised admittance is 1 jx .
Then stub reactance is selected as jx so that the matched condition is obtained.
This is further clarified by the following examples .
Example 6.10a. If a load impedance Z L 15 j10 is connected to a line of
characteristic impedance Z 0 50 design shunt matching circuit using a single
short-circuited stub at a frequency 3 GHz.
256
Smith chart & impedance matching
Solution:
Z L 15 j10
Z
(a) Therefore normalised impedance Z L L 0.3 j 0.2
Z0
(b) Locate this point in the Smith chart.
(c) Extend the line joining this point to centre on the other side (YL) by
the same distance at the point
(d) Draw the circle which passes through these points.
(e) This circle intersects the unit circle (i.e., the circle for which g = 1)
(f) The load/admittance circle cuts unit circle at points y1 and y2. The distances of
these points from the load end (it can be read on the Smith chart) are
d1 0.328 0.284 0.044
d 2 0.5 0.284 0.171 0.387
Remember the notation is in the direction of generator. All these points in
the normalised admittances are
y1 1 j 1.33
y 2 1 j 1.33
This means that the first solution requires a stub of susceptance of
j 1.33. If the stub is short-circuited, the lengths of two stubs for the two cases is
found by starting at points y and then moving towards the generator to the
point for which j = 1.33 point.
l1 0.25 0.147 0.397
Similarly, for the second case
l 2 0.352 0.25 0.102
3 1010
3 10 9
10 cm
Therefore,
d 1 0.044 10 0.44 cm
d 2 0.387 10 3.87 cm
and
l1 0.40 10 4 cm
l 2 0.102 10 1.02 cm
Therefore, d is the distance at which the stub is to be connected and l is the
length of the stub in the two cases. This is shown in Fig. 6.16. The actual circuit is
shown in Fig. 6.17.
257
Microwave Circuits & Components
0.11
(+
jX
/Z
5
45
1.0
0.9
55
1.4
0.8
1.8
1.6
0.2
0.
18
30
0.3
2
50
25
0.4
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75
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20
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3.
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0.6
0.3
y2
0.8
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AC
0
1.
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10
0.27
0.25
0.26
0.24
0.27
0.25
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0.26
0.23
REFLE CTIO N COE FFICIE N T IN DEG
RE ES
N SM ISSION COE FFICIE N T IN D
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0.
8
0.6
0.2
0.1
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20
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.2
50
B
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
50
0.2
20
0.4
0.1
10
1.
0
E
OR
1.8
2.0
dB
]
0.7
-11
0.37
CT
A
(-
0
4
-1
T
-70
EN
0.
0
-1 7
30
0.
43
0.4
2
-12 0.0
8
0
0.0
9
0
N
0.6
1.6
5
-4
0.12
RE
A
0.4
1
0.1
0.11
-100
-90
0.13
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VE
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1.4
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0
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-25
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20
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3
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2
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4
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10
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4
0.5
0.4
5
2
8
6
0.3
7
8
0.2
1.8
6
5
9
10
0.1
1.6
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4
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12
14
0.05
1.2 1.1
2
15
TOWARD LOAD Ð>
10
7
5
1
20
30
0.1
0.01
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.99
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CENTER
1
1.1
4
1.2
1.3 1.4
0.4
0.6
1.2
1.3
0.95
1.2
1.4
0.8
1.5
0.9
1.3
<Ð TOWARD GENERATOR
2
1
3
1.1
0.2
1.6
1
1.4
1.8
1.5
2
3
2
0.7
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0.6
1.6
4
3
1.6 1.7 1.8 1.9 2
0.8
A
TT
EN
.[
40 30
0
5
5
4
10
5
2.5
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3
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20
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4
0.3
0.2
1.9
5
I
IN
0.6
),
Zo
3.0
0 .3
0.
3
0.1 1
9
l2
RADIALLY SCALED PARAMETERS
10
or
C
AN
PT
E
SC
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ES
IV
CT
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y1
-4
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4
0.
TR
A
N
SM
.C
O
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F,
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)
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( -jB
0.8
-75
-1 5
4.0
6
0.4
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0
-15 -80
1.
0
8
0.2
9
0.2
1
-30
0.3
0.
2
45
0.
5
0.0
5.0
7
-10
d1
0.2
d2
2
E
VE L
WA
0
<Ð
-16
-85
8
0.
0.2
0.4
y2
0.6
-20
0.2
10040 20
1
0.23
10
L E OF
TRA
L E OF
AN G
ANG
2L
20
IN D
U CT
I VE
8
5.0
2
0.2
1.0
0.2
TA
N
80
0.1
7
0.3
3
60
Yo)
jB/
2.0
6
0.5 5
70
0.
06
0.
44
0
14
0.0
5
0 .4
0.4
TA
0.3
4
35
1
0.2
9
0.2
30
0.0
4
0.4
6
15
0
EP
SC
E (+
NC
0.1
6
70
0.3
85
0 .15
0.35
40
0.2
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0.49
N GTH S
TOW
ARD
0.48
0.0
<Ð
0.49
GE N
RD L OAD
E RA
OW A
0.48
± 180
HS T
TO
170
N GT
RÐ
-170
0.4
>
7
160
-90
90
P
CA
R
l1
0.14
0.36
80
40
O
0.37
19
0.
31
0.
,
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U
ES
IV
IT
AC
0.13
0.38
0.7
0.6 60
0
12
07
0.
43
0.
0
13
110
1
0.4
0.12
50
0.4
9
0.0
8
0.0
2
0.4
0.39
100
1.2
0.1
10
0.1
0
2
Figure 6.16. Solution to the Examples 6.10(a) and 6.10(b)
Example 6.10b. Carry out same design with shunt open-circuited line.
Solution: In this case also the distance d from the load to stub is given by either of
the two intersections.
The values of d 's are
d1 0.044
d 2 0.387
One can say that there are infinite number of d's which can be used.
However, to improve the bandwidth of the match, the matching stub is connected
to the points nearest to the load. The points written above are nearest to the load.
The two intersection points with unit circle having normalised
admittances are,
258
Smith chart & impedance matching
4c
m
4.5cm
Figure 6.17. Solution to the Examples 6.10(a) and 6.10(b)
y1 1 j 1.33
y 2 1 j 1.33
The first matching requires a stub of susceptance as in the
example 6.10(a), with a susceptance of j1.33. This length can be found since the
stub is open-circuited on the Smith chart by starting at y 0 and moving along the
outer edge towards the generator. The length that gives susceptance of j1.33 is
l1 0.147
Similarly the stub length of second solution is
l 2 0.353
0.44
15
50
50
0.796 nH
0.387
50
15
0.147
50
50
0.796 nH
50
0.353
(a) Solution 1
(b) Solution 2
Figure 6.18. Two open circuit of shunt stub solutions
259
Microwave Circuits & Components
This completes the design. The resulting stub is shown in Figs. 18 (a and b).
Example 6.11. A load impedance of Z L 150 j 120 terminates at a line of
characteristic impedance of 75 using a single series open-circuit stub. The
frequency of operation is 2 GHz. Calculate position and length of the stub.
Solution: The load impedance is
Z L 150 j 120
The normalised impedance is
Z
150 j 120
ZL L
2 j 1.6
Z0
75
The
circle
which
passes
through
the
load
impedance
Z L 2.0 j 1.6 cuts the unit circle for which r = 1.0 at two points. The impedance
at these points are
Z1 1 j 1.33 and
Z 2 1 j 1.33
are
The distances of the points from the load-end found from the Smith chart
d1 0.328 0.208 0.120 and
d 2 0.5 0.208 0.463
So the open-circuited stubs are to be connected at these points
(Fig. 6.11). In the first case, the reactance of + j 1.33 is introduced by open
circuited stub, whose length is obtained on the following principle. The impedance
of open-circuited line is to start from the point for which Z L to the point
where reactance is j 0.133. Therefore, the length of the line which gives reactance
of 1.33 is found by moving on the edge from the point Z L to point where
reactance is + j 1.33. The length found from Smith chart is
l1 0.25 0.147 0.397
Similarly, to find the length which gives reactance of – j 1.33 starting
from point Z L and moving towards generator. Thus,
l2 0.352 0.25 0.102
Thus the design is complete. The two solutions are given in Fig. 6.19 and
6.20(a and b).
This type of matching is suitable in many cases like (a) two-wire lines
(b) waveguide matching/tuning to bring VSWR close to one (c) it may be useful for
microstrip matching where both open-circuited and short-circuited stubs can be used.
260
Smith chart & impedance matching
14
7
45
1.0
50
0.9
55
1.4
0.8
1.6
0.6 60
1.8
2 .0
6
0.5 5
N
AC
TA
N
Ð>
0.3
U CT
IVE
0
1.
RE
0.2
85
PO
NE
CE
CO
M
0.0
4
0.4
6
15
0
80
75
T
(+
jX
/Z
0.
4
10
IN D
0.
8
0.6
RA
TO
R
5.0
L E OF
170
10
0.1
0.49
0.4
20
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
± 180
50
Ð
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
AD <
A RD L O
S TOW
-170
0.2
20
0.4
0.1
1.
0
2.0
1.8
1.6
1.4
1.2
1.0
5
-4
dB
]
0.37
-70
0
4
-1
0.
0
-1 7
30
0.
43
0.4
2
-12 0.0
8
0
0.0
9
0
(-
0.4
1
0.4
0.39
0.38
0
1
1
0.0
1
0.9
0.8
0.9
0.1
5
2
0.7
4
3
15
0.6
0.8
0.2
10
3
4
0.5
0.4
0.7
0.3
5
6
0.3
0.6
0.4
2.5
2
1.8
1.6
8
6
5
4
3
9
10
12
14
7
8
0.2
0.5
0.5
0.1
0.4
0.6
0.3
0.7
0.05
0.2
0.8
1.4
1.2 1.1
2
15
30
0.01
0.1
1.1
0.1
0.9
TOWARD LOAD Ð>
10
7
5
1
20
0.99
CENTER
1
1.1
4
1.2
1.3 1.4
0.4
0.6
1.2
1.3
0.95
1.2
1.4
0.8
1.5
0.9
1.3
<Ð TOWARD GENERATOR
2
1
3
1.1
0.2
1.6
1
1.4
1.8
1.5
2
3
2
0.7
1.5
0.6
1.6
4
3
1.6 1.7 1.8 1.9 2
0.8
A
TT
EN
.[
20
5
4
0.5
1.7
10
5
2.5
3
0.4
1.8
6
0.3
20
10 15
4
0.2
1.9
5
I
0.9
0
-5
0.12
0.6
-60
0.7
0.13
0.36
-11
0.1
0.11
-100
-90
CT
A
T
0
-65 .5
RE
A
EN
10
40 30
or
OR
0.14
-80
-4
0
0.15
0.35
0.8
6
0.1
4
0.3
5
-5
-70
-3
5
ITI
VE
N
RADIALLY SCALED PARAMETERS
10040 20
TR
A
N
SM
.C
O
EF
F,
E
D
IN
CA P
AC
PO
G TH
o)
jB/ Y
E (NC
TA
EP
),
Zo
3
0.3
7
0.1
0.2
-60
-30
OM
5
0.0
/
jX
2
WA
0
<Ð
-16
-85
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US
ES
IV
CT
U
0.3
NC
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44
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06
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0.4
0.
3
0.1 1
9
18
0.
0
-5
-25
EN
VE L
-75
0.6
45
0.
0.8
-20
3.0
4
0.0
0
-15 -80
-1 5
4.0
6
0.4
1.0
8
0 .3
0.
2
-4
0
4
0.
0.2
2
Z1
0.2
9
0.2
1
-30
0.3
0.2
7
8
0.
5.0
0.2
-10
-90
10
0.6
Y2
-20
0.4
50
0.0
0.2
0.48
2
0.4
15
0.2
EN E
4.0
1.0
8
90
Z2
0.8
0.2
160
20
3.0
0.6
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLE CTIO N COE FFICIE N T IN DEG
L E OF
RE ES
AN G
SSIO N COE FFICIE N T I
TRAN SM I
N DE G
RE E S
ARD
G
25
0.4
ANG
0.48
0.
18
0.
32
50
20
S TOW
AC
0.2
0.1
7
0.3
3
30
1
0.2
9
0.2
30
N GTH
P
CA
R
TA
60
0.3
Ð > W A V EL E
O
P
CE
0.3
4
35
0 .2
,
o)
US
ES
IV
IT
Yo)
jB/
E (+
NC
0.1
6
70
40
0 .0
5
0.4
5
0
70
0.0
6
0.
44
43
0.
0
13
0.35
40
19
0.
31
0.
0.49
0
12
7
0.15
0.36
80
0.7
8
0.0
2
0.4
0.0
110
1
0.4
0.37
0.38
0.39
100
0.4
1.2
0.1
9
0.0
10
0.1
0
2
Figure 6.19. Solution to Example 6.11
However, in many case like coaxial system, single-stub tuning is very inconvenient
and double-stub tuning has to be used as described in the following paragraph.
6.4.3.2 Double stub matching
On coaxial lines, the use of matching system just described requires a
variable point where the stub is to be located to provide adjustment of distance d.
Because of impedance discontinuity at the sliding junction there is a possibility of
poor contact, hence variable length is not desirable. Even in waveguide system, it
is difficult to fabricate a single-stub matching circuit. Although somewhat difficult
to understand double-stub matching it is very convenient to use. Sometimes, a
waveguide system consists of three properly spaced tuning stubs. A single-stub
and double-stub matching coaxial system is shown in Figs. 6.21(a) and 6.21(b).
261
Microwave Circuits & Components
50
30
0.397
50
ZL =(150+j120)
30
50
0.102
50
0.120
0.463
(a)
(b)
Figure 6.20. Two series stub tuning solution
The operation of a double-stub tuner is illustrated in Figs. 6.21(b) and
6.22. Any spacing between stub 1 and load may be used. In the present example
the first stub is located at the load itself. The quarter wavelength spacing of the two
stubs has been taken arbitrarily. It can have any value less than 2 like
3 8 , 5 8 , etc. In the Fig. 6.22, point A represents the normalised load
admittance. To locate point B is the objective of the transformation process, i.e.,
one wish is to move point A to point B by double stub matching. Stub 1 adds
susceptance to admittance A, moving the point along the circle of constant g = 0.2.
L
Y0
Yr
d
Figure 6.21(a). Single stub impedance matching coaxial system
I2
I1
Figure 6.21(b). Double stub impedance matching
262
Smith chart & impedance matching
The proper amount of susceptance must be added by stub 1 so that the resulting
admittance may be transformed by the quarter wavelength line to a point on g = 1
circle. Since the quarter wavelength line transforms admittances to the opposite
end of the diameter of a circle, the locus of admittances can be transformed to g = 1
circle. Circle by such a line is easily seen to be a similar circle on left-hand portion
of the Chart. As a matter of fact the circle for which g = 1 is rotated by an angle of
2 corresponding to 4 in the direction of load. Call it a circle for which g' = 1
(rotated by 90°). In this case, since stub 1 is at the load-end itself, adding any
susceptance would move A on the circle of constant g, it should be moved as shown
in the Fig. 6.22 from A to E (or E') by adding susceptance or moving short circuit in
the stub 1. Then the circle is rotated by 90° towards generator so that point E (or
E') moves to F (or F'). The susceptance at these points are cancelled by adding
negative susceptance by moving the short circuit of stub 2. The resulting
impedance would be 0.1 or the A has moved to B and the load is matched to line of
any characteristic impedance.
Generally, the load is not connected at the point where the first stub is
situated. In that case the point A is first moved to some point on the circle which
0.1
5
0.10
1.0
2
0.5
20
0.
0.
05
A
E
F'
C
0.2
0.25
5
2
1.0
0.5
B
-5
d
To loa
Y
-jB/ 0
2
-0.
F
E'
5
0.4
0.40
-1.0
-0
.5
-2
D
0.3
0
G/Y0
0.2
0 To
gen
0 +jB
/Y
0
S=10
5
0.35
Figure 6.22. Single and double stub impedance matching
263
Microwave Circuits & Components
has the radius equal to the distance between the centre and the point A. Then to
obtain point E (or E') one should move on circle of constant g. This would become
clear when one looks into the following solved examples.
Example 6.12. A line of characteristic impedance 50 is terminated in impedance
Z l 100 j 100 . A double stub is to be used to match the load to line. The
3
separation between two stubs is . Determine the lengths of two stubs when the
8
match is achieved. The first stub is situated at 0.40 .
Solution: As given in the example, the value of load impedance (Fig 6.23) is
Z l 100 j 100
Characteristic impedance =50
Normalised impedance
100 j 100
2 j2
50
This impedance is plotted in the Fig. 6.21 (Smith chart) as point Zl. By
rotation of 180°, the normalised admittance is found to be
y l 0.25 j 0.25
The g' = 1 constant conductance unity circle is rotated by
2 d 2
2
3 3
8
2
It can also be considered as rotation of 0.375 towards the load (Fig 6.24
and Fig 6.25). Call this circle g' = 1 circle. The point yl is moved by 0.4 by
moving this point from 0.458 to 0.358 . Let this point be yd1. The normalised
admittance at yd1 is
STUB 1
STUB 2
I1
I2
a
l
Figure 6.23. Solution to the Example 6.12
264
Smith chart & impedance matching
0.4
0.375
50
50
100 + j 100
I
STUB 2
I2
STUB 1
Figure 6.24. Double stub matching for Example 6.12
yd1 0.55 j 1.08
This point is moved on constant conductance circle (0.55) and this
movement cuts new circle at points y11 and y'11. The values of this admittance are
y11 0.55 j 0.11
0.55 j 1.88
y11
This is possible only if the difference susceptance is added by the stub 1.
Therefore, the susceptance to be introduced by stub 1 is
y S 1 0.55 j 0.11 0.55 j 1.08 j 0.97
The other point gives the other susceptance
y S 1 0.55 j 1.88 0.55 j 1.08 j 0.80
The length of the stubs which give these susceptances are found by
starting from the point y l (short-circuited stub) to points of these
susceptances. The lengths obtained are
l1 0.25 0.123 0.373
l1 0.25 0.107 0.143
The unit circle g' = 1 is rotated back to original circle g = 1. This rotation
transforms y11 to yd2 and y'11 to yd'2 along their constant VWSR circles. The
admittances at points yd2 and yd'2 are
yd 2 1 j 0.61
yd 2 1 j 2.60
265
Microwave Circuits & Components
Figure 6.25. Solution to the Example 6.12
The stub 2 therefore contributes
y S 2 j 0.61
y S 2 j 2.60
The lengths of stub 2 are found as
l 2 0.25 0.087 0.337
l 2 0.308 0.25 0.058
It can be seen from the Fig. 6.21 that if normalised admittance yl is located
inside the hatched area, it cannot be matched by double-stub technique. It cannot
be brought to lie on the locus of y11 or y'11 for a possible match by the parallel
connection of any short-circuited stub because the space-unit circle
(g' = 1) and g = 2 circle are mutually tangent. Thus, g = 2 circle is known as one
266
Smith chart & impedance matching
which encloses forbidden region. This point will be discussed in detail in the next
paragraph.
In the present example, the characteristic impedance of two stubs have
been taken to be 50 , which may not be true for all cases. Some stubs have
different Z 0 . The example becomes more complicated.
6.4.3.3 Forbidden regions in double stub matching
Further consideration of Fig. 6.22 and the process involved in doublestub matching will show that with quarter wavelength spacing of stubs with stub 1
located at the load for which 0 g 1 can be matched to the line. Any normalised
load admittance that falls within the g = 1 circle cannot be matched by this system
because stub 1 cannot transform points within the g = 1 circle on to the left-hand
circles of the Fig. 6.26. In the figure, these are labelled I and II. If the distance
from the load to stub 1 is increased with the stub spacing held at quarter
wavelength, the area on the chart cannot be matched. If the distance from the load
1 1 3
, , wavelength, then areas that cannot be
to stub is successively made
8 4 8
matched are shown by circles III, II, IV respectively, in Fig. 6.26.
1
Another possibility is to change the spacing of stubs to wavelength and
8
place stub 1 at the load. Now stub 1 must transform the load admittance onto circle
III, the eight wavelength line between the stub transforms points on circle III to
circle I and stub 2 transforms points on circle I to the centre of the chart. With this
system, the load-admittance points cannot be matched as these lie within circle V,
a smaller area. A stub spacing of 3/8 wavelength with stub 1 at the load would have
circle V a smaller area. A stub spacing of 3/8 wavelength with stub 1 at the load
would have circle V as the unmatched area and would be more practical because
the greater spacing between stubs would provide more room for manipulation.
6.4.4
Matching with Three Stubs
In the waveguides, it is very common to use three stubs for tuning or
matching. In principle, three stubs at quarter wavelength intervals can match any
load to line. The stub at the load then needs to provide only two susceptance values
0 or –1, i.e., this stub is set to add a susceptance of –1, if the load admittance falls
within circle 2 in Fig. 6.22 and susceptance is 0 for any other load. The other two
stubs are then tuned until an impedance match is obtained.
It can also be understood that the pair of any stubs like stub1 and stub 2 or
stub 2 and stub 3 can be used for impedance matching like two-stub matching, the
third one is unused or is adding 0 impedance in terms stated above. Alternatively,
those impedances which cannot be matched by one and two pair, or two and three
pair can be matched by the adjustment of the second pair.
Another method is to gang the outer stubs so that only two tuning or
matching adjustments are used. The difficulty encountered in three-stub matching
267
Microwave Circuits & Components
0.10
0.1
5
1.0
2
20
0.
0.
05
0.5
0.2
III
I
II
0.25
5
2
1.0
0.5
0.2
+jB/Y0
0
V
-jB/Y
0
G/Y0
-5
d
To loa
0
To ge
n
5
0.3
0
IV
0.4
0
-1.0
-0
.5
45
0.
-2
2
-0.
5
0.3
Figure 6.26. Areas that cannot be matched to the centre of the Smith chart by
various double-stub tuners.
is so much that a double-stub tuner with 3/8 wavelength spacing is preferred,
though in principle the three-stub matching can be used for matching any two
impedances.
It may be further stated that these matching systems are narrow-band
matching system. The bandwidth sometimes is required to be studied for practical
applications. Though some studies have been carried as for open-circuited stubs
which are useful for many systems like microstrip lines, most of the ideas
developed here can be extended to open-circuited stub for matching.
6.5
COMPRESSED SMITH CHART
The normal Smith chart is a plot of reflection coefficient for values either
equal to or less than one. In the cases of active devices for which the concept of
negative-dynamic impedance has been developed, the reflection coefficient may
be greater than one. Such a plot which can be used to represent reflection
coefficient greater than one, is known as compressed Smith chart as shown in the
Fig. 6.27.
It can be seen that the compressed Smith chart includes the normal Smith
chart plus additional negative-impedance portions. Very often, it is plotted for
values of reflection coefficient less than –3.16 (or 10 dB of return gain). The chart
268
Smith chart & impedance matching
is useful for plotting the variations of in and out for the design of oscillators.
The impedance and admittance characteristic of normal Smith chart is retained for
the compressed Smith chart. One or two examples may clarify many concepts.
Example 6.13. An amplifier has load reflection coefficient greater than unity. Its
value is
l 1.50 155 . Determine the normalised load impedance and
admittance by using compressed Smith chart.
Solution: Though the calibration is not clear in the compressed Smith chart shown
in the Fig. 6.27, the load reflection coefficient l is plotted in the Fig.6.27 as point
-0.8
j0.5
-1.0
-1.2
j1
-1.5
j0.2
j2
-j0.5
A
j3
- 0.5
316
.
- 0.4
- 0.2
0 0.2 0.5 1
C
2
-4 -3 - 2.5
-2
316
.
1
j3
j2
D
B
-j0.5
-j0.2
j1
-1.5
-1.2
-j0.5
-1.0
-0.8
Figure 6.27. Compressed Smith chart
A in the figure. The value of Zl can be read from the Fig. 6.27. Its value comes out
to be
Z l 0.2 j 0.2
The conjugate normalised impedance is
Z l 0.2 j 0.2
and is shown as point B in the Fig. 6.27. The normalised admittance is
y l 2 .5 j 2 .5
269
Microwave Circuits & Components
It is shown as point D in the Fig. 6.27. The conjugate normalised
admittance is
y l 2.5 j 2.5
and is shown as point C in the Fig. 6.27. An alternative way of handling negative
1
resistances ( i.e., 1 ) is to plot in Smith chart and take the values of
resistance circles as being negative and the reactance circles as labelled.
Example 6.14. Find an impedance whose reflection coefficient is 2.236 e j 26.56 .
Solution: If one plots in Smith chart as shown in Fig. 6.27, the quantity
1
0.447 e j 26.56
The resulting Z is –2 + j 1
Using the relation for the reflection coefficient
Z 1
Z 1
where, Z is the normalised impedance. Therefore
2 j 1 1
2.236 e j 26.56
2 j11
The use of the Smith chart in a transmission line calculation follows from
the equations already derived in the Chapters 2 and 6. The reflection coefficient
for d = 0 is
Z 1
0
Z 1 d 0
in d 0 e j 2 d
Z in d
1 in d
1 in d
Thus, the transmission line input impedance calculation involves the
following steps:
(b)
ZL
Z0
Rotate 0 by 2 d to obtain in d . Observe that the rotation is along
(c)
Read the value of the normalised Z in d associated with in d .
(a)
Locate 0 in the Z Smith chart for a given Z
a vector of constant magnitude, namely 0 in d
270
Smith chart & impedance matching
6.6
THE NORMALISED IMPEDANCE & ADMITTANCE SMITH
CHART
Some of the concepts developed earlier are reproduced here for
introducing some new concepts. These have been introduced in such a way that the
new concept becomes easy to comprehend and understand. As mentioned earlier,
the conversion of a normalised impedance to normalised admittance can be done
easily in the Smith chart. Since
Z
1
1
where Z is the normalised impedance and is the reflection coefficient which is
a complex quantity. The expression for normalised admittance is given by
y
1 1
Z 1
Remember that in these expressions is voltage reflection coefficient. It
can be seen that rotating by results in
Z
1 e j
1 e
j
1
1
This results in a value which is identical with the value of admittance
y
1
1
Therefore, Z-Smith chart can be used to find the normalised value of y
by rotating the value of Z by .
The impedance to admittance conversion thus can be obtained by rotating
the entire Smith chart by 180° and the rotated Smith chart can be called admittance
chart. The superposition of the original Smith chart which is the
Z-Smith chart and the rotated Smith chart which is the Y-Smith chart is known as
the normalised impedance and admittance coordinates Smith chart. This is
normally referred to as Z-Y Smith chart.
6.7
THE NORMALISED Z-Y SMITH CHART
The Z-Y Smith chart is shown in Fig. 6.28. Observe that the upper-half of
the chart for admittance coordinate represents normalised admittances having
negative susceptibilities (i.e., – jb) and lower -half represents positive susceptances
(i.e., + jb). The impedance coordinates are the same as in the Z-Smith chart. This
chart has a special feature. For any impedance say Z 1 j 1 marked in Smith
chart can give values of y if read on the Y-Smith chart. It can be seen that it is
y 0.5 j 0.5 .
271
Microwave Circuits & Components
Figure 6.28. Z-Y Smith chart
Some equivalent circuits shown can be obtained from the Smith chart
impedance plot of the S-parameters.
6.8
IMPEDANCE MATCHING USING LUMPED ELEMENTS
The lumped elements have been fabricated to operate in the microwave
region up to a frequency of about 10 GHz. It is a very useful microwave region.
Impedance matching is very much desirable. Standing waves lead to increased
losses. Shift in frequency sometimes leads to malfunctioning of the source. A line
terminated in its characteristic impedance and matched with VSWR equal to one,
transmits the power without reflections. In transmission line example matching
means simply terminating the line in its characteristic impedance. A matched
transmission line system is shown in Fig. 6.29.
Since the matching, in general, involves parallel connections on
transmission line, it is desirable to work out examples with admittances rather than
272
Smith chart & impedance matching
TRANSMISSION
DEVICE
MATCHING
DEVICE
Zg
Z0
Zg
Z0
Z0
MATCHING
DEVICE
ZL
Z0
ZL
Figure 6.29. Matched transmission line system
impedances. The Smith chart as stated earlier can convert the normalised
impedance to admittance by rotation of 180°.
Example 6.15. A section of 50W line is terminated in a normalised load of
1 j 1 and its input normalised impedance is 1 j 1 . Determine the element
to match the two ports by using Smith chart.
C
B
A
zin= 1 + j1
-j2.0
zin= 1 - j1
(b)
(a)
Figure 6.30. (a) Graphical solution to the example 6.15 (b) Equivalent circuit
Solution: Plot Z L 1 j 1 and Z in 1 j 1 at points A and B respectively on
Smith chart as shown in Fig. 6.30(a).
Read the series element C as a capacitive reactance
j X C j 2 .0
The equivalent network is shown in Fig. 6.30(b).
273
Microwave Circuits & Components
Example 6.16. A section of 50W line has the following load and input admittance
Y L 0 .5 j 2 .0
Yin 0.5 j 2.0
Determine the shunt element L to match the two ports using Smith chart.
Solution: Plot YL and Yin as points A and B respectively on Smith chart as shown
in Fig. 6.31(a). Read the shunt element L as inductive susceptance
YL j 0.25
The equivalent network is shown in Fig. 6.31(b).
L
yL= -j 4
yR= 0.5+ j 2.0
A
B
-10
Yin
yin= 0.5 - j 2
-j5
-j0.20
(b)
(a)
Figure 6.31. (a) Graphical solution to example 6.16 (b) Equivalent circuit
Example 6.17. A microwave device has output impedance Z out 15 j 15 .It
is necessary to design a matching network to transform the device output
impedance to a 50 line. Determine the matching network by using Smith chart.
Solution: The normalised device output impedance is shown as point A in the Smith
chart as shown in Fig. 6.32(a)
Z out 0.3 j 0.3 at point A
The circle intersects the 0.3 constant-resistance circle at point B. The
impedance at B is
Z B 0.3 j 0.45
274
Smith chart & impedance matching
B
L
+j 0.15
C
C
-j5
-j0.20
A
Zout = 0.3 + j 0.31
-10
Z0 = 1
(a)
(b)
Figure 6.32. (a) Graphical solution to Example 6.17 (b) Equivalent circuit
The series element L is
Z series j 0.15
The point B corresponds to the point at admittance chart at
YB 1 j 1.60
The value of shunt element C from B to centre C is
YC j 1.60
This means that shunt element is a capacitor as shown in Fig. 6.32(b).
jX C j 0.63
The design is complete.
Example
6.18.
A
certain
microwave
device
has
output
impedance
Z out 100 j 100 . Design a network to match the device admittance to a 50
load by using Smith chart.
Solution: The normalised output impedance is
Z out
100 j 100
2 j2
50
Note that normalisation is done wrt 50 load. It is plotted as a point D
on the Smith chart as shown in Fig. 6.33(a).
Read the device admittance at the point C
Yout 0.25 j 0.25 ( point C )
275
Microwave Circuits & Components
L
B
A
+ j 1.90
– j 1.49
C
Z0 = 1
(50)
C
y = 0.25 + j 0.25
(b)
(a)
Figure 6.33. (a) Graphical solution to the Example 6.18 (b) Equivalent circuit
Move towards the load on a circle A as centre and AC as radius. The circle
cuts the unit conductance circle at point B. It can be seen that
Z series j 1.90 from impedance chart
YB 0.25 j 0.42 from admittance chart
The value of point B to point C is
Yshunt j 0.67
Z shunt j 1.49
which is naturally a capacitance. The matching network is shown is Fig.6.33(b).
As can be seen from the solved examples to design matching circuits the
same Smith chart can be used both as Z- and Y-Smith chart. Obviously, solving
matching example Z-Y Smith chart is useful. It has been demonstrated that the
impedance to admittance conversion can be obtained by rotating the Smith chart
by 180° and superimposing the original on the rotated chart Z-Y Smith chart.
Before describing the utility of Z-Y Smith chart to design matching
network using passive elements is described, it is important to discuss various
possible networks that can be used for this purpose.
6.8.1
Impedance Matching Networks
The need for matching networks arises to enable amplifiers to deliver
maximum power to load or to perform in certain specific manner. This is already
illustrated to some extent in Fig. 6.29 as a typical situation in which a transistor
276
Smith chart & impedance matching
must have input impedance equal to source impedance and the output impedance
equal to ZL. This can be done using matching networks on both sides, i.e., at the
input and the output ends. Theoretically it is possible to design this by the infinite
ways. However, it has been found that the eight sections shown in Fig. 6.34 are not
only simple to design but are practical also.
The Z-Y Smith chart can be used conveniently in design of matching
networks. The eight circuits which are practical are shown in Fig. 6.34. In the first
one, element L is connected in parallel and C is connected in series. The other
circuit is obtained using both elements as capacitance. In the next two networks, C
is connected in parallel and L is connected in series. The same situation is also
created using both elements as inductance.
In the next two figures, the series reactance element, which is inductance
connected in series and the capacitance C connected parallel to the load. The same
condition is created by using both elements as capacitance. In the last combination,
the series element capacitance C and inductance L is connected in parallel to the
load impedance ZL. The same situation is obtained using both elements as
inductance. The utility of Z-Y Smith chart is shown by the examples discussed in
the solutions given below. Designing a matching network in the Z-Y Smith chart
consists of moving on a constant resistance or constant conductance circles from
one value of impedance or admittance to another.
Example 6.19. Design two matching networks to match a Z Load 10 j 10 to
a 50 line. Specify the values of L and C at 1 GHz.
Solution: Let L shunt C network be designed. The Z Load 10 j 10 .
Therefore, the normalised impedance is
C
L
C
L
ZL
L
C
L
L
ZL
C
C
C
ZL
C
ZL
L
ZL
C
ZL
Figure 6.34. Matching of networks
277
L
ZL
Microwave Circuits & Components
Z Load
Z Load
0.2 j 0.2
Z0
The design circuit is shown in Fig. 6.32(a).
The normalised impedance is shown as point A in the Fig. 6.32(a). To this
a series impedance L is to be added by moving to the point B. The motion is on the
circle of resistance 0.2 so that point B lies on unit conductance circle. The reading
of inductance is 0.4. Therefore, the magnitude of normalised inductance to be
added in series is
L B j 0.4 j 0.2 j 0.2
The exact value LB 0.2 50 j 10 . The conductance at point B is
YB 1 j 0.5
This value is obtained by reading the value of susceptance at the point B.
To obtain the matched condition, this susceptance is reduced to zero by adding
susceptance (capacitance) of magnitude j 0.5 in shunt. The point B moves to
point C.
Therefore, the value of capacitive reactance is
0.5 50 25
The series inductance added at 1 GHz has magnitude of
L
10
2 1 109
10
10 9 1.58 nH
6.28
Similarly, the value of capacitance is
C
1
9
2 10 25
20
10 12 F 6.37 pF
3.14
The other matching circuit that can be used for matching is shown in
Fig. 6.33(a). The normalised impedance is again shown as point A in the Z-Y Smith
chart. In this circuit since capacitance C is added in series, the point A is to be
moved in opposite direction on a constant-resistance circle to cut the unit
conductance circle at the point B. The admittance point B is
Y B 1 j 0 .5
Therefore, inductance of magnitude – j 0.5 is to be added whose
magnitude in Z chart is 0.5. The magnitude of series capacitive reactance is
j 0.6 30
278
Smith chart & impedance matching
The magnitude of shunt inductive reactance
j 0.5 25
Therefore, capacitance to be added is
1
2 109 30
1010
6.28
100
1012 F
18.84
5.3 pF
C
The inductance added in shunt is
25
2 109
25
109 H
6.28
3.975 nH
L
Example 6.20. Design a matching network to match 50 load to admittance
Y 8 j 12 mho .
Solution: Figure 6.34 illustrates one of the possible network. The normalised
admittance can be found by finding out the reference admittance. The load
1
2 102 mho . This is the
impedance is 50 . Therefore load admittance is
50
reference admittance. Therefore the admittance to be matched can be normalised
wrt this admittance. Therefore, it is
Y
Y
2 10
2
8 j 12 10 3 0.4 j 0.6
2 10 2
This is shown as point C in the Fig. 6.35. The 50 point is shown as point
A in the figure. With centre denoting the centre of Smith chart and radius AC, draw
a circle which cuts the unit resistance circle at the point B. Move from point A to B
on unit resistance circle. For this the capacitance to be added has the magnitude
j 1.21 j 60.5
Read the value of conductance at this point. Since the inductance is
connected parallel, the motion from B to C produces a shunt inductor having an
admittance
279
Microwave Circuits & Components
NAME
DWG. NO.
TITLE
DATE
SMITH CHART FORM ZY-01-N
Microwave Circuit Design - EE523 - Fall 2000
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
1.6
0.6 60
45
1.2
1.4
1.0
50
0.9
55
0.8
2.0
0.3
CT
AN
C
1.0
RE A
0.2
8
0.
A NG
0.6
0.6
10
10
0.1
0.1
0.4
0.4
20
20
0.2 0.2
50
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1.0
50
50
20
20
0.4
0.1
0.4
10
CE
SU
IV
CT
DU
IN
1.0
0.8
E
0.8
OR
,
o)
/Z
2.0
1.8
0.13
1.4
.[
dB
]
0.36
N
T
0.0
0.0
-1 7
30
0.4
3
0.4
2
-12 0.08
0
9
-70
NE
6
0.0
PO
3.0
0.4
1
0.4
0.39
0.38
1
1
0.9 0.8
1
0.9
0
0.1
2
0.7
3
0.6
0.8
0.2
10
3
4
0.5
0.4
0.7
0.3
5
6
0.3
0.6
0.4
2.5
2
1.8
1.6
1.4
8
6
5
4
3
9 10
12
14
0.1
0.05
7
8
0.2
0.5
0.5
0.4
0.6
0.3
0.7
0.2
0.8
1.2 1.1
2
15
1.1
20
30
0.01
0.1
1.1
0.1
0.9
TOWARD LOAD —>
10
7
5
1
0.99
CENTER
1
1.1
1.2
0.2
1.2
3
1.6
0.4
1.3
0.95
1.2
4
1.3 1.4
0.6
1.4
0.8
1.5
0.9
1.3
1
1.4
<— TOWARD GENERATOR
2
1
1.8
1.5
2
3
2
0.7
1.5
0.6
1.6
4
3
1.6 1.7 1.8 1.9 2
0.8
AT
TE
N
4
5
4
0.5
1.7
10
5
2.5
3
0.4
1.8
6
0.3
20
10 15
4
0.2
1.9
5
I
B]
[d
0
5
15
or
SS
LO
20
10
0.1
0
TR
A
NS
M
.C
N.
RT
10
40 30
2
ORIGIN
Figure 6.35(a).Forbidden region in Z-Y Smith chart to match 50 using a series L
shunt C matching network.
YL j 0.6 j 0.49 j 1.09
Therefore,
ZL
1
j 0.917
j 1.09
280
F,
E
-11
0
CO
M
X
RADIALLY SCALED PARAMETERS
10040 20
O
EF
0.12
0.37
EA
0.1
0.11
-100
-90
0.14
ER
E
0.6
1.6
1.2
1.0
5
-4
0.15
0.35
0.9
-4
0
6
4
0.3
IV
0.7
CAP
AC
IT
0
-5
-3
5
0.8
0.1
0.2
0.2
-55
3
0.3
7
0.1
-60
-30
0.9
1.0
1.2
0.8
1.4
0.7
-60
2
0.3
CT
AN
C
(-j
40
-1
0.4
0.4
8
0.1
0
-5
-25
1
1.6
0.6
1.8
-65 0.5
0.
3
9
0
4
0.1
2.0
0.5
4
.4
0.
0.3
4
0.
-20
5.0
-75
0.6
0.6
3.0
4.0
5
9
0.0
-15
4.0
N
TA
EP
SC
1.0
1.0
1
0 .2
-30
0.3
0 .2
0.3
0.28
0 .2
2
6
0.4
4
0.0
0
-15 -80
0.2
5.0
-85
8
0.2
5
0.
-4
0
0.4
0.6
0.
8
1.0
0.2
-10
Yo)
(- jB
0.6
-20
0.49
0.48
AD <—
ARD L O
7
S TOW
0.4
GTH
0
N
-17
E
VEL
WA
<—
-90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2 0.2
0.8
0.9
1.0
1.2
1.4
1.6
1.8
2.0
3.0
4.0
5.0
10
20
50
180
0.1
50
0
0.25
0.26
0.24
0.27
0.23
0.25
0.2
0.26
4
0.23
0.27
FL E CTION COEFF CIE N T IN DE
E OF RE
GRE E
S
AN G L
ISS ON COE FFICIE N T IN
TRAN SM
DE GR
LE OF
EE S
0.
8
80
1.8
65
0.5
NT
PO
NE
75
EC
OM
0.0
4
0.4
6
15
0
o)
,
70
0.0
6
0.
44
5
14
0
0.4
5
(+
jX
/Z
10
20
IND
U CT
IVE
5.0
0.2
0.0
0.
4
RA
TOR
—>
15
2
0.2
0.47
4.0
1.0
8
0.2
E NE
0.8
9
0.8
1.0
1
90
20
3.0
0.6
0.2
160
85
4
0.6
0.2
ARD
G
25
0.4
30
0.48
0.4
0.2
S TOW
P
CA
0.3
170
R
40
O
0.
18
0.3
2
50
9
0.1 1
3
0.
N G TH
0.2
30
0.5
0.49
0.1
7
0.3
3
60
0.6
—> W AVE LE
6
0.3
4
35
)
/ Yo
(+jB
0.2
0.1
70
0.7
0
0.15
0.35
40
1.0
5.0
0.14
0.36
80
0.8
4.0
0.37
0.9
TA
0.13
0.38
1.0
1.4
P
CE
US
ES
IV
IT
AC
E
NC
0.12
2.0
3.0
1 .8
3
0.4
0
13
1.6
7
0.0
110
1
1.2
0.4
8
0 .0
2
0 .4 120
0.7
0
0.39
100
0.4
0.
0.11
0.1
.09
Smith chart & impedance matching
The magnitude of impedance due to inductance is thus equal to
j 0.917 50 j 45.8
The complete design is shown in Fig. 6.34(b). Another matching circuit
can also be designed similarly.
Sometimes, matching network cannot be used to accomplish a given
match. In the Fig. 6.35(a), impedance falling in the shaded region cannot be
matched to 50 . In the network shown in the Fig. 6.35(b) adding an inductance in
series produces a motion in the clockwise direction away from constantconductance circle that passes through origin.
6.8.2
Microstrip Matching Network
Microstrip lines are now used extensively in designing microwave planar
circuits. Interconnections and placement of lumped elements can easily be made.
Microstrip circuits are easy to fabricate. The various parameters of microstrip
circuits are discussed in Chapter 2. Many useful design formulae have been
mentioned. Though the number of useful design relations are many, some of the
commonly used relations are again mentioned. A microstrip line is shown in
Fig. 6.35(b).
The electromagnetic field lines in microstrip are not contained entirely in
the substrate. Therefore, propagating mode in microstrip is not a pure transverse
electromagnetic mode (TEM) as mentioned earlier but in quasi-TEM mode. The
phase velocity is given by
vp
c
(6.28)
eff
where, c is the velocity of light and eff is the effective relative dielectric constant.
S TR A P
CONDUCTOR
t
DIELECTRIC
SUBTRATE
W
h
GROUND
PLATE
Figure 6.35(b). Microstrip geometry and field configuration
281
Microwave Circuits & Components
As already mentioned, the effective relative dielectric constant is related to the
relative dielectric constant of the substrate. The characteristic impedance of
microstrip line is given by
Z0
1
(6.29)
vp C
and the wavelength in the microstrip line is
vp
0
c
f
eff
f eff
(6.30)
Useful relations for characteristic impedance assuming zero or negligible
t
thickness of strip conductor 0.005 are9
h
For
W
1
h
W
h
ln 8 0.25
h
W
eff
60
Z0
(6.31)
where,
1
2
h 2
W
1 12 W 0.041 h
eff
For
r 1 r 1
2
2
(6.32)
W
1
h
Z0
120
eff
W
W
1.393 0.667 ln 1.444
h
h
where,
eff
r 1 r 1
2
2
h
1 12
W
1
2
(6.33)
(6.34)
Based on these and the experimental results, following are some other
relations suggested 11
For
W
6.6
h
282
Smith chart & impedance matching
1
2
r
0
0.1255
r
W
1 0.63 r 1
h
For
(6.35)
W
0.6
h
1
2
r
0
0.0297
r
W
1 0.6 r 1
h
F
o
r
d
e
s
i
g
n
p
u
r
p
o
s
e
s
a
s
e
t
o
f
e
q
u
a
t
i
o
n
s
(6.36)
r
e
l
a
t
i
n
g
Z0 and r to ratio of
W
2 [1]
h
8e A
W
h e2A 2
For
(6.37)
W
2
h
1
W 2
0.61
B 1 ln2 B 1 r
lnB 1 0.39
h
2 r
r
(6.38)
where,
A
Z0
60
r 1
2
r 1
0.11
0.23
r 1
r
and
B
377
2Z 0 r
The zero or negligible conductor thickness formulae can also be modified
to include thickness. The strip conductor of finite thickness introduces capacitance.
So the first-order effect is to replace W by effective width Weff. The new relations
are
283
Microwave Circuits & Components
For
W 1
h 2
Weff
h
For
W
t
2h
1 ln
h h
t
(6.39a)
W
t
4 W
1 ln
h h
t
(6.39b)
W 1
h 2
Weff
h
W
are usually satisfied since dielectric
2
substrate t 0.002 . These relations are valid for those frequencies for which
microstrip line is non-dispersive and quasi TEM mode approximation is valid.
The condition t < h and t
Otherwise, Z 0 , eff , Weff all become functions of frequency. The frequency below
which dispersion can be neglected is given by
f 0 GHz 0.3
Z0
(6.40)
h r 1
where, h must be expressed in cm.
The effect of dispersion9 in eff f is
eff f r
where
fp
Z0
8 h
r eff
f
1 G f f p
h in cm.
and
2
in GHz
G 0.6 0.009 Z 0
when f p f the eff f eff . In other words, high impedance lines on this
substrates are less dispersive.
The expression for dispersion9 is
Z0 f
377 h
Weff f eff
284
Smith chart & impedance matching
where,
Weff 0 W
Weff f W
1 f f p
2
and
Weff 0
377 h
Z 0 0 eff 0
Another important quantity is attenuation. The expression for dielectric
loss9 (d) is described as
r eff 1 tan
dB
d 27.3
cm
eff r 1 0
where, loss tangent given by
tan
For dielectric with 0
d
4.34
eff 1
eff
r 1
1
0
0
2
dB
cm
A set of expressions9 for calculating conduction loss ( C ) is
For
W
h
where,
For
C
8.68
RS
Z 0W
RS
f 0
W
1
h 2
C
For
8.68 RS P
2 Z 0 h
h
h 4 W
t
1
ln
t
W
Weff Weff
1
W
2
2
h
C
8.68
PQ
2 Z 0 h
285
Microwave Circuits & Components
For
W
2
h
C
where,
Weff
8.68 RS Q Weff
2
ln 2 e
0.94
2h
Z0 h
h
Weff
P 1
4h
Q 1
2
Weff
Weff
h
Weff
h
0.94
2h
2
h
h 2h t
ln
Weff Weff t
h
The quality factor Q of a microstrip line is calculated from
2
2
Q
Q
where,
or in dB as
Q
8.686
dB
27.3 dB
,
where, 1 dB 8.686 nepers .
A microstrip line also has radiation losses. The effect of radiation9 losses
is
Qr
F
Z0
h
480
0
eff f 1
eff f
F
eff f 12 ln
2
2 eff f 3
The total Q , i.e.,
QT
1
1
1
QC Qd Qr
286
eff f 1
eff f 1
Smith chart & impedance matching
where,
Qd
d
QC
C
and
Example 6.21. Design a microstrip matching network for amplifier shown in
Fig. 6.36 whose reflection coefficients for a good match in 50 system are
S 0.614 160
L 0.682 97
Solution: Amplifier block diagram is shown in Fig. 6.36. The normalised
impedances and admittances associated with S and L can be read to reasonable
accuracy from the Z-Y Smith chart, viz.,
YS
1
1
2.8 j 1.9
Z S 0.245 j 0.165
YL
1
1
0.4 j 1.05
Z L 0.325 j 0.83
50
+
V0
-
INPUT
MATCHING
NETWORK
TRANSISTOR
L 0.614160
YL 2.8 j19
.
OUTPUT
MATCHING
NETWORK
50
L 0.68297
YL 0.4 j105
.
Figure 6.36. Amplifier block diagram
We can now use Z and Y Smith chart separately in this example. In
Fig. 6.37, YS can be located in Y Smith chart. The location of open-circuited stub
and its length can be easily found in the same way as it is done in many earlier
examples. It can be seen that
Length of open-circuited stub is 0.159 and position of stub from the
load-end is 0.099 . One can follow the same procedure for Y L as shown in Fig.
6.38.
•
•
Length of short-circuited stub 0.077
Position from load end of the stub 0.051
287
Microwave Circuits & Components
A
Y
Figure 6.37. Input matching network design
The complete design with transistor is shown in Fig. 6.39. The
characteristic impedance of all lines is 50 .
Practical Design: The capacitors CA are the coupling capacitors. Chip
capacitors which are available in the market have values between 200 –1000 pF
which have high quality factor. The bypass capacitors CB , available as chip
capacitor have values between 50 to 500 pF. These provide ac short circuit for
0.077 and 4 short-circuited stubs. The 4 short-circuited stub provide high
impedance. But dc path for VBB is available. The narrowest practical line should be
used for 4 short-circuited stub.
To reduce transition interaction between shunt stub and series
transmission line, the shunt stub are usually balanced along the series transmission
line. The scheme is shown in Fig. 6.40. Lines of 50 are added on both sides of
CA to provides soldering area. Two parallel shunt stub in Fig. 6.40 provide the
same admittance as the single stub in Fig. 6.39. Therefore, the admittance of each
side of balanced stub must be equal to half of the total admittance.
288
Smith chart & impedance matching
14
0 .0
5
5
(+
jX
/Z
0.
4
45
1.0
50
0.9
55
1.4
0.8
1.6
0.6 60
2 .0
20
N
75
PO
NE
CE
CO
M
4.0
15
10
0.
8
0.2
IN D
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
0.27
REFLE CTIO N COE FFICIE N T IN DEG
L E OF
RE ES
AN G
SSIO N COE FFICIE N T I
TRAN SM I
N DE G
RE E S
U CT
IVE
0
1.
5.0
0.6
L E OF
10
0.1
0.4
20
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.2
50
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
50
0.2
20
0.4
0.1
E
1.
0
C
AN
PT
SU
VE
TI
UC
2.0
1.8
dB
]
0
-65 .5
0
06
0.
-70
4
-1
(-
0.4
T
0.
0
-1 7
30
0.
43
0.4
2
-12 0.0
8
0
0.0
9
0
EN
0.6
1.6
1.4
1.0
1.2
0.8
0.9
0.37
-60
0.7
-11
0
-5
5
-4
0.36
0.12
CT
A
0.4
1
0.1
0.11
-100
-90
0.13
RE
A
N
0.4
0.39
0.38
1
0.0
0.9
0.8
15
2
0.7
0.9
0.1
3
0.6
0.8
0.2
10
3
4
0.5
0.4
0.7
0.3
5
0.4
2
8
6
0.3
0.6
2.5
7
8
6
5
9
10
0.2
0.5
0.5
1.8
0.1
0.4
0.6
1.6
4
3
12
14
0.05
0.3
0.7
1.4
0.2
0.8
1.2 1.1
2
15
TOWARD LOAD Ð>
10
7
5
1
20
30
0.1
0.01
1.1
0.1
0.99
0.9
CENTER
1
1.1
4
1.2
1.3 1.4
0.4
0.6
1.2
1.3
0.95
1.2
1.4
1.6
0.8
1.5
1
1.8
1.5
2
3
2
0.8
1.4
0.7
1.5
4
3
1.6 1.7 1.8 1.9 2
0.9
1.3
<Ð TOWARD GENERATOR
2
1
3
1.1
0.2
A
TT
EN
.[
20
1
4
0.6
1.6
5
4
1.7
10
5
2.5
0.5
3
0.4
1.8
6
20
10 15
4
0.3
0.2
1.9
5
I
OR
0.14
-80
-4
0
0.15
0.35
ITI
VE
5
-5
6
0.1
4
0.3
AC
PO
40 30
0
5
or
D
IN
CA P
-70
-3
5
-80
-75
),
Zo
3
0.3
<Ð
E
SC
0.2
7
0.1 -60
-30
2
OM
5
0.0
/
jX
0.3
NC
EC
4
0.4
RADIALLY SCALED PARAMETERS
10
TR
A
N
SM
.C
O
EF
F,
E
)
/ Yo
( -jB
0.6
0.
3
0.1 1
9
18
0.
0
-5
-25
6
0.4
4
0.0
0
-15
0.8
-20
3.0
0
0 .3
-4
0
4
0.
0.
2
5
.4
-1 5
4.0
WA
1.0
8
0.2
9
0.2
1
-30
0.3
0.2
0.2
2
5.0
VE L
0
-16
-85
8
0.
-20
0.2
-10
-90
10
0.6
10040 20
1
2
RE
1.0
0.2
AC
TA
N
80
0.3
0.8
ANG
S TOW
3.0
0.6
20
0.48
25
0.4
8
TH
EN G
0.
32
50
0.2
7
0.
18
30
30
0.4
1.8
6
0.5 5
0.2
T
0.4
TA
0.1
7
0.3
3
1
0.2
9
0.2
0.0
4
0.4
6
15
0
P
CA
R
P
CE
60
0.3
85
O
US
0.3
4
35
0 .2
,
o)
ES
IV
IT
AC
Yo)
jB/
E (+
NC
0.1
6
70
40
0
70
0.
06
0.
44
43
0.
0
13
0.35
40
19
0.
31
0.
Ð > W A V EL E
0.49
N GTH S
TOW
ARD
0.0
A D <Ð
0.49
GE N
A RD L O
E RA
0.48
± 180
TO
170
0
RÐ
-17
0.4
>
7
160
90
0
12
7
0 .15
0.36
80
0.7
8
0.0
2
0.4
0.0
110
1
0.4
0.37
0.38
0.39
100
0.4
1.2
0.1
9
0.0
10
0.1
0
2
Figure 6.38. Design of output matching network
+
Vi
CA
/4
0.159
50
0.077
CA
50
RFC
CB
-
CB
RFC
V CC
V BB
Figure 6.39. Complete amplifier schematic. The characteristic impedance of
microstrip line is 50
289
Microwave Circuits & Components
+
CB
/4
Vi
CA
50
50
0.129
CA
0.129
50
0.105
50
0.105
CB
RFC
-
CB
V CC
RFC
V BB
Figure 6.40. Complete amplifier using balanced shunt stub, the characteristic
impedance of microstrip line is 50
Therefore Y of each stub y
Length of each side
j 1.55
j 0.775
2
0.105
If diode is used for amplifier design, ( r 2.23 and h 0.7874 mm )
for obtaining 50 characteristic impedance of
W 2.42 mm , eff 1.91
for 1 GHz
0
1.91
0.72360
0 30 cm
For characteristic impedance of 100 in 4 line W = 0.7 mm. The line
lengths are
0.105 2.28 cm
0.099 2.15 cm
0.051 1.10 cm
0.129 2.80 cm
5.43 cm
4
It must be mentioned here that this is not the only possible design. There
can be several other designs also.
290
Smith chart & impedance matching
6.9
CONCLUSION
Although the tremendous amount of available computation speed is
available today, the Smith chart still retains its popularity because, it easily allows
the user to have a quick physical interpretation of what is happening at any point
along the transmission line. The reflector coefficient VSWR and location of stub
tuner can easily be determined using the Smith chart. Normally, the Smith chart is
used for lossless lines whether it is waveguide or microstrip line, coplanar lines or
finlines.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Wheeler, H.A. Reflection charts relating to impedance matching. IEEE
Trans on Microwave Theory and Techniques, 1984, MTT-32,
1008-021.
Smith, P.H. Transmission line calculator. Electronics, 1939, 12, 29-31.
Smith, P.H. An improved transmission line calculator. Electronics, 1944,
17, 130.
Barlow, H.M. & Cullen, A L. Microwave measurements. Constable and
Co., London, 1966, p.165.
Collin, R.E. Foundation of microwave engineering. (2nd Ed) 1992, p. 343.
Collin, R.E. & Brown, J. The design of quarter wave matching layer for
dielectric surfaces. Proc. IRE. 1956, 103, 153-58.
Collin, R.E. Theory and design of wideband multi-section quarter wave
transformers. Proc. IRE. 1955, 43, 179-85.
Arnold, R.P. & Bacley, W.L. Match impedances with tapered lines.
Electronics Design, 1974, 12, 136-39.
Bahl, J. & Trivedi, D.K. A designer’s guide to microstrip line. Microwaves,
1977.
Sobol, H. Applications of integrated circuit technology to microwave
frequencies. Proc IEEE. August, 1971.
Sobol, H. Extending I.C. technology to microwave equipment. Electronics,
March, 1967.
291
CHAPTER 7
DESIGN OF MICROWAVE NETWORK,
MODELLING & COMPUTER-AIDED DESIGN
7.1
INTRODUCTION
The importance of computer-aided design (CAD) techniques in
microwave network is very well understood. The advance in monolithic microwave
circuit (MMIC) technology in the last two decades certainly would not have been
possible without sophisticated CAD procedures and programmes. In CAD
procedures, the initial design is simulated and optimized. Use of computer-aided
design software for microwave analysis and design is well established. Until about
20 years back, computational methods found little use in analysis and design of
networks. With the use of integrated circuitry, it became essential to use computers
for simulation before designing and fabrication of circuits. Cheap micro and mini
computers are available which institutions and in many cases individuals can
afford. Technological progress has made possible design of large functional
circuits. This has made possible the design of large functional block, containing
thousands of inter connections and transmitters on one chip. Such designs cannot
be carried out by experimenting on bench. In addition to progress in computer
hardware, major innovations in numerical mathematics have created impact on all
aspects of computer-aided network analysis and design. The numerical analysis
innovations are sparse matrix methods, linear multi-step methods for the solution
of sets of algebraic differentiation system, adjoint techniques are used for similarity
computation. The sequential quadratic programmeming has been used for
constrained optimisation.
During the last ten or fifteen years, a number of programmes for network
analysis using these advanced features have become available. Some of these
programmes are quite efficient when used for specific applications for which they
were designed. In order to write new programmes or make modifications in existing
programmes, the engineers must have knowledge of the basic theory on which
computer-aided design relies on.
This chapter includes a computer-oriented microwave circuit analysis
based on the scattering matrix. The analysis involves evaluation of scattering
parameters of individual elements of connections. Computational algorithm are
presented in detail. The theory related to transposed matrix for sensitivity analysis
of microwave networks has also been discussed in this chapter. Sparse matrix
Microwave Circuits & Components
techniques are also discussed briefly. The optimisation techniques are also
described. This chapter also addresses the time domain analysis of linear
microwave network.
Three principal categories have been known in electrical circuitry.
Lumped-constant (0-dimensional) circuit, distributed-constant (1-dimensional)
circuit and waveguide (3-dimensional) circuit. The planar circuit to be discussed in
general should be considered as a two-dimensional circuit. Planar circuit can be
defined as electrical circuit having dimensions comparable to the wavelength in
two directions, but much less thickness in one direction. It can be seen that
function of wave equation can provide circuit derivation of circuit parameters.
7.2
FREQUENCY DOMAIN ANALYSIS OF MICROWAVE
NETWORKS
The computation of response function of a microwave network in terms
of element parameters is frequency domain analysis. A microwave network may
contain both lumped elements and distributed elements. Circuit elements are
considered as multiports, characterised in terms of their scattering matrices.
The conduction scattering matrix method is a very convenient approach
to CAD microwave network analysis. Coefficients are determined at certain points
or ports and also transmission coefficients between selected ports. The circuit
topology then dictates how these matrices from overall circuit connection matrix
can be derived.
7.2.1
Connection Scattering Matrix in Circuit Analysis
Figure 7.1 shows a microwave network inter connection of m multiport
elements.
For Rth circuit element there is a set of linear equations.
(7.1)
b( k ) S ( k ) a ( k )
where, S(k) in its scattering matrix and a(R) b(R) are vectors of ingoing and outgoing
wave variables respectively, at its ports.
An independent signal generator is described by
(7.2)
bG SG aG C
as illustrated in Fig. 7.2. Substituting this into the equation
VG EG I G Z G
(7.3)
gives
VG
1
Z R aG Z RbG
Zq
(7.4a)
294
Design of microwave network modelling & computer aided design
CIRCUIT ELEMENT
PORT
CO NNE CTI ON
Figure 7.1.
Microwave network which contains interconnected multiports and
independent signal generator.
IG
1
ZR
aG bG
(7.4b)
The relation S relates voltage VG across and the current IG flowing into
independent generator with incoming wave aG and the outgoing wave bG at that
port, we get
bG
Z R EG
ZG Z R
aG
ZG Z R
ZG Z R
(7.5)
where, ZR is the reference impedance of generator. Looking at Eqns. 7.2 and 7.5, it
can be seen that
SG
ZG Z R
ZG Z R
(7.6)
is the reflection coefficient of the generator and
C
Z R EG
ZG Z R
(7.7)
This is the complex wave sent by the generator. The magnitude of C set to
the second power equals the power transferred from the generator to the load whose
impedance equals the reference impedance ZR. Considering a whole microwave
network with m elements, the linear equations
(7.8)
S acb
295
Microwave Circuits & Components
IG
Z0
AG
VG
EG
BG = SGGG+CG
(B)
(A)
Figure 7.2.
where,
Independant signal generator showing (a) Incoming and outgoing
waves, and (b) Equivalent sinusoidal signal voltage source.
a 1
b 1
c 1
2
2
2
a
b
c
.
.
.
.
.
.
a k ; b k ; c k
a
b
c
.
.
.
.
.
.
m
m
m
a
b
c
and
S
S 1
0
0
0
0
2
S
0
0
k
S
0
0
0
0
m
S
(7.9)
(7.10)
S(1), S(2), ……. S(m) are scattering coefficient or the reflection coefficient of
independent generator. The vectors a(1), a(2) …… a(m). b(1), b(2), ….. b(m) and c(1), c(2),
…. C(m) represent wave variable of these elements. The vectors b and a are related
by iteration.
b a
(7.11)
where, is the connection matrix of the network. Substituting of Eqn. 7.11into
Eqn. 7.8 we obtain
(7.12)
Wa C
296
Design of microwave network modelling & computer aided design
where,
W
(7.13)
S
The coefficient matrix W in Eqn. 7.13 is called the connection scattering
matrix of the network.
The vector C is the vector impressed waves of the independent signal
generators. It can be seen that
(7.14)
a W 1 C
from Eqns. 7.11 and 7.14 it can be seen that
(7.15)
b W 1C
The set of linear Eqns. 7.12 with connection scattering matrix W as the
coefficient matrix allows computing of the incoming and outgoing waves at all
network ports if the excitation C of the network is known. The connection
scattering matrix represents the full description of network because values of the
scattering parameters of all network elements are given in matrix S and information
about the network topology is given in the matrix .
7.2.2
Formation of Connection Matrix
Let us consider the simple case of connection of two ports as shown in
Fig. 7.3. The following of relation holds good for ith and jth ports.
(7.16)
Vi V j and I i I j
When substituted expressions of incoming and outgoing waves is a
function of currents and voltages or vice versa, the relation obtained is
bi
b j
I
1
Zi Z j
Z j Z i
a
i
Z i Z j a j
Z j Zi
2 Z Z
j
i
2
I
I
j
I
i
Vi
Figure 7.3.
(7.17)
Vj
ai
aj
bi
bj
Connection of two ports in a microwave circuit. The reference
impedance are different for the two ports.
297
Microwave Circuits & Components
The coefficient matrix in above relation is the connection matrix for
one pair of connected ports. If Zi = Zj the connection of two ports is a non-reflection
of connection, which means that outgoing and incoming waves at ports i and j
connected together satisfy the relations
a i = b j and a j = b i
(7.18)
Thus when reference impedances of connected ports satisfy condition for
impedances (Zi = Zj) the connection matrix reduces to
0 1
1 0
(7.19)
As it can be seen that connection matrix of whole network is a sparse
matrix in which most of the elements rij and rji corresponding to pairs of the
connected ports are 1. The reference impedances are real to each other. The
algorithm of formation of connection matrix of microwave circuit is not
complicated in case it only involves connection power of ports. Other cases are
complicated. Therefore, if possible, they should be reduced to pair of ports. It is
more convenient to limit the number of connected pairs to two. Let us take the case
of parallel connectors of three ports in one plane as shown in Fig. 7.4. Parallel
connection of three ports in one plane treated as a network element the scattering
matrix is given by
S
1 2 2
1
2 1 2
3
2 2 1
(7.20)
In this expression the reference impedances of all ports are real and equal
to each other. We also consider three ports connected in similar way in series. In
this case
V2
V2
2
2
V1
Figure 7.4.
1
3
V1
V3
1
3
V3
Parallel and series connection for connecting three ports in one plane.
298
Design of microwave network modelling & computer aided design
S
2
1 2
1
2
1
2
3
2 2 1
(7.21)
The computer aided analysis method based on connection scattering
matrix requires multiple computation of the solution of a system of linear equations
in which the order of coefficient matrix equals to number of ports of all network
elements. Convention at numerical procedures for the solution of sets of linear
n3
n
equation requires
n 2 multiplications and divisions where, n is the order
3
3
of the coefficient matrix. Sparse matrix technique is very convenient for reduction
and simplification for solution of Eqn. 7.12. Sparse matrix techniques also allow
reduction in memory sparse to store non-zero entries of the coefficient matrix of
the system of linear equations so be solved.
Problem 7.1. Consider the microwave transistor amplifier shown in the Fig. 7.5.
The scattering matrix S and connection matrix are
S11 S12
S 21 S 22
0
0
0
0
S
0
0
0
0
0
0
0
0
S 43
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
S33
0
0
S34
S 44
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
S55
S 65
0
0
0
S56
S 66
0
0
0
0
0
S 77
S87
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
Find (W) and (C)
299
0
0
0
0
0
0
S 78
S88
(7.22a)
(7.22b)
Microwave Circuits & Components
7
Figure 7.5.
4
3
1
2
5
6
8
Microwave transistor amplifier with three ports and two line port
connected by their ports to create the network.
Solution: The Eqns. 7.13 and 7.14 can be used for estimations (C) which have
following forms
S11 S12
S 21 S 22
0
0
1
0
W
2
0
0
0
0
0
0
0
0
0
S33
S 43
0
0
3
0
1
0
S34
S 44
0
0
0
0
0
1
0
0
0
0
0
0
1
0
S55
S 65
0
0
0
S56
S 66
0
1
0
0
0
S 77
0
0
0
0
0
0
1
0
S88
(7.22c)
and
C
0
0
0
0
0
0
1
0
(7.22d)
Note that excitaton vector C has only one non zero element C, which
corresponds to generator port 7.
7.3
COMPUTATION OF NETWORK RESPONSE FUNCTION
The connection scattering matrix allows estimation of incident and
reflected waves at all network ports. All response functions of network may be
derived.
7.3.1
Input Port Reflection Coefficient
Let us take a signal generator which has internal impedance of ZG
connected to a network with scattering matrix S which is terminated impedance ZL
300
Design of microwave network modelling & computer aided design
G IN
ZG
EG M
ZL
M
aL
aG
bG G aG c
bL
aIN
bIN
Figure 7.6.
A two-port driven by a signal generator and terminated in load
impedance ZL with reflection coefficient L. The network model can
be used for computing IN which is function of S and L .
as shown in Fig. 7.6. The incident and reflected waves at generator port are related
by equation
(7.23)
bG G aG c
where, the reflection coefficient
G
ZG Z R
ZG Z R
(7.24)
ZR is the reference impedance, c is the impressed wave of the generator.
The reflected wave and the incident wave at the load port are related by equation
(7.25)
bL La L
where,
L
ZL ZR
ZL ZR
(7.26)
If the reference impedances of the connected ports j and R are real,
positive and equal to each other, the outgoing and incoming waves at these ports
must satisfy the relations
a j bR
(7.27)
and a R b j
The input port of overall circuit is referred as port 1 and output as port 2.
Thus it can be seen that
a G b1
and
(7.28)
a1 bG
301
Microwave Circuits & Components
and for output and load ports
a 1 b2
and
(7.29)
a 2 bL
According to these relations, the input port reflection coefficient is
in
b1 aG
a1 a1
(7.30)
The complex impedance looking into input port is given as
1
1 in
Z in Z R
ZR
1 in
1
aG
a1
aG
a1
ZR
a1 aG
a1 aG
(7.31)
The output port reflection coefficient out is computed as
out
then
b2 a2
a2 a2
(7.32)
The output impedance seen at the output port of the network (Fig.7.7) is
Z out
1
1 out
ZR
ZR
1 out
1
aL
a2
aL
a2
ZR
a2 a L
a2 a L
(7.33)
Following the same technique (Eqn. 7.33) Dobrowolski 1 has found
overall scattering parameters, stability criteria, gains, etc. The simultaneous
conjugate match and maximum power gain have also been studied following same
technique. One topic which has not been studied will be described here. It is
obtaining expression for the insertion loss.
OUT
G
ZL
IG
S
a OUT
aL
b OUT
bL= L QL +C
Figure 7.7. A two port excited output port and terminated output port by signal
302
Design of microwave network modelling & computer aided design
7.3.2
Insertion Loss
The insertion loss of a network is defined as
Insertion loss IL =
PLO
PL
(7.34)
where, PLO is the power delivered to the load connected directly to the generator
port and PL as the power delivered to the load connected with circuit in between the
generator and the load. Figure 7.8 illustrates the definition of IL. Power delivered
to the load directly to the generator is
PLO
2
2
C 1 L
1 G L
(7.35)
2
The power delivered to the load in the presence of the circuit is
2
C
2
2
2
2
S 21 1 L
(7.36)
PL a L 1 L
1 S S 2
11 G
22 L
s G L
where, S11 S 22 S12 S 21
G and L are defined in the figure. Therefore insertion loss expressed in
terms of two port scattering parameters and generator and load port reflection
coefficient is (C-1).
(a)
G
PL
S
G
aL
bL
bG =
G G G + C
(b)
Figure 7.8.
S
ZL
P LO
(a) Two port with network with scattering of matrix S inserted
between generator and load (b) Generator directly connected to the
load network for insertion loss computation and definition.
303
Microwave Circuits & Components
IL
7.3.3
1 S11 G S 22 L s G L
S 21
2
1 G L
2
(7.37)
2
Group Delay
The group delay of microwave circuit is defined as
D
21
W
(7.38)
where, 21 ang S 21 . S21 is overall transmission scattering parameter between
the input port 1 and the output port 2. The group delay may be computed using the
relation
21
(7.39)
W
In this incremental ratio approximation the smaller the frequency
TD
step W , the better is the approximation.
7.3.4
Voltage Transfer Gain
Voltage transfer gain referring to Fig. 7.9 is
VL
(7.40)
EG
where, VL is the voltage at the load and EG is the signal generator voltage. Note that
GV
EG Z R a1 b1
ZG
ZR
a1 b1
(7.41)
and
VL Z R a L bL
(7.42)
From the above equations, the voltage transfer gain expressed in terms of
wave variables is equal to
ZG
S
EG
ZL
L
a GaL
a1aL
Figure 7.9. Circuit for simple computation of voltage transfer gain
304
Design of microwave network modelling & computer aided design
Z R a L bL
Gv
Z R a1 b1
ZG
ZR
a1 b1
Z R a L a2
Z R a L aG Z G a L aG
(7.43)
Note that ZR is the real and positive reference impedance for all circuits
involved. Note that ZG is
ZG Z R
1 G
1 G
ZR
1 SG
1 SG
(7.44)
Substituting Eqn. 7.44 into 7.43
GV
Z R 1 2
b2
Z R 1 2 Z R 1 2 a1
and using the relation
b2
S 21
a1 1 S 22L
(7.45)
(7.46)
Thus,
GV
Z R 1 2
S 21
Z R 1 2 Z G 1 2 1 S 22L
(7.47)
The transmission phase shift is the argument of
GV a
a
EG
rg U
rg
GV
(7.48)
and is using Eqn. 7.43.
7.4
SOLUTION OF SYSTEMS OF CIRCUIT EQUATIONS USING
SPARSE MATRIX TECHNIQUES
The matrix equations are used in the analysis of microwave circuits. The
scattering matrix approach using connection matrix is described. W is the
connection scattering matrix of the order n n matrix of complex coefficients. The
three equations in this approach are
(7.49)
Wa c
(7.50)
WT er
(7.51)
W A el
Equation 7.49 corresponds to the frequency domain analysis. The other
two are related to sensitivity analysis and computer aided noise analysis which we
do not intend to discuss here.
305
Microwave Circuits & Components
Computer aided analysis and design equations are solved many times. It
can be shown, as mentioned earlier, the total number of multiplications and
divisions requested for solution of a system of linear equations 2-6 is
n3
n
(7.52)
n2
3
3
As mentioned earlier, n is the order of the coefficient matrix. The
operation count is for a system Ax = b. A is assumed to be full. The full matrix also
requires shortage for n2+n complex numbers. It may be noted that the connection
scattering matrix W is sparsely filled with entries of non-zero values. Special
techniques are used to take advantage of sparsity as sparse matrix technique. They
reduce the number of arithmetic operations and storage requirement.
Nt
There are three basic ideas in sparse matrix techniques. The first one is
not to store matrix entries which have zero values. The second approach is not to
perform torrid arithmetic with non-zero elements. Further, to exploit the sparsity of
matrix, the sparsity must not be destroyed during decor position procedure. The
number of non-zero elements in the matrix may grow during its reduction. The new
elements are known as fill-ins. The growth of the number of fill-ins can be
minimized by proper pivoting strategy. Maintaining sparsity during reduction is
essential. One of the method is basic factorization technique with applications of a
looping indexed code approach.
7.4.1
Bifactorization Techniques
In this method the increase of three coefficient matrix A in the system of
linear Eqn. 7.49 is expressed as a multiple product of 2n factor matrices 7 as
A1 R 1 R 2 R n 1 R n Ln Ln 1 L2 L1
(7.53)
where, L(i), i =1,2 …. n are left hand factor matrices and R(i), i =1,2 …. n are right
hand factor matrices. The left-hand factor matrices L(R) differ from unity matix
R of L(R) is not equal to 1.
only in the Rth column. Note that the diagonal term l RR
(R)
The right hand factor matrices R differ from the unity matrix only in the Rth row,
and all original elements are equal to 1, thus, R(n) = 1. The matrices L and R(i),
I = 1,2 ….. n can be found from the equations
Ln Ln 1 ...... L2 L1
AR 2 ....... R n 1 R n J
(7.54)
which can be written as a sequence of the intermediate matrices.
A0 A
A1 L1 A0 L1
A2 L2 A1L2
AR LR AR 1LR
(7.55)
An Ln An 1Ln
306
Design of microwave network modelling & computer aided design
The above equations represent step by step transformation of initial
matrix A = A(0) to identity matrix I. This is achieved by forming successive inner
products L(R) A(R-1) R(n) for R = 1, 2, ……n. Using these relations, it can be found
that the elements of the reduced matrix are computed as
R 1, a R 0, a n 0
a RR
ry
iR
(7.56)
R 1 a R 1
aijR aijR aiR
kj
(7.57)
where, R is the pivotal index I, j = (R+1), ….. n.
The elements of the left hand factor matrices are
R
lkR
1
a RR
(R)
liR
as
(7.58)
R 1
R 1
a iR
R 1
a RR
i n 1, ......... n
(7.59)
Similarly, we can compute the elements of right hand side factor matrices
R 1
rRR
R 1
a Rj
rRj R 1
a RR
(7.60)
i R 1,...........n
(7.61)
a R 1
As in the case of other methods, the pivots a RR
, R = 1,2………n
must be different from zero.
The vector unknowns, i.e., the solution vector, must be different from 0.
The solution vector may be computed from the relation
x A 1 b R S R 2 .........R n 1 Ln Ln 1 ........L2 L1b
(7.62)
The computer algorithm can be written directly using these relations.
7.4.2
Algorithm for Bifactorisation
Step 0; Set R = 1
Step 1; Compute Rth column of L(R) by setting
(R)
lRR
1
R 1
a
(7.63a)
307
Microwave Circuits & Components
(R)
( R 1 )
(R)
liR
aiR
lRR
i R 1..........n
(7.63b)
( R 1 )
where, aij
are entries of reduced matrix A(R-1) and A(0) = A.
Step 2; If R = n, then stop
Step 3; Update the reduced matrix from A(R-1) to A(R) by setting
(R)
aij( R ) aij( R 1 ) a (RjR 1 )liR
(7.64a)
where, Lij = R + 1, R + 2 …….n
Step 4; Obtain Rth row and R(R) by setting
(R)
(R)
rRj
s (RjR 1 )l RR
Step 5; R = R +1 and go to 1.
7.4.3
Computation of Solution Vector
Set the solution vector to the right hand side vector
xi = bi , I=1,2, ……. N
The left hand factor matrix by vector multiplications;
Step 0; Set R = 1
matrix
Step 1; Update the solution vector by multiplying it by left hand factor
R)
x R l (RR
xR
(R )
x i x i liR
xR
i R 1,..........., n
(7.64b)
Step 2; If R = n, then go to step 4
Step 3; R = R + 1 and go to step 1
Right hand factor matrix by vector multiplications,
Step 4; R = n – 1
matrix
Step 5; Update the solution vector by multiplying it by right hand factor
xR xR
n
j R 1
(R)
rRj
xj
Step 6; If n=1, then stop
Step 7; R=R-1 and go to step 5
7.5
MICROWAVE NETWORK SENSITIVITY ANALYSIS
Information for sensitivity of network functions with respect to network
parameters is very important for design of microwave network. Sensitivities allow
the designer to estimate the effect of the network parameter behaviour on response
of the network. Sensitivity analysis is important from the point of view of
308
Design of microwave network modelling & computer aided design
optimisation of network performance which is discussed briefly in the next section.
It allows a more design algorithm, design procedure and also automation of circuit
design. Sensitivity studies also permits to choose with some specific performance
requirements. A network with least performance sensitivities with respect to
network parameters is the best. Sensitivities can also be used for tolerance analysis
and yield driven procedures for design of microwave networks 7-11.
form
Sensitivity parameters are complex quantities and can be expressed in the
(7.65)
f f i
taking the natural logarithm as
(7.66)
lnf ln f jq
and differentiating with respect to a parameter p
1
f
1 f
f
.
j
p
f p
p
(7.67)
The complex equation can be split into two parts.
f
p
1
Re
f
f
f
p
(7.68)
and
1 f
(7.69)
f p
Very often the magnitude is measured in decibels (dB). To obtain
corresponding formula
q
Im
p
In f
(7.70)
A i
where, A ln f
Differentiating with respect to p gives
1 f
f p
A
j
p
p
The sensitivity
(7.71)
A
is expressed in nepers. In order to convert to decibels
p
A
e
is multiplied by 20 log10
p
8.686 which
1 f
A
A
dB 8.686
8.686 Re
p
p
f p
309
(7.72)
Microwave Circuits & Components
7.5.1
Transposed Matrix Method for Sensitivity Analysis
Remembering that the expression for connection scattering matrix is
Wa=C
Differentiating this expression with respect to p
a
w 1
S
W 1
w c W 1
a
p
p
p
(7.73)
c
0
p
Then sensitivity of the incoming wave variable ar at the rth port of the
network may be computed by multiplying the left hand side of Eqn. 7.72 by a row
In the above expression it has been assumed that
vector erT all of whose elements are equal to null except for the rth element that is
equal to 1, i.e.,
erT 0,0, ................. 0,1,0...........,0
(7.74)
Performing the operation yields
T
1 s
s
ar
a 1 s
w
a wT er
a aT
a
erT
dp
p
p
p
p
(7.75a)
where,
WT
1
(7.75b)
er W 1er
is a solution vector of the transposed or the adjoint matrix equation.
(7.76)
W T er
The above matrix equation may be treated as the equation of the adjoint
network excited as rth port by a signal generator with impressed wave equal to 1.
According to Eqn. 7.76, the connection scattering matrix W of adjoint network
must be equal to the transposed connection scattering matrix W of the original
network because
(7.77)
W W T S T S T
It means that the topologies of original and adjoint matrices are same
whereas the elements of adjoint network has the scattering matrices
Ŝ 1 , Ŝ 2 .........Ŝ m equal to the transposed scattering matrices of corresponding
elements of the original network.
The Eqn. 7.74 which defines the sensitivities of incoming wave variable
at the rth port ar with respect to many network parameters has the form
310
Design of microwave network modelling & computer aided design
G ar
T
T
T
T
S
a
R
P1
R E1
S
a
T
P2
R
RE
2
T
S
a
R
Pn
RE
n
S R R
a
P1
R
S
a R
P2
S R R
a
Pn
(7.78)
In this expression, E1, E2 …..En represent sets of circuit elements that
depend on the parameters p1, p2………pn respectively.
S
For each parameter Pi the matrix P is formed and the products indicate
i
the right hand side of the Eqn. 7.78 must be evaluated. As the vectors a are
independent of the parameter index U, we see that the application of Eqn. 7.78
requires solution of only two sets of linear Eqns. 7.73 and 7.76 irrespective of the
number of parameters Pi .
7.5.2
Sensitivity Computation of the Overall Scattering Parameters
i
Let us compute the transfer function S ij S ij e ij and then evaluate the
sensitivities with network parameters. Assume a matched generator impresses a
wave for which c = 1. All the other ports are terminated in matched load. The
transfer function is then
S ij
i j
(7.79)
bi / a j 1 al
where, al is the input wave at the port l. The condition e = 1 makes aj = 1 connected
at the port j.
If the adjoint network with Ŵ W T is executed at the port l by a matched
Sij
i
generator which provides the impressed wave el=1 (aI=1). A
P P
S ij
P
T
S
a
P
R T
RE
S R R
a
P
(7.80)
The summation in taken over all network elements which are dependent
on parameter p. Figure 7.10 illustrates network models for sensitivity computation
of transfer function Sij of microwave network. In Fig. 7.10(a) the network
corresponds to
Wa=c
(7.81)
where, element WRR = SG and Wll = SL are set to zero. Since the generator is
311
Microwave Circuits & Components
SL= 0
SG= 0
MATCHED
LOAD
S
k
MATCHED
j
i
GENERATOR
a1= 0
b1
(a)
i
a1
SL= 0
S
MATCHED
i
j
k
i
LOAD
(b)
MATCHED
GENERATOR
Figure 7.10. Sensitivity computation for (a) Original network with scattering
matrix S excited at input port (b) Adjoint network with matrix S
equal to transposed scattering matrix ST of original network excited
at output port.
matched the scattering parameter SG = 0. The adjoint matrix corresponds to
W T a = ll
(7.82)
For vector el all the elements are zero except lst, for which el = 1.
The Table 7.1 shows scattering matrices for some basic parameters for
microwave networks. It is now proposed to describe the microwave network
optimisation technique in brief. The references 12-16 may be referred for further
details.
7.6
MICROWAVE NETWORK OPTIMISATION
The circuit parameters of a network must be adjusted to minimize the
actual circuit performance and the one for which the network has been designed.
This can be done by the method which as known as circuit optimisation technique.
Optimisation means the process of determining the extreme value of the
mathematical function. The optimisation means minimisation of scaler
functions12-16
U U x
subject to inequality constraints.
C x 0
and equality constraint
h(x) = 0
312
Design of microwave network modelling & computer aided design
Table 7.1. Differential scattering matrices for some basic elements of networks
S
p
Circuit element
p
2Z R
Z ZR
Z
Z
Z
2Z R
Z 2Z R
2
2 Z R
Y
1 -1
-1 1
1 Z RY
2YR
Y
Y 2YR
2
Y
2
1 1
1 1
Y
1
Z 02 Z R2 2Z 0Z R coth( r1 )
4Z 0Z R2 2Z R Z 02 Z R2 coth( r1 )
2Z R Z R Z 02 csc h( r1 )
1
Z0, r
2 zR
Z
4Z 0Z R2
Z csc h( r1 )
2Z Z Z coth( r1 )
R
2
0
2
0
R
r j ,
2n
g
2
Z 02 Z R2 2Z 0Z R coth( r1 )
Z 02 Z R2
2
2
2Z 0Z R sinh( r1 ) Z 0 Z R cosh( r1 )
2Z 0Z R sinh( r1 ) Z 02 Z R2 cosh( r1 )
Z 02 Z R2
313
Z0
2
R
2Z 0Z R r csc h 2( r1 )
TRANSMISSION LINE SECTION
Z
1
contd...
Microwave Circuits & Components
S
p
Circuit element
1:p
IDEAL
2p
2
2
1 p 2 1-p
2 Z R
Z 2R - 2
2
2
2
2 Z R
2 Z R2 Z R -
p
1
GYRATOR
0
2
kV
VOLTAGE CONTROLLED
VOLTAGE SOURCE
V
1-p 2
2p
2
TRANSFORMER
V
p
0
2 Z R
g mV
VOLTAGE CONTROLLED
CURRENT SOURCE
I
0
2
Z R
rmI
CURRENT CONTROLLED
VOLTAGE SOURCE
0
0
k
0
0
gm
0
0
rm
I
I
0
2
CURRENT CONTROLLED
CURRENT SOURCE
0
0
2YRcot h( r1 )
1 1
1 1
2YR Y0cot h( r1 )
2
Y0
PL
Z
2YR r
1 1
2 1 1
sin h 2YR Y0cot h( r1 )
2
SHORT ENDED
PARALLEL STUB
1
contd...
314
Design of microwave network modelling & computer aided design
S
p
Circuit element
2YR tan h( r1 )
p
1 1
2YR Y0tan h( r1 ) 1 1
2
Y0
Z
PL
2YR r
1 1
cos h ( r1 ) 2YR Y0cot h( r1 ) 1 1
2
OPEN ENDED PARALLEL STUB
2Z R tan h( r1 )
1
Z0, r
SHORT ENDED SERIES
STUB
1
Z0, r
OPEN ENDED SERIES
STUB
2
1 -1
2Z R Z 0tan h( r1 ) -1 1
2
1
Z0
2Z R r
1 -1
2
1
cos h 2( r1 ) 2Z R Z 0tan h( r1 ) -1 1
2Z Rcot h( r1 )
1 -1
2Z R Z0cot h( r1 ) -1 1
2Z Rr
2
1 -1
sin h ( r1 ) 2Z R Z0cot h( r1 ) -1 1
2
2
Z0
1
It should be noted that the dependence of functions U(x), C(x) and h(x) on
design parameter p = x is implicit through the circuit equations. It is required to run
on optimisation algorithm which very often involves an expensive simulation of
microwave circuit to be designed.
The desired performance of microwave circuit can be expressed by
specifications which are functions of independent variable frequency. The
frequency response curve can be computed at finite number of frequencies.
Figure 7.11 represents an example of lower and upper specifications for insertion
loss characteristic of band pass filter. The response function shown in the figure
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Microwave Circuits & Components
F(p,f)
S(f)
S(L)
S(f)
F(p,f)
Su(f)
f
i
f
i
fu
e(p,f)
(a)
e1(p,f)
e1(p,f)
eu(p,f)
f
i
f
i
fu
(b)
Figure 7.11. (a) Lower S l (f) and upper Su(f) specifications in response function
F(p,f) of band pass filter (b) Error function defined as a difference
between calculated response and given specification.
violates this characteristic. Set fI and fII denote respectively the lower bound and
upper bound of frequency interval for which the problem is considered. Figure 7.12
shows an example for lower and upper specifications for the insertion loss
characteristic of a band pass filter and response function that violates these
specifications. fI and fII denote respectively the lower bound and upper bound of
the frequency internal for which the problem is considered. In this figure response
specifications for which there are lower and upper specifications defined in the
same interval of independent variable (i.e., frequency).
Another type of specification is a single specification, which is two-sided
specification. A single specification may be interpreted as window specification
with zero width. An example of a single specification for design problem of an
F(p,f)
S(f)
f
fu
i
Figure 7.12. The lower and upper specifictions for the response function defined
for the same interval of frequency.
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Design of microwave network modelling & computer aided design
S(f)
F(p,f)
S(f)
F(p,f)
S(f)
S(f)
f
fu
i
Figure 7.13. The single specifications S(f) and a response function F(p, f) for an
amplifier design problem.
amplifier is shown in Fig. 7.13 and describes the requirements on the shape and
magnitude of transducer power gain of the amplifier. Response functions for
microwave design network are
Magnitudes of overall network scattering parameters
S11net , S12 net , S 21net , S 22 net ;
arguments of the overall network scattering parameters are
arg S11net ,arg S12 net ,arg S 21net ,arg S 22 net ;
real parts of overall network scattering parameters.
Re S11net , Re S12 net , Re S 21net , Re S 22 net ;
and imaginary parts
In S11net , In S12 net , In S 21net , In S 22 net ;
The group delay of overall network S21 transmission coefficient Toj and
noise figure F.
Because response functions are calculated at discrete frequencies, it is
convenient to define set of the sampled network set response functions.
F j p
(7.83)
jJ
and their corresponding set of exampled specification
Sj
(7.84)
j p
In these equations the subscript j refers to quantities (already defined)
evaluated at discrete values of independent variables (i.e., discrete frequency
points). Error functions arise from the difference between the actually calculated
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Microwave Circuits & Components
responses and the given specifications. The error functions are defined for lower
and upper specifications as
euj p Wuj F j p Suj , j J u
elj p Wlj F j p Slj , j J l
(7.85)
(7.86)
where, Fj (p) is the response functions, p represents the network parameters, Suj is
an upper specification, Slj is a lower specification, Wuj is non-negative weighting
factor for Slj. These weighting factors Wuj and Wlj can be thought out as giving the
relative importance of e u, lj (p). Simultaneously Wu, lj must take into account the
relative magnitudes of scales associated with euj, lj which may be defined for
different kinds of responses and in different units of measure. The index sets ju and
jl are defined as
ju j1 , j2 ,............, j R
(7.87)
jl j R 1 , j R 2 ,.......... jm
(7.88)
By reducing the error functions as
euj p j ji i 1,2.........R
ei( p )
eli ( p ) j ji i R 1, R 2...........,m
where, one obtains a set of uniformly indexed error functions. The response
functions F(p) corresponding to the above defined error functions ei indicates a
violation of corresponding specifications. If we define
M e ( p ) max ei p
I 1,2 ,.........,m
(7.89)
then the sign of Me(p) indicates whether all the upper and lower specifications are
satisfied; that if
Me(p) > 0, the specifications are violated
Me(p) = 0, the specifications are exactly matched
Me(p) < 0, the specifications are satisfied with excess
In case of designs with single specifications the definitions of error
functions is much simpler;
ei Wi Fi ( p ) Si
i 1,2........m
(7.90)
where, Wi is a non negative weighting factor.
In Eqn. 7.90 the response functions and the specifications can be real or
complex. The error functions ei, i = 1,2,……m can be treated as a component of a
vector defined as
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Design of microwave network modelling & computer aided design
e1 p
e p
e 2
em p
7.6.1
(7.91)
Philosophy & Methods of Optimisation
Circuit characteristics obtained from specification are compared with
given specifications. If the results fails to satisfy desired specification, the
designable parameters of circuit are altered in a systematic method. The sequence
of circuit analysis comparison with desired performance and parameter
modification, is performed iteratively till the optimum performance of the circuit is
achieved. As mentioned earlier this process is known as optimisation. There are
two different ways of carrying out modification. There are (a) gradient method and
(b) direct search method.
Gradient methods use information about derivatives of performance
functions (with respect to designable parameters) for arriving at modified set of
parameters. This information is obtained from the sensitivity carried out using the
method described earlier. One more step is added which is the sensitivity analysis.
In the direct method gradient is not used and parameter modification is carried out
by searching for the optimum in the systematic method. Various methods are
documented in references16-20.
7.6.1.1 Objective functions & constraints
The problems of optimisation, as mentioned earlier, can be expressed as
minimisation of objective function which is error function ei(p) which represents
the difference between the performance achieved at any stage and the desired
specifications say, for microwave transistor amplifier, the formation of ei(p) will
involve the specified and achieved values of the gains, the bandwidth and perhaps
the input impedence, noise figure and the dynamic range. The objective function
ei(p) is also called cost function, index or error criterion.
The optimisation problems are usually formulated as minimisation of
ei(p). This does not cause any loss of generality since the minima of function ei(p)
is maxima of –ei(p). Usually for the solution to be feasible the element of p are
subject to certain inequality constraints given by gi(p)³ = 0 and equality constraint
like hi(p) = 0. For microstrip circuits, element of p would be lengths and
characteristic impedance of various microstrip sections, values of lumped elements
and parameters of the active devices used.
The elements of p define a space. A portion of this space where all
constraints are satisfied is called feasible region R or the design space R, which
may be expressed as
R P g p 0,h p 0
(7.92)
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R is said to be closed if equalities are allowed and is said to be open if no
equalities are allowed. Equation 7.92 defines a closed space. The optimum value
of P inside R has to be found out. A global minimum of ei(p) located by a vector
Pmin on the response hyper surface generated by ei(p) is such that
lmm e pmin ei p
(7.93)
for any feasible p not equal to pmin commonly known methods do not generally
guarantee in finding a global minimum but yield a local minimum which may be
defined as
e p min min e p
R
(7.94a)
l
where, Rl is a part of R in the local vicinity of pmin.
As mentioned earlier, the aim of optimisation process is to reduce the
difference between the performance realised from specifications for the circuit
designed. The function which specifies this difference is called the objective
function.
In order to express the error as a single quantity it is possible to use the
norm of weighted error function e(, ). The pth norm when is continuous
variable is defined as
1 p
1
e 1e ,
1
,1 p
(7.94b)
In this expression e(, ) the weighted non-function is defined as
e , W F a , S
(7.95)
where, W() is the weighting function, F(, ) is approximating function and
S() represents the specified response function. Circuit response is independent
variable which can be frequency, time or other similar variable.
Two important types of objective functions are obtained. These are known
as the least pth approximation and minimised approximation.
Least pth approximation: In this case the objective function is written as
ei p
2
1
p
e p, d
(7.96)
or in discrete case
e p ei p
q
(7.97)
The minimisation of ei(p) given by above two expressions is known as the
least pth approximation. A value of q = 2 leads to commonly used least square
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Design of microwave network modelling & computer aided design
objective functions. When p = 2 the norm defined by above equations is called an
Euclidean norm as it basically measures the distance from origin in the
multidimensional error space to the tip of the vector e(p). For the least square case
the objection becomes
n
eE p ei p
2
(7.98)
i 1
where, n is the number of sample points. Further details are available in literature 17.
7.6.1.2 Minimax approximation
When q is made infinitely large in Eqns 7.96 and 7.97 the following
relation governs the behaviour of objective function. For well-behaved functions
we have
1
q
max l , u e p, lin
1 e p, l dx
q u l
l
and for discrete case functions
(7.99)
1
q q
maxli p linq e p ,i J
i
For well-behaved functions we have
p max , Wu F p, Su
Wl x F p, Su
l
u
(7.100)
(7.101)
where, Su() is the desired upper response specification, Sl() is the desired lower
response specification, Wu() is the weighting function for Sl(). These quantities
satisfy the following restrictions.
S u S l ,Wu 0 and Wl 0
where, these conditions
Wu E , S u and Wl F , S l
are positive if the specifications are not met, quantities are zero when specifications
are met and are negative when specifications are exceeded. We will now discuss
briefly the direct search optimisation methods and gradient method of optimisation.
7.6.2
Direct Search Optimisation Method
Broadly speaking, these methods rely on the sequential examination of
trial solutions in which each solution is compared with best available up to that
time, using a strategy generally based on past experience.
One of the simplest approach is to let one parameter vary till the limits of
improvement has reached, and then try, the other parameter. Progress is very slow
in this method if the parameters vary in the direction of coordinate axis. This is
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Microwave Circuits & Components
known as pattern search method. Hooke and Jeeves gave one method which is a
sequential technique. The exploratory move is used to explore the local behaviour
of objective function. The pattern move takes advantage of the pattern directions
identified in the earlier step.
The Powell's method is an improvement over basic search method 16-17. It
is a method of conjugate directions. Consider a quadratic function of dimension
and two parallel hyperplanes 1 and 2 of dimension R < n. Suppose that constrained
stationery points of quadratics in hyperplanes are 1 and 2 respectively. Then line
joining and2 are two different minima obtained by searches along directions
starting from two different points, then vector ( 1 ,2) is conjugate to direction S.
Thus if S is chosen along the line joining two minima obtained by searching along
Si-1, the directions (Si-1, S) are conjugate.
Powell’s method for pattern search consists of minimisation along the set
of direction in the -plane/space.
7.6.3
Razor Search Method
The minimum for many minimax types of objective function has along
the path of discontinuous derivative of e(p) function.
An example of the objective function of this type is
e p min p max, l P p, f
(7.102)
fl , fu
where, p represents the complex reflection coefficient of multisectional
inhomogeneous impedance matching transformer. The frequencies fl and fu
represent the lower and upper limits of operating frequency range. P is set of
designable parameters consisting of lengths and impedances (or dimensions) of
various sections. Objective function of this type is found to have a path of
discontinuous derivative situated along a narrow curved valley. The discontinuity
in derivative arises when e(p) jumps from one response ripple extremism to
another.
7.6.4
The Simplex Method
The geometrical figure formed by a set of (n+1) points in an one
dimensional space is called a simplex. The (n+1) points are called vertices of
simplex. Thus it is topological generalisation of a triangle which may be called a
simplex in two dimensional space. A tetrahedron is a three-dimensional simplex.
When vertices are equidistant, the simplex is said to be a regular simplex. The
simplex method 19-20 for minimisation of multidimensional objective functions
starts with arbitrary shaped simplex in a n-dimensional p-space. The function is
evaluated at the (n+1) vertices of the simplex. The simplex modifies and gradually
moves towards and shrinks around the optimum point during the iterative process.
The manipulation of simplex is achieved using various operations known as
reflection expansion, contraction and shrinking.
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Design of microwave network modelling & computer aided design
7.6.5
Gradient Methods for Optimisation
This method makes use of derivative of objective functions e(p). The
primary reason for the use of derivatives or gradient techniques in circuit
optimisaton is that at any point in the design space, the negative gradient direction
indicates the direction of the greatest rate of decrease of the objective function at
that point.
The Steepest Descent Method 21-22 is gradient technique available for
function minimisation. It works by determining a minimum of objective function
along a line in the direction of gradient. A trial point is selected and iteratively
moves towards the optimum point according to the rule.
(7.103)
i 1 i i Si
where,
Si e p Pi II e P / Qi
(7.104)
i.e., S1 is given by the negative of the normalized gradient vector at the current
value of P in the ith iteration. The function e is defined as
e e
e
e
,
, ........
eR
P1 P2
t
(7.105)
where, p is defined as
p p1 , p 2 , ............ p 3 t
(7.106)
is called the increment vector, e is known as the gradient vector. This method
uses multidimentional Taylor series expansion. The expansion of e p p may
be written as
1
p t H p ............
2
where, H is the Hessian matrix which is R x R matrix containing second order
partial derivatives.
p p e p p
For complete understanding of the method one will have to refer to
original papers. This expansion is also made use in the generalized Newton-Rap
son Method 24. The main difficulty in this method is evaluation of H and its inverse
at each stage of iteration. The Davidon-Fletcher-Powell 25-26 method overcomes
this difficulty by an approximation to the inverse of Hessian in place of H-1(p). This
approximation is gradually improved in successive stages of iteration.
The optimisation methods discussed earlier are useful for minimizing any
scalar objective function. A least square optimisation requires that the sum of
squares of deviation be minimized.
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7.7
MICROWAVE CAD PROGRAMMES
K.C.Gupta and R Chadha have developed a general purpose microwave
circuit analysis programme called MCAP. This programme evaluates overall
scattering matrix of a multiport circuit from known S-matrices of the constituent
components of the circuit. The subroutine incorporated into the programme can
handle various stripline and microstrip components. The circuit analysis uses the
connection-scattering matrix approach as discussed earlier. MCAP can also be
extended to perform the sensitivity analysis using the adjoint matrix method.
A common grid flow chart for analysis programme is shown in Fig. 7.14.
The circuit is divided into basic components, whose characterization are stored in
literary of subroutines. The data read includes description of components, inter
SKEW GRID
RECTANGULAR GRID
TRIANGULAR GRID
CIRCULAR GRID
Figure 7.14. Common grid pattern
connection between ports, and frequency of operation. The programme stores the
expressions for evaluation of scattering matrix for all basic components which
constitutes the circuit to be analysed. The interconnection scattering matrix is
computed from inter connection pattern and component scattering matrices. It is
factorized into a lower triangular matrix and an upper triangular matrix.
Source vector c is setup with c(i) = 1 for external port i. The solution for
Wa = C is obtained by forward backward substitution and process is repeated for
any external port.
7.7.1
Subroutines
(a)
REDATA: The subroutine reads into the input data. The first line read
indicates whether a strip line or microstrip circuit is being analysed. The
other data consists of description of elements, the inter connection of
various ports and the frequencies at which the scattering matrix is to be
computed. It also reads information about whether the discontinuity
effects are to be taken into consideration for analysis. It is called by the
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Design of microwave network modelling & computer aided design
(b)
(c)
(d)
(e)
(f)
(g)
(h)
7.7.2
main programme.
ELCODE: It converts the mnemonic code deviating the type of element
into numeric code. It is called by subroutine REDATA.
SETWNS: Set up the W matrix S by putting unity elements in
corresponding to inter connections and inserting the scattering matrices
of the components in the network in their proper place. It is called by the
main programme.
SETUPS: Called repeatedly by the subroutine SETWNS and inserts the
scattering matrix at one of the components at a time into W matrix.
IMFWD: Computes the characteristic impedance of a stripline or
microstrip line from the width of the strip other parameters being kept
constant. It is called by subroutine REDATA.
WIDTH: This subroutine computes the width of the conducting strip
required to get a particular characteristic impedance.
FACTLU: Factorises the W matrix into lower triangular matrix and upper
triangular matrix. It is called by the main programme.
FORBAK: This subroutine computes the solution by a pair of forward
and backward substitution using LU factors obtained by FACTLU. It is
called by the main programme.
An Overview of Available CAD Programmes
An ideal design manufacturing cycle of a microwave amplifier is
illustrated here. Most of the routine decisions and tedious computations are
performed by computing the human interaction and provided in the form of
interactive graphics. Such conception is called an Integrated Design Manufacturing
System (IDMS) as shown in Fig. 7.15.
Amplifier performance requirements are specified by system engineers.
They use graphic and test processing features of IDMS to prepare and record the
amplifiers specifications in segments of IDMS database. They can search the
f(x)
P
B
A
x0 – Ax
x0
x0 + Ax
Figure 7.15. Estimation of derivative of f(x) at p
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x
Microwave Circuits & Components
IDMS for such a module which meets the requirements of new system and it can be
included in the system design. If the system designer concludes that a new amplifier
is needed the task is turned over to component designers. The microwave engineers
assigned to amplifier design retrieve the specifications from IDMS database. In
many cases an existing amplifier can be modified to perform a specified
programme. If it is decided to start from scratch rather than to use or modify a
known amplifier, the engineer consults the IDMS library of transistor
characteristics to select devices that show the best promise of meeting design
requirements. This is done by specifying the gain and bandwidth requirements and
the databank provides a list of alternative devices that can possibly meet
specifications when transistors are selected; the programme generates equivalent
circuits to model the devices27. Next the engineer consults the IDMS library of
matching network schematics to select, input, output and inter stage network
topologies that appear appropriate to the design. If a set of promising networks is
located in the library they are used as the unhide point in synthesis to follow. If a
new design topology is needed the engineer invokes the IDMS network design and
to create and verify the needed schematics. If network design aid fails to produce
a useable topology the engineer consults the IDMS graphic design aids to create
and capture a schematic by manual methods. Then the engineer uses the
optimisation techniques to reform the electric characteristics.
7.7.3
Some CAD Programmes
The compact servers offer analysis optimisation of active and passive
circuits. AMPSYN and CADSYN provide matching network synthesis while
FILSYN is a general purpose filter design programme.
(a)
(b)
(c)
HANDY COMPACT: It can be used on the HP-41C handheld calculator.
This is available in a form of plugin, ROM provides user with capability
to examine complex circuit using two-port or ladder analysis.
MICRO-COMPACT: It can be used with HP-9845 B/T Desktop
computer. It analysis or optimizes two port linear circuit in the frequency
domain. The basic block, selected from their compact’s circuit library are
combined into subcircuits that are interconnected from the final circuit.
The optimisation is based on an adaptive random search technique that
seeks to find the global minimum. MICRO-COMAPCT is self
documenting by means of “HELP” messages and internal editor is
provided for easy circuit modifications.
SUPER COMPACT: This programme replaces circuit optimisation
programme, COMPACT, that served worldwide needs of microwave
industry. SUPER COMPACT offers unrestricted simple data input and
flexible inter active, optimisation of one, two, three or four port circuits
users may specify a tailored optimisation error functions which may
include combination of S-, Y- and Z- parameters as well as nodal
impedances or admittances with arbitrary terminations. Specifications
may be different for various input-output pairs allowing simultaneous
optimisation for desired characteristics.
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Design of microwave network modelling & computer aided design
(d)
(e)
(f)
(g)
AMPSYN: This programme synthesizes lumped element matching
networks and provides for transformation of lumped design to
approximate transmission line equivalents. AMPSYN allows user to
select any topology to absorb the parasitic elements. Impedance
transformations are implemented by programme to provide the proper
networks for specified terminating impedance levels.
CADSYN: This programme is a distributed matching network synthesis
computer programme that offers commensurate equal length transmission
line network design. Bandwidth gain, slope transmission line length,
insertion loss and ripple along with circuit topology is selected and may
be modified as desired. Theoretical gain-bandwidth limitations are
calculated to predict device performance prior to synthesis. Interactive
use of Kuroda’s identity, Norton’s transformation duality and impedance
sealing provide the capability to design matching networks with arbitrary
impedance transformation and recallable element values.
A general purpose programme for microwave network analysis and
design JADMIC 2: JADMIC 2 is the personal computer based software
development for analysis and design of linear microwave circuits. The
development of JADMIC 2 followed four objectives.
• JADMIC 2 may run on PC-platforms commonly found in
engineering environments.
• JADMIC 2 should be intuitive, mean driven and easy to use.
• JADMIC 2 should correctly solve realistic linear microwave circuits
design problems of interest to educational, industrial and research
communities.
• JADMIC 2 documentation should be very extensive and highly
understandable to the user who may want to introduce his own ideas
into software.
SPICE: SPICE is an acronym for a ‘Simulation Programme with
Integrated Circuit Emphasis’. It was developed in University of
California, Berkeley. SPICE is a computer aided simulation programme
that simulates electric circuits. This programme is capable of simulating
analog and digital circuits, non linear circuits and transmission lines. The
basic SPICE circuit simulator is used in number of software packages that
includes SPICE in arcade PSPICE. In addition to having the ability to
model a wide selection of electronic components, SPICE can perform a
variety of type of analyses including DC analysis, transient analysis,
steady state analysis, etc.
Simulation can take on various levels of device and component
modelling depending on the objective of simulation. Most of the
simulation utilises idealised or default component models, making first
order approximations. Writing a circuit file for PSPICE simulation is
quite easy. First of all a circuit diagram must be drawn and all nodes
numbered. All special devices like diode, SCRs or transistors must be
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Microwave Circuits & Components
modelled before using them. Sometimes it is possible to use the default
models provided with the software package.
Probe is a separate programme that comes with PSPICE. It allows
the user to look at the waveforms of different current and voltages. After
running the PSPICE file, the output required for running probe is written
to a data file if probe command was included in the original PSPICE
CIRCUIT file. Probe is also capable of mathematical computations
involving currents and or voltages, including numerical determination of
rms and average values, and fouler analysis.
Component values may be specified as an integer (4, 12, -8) or a
real number (25, 34179, -1414). Integers and real numbers may be
followed by either an integer exponent (7E-6, 2.136E3) or a symbolic
scale factor (7U, 2.136k).
(h)
High Frequency Structure Simulator ANSOFT HFSS: There is 3DEM
simulation software for RF & Wireless design. A soft HFSS is an
interactive software package that computes S-parameters and full wave
fields for arbitrary shaped 3D passive structures. It offers an intuitive
interface to simply design entry, a field solving engine with accuracy,
driven adaptive solutions and a powerful post processor for
unprecedented insight into electrical performance. The software
eliminates traditional ‘cut-and-try’ prototyping, reducing developing
costs and speeding time-to-market’.
It is a 3D high-frequency full-wave electromagnetic field solver
based upon the Finite Element Method (FEM). Through improvement in
technology, innovative features and robust/expansive functionality,
Ansoft offers to design better and faster. It supports For-Wave TM Spice
and sensitivity analysis. It includes modes-to-nodes, fast frequency
sweep and SPICE model. It also provides multiprocessing for UNIX
platforms.
(i)
EESOFF: EESOF RF AM/S (Analog Mixed Signal) has simulation
technologies for AC/S parameters HF Spice, New Harmonic Balance,
Convolution, Berkelay Ptolemy Dataflow. They generate circuit
envelope, HP Ptolemy synchronous Dataflow. Finally one can obtain HP
Ptolemy Tuned Synchronous Dataflow (TSDF). It can simulate in both
time and frequency domains. It can provide behaviour model for mixer
and amplifier. Modelling with EESOF can provide effects of bond wires
in REIC simulation. New Engine Technology (Momentum RF)
significantly enhances electromagnetic simulation performance at RF
frequencies. Microwave Low-Pass filter (frequency < 12.5 GHz) can be
designed by EESOF. It has a design guide developer studio. It provides
both (i) power amplifier design guide and (ii) Oscillator design guide. It
can also provide performance testing using design guide. It can also be
used for designing input and output matching network and study of the
performance. One can study using EESOF spectrum, gain, harmonic
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Design of microwave network modelling & computer aided design
distortion of US power (W/PAE). It provides S-parameters, Noise figure,
stability, group delay versus frequency information. Advanced Design
System (ADS) provides simulation Technology for filters, amplifiers,
mixers, oscillators, passive components and systems. Standalone ADS
front end (SAFE) consists of system scheme editor, system simulator,
circuit schematic editor, circuit simulator and also provides Design Rule
Clerk (DRC). It helps in designing Wireless Lan Applications (WLAN).
It stands to design challenges for 3G Wireless Hand Power amplifiers. It
minimizes the need for wager design and test systems. It can study
switching between simulation (virtual) and measurement (actual).
7.8
ELECTROMAGNETIC TECHNIQUES IN COMPUTER
AIDED DESIGN OF MICROWAVE COMPONENTS &
CIRCUITS
So far we have confined to circuit approach for computer aided design of
microwave components and circuits; now we take up other approaches 28-29. There
are excellent books written by authorities on these disciplines. Several techniques
are used in the electromagnetic approach of solving problems and writing computer
programmes. They can be classified into two
(A) Analytical methods (exact solutions) which include
(a) Separation expansion
(b) Series mapping
(c) Conformal mapping
(d) Integral solutions i.e., Laplace and Fourier Transforms
(e) Perpetration methods
(B) Numerical methods
(a) Finite difference method
(b) Method of weighted residuals
(c) Moment method
(d) Finite element method
(e) Transmission line modelling
(f) Monte Carlo method
(g) Method of lines
Application of these methods is not only for EM related problems but
they can also be used for continuum problems like fluid, heat changer and
acoustics. Some of the numerical methods are related and they all generally gave
approximate solutions with sufficient accuracy for engineering purposes. The
various methods belonging to numerical methods will be discussed first. The first
one is the finite difference methods. Let us discuss this in detail.
7.8.1
FINITE DIFFERENCE METHODS
Finite Difference Method (FDM) was first developed by Thom30 in 1920s
to tackle and find solution of nonlinear hydrodynamical equations. This method,
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Microwave Circuits & Components
however, has been extended to solve many other problems. This method is based
on approximations which permit replacing differential equations by finite
difference equations. Finite difference approximations relate the value of the
dependent variable at a point in solution region to values at some neighbouring
points. Following three steps are involved in the finite difference method.
(a) Dividing the solution region into grid of nodes.
(b) Approximating the given differential equation by finite difference equivalent
that relates dependant variable at a point in solution region to its values as
neighbouring points.
(c) Solving the difference equations subject to prescribed boundary conditions
and /or initial conditions.
The action taken in three steps is dictated by nature of problem to be
solved, the solution region and boundary conditions. The most commonly used
grid patterns are shown in Fig. 7.14. The three dimensional method can be
considered as extension of two dimensional one shown in Fig. 7.14.
How finite difference approximations are constructed from Partial
Differential Equations (PDE)?
Given a function f(x) shown in Fig. 7.15, its derivative is approximated at
p by slope of arc PD, giving a forward difference formulas.
f x0 x f xi
x
or slope of arc AP yielding the backward difference
f ' x0
f x0 f x0 x
x
or the slope of arc AB, resulting in central difference formula,
f ' x0
f x0 x f x0 x
x
Also the second derivative f(x) at p can be estimated by
f ' x0
f ' x0 x 2 f ' x0 x 2
x
1 f x0 x f x0 f x0 f x0 x
x
x
x
f x0 x 2 f x0 f x0 x
or f " x0
x 2
(7.107)
(7.108)
(7.109)
f " xi
(7.110)
Any approximation of a derivative in teams of values at derivate set of
points is called finite difference approximation. The same result can also be
obtained by using Taylor’s series. Higher order finite difference approximations
can be obtained by taking reverse terms in Taylor series expansion. If infinite
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Design of microwave network modelling & computer aided design
t
jAt
At
Ax
jAx
x
Figure 7.16. Finite difference mesh (variables at x and t)
Taylor series is refrained, an exact solution would be realized. However, for
practical reasons the infinite series is usually truncated after second order term.
This impresses an error which exists in all finite difference solutions.
To apply difference method for solution of function x ,t we divide the
solution region in the x-t plane into rectangles of meshes of sides x and t in the
Fig. 7.16 with the coordinates (x,t) of a typical grid point or node be
x ix ,
i 0 ,1,2........
t jt ,
j 0 ,1,2
(7.111a)
and the value of at p be
p ix , jt i , j
f at
(7.111b)
with this notation, the central difference approximations of derivatives of
x i, j
t i , j
i 1, j i 1 j
i j 1 i , j 1
xx i , j
tt
(7.112a)
2x
2t
i 1, j 2 i , j i 1, j
x 2
i , j 1 2 i , j i , j 1
i, j
t 2
(7.112b)
(7.112c)
(7.112d)
This would become clear by solving a diffusion equation which is
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Microwave Circuits & Components
2
x
2
,
t
(7.113)
0 x t
subject to boundary conditions
o ,b 0 l ,t 0, t 0
(7.114a)
and initial condition
0, x 100
(7.114b)
This problem may be regarded as a mathematical model of the
temperature distribution in a rod of length L = 1m with its end in contact with ice
block (held at 0 oC) and rod initially a temperature 100 oC. Therefore the problem
is to find the maternal temperatureas a function of position and time. Let us solve
this problem by both explicit and implicit methods.
7.8.1.1 Explicit method
Let us choose x 0.1 , r
1
so that
2
r x 2
0.05
R
This relation is for a parabolic (or differentiation) partial differential
equation of the type
t
d 2
2
t
x
where, R is a constant. The equivalent finite difference approximation is
R
R
i , j 1 i , j
t
i 1, j 2 i , j i 1, j
x 2
(7.115)
Equation 7.115 can be written as
i , j 1 r i 1, j 1 2r i , j r i 1, j
(7.116)
This explicit formula can be used to compute x , t t explicitly in
terms of x ,t . Thus the values of f along the first time row (Fig. 7.17), t t ,
can be calculated in terms of the boundary and initial conditions, then the value of
f along the second time row, t 2t are calculated in terms of first time row..
It must be remembered that value of r must be choosen carefully.
The fact that obtaining stable solution depends on r or the size of the time
step t renders the explicit formula Eqn.7.116 inefficient. An implicit formula is
proposed by Crank and Nicholson is valid for finite values of r, if 2 x 2 in
Eqn. 7.115 is replaced by average of central difference formulas on j th and (j+1)th
time rows so that
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Design of microwave network modelling & computer aided design
j+1
ERANIC-NICHOLSON
METHOD
j
i–1
i+1
i
Figure 7.17. Computational technique for Crank-Nicholson method
i 1, j 2 i , j i 1, j
2
x
i , j 1 i , j 1
R
t
2 i 1, j 1 2 i , j 1 i 1, j 1
x2
This can be written as
r i 1, j 1 21 r i , j 1 r i 1, j 1
r i 1, j 21 r i , j r i 1, j
(7.117)
Right hand side of this equation consists of three known values while left
hand side has three unknown values of . This is shown in Fig. 7.17. Thus, if there
are n five nodes along each of five row, I=1,2, …..n results in n simultaneous
equation with n unknown value of f and known initial boundary value of .
Similarly, for j = 1, we obtain n simultaneous equations for n unknown values of
in terms of known values of j = 0, and so on. The combinational accuracy and
additional stability allows use of much larger time step with Crank-Nicholson
method than is possible with explicit method. For r = 1
i 1, j 1 4i, j 1 i 1, j 1
i 1, j i 1, j
(7.118)
Let us now continue with solution of the problem. We need solution for
only 0 x 0.5 due to the fact that the problem is symmetric with respect to
x = 0.5. First we calculate the initial and boundary values using Eqn. 7.114. These
values of f at fixed nodes are x = 0, x = 1 and r = 0. Notice that values of (0,0) and
(1,0) are taken as average of 0 and 100. at the free nodes can be calculated
using Eqn. 7.116.
7.8.1.2 Implicit method
Let us choose x 0.2, r 1 so that t 0.04 . The values of at the
fixed nodes are calculated as in part (a). For free nodes we apply Eqn. 7.116 or
scheme of Fig. 7.18. If we denote i , j 1 by i i 1,2,3,4 the values of for
333
Microwave Circuits & Components
first step can be obtained by solving following equations.
0 4d1 d 2 50 100
d1 4d 2 d 3 100 100
d 2 4d 3 d 4 100 100
d 3 4d 4 0 100 50
we obtain
1 58.13, 2 82.54, 3 72, 4 55.5 at t 0.04
Using these values of we apply Eqn. 7.116 to obtain another set of
simultaneous equation for t = 0.08 as
j+1
FINITE DIFFERENCE
METHOD
j
i–1
i
i+1
Figure 7.18. Computational technique for finite difference method
0 41 2 0 82.54
1 4 2 3 58.13 72
2 43 4 82.54 55.5
3 4 4 0 72 0
which results in 1 34.44 , 2 55.23, 3 56.33, 4 32.08 . This procedure can
be programmed and accuracy can be increased by choosing more points for each
time step. This ends the problem.
In addition, finite differencing of hyperbolic PDEs, finite differencing of
elliptic PDEs, interactive methods, accuracy and stability of FD solutions are
another topics which should also be considered. Any standard text may be
consulted for this purpose. However, finite-difference time-domain (FDTD)
formulation of EM field problem is explained here.
7.8.2
Finite-Difference Time Domain Technique
EM field problem formulated in finite difference and time domain is a
convenient way of solving scattering problems. Finite-Difference Time
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Design of microwave network modelling & computer aided design
Domain (FDTD) was first introduced by Yee31 in 1966 and Taylore and others32-36.
The scheme treats the irradiation of the scatterer as an initial problem.
7.8.2.1 Yee’s finite difference algorithm
In isotropic medium the Maxwell’s equation can be written as
E H
(7.119a)
E
(7.119b)
E
t
The vector Eqn. 7.119 can be expressed as six scalar equations, which
expressed in rectangular coordinates are
H
Η x 1 E y Ε 2
y
t
z
Η y
1 E z Ε x
z
x
(7.120a)
(7.120b)
Η z 1 E x Ε y
t
x
y
(7.120c)
E x 1 H z H y
Ε x
t
y
z
(7.120d)
t
E y
t
1 H x H z
Ε y
z
x
(7.120e)
E z 1 H y H x
Ε z
(7.120f)
y
t
x
Following Yee’s notation, we define a grid point in solution region as
i , j , R ix , jy , Rz
(7.121)
and any function of space and time as
(7.122)
F n i , j , R F i , j , R ,nt
where, x y z is the space increment, t is time increment, where i,
j, R and n are integers using central finite difference approximation for space and
time derivatives that are second order accurate,
n
n
1
1
F n i , j , R F i 2 , j ,k F i 2 , j ,k
0 2
x
335
(7.123)
Microwave Circuits & Components
F n i , j , R F
t
n 1
2
i , j , R F
n 1
2
t
i , j , R 0t 2
(7.124)
In applying Eqn. 7.123 to 7.124 derivatives in 7.120, Yee positions the
components of E and H about unit cell of lattice as shown in Fig. 7.19. Yee
positions components of E and H about unit cell of lattice as shown in Fig. 7.19. To
incorporate Eqn. 7.124 at alternate half time steps, thus, we obtain the explicit time
difference approximation of Eqn. 7.120 as
n 12
Hx
i, j
1
2
n 12
,R 12 H x
i, j
1
2
,R 12
E n i, j 1 ,R 1 E n i, j 1 ,R
y
2
2
y
n
n
1
1
1
i, j 2 ,R 2 E2 i, j,R 2 E2 i, j 1,R 12
t
n 12
Hy
i
1
2
n 12
, j,R 12 H y
i
t
1
2
, j,R 12
(7.125a)
E n i E , j,R 1
2
z
E zn i, jiR 12 E xn i, j,R 12
E xn i 12 , j,R E xn i 12 , j,R 1
i 12 , j,R 12
(7.125b)
SIX NODE TRIANGLE
THREE NODE TRIANGLE
SIX NODE TRIANGLE
FIVE NODE TRIANGLE
FOUR NODE QUADRILATERAL
FOUR NODE TRAPEZIUM
Figure 7.19. Typical finite elements
336
EIGHT NODE HEXAHIDRON
Design of microwave network modelling & computer aided design
n 12
Hz
i
1
2
i
n 12
, j 12 ,R H z
t
1
2
, j 12 ,R
E n i 1 , j 1,R
x
2
i , j ,R
1
2
1
2
i, j
,R E i 1, j
Exn i 12 , j,R
E yn
n
y
1
2
1
2
,R
(7.125c)
i 1 , j,R t
2
E n i 1 , j,R
E xn 1 i 12 , j,R 1
2
1
i
,
j,R
2
t
H n 12 i 1 , j 1 ,R
z
2
2
i 1 , j,R
1
2
2
(7.125d)
i , j ,R
H
i , j,R
H
i , j,R
i, j ,Rj, t
E i, j ,R
,R 1
i, j ,R
t
H
i, j ,R
i, j ,R
H
i, j ,n H i , j ,R
H
i , j ,R
n 12
1
2
n 12
y
1
2
1
2
n 12
y
1
2
1
2
Hz
E yn1 i, j
1
2
1
2
n
y
1
2
n 12
x
1
2
n 12
z
1
2
n 12
z
1
2
1
2
1
2
1
2
n 12
z
1
2
1
2
1
2
1
2
(7.125e)
i, j,R 1 t
2
E n i, j,R 1
E yn1 i, j,R 12 , 1
z
2
i, j,R 12
t
H n 12 i 1 , j,R 1
y
2
2
1
i, j,R
n 12
Hy
n 1
Hx 2
2
i , j,R H i
i, j ,R
1
2
1
2
1
2
1
2
n 12
y
1
2
, j,R
1
2
(7.125f)
It can be seen that all components are present in the quarter of unit cell as
shown typically in Fig. 7.20a. Figure 7.20b illustrates typical relations between
field components. The Figure can be inferred from Eqn 7.120d or 7.125d.
337
Microwave Circuits & Components
Y
X
ACTUAL BONDING
Y
APPROXIMATE BONDING
i NODE NUMBER
j ELEMENT NUMBER
X
Figure 7.20. Solution region with its finite element characteristics
In translating the hyperbolic system of Eqns. 7.125a to 7.125f into
computer code, one must make sure that within the same time loop, one type of
field components is calculated first and the results obtained are used in calculating
another type. Other details may be obtained from standard text.
7.8.3
Moments Method
The EM problems can be stated in terms of an inhomogeneous equation.
L g
L is a operator which can be either differential integral or integro
differential, g is the source function, f is the unknown function whose value is to be
determined.
The method of moment (MOM) is a general procedure for solving the
above equations. It is called method of moment because it involves taking moment
by multiplying by appropriate weighting factor and integrating business37, 38. After
many authors39-41 wrote books and articles, the method of moment became popular
since the work of Richmond42 and Harrington43. This method since then has been
applied to wide variety of problems in E.M. Theory like elements of arrays,
scattering problem S, microstrip and lossy structures. The procedure for applying
MOM to solve above equations involve following steps.
(a) Derivation of appropriate integral equation (IE).
(b) Conversion of IE equation into a matrix equation using basis function and
weighting function.
338
Design of microwave network modelling & computer aided design
(c) Evaluation of matrix elements and
(d) Solving the matrix equation and obtaining the parameters of interest.
An integral equation is any equation involving unknown function f under
the integral sign. The integral equation may be either Fourier or Laplace or Hankel.
Linear integral equations that are most frequently studied fall into two categories.
They are named after Fredholm and Volterra. The Fredholm equations of first,
second and third kind, are
b
f x K x,t b dt
(7.126a)
a
b
f x x K x,t b dt
(7.126b)
a
b
f x a x x K x,t b dt
(7.126c)
a
where, l is a scalar (or possibly complex) parameter. Functions k(x,b) and f(x), the
limits a and b are known white f(x) is unknown. The function k(x,t) is known as
Kernel of integral equation. The parameter l is sometimes equal to unity.
The Volterna equations of first, second and third kind are written on the
next page. These equations have a variable upper limit of integrations. The
equations are
x
f x K x,t t dt
(7.127a)
a
x
f x x K x,t t dt
a
x
f x a x x K x,t t dt
(7.127b)
a
If f(x)=0, the integral Eqns. 126 and 127 become homogeneous. They are
all linear equations because enters in a linear fashions. An integral equation is
nonlinear if appears in the power of n>1. Equation given below
l
f x x K x,t 2 t dt
is a nonlinear equation.
a
7.8.3.1 Green’s function
A more general method of attaining IE from PDE is by constructing
Green's function. The Green's function also known as the source function or
influence function, is the Kenel function obtained from a lower boundary value
339
Microwave Circuits & Components
problem. Green's function v provides a method of dealing with source terms (g in
LQ = g) in a PDE. To obtain the field caused by distributed source by the Green's
function technique, we find efforts of each elementary portion of source and sum
them. If G(r1,r1) is the field at the observation point or caused by a unit point
source at source point r, then the field at r by a source distribution g(r1) is the
integral of g(r1) G(r1,r1) over the range of r1 occupied by the source. G is the
Green's function.
Physically, the Green's function G(r1,r1) represents the potential due to
unit point charge at r1. The Dirchlet problem has solution
2 g in R
f in B
given by
G
g r1 G r1r1 dv1
f n ds
R
(7.127c)
where, n is the outward normal to boundary B of the solution region R.
Equation 7.127 indicates that solution can be obtained provided Green's function
G is known. Thus it can be seen that constructing the Green's function is very
important perhaps more important that finding the real solution.
A linear second order PDE, if is L = g.
The Green's function corresponding to differential operator L can be
obtained by finding solution of point source in homogeneous equation.
L G (r1r1) = (r1r1)
(7.128)
where, r and r1 are the position vectors of the field (x, y, z) and source point (x1, y1,
z1) . (r1r1) is Dirac delta function which is zero if r1r1 and satisfies
1
1
1
S r1r g r dv g r
(7.129)
It can thus be inferred that G(r1r1) physically represents the response of
the linear system to a unit in pulse applied at the point r = r1. We must remember
that the Green's function has following characteristics42.
(a)
G satisfies the equation LG = 0 except at source point r1, i.e.,
LG r1r1 r1r1
(b)
(c)
(d)
1
1
G is symmetric in the sense that G r1r G r ,r
G satisfies the boundary value f on B u G = f on B
The directional directive G has discontinuity at r1 which is specified by
n
the equation.
340
Design of microwave network modelling & computer aided design
G
n ds 1
0
where, n is the outward normal to the sphere of radius as shown in the Fig. 7.21
i.e., r r1 2 .
The property (b) expresses the principle of reciprocity.
In the brief description given here it is not possible to go into the various
methods used for the construction of Green's function for different conditions. For
this purpose standard text should be consulted. For a biological body illuminated
by a plane EM waves following tensor integral equation E r is
r
1
1
1
1
j
1
E r pv r E r .G r1r dv E r 8
3
jw
0
v
(7.130)
all symbols used here are explained in reference43. r r jw r 0 and
the initial electric field E i are known quantities. In the next section for the present
case we will discuss the Transformation to Matrix equation.
y
3
1
Ve1 (x1, y1)
2
Ve2 (x2, y2)
x
Figure 7.21. Typical triangular element
7.8.3.2 Transformation to matrix equation
The inner product E r .G r1r1 (Discretisation) in Eqn. 7.130 may be
written as
G xx r1r1 G xy r1r1 G xz r1r1 E r 1
x
1
1
1
1
E r .G r1r G yx r1r G yy r1r G yz r1r
E y r1
G r r1 G r r1 G r r1 E r 1
zy 1
zz 1
zx 1
z
341
(7.131)
Microwave Circuits & Components
showing that G r1r1 is a symmetric y ad. If we let x1=x, x2=y, x3=z, then
E r .G r1r1 can be written as
2
1
1
G xp .xq r1r 1 jw 0 pq 2
G0 r1r , pq 1,2,3
R0 xq xp
(7.132)
Let us now apply MOM to transform Eqn. 7.132 into a matrix equation.
We partition the body into N subvolumes or cells, each denoted by Um(m=1,2,….,
N) and assume E r and r are constant with each cell. If rm is the centre of the
mth cell requiring each scalar component of Eqn. 7.130 be satisfied at ( rm ), this
leads to
3
r
1
1
i
1
E xp rm rn pv Gxp xq rm r dv .E xq rm E xp rm
3
jw
1
q
0
vm
(7.133)
If we let G xp xq be N N matrix with elements
1 r
mn
rn pv Gxp rq rm1r1 dv1 pq mn
Gxp.xq
xn
3 jw 0
(7.134)
i
where, m, n = 1,2,…….N, p, q=1, 2, 3 and let [Exp] and E xp be column.
E xp
E xy r1
E xb r1
, E xp
E r
E r
xp n
xp 1
(7.135)
then from Eqns. 7.130 and 7.133, we obtain 3N simultaneous equations for Ex, Ey
and Ez at the centres of N cells by the point matching technique. These
simultaneous equation can be written as
E xi
G xx G xy G xz E
x
i
G yx G yy G yz E y E y
E i
G zx G zy G zz E z
z
(7.136a)
or simply G E E i
(7.136b)
where, [G] is 3N 3N matrix and [E] and [Ei] are 3N column matrices.
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Design of microwave network modelling & computer aided design
7.8.3.3 Evaluation of matrix elements
The elements of Eqn. 7.136 are yet to be found out. These elements can
be found out by solving Eqn. 7.134. For the off diagonal elements of Gxp xq is not
in the n cell rm is not in v n so that Gxp xq rm r is continuous in vn and the
principal value operation can be dropped. Equation 7.134 becomes
1
th
mn
1
1
Gxp
xq rn G xp.xq rm , r dv m n
vn
(7.137)
As a first approximation
mn
1
Gxp
xq rn Gxp xq rm , r vn , m n
(7.138)
where, vn is the volume of cell vn. Incorporating the relation
Gox r1r1 jw 0 1
1
R2
(7.139a)
G0 r , r1
and Eqn. 7.132 into the Eqn. 7.138 yields
mn
Gxp
xq
jw R0 U n rn exp j mn
3
4 mn
1 j w mn 3 2 ,m n
m
pq
xp
mn
mn
(7.139b)
where, mn R0 Rmn , Rmn r m r n
mn
cos xp
rm
x mp x np
Rmn
m m m
x1 ,x2 ,x3
mn
,cos xq
,r
m
xqm xqn
Rmn
x1n ,x2n ,x3n
This approximation yields adequate results provided N is large. If greater
accuracy is desired Eqn. 7.136 should be numerically evaluated.
For diagonal terms (m=n) Eqn. 7.135 becomes
r
G xp .xq rn pv G xp .xq rn , r 1 dv1 pq 1
3 jw 0
vm
(7.140)
To evaluate this integral, vn cell is approximated by an equivolume I
sphere of radium Gn centered at rn1 i.e.,
343
Microwave Circuits & Components
v
or
4 3
an
3
12
3v
an
4
After lengthy calculation one obtains
2 w r
rn
0 n
mn
Gxp
exp jR0 an 1 jR0 an 1 1
xq
pq
3
3 jw 0
3R 0
(7.141)
, m n
(7.142)
In case, the shape of cell vn differs considerably from that of a sphere, the
approximate Eqn. 7.142 may yield poor results. For greater accuracy vn may be due
to either a cube or a cylinder or sphere created around rn . The integration is then
commanded numerically for the rest of the volume.
The method of moments is a powerful numerical method capable of
applying weighted residual techniques to reduce an integral equation to a matrix
equation. Then the matrix equation is solved using inversion, illumination or
iterative technique. The method moment is generally applied to open structures
such as radiation and scattering, it has been successfully used for closed structures
like waveguides and cavities. The background and reference material are covered
but of in-depth studies literature must be studied carefully. General concepts of
MOM are covered in references44-45.
7.8.4
Finite Element Method
Although the mathematical treatment of finite element method (FEM) was
given by Courant46 in 1943 but it was applied to the electromagnetic problem in
1967. The finite difference method (FDM) and method of moment (MOM) are
easier to programme than finite element method (FEM), it is more powerful and
has its versatile numerical technique for solving complex geometries and
inhomogeneous method. Several FEM tests texts have been written on GEM. A
few of them on electrical engineering are by Slvester and Ferrari 47, Chavi and
Slvester48 Steele 49, Hoole50 and Itoh51. Due to its flexibility and versatility, the
finite element method has become a powerful tool throughout engineering
discipline. It has been applied with great success to numerous EM related
problems. The systematic generality of method makes it possible to construct
general purpose computer programmes for solving a wide range of problems 52.
The finite element analysis of any problem involves53
(a)
(b)
(c)
(d)
Discretizing the solution region into finite number of superegious or elements.
Deriving governing equations for a typical elements,
Assembling of all elements in the solution region, and
Solving system of equation obtained.
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Design of microwave network modelling & computer aided design
Discretization of continuum involving dividing up the solution region into
subdomain called finite elements. Figure 7.21 shows a typical elements for one,
two and three dimensional problems.
Solution of Laplace’s Equation
As an example let us take a Laplace equation problem.
2v 0
This can be strictly treated as four step problem and is discussed briefly.
7.8.4.1 Finite element discretisation
To find the potential v(x,y) for two dimensional solution region shown in
Fig. 7.22, the region can be divided into number of finite elements as illustrated,
the solution region is subdivided into more non-overlapping finite elements;
elements 6, 8 and 9 are four node quadrilaterals while others are three node
3
1
2
Figure 7.22. Shape of factor
triangles. In practical solutions, however it is preferred for case of competition, to
have elements of same type throughout the region. The four quadrilaterals can be
split into triangles so that there are 12 triangular elements.
An approximation for Ve within an element e and then interrelate the
potential distribution in various elements such that potential is continuous across
inter element boundaries. The approximate solution for the whole region is
v x, y
N
Ve x, y
(7.143)
e 1
where, N is the element of triangular regions
The most common form of approximation for Ve within an element in
polynominal approximation namely
Ve x , y a bx cy
(7.144)
for triangular element and
Ve x , y a bx cy dxy
(7.145a)
345
Microwave Circuits & Components
for quadrilateral element. The constants a, b, e, d are to be determined. The
potential Ve in general is non zero within element e but zero outside e. Remember
that quadrilateral elements do not conform to curved boundary as easily as
triangular boundary and therefore they are used more often. Assumption of linear
potential within triangular element is similar to assuming that electric field is
uniform in the element, i.e.,
Ee Ve bax Cay
(7.145b)
7.8.4.2 Elements governing equations
Consider a typical triangular element shown in Fig. 7.23. The potential
Ve1 Ve2 and Ve3 at nodes 1, 2, 3 respectively are obtained using the relation
Ve1 1 x1
V 1 x
2
e2
Ve3 1 x3
y1 a
y2 b
y3 c
(7.146)
The coefficients a, b, c is determined using Eqn. 7.146 as
a 1 x1
b 1 x
2
c 1 x3
y1
y2
y3
1
Ve1
V
e2
Ve3
(7.147)
Substituting this into Eqn. 7.145
x2 y3 x3 y2
1
y2 y3
Ve 1xy
2A
x3 x2
x3 y1 x1 y3 x , y2 x2 y1 Ve1
y3 y1
y1 y2 Ve2
x1 x3
x2 x1 Ve3
(7.148a)
or
3
Ve i x, y Vei
(7.148b)
1
where,
1
1
x 2 y3 x 3 y2 y2 y3 x x3 x2 y
2A
(7.149a)
2
1
x 3 y1 x1 y3 y3 y1 x x1 x3 y
2A
(7.149b)
346
Design of microwave network modelling & computer aided design
y
MEDIUM 1
MEDIUM 2
x
Figure 7.23. Discretisations of inhomogeneous solution region
3
1
x1 y2 x 21 y1 y1 y2 x x2 x1 y
2A
(7.149c)
and A is the area of element, i.e.,
1 x1
2 A 1 x2
1 x3
or A
y1
y 2 x1 y 2 x 2 x 3 y1 x1 y 3 x 2 y 3 x 3 y1
y3
1
x2 x1 y3 y1 x3 x1 y2 y1
2
(7.150)
The value of A is positive if nodes are numbered counter-clockwise from
any node as shown in Fig. 7.23. The Eqn. 148 gives potential at any point (x,y)
within the relevant provided. Note that potential at vertices are known.
Note that i are linear interpolation function and they have following
properties 54.
1 i j
0 i j
i
3
i x, y 1
i
The shape functions are illustrated in Fig. 7.22 (a, b, c).
The functional corresponding to Laplace’s equation 2V 0 is given by
347
Microwave Circuits & Components
We
2
2
1
1
Ee d s Ve d s
2
2
(7.151)
We is energy per unit length associated with element e. From Eqn. 7.148b.
3
(7.152)
Ve Vei i
i 1
Substituting 7.152 in 7.153
We
1 3
2 i
1
3
Vei xi , j ,ds Vej
i 1
(7.153)
If we define the term in brackets as
Cijie i , j ,ds
then
We
(7.154)
1
Ve t C e Ve
2
(7.155)
where, the subscript denotes
Ve1
Ve Ve2
(7.156)
Ve3
and
C e C e C e
12
13
11
e C e C e
C e C 21
(7.157)
22
23
e
e C e
C
C
32
33
31
e
is usually called the element coefficient matrix. The
The element C
element Cijie of the coefficient may be regarded as the compiling between nodes i
and j its value is obtained from Eqns. 7.149 and 7.154
As an examples we can take
V .V ds
C12
i 2
1
y2 y3 y3 y1 x3 x2 x1 x3 ds
4 A2
1
y y3 y3 y1 x3 x2 x1 x3
4A 2
e
(7.158a)
Similarly
e
C13
1
y2 y3 y1 y2 x3 x2 x2 x1
4A
348
(7.158b)
Design of microwave network modelling & computer aided design
e
C23
1
y3 y1 y1 y2 x1 x3 x2 x1
4A
e
C11
1
y2 y3 2 x3 x2 2
4A
e
C22
1
y3 y1 2 x1 x3 2
4A
(7.158e)
(7.158f)
1
y1 y2 2 x2 x1 2
4A
Further it can be seen that
e
C33
(7.158c)
e C e ,C e C e ,C e C e
C 21
12
31
13
3e
23
(7.158d)
(7.159)
7.8.4.3 Assembling all elements
Having considered a typical element, the next step is to assemble all such
elements in the solution region. The energy associated with the assemblage of
elements is
W
N
1
We 2 V t C V
(7.160)
e 1
where,
V
1
V V2
V3
V
n
(7.161)
n is the number of nodes and N is the number of elements. C is called the
overall or global coefficient matrix, which is the assembly of individual element
coefficient matrices. For obtaining 7.154 it has been assumed that whole solution
region is homogeneous so that is the constant. For an inhomogeneous region
such as shown in Fig. 7.23, the region is discretized in such a way that each finite
element is homogeneous. In this case the equation
We
2
2
1
1
E ds Ve ds
2
2
349
Microwave Circuits & Components
2
4
5
3
2
2
3
3
1
2
1 1
2 1
3
1
Figure 7.24. Assembly of three elements i, j, k corresponding to local numbering
2-2-3 of elements.
holds but the Eqn. 7.155 does not apply since r 0 varies from element to
element. Equation 7.155 should be replaced by 0 and the integer in equation
given below.
Cijie i . i ds
multiply the integer by r .
Let us clarify this by taking an example. Let us consider finite element
consisting of three finite element as shown in Fig. 7.24. The numbering 1, 2, 3, 4, 5
is called global numbering. I-j-r is called elements of Fig.7.23. For example, for 3
in the Fig. 7.26 the global numbering is 3-5-4, which corresponds to local
numbering 1-2-3 of element of Fig. 7.23. For example, for element 3 in Fig. 7.24
the global numbering 3-5-4 corresponds to local numbering 1-2-3 of Fig. 7.23.
Numbering must be counter-clockwise for number 3, 4-3-5 instead of 3-5-4 to
correspond with 1-2-3 of element in Fig. 7.23. Thus, whichever numbering is used
the global coefficient matrix remains the same. Assuming the particular numbering
in Fig. 7.24 the global coefficient matrix is expected to have the form
C11
C
21
C C31
C41
C51
C12
C22
C32
C42
C52
C13 C14
C23 C24
C33 C34
C43 C44
C53 C54
C15
C25
C35
C45
C55
(7.162)
which is 5x5 matrix since five nodes (n = 5) are involved. Again, Cij is the coupling
between nodes i and j. We obtain Cij by using the fact that potential difference must
be continuous across inter-element boundaries. The to i, j for example, Fig. 7.24
elements 1 and 2 have node 1 in common hence
1 C 2
C11 C11
11
(7.163a)
350
Design of microwave network modelling & computer aided design
Node 2 belongs to element 1 only j hence
1
C 22 C33
(7.163b)
Node 4 belongs to elements 1, 2 and 3 consequently
1 C 2
C 44 C 41 C12
13
(7.163c)
Nodes 1 and 4 belongs simultaneously to elements 1 and 2; hence
1 C 2
C14 C 41 C12
13
(7.163d)
Since there is no coupling 1 or direct link between nodes (2) and (3).
C23 C32 0
(7.163e)
Coupling in this manner, all the term in global coefficient matrix is
obtained and Eqn. 7.156 becomes
2
1 C 2
C 1 C 2 C
C12
C12
0
13
13
11 1 11
1
1
C33
C32
0
0
C31
2
2 C 3
2 C 3
3
C23
C13
0 C 22
C21
11
13
C 1 C 1 C 1 C 2 C 3 C 1 C 2 C 3 C 3
31
23
32
31
22
33
33
32
21
3
3
3
C21
C23
C22
0
0
(7.164)
Note that element coefficient matrices overlap at nodes shared by
elements and that there are 27 terms (9 for each of the 3 elements) in the global
coefficient matrices which overlap at nodes shared by elements and that there are
27 terms (9 for each of 3 elements) in the global coefficient matrix C. Also note the
following properties of matrix C.
(a) It is symmetric (Cij = Cji ) just as element coefficient matrix
(b) Since Cij= 0 if no coupling exists between node i and j, it is expected that for
a large elements of elements c becomes sparse matrix c is also banded if the
nodes are carefully numbered. It can be shown using Eqn. 7.159 that
3
3
i 1
i 1
e
e
Cij 0 Cij
(c) It is singular. Although this is not so obvious, it can be shown using the
element coefficient matrix of 7.152.
7.8.4.4 Solving the resulting equation
It can be shown that Laplace’s equation is satisfied when the total energy
in the solution region is minimum. Therefore, the requirement is that the partial
derivative of W with respect to each nodal value of the potential be zero, i.e.,
351
Microwave Circuits & Components
or
W W
W
0
V1 V2
Vn
W
0 , R 1,2 ,n
VR
(7.165)
W
0 for finite element mesh, we substitute Eqn.
V1
7.126 into 7.154 and take partial derivation of W with respect to V1 we obtain
For example to get
O
W
2V1C11 V2C12 V3C13 V4C14 V5C15
V1
V2C21 V3C31 V4C41 V5C51
or
O V1C11 V2C12 V3C13 V4C14 V5C15
In general,
(7.166)
W
0 leads to
VR
n
(7.167)
O CiCiR
i 1
where, n is the number of nodes in the mesh. Writing Eqn. 7.161 for all nodes
R = 1, 2,…….n we obtain a set of simultaneous equations from which the solution
of [V]1 = [V1, V2 …..Vn] can be found. It can be done in two ways.
7.8.5
Iterative Method
Let us choose node 1 in Fig. 7.26 as free node. From Eqn. 7.160
V1
1 5
ViCii
C11 i
2
(7.168)
Thus, in general, a node R in a mesh n nodes
VR
1
CRR
n
ViCRi
(7.169)
i 1,i R
where, node R is a free node. Since CRi = 0 if node R is not directly connected to
node i, only nodes that are directly linked to node R contribute to VR in Eqn. 7.161.
Equation 7.161 can be applied tentatively to all free nodes. The iteration process
begins by setting the potentials of fixed nodes (where potentials are prescribed or
known) to their prescribed values and the potential at free nodes equal to or to
average potential55.
1
(7.170)
Vmin Vmax
2
where, Vmin and Vmax are minimum and maximum value of V at fixed nodes. With
Vave
352
Design of microwave network modelling & computer aided design
these initial values the potentials at free nodes are calculated using Eqn. 7.161.
After the first iteration when new values are calculated for all the free nodes, they
become old values for second iteration. This is repeated many times and the change
between subsequent iteration is negligible.
7.8.6
Band Matrix Method
If all nodes are numbered first and the fixed nodes last, Eqn. 7.155 can be
written as47
W
1
V f Vp
2
CC ff
pf
C fp
C pp
V f
V
p
(7.171)
where, f and p, respectively refer to nodes with free and fixed 1 or prescribed
potential. Since Vp is constant (if consists of known fixed values). We only
differentiate with respect to Vf so that applying Eqn. 7.159 to 7.165 yields
or
C ff C fp VV f 0
p
C ff V f C fp V p
(7.172)
This equation can be written as
or
AV B
(7.173a)
(7.173b)
V A1 B
V V f , A C ff ,B C fp V p since A in general is non singular,,
where,
the potential at the free nodes can be found using Eqn. 7.173. We can solve for [V]
in Eqn. 7.173a using Gaussian elimination technique. We can also solve for [V] in
Eqn. 7.173b using matrix in region of the size of matrix to be inverted is not large.
It is sometimes necessary to impose Neumann condition at the line of symmetry
when we take advantage of symmetry of the problem. Suppose the solution region
is symmetric along Y axis, we impose the condition V 0 along y-axis by
x
making
(7.174)
V1 V2 ,V4 V5 ,V7 V8
Notice that Eqn. 7.151 onwards the solution has been restricted to a two
dimensional problem involving Laplace’s equation, 2V 0 . The basic concepts
developed will be extended to finite element analysis of problems involving
pv 2
Poisson’s equation 2V
, A j or wave equation 2 r 2 0
353
Microwave Circuits & Components
As can be shown the basic concepts and applications of finite element
method has been introduced in this chapter without going into details. In no way it
can be considered to be complete.
7.9
SOLUTION TO SOME PROBLEMS
The numerical technique involved in solving of problems has not been
discussed separately in the text. This aspect is used and discussed in the present
relation. We will start with simple problems but take up more difficult ones towards
the end emphasizing the numerical techniques involved.
Problem 7.2: A transmission line of length 10 meters at a frequency of 100.0 MHz
attenuation per length of 0.002 nepers/m. If the phase velocity of the line is
2.8 108 m/s and the line has characteristic impedance of 50 ohms, what is the
value of terminal load impedance looking into the line is measured to be 30 – j10
ohms.
Solution: Using the transmission line relations it is found that
Zs
tan h ( l )
ZT Z o
Z o 1 Z s tan h l
Zo
(7.175)
where, ZT is the terminating impedance to be found
Sending end impedance Zs = 30 - j10 ohms
Characteristic impedance Zo = 50 ohms
l = 10 meters
= 0.002 nepers/m
p= 2.8 108 m/s
so that,
Zs
0.6 j0.2
Zo
l = 2 10-3 10 = 0.02 nepers
2l = 0.04 nepers
l
l 2 100 106 10
22.43 radians
p
2.8 108
tan h (l) = tan h ( l + j 22.43)
tan h (0.02 + j 22.43)
354
Design of microwave network modelling & computer aided design
sin h (0.04) j sin (44.83)
cos h (0.04) cos (44.83)
sin h (0.04)
exp (0.04) exp ( 0.04)
0.04
2
cos h (0.04) =
1.0408+ 0.96079
= 1.0008
2
sin (44.86) = sin (44.86 -14) = sin (44.86 – 43.96)
= sin (0.9)
= 0.62
cos (0.9) = 0.79
tan h ( l) =
0.04 + j 0.62 0.02 j 0.62
0.1 j 2.93
1.0008 - 0.79
0.211
Therefore,
ZT
0.6 j 0.20.1 j 2.93
0.5 j 2.73
Z o 1(0.6 j 0.2)(0.1 j 2.93) 0.36 j1.76
ZT 2.73 j 4.55
1.55 j 0.34
Z o 1.76 j 4.89
1.46 j0.53
ZT 73 j 26.5 ohms
A programme for these calculation has been written in BASIC. This
programme is framed in terms of reflection coefficients
] [ FORMATTED LISTING
FILE:
PROGRAMME Z TERM
PAGE - 1
10 REM
20 REM ***Z TERM
30 REM
40 REM CALCULATES TERMINATING END
50 REM IMPEDANCE FOR A
60 REM UNIFORM TRANSMISSION
70 REM LINE, WITH OR WITHOUT
80 REM LOSS
90 REM -L GIVES MOVEMENT
100 REM TOWARDS LOAD
110 REM + L GIVES MOVEMENT
120 REM TOWARDS GENERATOR
130 REM
140 REM F = FREQ. (GHz)
150 REM LINE LTH. (cm)
160 REM Az = ATTENUATION (DB/cm)
170 REM Z0 = CHARC. JMP. (OHMS)
180 REM E = REL/EFF. PERM
190 HOME
355
Microwave Circuits & Components
200 PRINT ‘INPUT CHARC. IMP. (OHMS)’
2
1
0
I
N
P
U
T
Z
0
220 PRINT INPUT FREQ. (GHz)
230 INPUT F
240 PRINT "INPUT ATTN. (DB/cm)"
250 INPUT A2
260 PRINT INPUT REL. OR EFF PERM’
270 INPUT E
280 PRINT INPUT LINE LTH. (CM)
290 INPUT L
300 INPUT R 1
320 PRINT "INPUT LOAD REACTANCE (OHMS)"
330 INPUT X1
340 LET R = R1
350 LET X=X1
360 LET W1 = 30 /(F* SQR (E))
370 LET R2 = (R1 * R1 - Z0 * Z0 + X1 * X1 )/(R1 + Z0) * (R1 + Z0) + Y1 * X1
380 LET ´ 2 = 2 * ´ 1 * Z0/[(R1 + Z0) * (R + Z0) * (R + Z0) + X1 * X1
390 IF X2 < > 0 THEN 430
400 IF R2 > = 0 THEN 420
410 LET G = -3.1415927>
GO TO 053
420 LET G = 1 E - 20
GO TO 530
430 IF R< = 0 THEN 460
440 IF 2 = 0 THEN 390
450 LET G = ATN (X2/R2):
GO TO 530
460 IF R2 < > 0 THEN 500
470 IF X2 > = 0 THEN 490
480 LET G = -1.5707963:
GO TO 530
490 LET G = 1.5707963:
GO TO 530
500 IF R2 > = 0 THEN
530
510 IF X2 = 0 THEN
390
520 LET G = 3.1415927 + ATN (X2)
530 LET T1 = G
540 LET M1 = SQR R2 * R2 + X2 * X2
550 LET A2 = A2/8-680
560 LET T.2 = T1 - 4 * 3.1415927 * L/W1
570 LET M2 = M1 * EXP [- 2 (2 * A2 * L)]
580 LET D = 1 - 2 * M2 * COS (T2) + M2 * M2
590 IF D = 0 THEN
LET D = 1E - 20
600 LET R1= Z0 * (1 - M2 * M2)/D
610 LET X1 = Z0 * 2 * M2 * SIN (T2)/D
620 PRINT
630 PRINT
640 PRINT "XXXX RESULTS XXXX"
650 PRINT
660 PRINT “ LINE LT H . (CMS)” L
670 PRINT “ REL. OR EFF. PERM.” E
680 PRINT “OPERATING FREQ. (GHz)” F
690 PRINT “ ATT (DB/CM)” INT (A2 * 1000 * 8.686 + .5)/1000
700 PRINT “ CHARAC. IMP. (OHMS)” Z0
710 PRINT “ LOAD IMPEDANCE”
720 “ “ INT (1000 * R/1000 + .5)” “INT (1000/1000 + .5)” J OHMS”
730 PRINT
740 PRINT “SENDING END IMPEDANCE”
750 PRINT “ INT (1000 * R1/1000 + .5)” “IWT (1000 * X/1000 + .5)” jOHMS
760 PRINT
356
Design of microwave network modelling & computer aided design
770 PRINT “ ************”
780 PRINT
790 “ FINISHED? IF NO ENTER 1”
800 INPUT T
810 IF T + 1 THEN 100
820 PRINT
830 PRINT “ £ £ £ £ £ £ END OF PROGRAMME £ £ £ £ £ £"
840 END
850 F-LISTING
PUT
860 CHARC. IMP. (OHMS)
PUT
870 FREQ. (GHz)
PUT
880 ATTN (DB/CM)
PUT REL. OR EFF. PERM
PUT LINE LTH (CM)
PUT LOAD RES. (OHMS)
PUT LOAD REACTANCE (OHMS)
************ RESULTS *************
LINE LTH. (CMS). 75
REL. OR EFF. PERM. 1
OPERATING FREQ. (GHz) 10
ATT (DB/CM) 0
CHARAC. IMP. (OHMS) 50
LOAD IMPEDANCE
50
50J
OHMS
SEND END IMPEDANCE
25 - 25 J OHMS
‘FINISHED? IF NO ENTER 1
£ £ £ £ £ £ £ END OF PROGRAMME £ £ £ £ £ £ £
Problem 7.3. Design curves gives following of parameter for a 70 microstrip
line
Characteristic impedance
70
18
r
2.0
10
z0
99.0
57
1.6
4
Shape ratio (w/h)
The relative dielectric constant of substrate is 2. Find the characteristic
impedance for microstrip line for a shape ratio of 4 constructed on a substrate of
dielectric constant 10. Assume that the conductors have zero thickness.
Solution: A computer programme 4.2 single MIC has been developed for this
purpose. The relations which have been used are for a relative dielectric constant
less than 16. Generally thickness of conductor has been assumed to be finite. It
may be noted that for this programme the usable range of shape ratio is 0.65–20.
Normally this covers the practical range.
] [ FORMATTED LISTING
FILE PROGRAMME 4.2 SINGLE MIC
PAGE 1
10 REM
20 REM --SINGLE MICROSTRIP--
357
Microwave Circuits & Components
30 REM
40 REM THIS PROGRAMME CAN
50 REM BE USED FOR THE
60 REM ANALYSIS OR SYNTHESIS
70 REM OF SINGLE MICROSTRIP
80 REM LINE, PROVIDED ER IS
90 REM LESS THAN 16, AND
100 REM 0.65 < =W/H < =20
110 REM FINITE CONDUCTOR
120 REM THICKNESS IS INCLUDED
130 REM
140 REMW/H=WIDTH/HEIGHT
150 REM H=DIE. THICKNESS (MM)
160 REM W=LINE WIDTH (MM)
170 REM ER=REL. DIE. CONST.
180 REM EEF=EFFECTIVE DIE.
190 REM CONSTANT
200 REM T=COND. THICKNESS (MM)
210 REM
220 REM INPUT ROUTINE
230 HOME
240 PRINT “ENTER 1 FOR SYNTHESIS”
250 PRINT “ ENTER 0 FOR ANALYSIS ”
260 INPUT K
270 IF K < 0 OR K > 1 THEN 230
280 PRINT
290 PRINT “----- INPUT LINE GEOMETRY -----”
300 PRINT
310 PRINT “INPUT CONDUCIOR THICKNESS (MM)”
320 INPUT T
330 PRINT “INPUT DIELECTRIC HEIGHT (MM)”
340 INPUT H
350 PRINT “ENTER REL. DIE CONSTANT”
360 INPUT ER
370 LET TH = T/H
380 LET PI = 3.14159
390 IF K = 1 THEN 570
400 REM
410 REM ANALYSIS ROUTINE
420 PRINT
430 PRINT “INPUT LINE WIDTH (MM)”
440 INPUT W
450 LET WH=W/H
460 PRINT
470 PRINT “***** ANALYSIS RESULTS *****”
480 COSUB 1060
490 COSUB 740
500 PRINT “LINE WIDTH (MM)” W
510 PRINT
520 PRINT “EFFECTIVE DIELECTRIC CONSTANT ” EFF
530 PRINT “CHARAC. IMP. “ZO” OHMS”
540 PRINT
550 “*************************”
560 GO TO 1130
570 REM
580 REM SYNTHESIS ROUTINE
590 PRINT
600 PRINT “ ENTER CHARAC. IMP. (OHMS)
610 INPUT ZO
620 GO SUB 910
630 PRINT
640 PRINT “ ***** SYNTHESIS ROUTINE ***** ”
650 GOSUB 1060
358
Design of microwave network modelling & computer aided design
660 PRINT “ CHARAC. IMP. “ZO” OHMS ”
670 PRINT “LINE WIDTH (MM)” W
680 PRINT
690 PRINT “ ******************************* ”
700 GO TO 1130
710 PRINT
720 PRINT “ ******* END OF PROGRAMME ******* ”
730 END
740 REM
750 REM ANALYSIS ROUTINE
760 IF WH > = 1/2/PI THEN 790
770 LET WHE = WH + TH/PI * [1 + LOG (4 * PI * W/T)]
780 GO TO 800
790 LET WHE = WH + TH/PI * [1 + LOG (2/TH)]
800 IF WH > = 1 THEN
850
810 LET EEF = [1/SQR (1 + 12/WH) + 0.04 * (1 - WH) 2]
820 LET EEF = (ER + 1)/2 + [(ER - 1)/2) * EEF
830 LET ZO = 60/SQR (EEF) * LOG (8/WH + 0.25 * WH)
840 GO TO 880
850 LET EEF = (ER + 1)/2 + (((ER - 1)/2)/SQR (SQR (1 + 12/WH))
860 LET ZO = 120 * PI/SQR (EEF)
870 LET ZO = ZO/(WH + 1.393 + 0.667 * LOG (WH + 1.444)]
880 LET ZO = INT (ZO * 100 + 0.5)/100
890 LET EEF = INT (EEF*100+0.5)/100
900 RETURN
910 REM
920 REM SYNTHESIS ROUTINE
930 LET B = 377 * PI/2/ZO/SQR (ER)
940 LET A = [(ER - 1)/(ER + 1)] * (0.23 + 0.11/ER)
950 LET A = A + ZO/60 * SQR [(ER + 1)/2]
960 IF A < = 1.52 THEN
1020
970 LET WH = LOG (B - 1) + 0.39 - 0.61/ER
980 LET WH = (ER -1)/2/ER * WH
990 LET WH = WH + (B - 1 - LOG (2 * B - 1)]
1000 LET WH = 2/PI * WH
1010 GO TO 1030
1020 LET WH = 8 * EXP (A)/[EXP (2 * A) - 2]
1030 LET W = WH * H
1040 LET W = INT (W * 100 + 0.5)/100
1050 RETURN
1060 REM
1070 PRINT
1080 PRINT “CONDUCTOR THICKNESS (MMS) "T
1090 PRINT “DIELECTRIC HEIGHT (MMS) "H
1100 PRINT “REL. DIE. CONSTANT "ER
1110 PRINT
1120 RETURN
1130 REM
1140 REM SERVICE ROUTINE
1150 PRINT
1160 PRINT “DO YOU WANT ANOTHER GO ? ”
1170 PRINT “ENTER 1 FOR YES; 0 FOR NO”
1180 INPUT L
1190 IF L = 1 THEN
230
1200 IF L = 0 THEN
710
1210 GO TO 1130
1220 END OF LISTING
] RUN
ENTER 1 FOR SYNTHESIS
359
Microwave Circuits & Components
ENTER 0 FOR ANALYSIS
?1
----- INPUT LINE GEOMETRY ----INPUT CONDUCTOR THICKNESS (MM)
?0.001
INPUT DIELECTRIC HEIGHT (MM)
? 0.1
ENTER REL. DIE CONSTANT
? 2.3
ENTER CHARAC. IMP (OHMS)
? 50
***** SYNTHESIS ROUTINE *****
CONDUCTOR THICKNESS (MMS) 1E-03
DIELECTRIC HEIGHT (MMS) .1
REL. DIE. CONSTANT 2.3
CHARAC. IMP. 50 OHMS
LINE WIDTH .3 MMS
********************************
DO YOU WANT ANOTHER GO ?
ENTER 1 FOR YES ; 0 FOR NO
1
ENTER 1 FOR SYNTHESIS
ENTER 0 FOR ANALYSIS
----- INPUT LINE GEOMETRY ----INPUT CONDUCTOR THICKNESS (MM)
0.001
INPUT DIELECTRIC HEIGHT (MM)
0.1
ENTER REL DIE CONSTANT
2.3
INPUT LINE WIDTH (MM)
2.3
***** ANALYSIS RESULTS *****
CONDUCTOR THICKNESS (MMS) 1E-03
DIELECTRIC HEIGHT (MMS) .1
REL. DIE CONSTANT 2.3
LINE WIDTH (MM) .3
EFFECTIVE DIELECTRIC CONSTANT 1.94
CHARAC. IMP. 50 OHMS
***********************************
DO YOU WANT ANOTHER GO ?
ENTER 1 FOR YES ; 0 FOR NO
****** END OF PROGRAMME ******
Problem 7.4. Develop a computer programme 4.4 PSTRIP for analysis and
synthesis of parallel edge side coupled strip line circuits.
Before discussing the problem 7.4 some of the relation for coupled
stripline circuit is written below. For this case circuit shown in Fig. 7.25 is used
r 12 Z oe 30 (b t ) / w bc
Ae
(7.176a)
bc
Z ou 30 (b t ) / w
A
2
o
(7.176b)
r
1
2
2
where,
360
Design of microwave network modelling & computer aided design
S
t
W
b
W
+
–
+
EVEN
–
ODD
MODE
MODE
Figure 7.25. Field distribution for stripline closed for analysis (t < 0.1 and
w/b > 0.35).
Ae 1
ln (1 tan h )
l (1 cot h )
; Ao 1 n
0.6932
0.6932
(7.177)
s
2b
and
2b t ln t (2b t )
t
c 2ln
2
p
b t b (b t )
For closed form Synthesis equation
w
2
tan h 1 (ke ko )
b
(7.178)
1 k k
s 2
o
e
tan h 1
p
1 ke ko
1/ 2
361
(7.179)
Microwave Circuits & Components
with
exp(x) 2
k e ,o 1
exp(x) 2
and
1/ 2
for range 1 x
2
exp (π / x ) 2
ke,o
for range 0 x 1
exp (π / x ) 2
where,
x
Z oo ,e ( r )1 / 2
30
For stripline circuits using parallel edge coupling it is difficult to obtain
light coupling that is greater than –10 dB or at the most –6 dB. Loose coupling
means that only narrow band circuits can be constructed.
Solution 7.4:
] [ FORMATTED LISTING FILE PROGRAMME 4.4 P STRIP
PAGE - 1
10 REM
20 REM -- COUPLED STRIPLINE-30 REM
40 REM THIS PROGRAMME COMPUTES
50 REM THE PARAMETERS REQUIRED
60 REM FOR THE ANALYSIS OR
70 REM
SYNTHESIS OF
80 REM COUPLED STRIPLINES.
90 REM BEST ACCURACY OCCURS
100 REM FOR T/B < 0 .1 AND W > 0.35
110 REM
120 REM W = LINE WIDTH (MMS)
130 REM S = LINE SPACING (MMS)
140 REM B = GROUND PLANE SPACING
150 REM GIVEN IN MMS
160 REM ER = REL. DIE. CONST.
170 REM ZE = EVEN MODE IMP.
180 REM ZO ODD MODE IMP.
190 REM
200 HOME
210 PRINT “ FOR ANALYSIS ENTER 1”
220 PRINT “ FOR SYNTHESIS ENTER 0”
230 INPUT P
240 IF P < 0 OR P > 1 THEN
200
250 LET PI = 3.141592
260 PRINT
270 PRINT “INPUT GROUND PLANE SPACING (MMS)”
280 INPUT B
290 PRINT “INPUT CONDUCTOR THICKNESS (MMS)”
300 INPUT T
310 PRINT “INPUT RELATIVE DIELECTRIC CONST.”
320 INPUT ER
330 IF P = 0 THEN
362
Design of microwave network modelling & computer aided design
440
340 PRINT “INPUT LINE WIDTH (MMS)”
350 INPUT W
360 PRINT “INPUT LINE SPACING (MMS)”
370 INPUT S
380 PRINT
390 PRINT “***** ANALYSIS RESULTS *****”
400 PRINT
410 “LINE WIDTH “W” MMS”
420 “LINE SPACING “S” MMS”
430 GO TO 600
440 PRINT “INPUT REQUIRED COUPLING (DB)”
450 INPUT DB
460 “INPUT COUPLER IMPEDANCE (OHMS)”
470 INPUT Z
480 PRINT
490 “**** SYNTHESIS RESULTS ****”
500 PRINT
510 LET X = DB/20
520 LET COUPLE = 10^X
530 LET F1 = (1 + COUPLE)/(1 - COUPLE)
540 LET F2 = 1/F1
550 LET ZE = Z * SQR (F1)
560 LET ZO = Z * SQR (F2)
570 PRINT “REQUIRED COUPLING” INT (COUPLE * 10000 + 0.5)/10000” OR “DB” “DB”
580 PRINT “EVEN MODE IMPEDENCE” INT (ZE * 100 + 0.5)/100 “OHMS”
590 PRINT “ODD MODE IMPEDENCE” INT (ZO * 100 + 0.5)/100 “OHMS”
600 PRINT “ GROUND PLANE SPACING “B” MMS”
610 PRINT “CONDUCTOR THICKNESS “T” MMS”
620 PRINT “REL. DIELECTRIC CONSTANT “ER
630 IF P = 0 THEN GO SUB 740
640 IF P = 1 THEN GO SUB 1010
650 PRINT
660 PRINT “DO YOU WANT ANOTHER GO ?”
670 PRINT “ ENTER 1 IF YES ; 0 IF NO”
680 INPUT P
690 IF P < 0 OR P > 1 THEN
650
700 IF P = 1 THEN
200
710 PRINT
720 PRINT “**** END OF PROGRAMME ****”
730 END
740 REM
750 REM SYNTHESIS ROUTINE
760 FOR I = 1 TO 2
770 IF I = 1 THEN
LET Z = ZE
780 IF I = 2 THEN
LET Z = ZO
790 LET X = Z * SQR (ER)/30/PI
800 LET L = EXP (PI * X)
810 LET M = EXP (PI/X)
820 IF X < = 1 AND X > = 0 THEN – 850
830 LET K = SQR (1 - ((L - 2)/(L + 2)) ^ 4
840 GO TO 860
850 LET K = ((M - 2)/(M + 2)) ^ 2
860 IF I = 1 THEN
LET KE = K
870 IF I = 2 THEN
LET KO = K
880 NEXT I
890 LET WB = LOG ((1 + (SQR (KE * KO))/(1 - (SQR (KE * KO))))/PI
363
Microwave Circuits & Components
900 LET A = (1 - KO)/(1 - KE)
910 LET A = A * SQR (KE/KO)
920 LET SB = LOG [(1 + A)/(1 - A)/PI
930 LET W = WB * B
940 LET S = SB * B
950 PRINT
960 PRINT “LINE WIDTH” INT (W * 1000 + 0.5)/1000” MMS”
970 PRINT “ LINE SPACING” INT (S * 1000 + 0.5)/1000 “MMS”
980 PRINT
990 PRINT “******************”
1000 RETURN
1010 REM
1020 REM ANALYSIS ROUTINE
1030 LET CF = 2 * LOG ((2 * B - T)/(B - T))
1040 LET CF = CF - T/B * LOG ((T * (2 * B - T))/(B - T)/(B - T))
1050 LET TH = PI * S/2/B
1060 LET AO = 1 + LOG (1 + (EXP (- TH)/(EXP (TH) - (EXP (- TH)) * 2 + 1 /LOG(2)
1070 IF AE = 1 + LOG (1 + (- EXP (- TH)/(EXP (TH) + (EXP (- TH)) * 2 + 1 ))/LOG(2)
1080 LET Z = 30 * PI * (B - T)/SQR (ER)
1110 PRINT
1120 PRINT “EVEN MODE IMPEDENCE” INT (ZE * 100 + 0.5)/100 “OHMS”
1130 PRINT “ODD MODE IMPEDENCE” INT (Z0 * 100 + 0.5)/100 “OHMS”
1140 LET C = 20/2.303 * LOG (ABS ((ZE - ZO)/(ZE + ZO)))
1150 PRINT “COUPLING” INT (C * 100 + 0.5)/100 “DB”
1160 PRINT
1170 PRINT “*****************”
1180 RETURN
END OF LISTING
I RUN/?
FOR ANALYSIS ENTER 1
FOR SYNTHESIS ENTER 0
INPUT PLANE SPACING (MMS)
INPUT CONDUCTOR THICKNESS (MMS)
?0.005
INPUT RELATIVE DIELECTRIC CONST.
?2
INPUT LINE WIDTH (MMS)
?9
INPUT LINE SPACING (MMS)
?0.02
****ANALYSIS RESULTS****
LINE WIDTH 9 MMS
LINE SPACING .02 MMS
GROUND PLANE SPACING 10 MMS
CONDUCTOR THICKNESS 5E-03 MMS
REL. DIELECTRIC CONSTANT 2
EVEN MODE IMPEDANCE 59.35 OHMS
ODD MODE IMPEDANCE 22.49 OHMS
COUPLING -6.93 DB
******************************
DO YOU WANT ANOTHER GO ?
ENTER 1 IF YES; 0 IF NO
?1
FOR ANALYSIS ENTER 1
FOR SYNTHESIS ENTER 0
?0
INPUT GROUND PLANE SPACING (MMS)
?10
INPUT CONDUCTOR THICKNESS (MMS)
?0.005
INPUT RELATIVE DIELECTRIC CONST.
?2
364
Design of microwave network modelling & computer aided design
INPUT REQUIRED COUPLING (DB)
?-10
INPUT COUPLER IMPEDANCE (OHMS)
?50
**** SYNTHESIS RESULTS ****
REQUIRED COUPLING
.3162 OR -10 DB
EVEN MODE IMPEDANCE 69.37 OHMS
ODD MODE IMPEDANCE 36.04 OHMS
GROUND PLANE SPACING 10 MMS
CONDUCTOR THICKNESS 5E-03 MMS
RELATIVE DIELECTRIC CONSTANT. 2
LINE WIDTH 7.213 MMS
LINE SPACING .387 MMS
*******************************
DO YOU WANT ANOTHER GO ?
ENTER 1 IF YES ; 0 IF NO
?0
****END OF PROGRAMME****\
Problem 7.5. Write a programme from line specification shown in the Fig. 7.26
with or without dielectric substrate for calculating the capacitance and then line
impedance and then the guide wavelength.
The problem is for an encased microstrip line.
] [ FORMATTED LISTING
FILE: PROGRAMME 4.6 RELGRID
PAGE - 1
10
REM
20
REM
--- RELAXATION GRID--30
REM
40
REM
THIS PROGRAMME USES
50
REM
THE FINITE DIFFERENCE
60
REM
APPROACH FOR A STATIC
70
REM
SOLUTION OF LAPLACE’S .
80
REM
EQUATION
90
REM
FOR THE PUPOSE OF
100
REM
DEMONSTRATION THIS
110
120
130
140
150
160
REM
REM
REM
REM
REM
REM
PROGRAMME HAS BEEN
SET UP TO COMPUTE
CHARAC. IMP. AND
EFFECTIVE PERM.
FOR A SINGLE
MICROSTRIP LINE
j=50
.
r 10
12
11
r 2.3
j=1
i=1
i= 34
i=64
Figure 7.26. Microstrip line housed in metal enclosure
365
i=100
Microwave Circuits & Components
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
640
650
660
REM
HELD IN A METAL
REM
ENCLOSURE
REM
MODS.GIVEN FOR
REM
COUPLED LINES
REM
REM
ITRY = MAX NO OF ITS
REM
X1, Y1 = LOWER COND.
REM
CORNER
REM
X2, Y2 = UPPER COND.
REM
CORNER
REM
RES = DAMPING COEFF.
REM
DAMPING COEFF.
REM
ER = REL. DIE . CONST.
REM
ACC = CONVERGENCE ACC.
REM
REM
FOR THIS ARRAY
REM
MAX BOX SIZE
REM
IS 100 * 50
REM
FOR ZOE SET
REM
DIMV (101, 50)
REM
REM
PROGRAMME IS TERMINATED
REM
WHEN LINE CAP.
REM
IS SEEN TO CONVERGE
REM
V(100, 50)
HOME
LET MEW = 12.57E – 7
LET EO = 8.854E – 12
HOME
PRINT “INPUT RELATIVE DIELECTRIC CONSTANT”
INPUT ER
LET ACC = 0.01
LET D = 0
LET RES = 1.5
PRINT “INPUT MAX NO. OF ITERATIONS ALLOWED”
INPUT ITRY
REM
REM
SET UP METAL CASE
PRINT “I/P DIMENSIONS OF METAL CASE”
PRINT “FIRST X-COORDS THEN Y-COORD"
INPUT X, Y
REM
REM SET UP CENTER COND
PRINT "I/P DIMENSIONS OF CENTER CONDUCTOR"
PRINT "RELATIVE TO ENCLOSURE ORIGIN"
PRINT “FIRST X-COORDS"
INPUT X1, X2
PRINT “NOW Y-COORDS"
PRINT “INPUT Y1, Y2
670
680
690
700
710
720
730
740
750
760
770
780
790
800
HOME
PRINT
*******************************
PRINT
PRINT "RELATIVE DIELECTRIC CONSTANT' ER
PRINT “MAX. NO. OF ITERATIONS "ITRY
PRINT "CASE DIMENSIONS "X" X "Y:
PRINT
PRINT "CENTER CONDUCTOR: ---"
PRINT "X-COORDS "X1", "X2"
PRINT "Y-COORDS "Y1", "Y2"
PRINT
REM SET COND. TO 1 VOLT
FOR ZOO SET
366
Design of microwave network modelling & computer aided design
810
820
830
840
850
860
870
880
890
900
910
920
930
940
950
960
970
980
990
1000
1010
1020
1030
1040
1050
1060
1080
1090
1100
1110
1120
1130
1140
1150
1160
1170
1180
1190
1200
1210
1220
1230
1240
1250
1260
1270
1280
1290
1300
1310
1320
1330
1340
1360
FOR I = (X–1–(X2 – X1))
TO (X–1)
FOR I = X1 TO X2
FOR J = Y1 TO Y2
LET V(I,J) =1
NEXT J
NEXT I
REM
REM E = (1 + ER)/2
REM
REM
REM SET ITERATION LOOP
FOR L7 = 1 TO 2
IF L7 = 2 THEN
LET ER = 1
FOR L4 = 1 TO ITRY
LET E1 = 0
LET E2 = 0
REM FOR ZOE SET
REM FOR I = 2 TO X
REM AND INSERT NEW LINE
REM V(101, J) = V (99, J)
FOR I = 2 TO (X–- 1)
FOR J = 2 TO (Y – 1)
IF V(I, J) = 1 THEN 1140
IF (J – Y1) = 0 THEN 1070
GO TO 1090
LET VCAL = (V(I, J + 1) + ER * V(I, J - 1) + E * (V(I + 1,J) +
V (I – 1, J)))/(1 + ER + 2 * E) * RES + (1 - RES) * V (I, J)
GO TO 1100
LET VCAL = (V (I + 1, J) + V(I - 1, J) + V(I, J + 1) + V(I, J – 1))/4 * RES
+ (1 – RES) * V(I, J)
LET D1 = (VCAL - V(I, J)) ^ 2
IF D1 > E1 THEN
LET E1 = D1
LET E2 = E2 + D1
LET V(I, J) = VCAL
NEXT J
NEXT I
IF INT (INT (INT (L4/10) * 10 – L4) = 0 THEN
1180
GOTO 1340
LET CAP = 0
REM FIND CAPACITANCE
FOR I = 1 TO (X - 1)
FOR J = 1 TO (Y - 1)
LET L5 = (V(I, J) - V(I + 1, J + 1)) ^ 2
LET L6 = (V(I + 1, J) - V(I, J + 1)) ^ 2
LET LOT = L5 + L6
IF J < Y1 THEN
LET LOT = LOT * ER
LET CAP = CAP = CAP + LOT
NEXT J
NEXT I
LET CAP = CAP * EO/2
REM CONVERGENCE CHECK
IF L4 < 20 THEN
1340
IF ABS (CAP - D) < ACC * ABS (CAP) THEN
1350
LET D = CAP
NEXT L4
LET D = CAP
367
Microwave Circuits & Components
1370
1380
LET CFINAL = D
IF (L7 - 1) < = 0 THEN
1460
LET ZO = SQR (EO * MEW/CFINAL/CKEEP)
LET EEF = CKEEP/CFINAL
PRINT
1390
1400
1410
PRINT “CHARACTERISTIC IMPEDANCE = “ INT (ZO * 100 + 0.5)/
1420
1430
1440
1450
1460
1470
1480
1490
1500
1510
100” OHMS
PRINT
PRINT “EFFECTIVE DIELECTRIC CONST. = “ INT (EEF * 100 +
0.5)/100
GO TO 1480
LET CKEEP = D
NEXT L7
PRINT
PRINT “***********************”
PRINT
END
END-OF-LISTING
]RUN
INPUT RELATIVE DIELECTRIC CONSTANT
?2.3
INPUT MAX NO. ITERATIONS ALLOWED
?200
I/P DIMENSIONS OF METAL CASE
FIRST X-COORD THEN Y-COORD
?100
??50
I/P
DIMENSIONS OF CENTER CONDUCTOR
RELATIVE TO ENCLOSURE ORIGIN
FIRST X-COORDS
?32
??42
Y-COORDS
************************
RELATIVE DIELECTRIC CONSTANT 2.3
MAX NO. OF ITERATIONS 200
DIMENSIONS 100 X 50
CENTER CONDUCTOR :--COORDS 32, 42
COORDS 12, 13
CHARACTERISTIC IMPEDANCE = 86.32 OHMS
RELATIVE DIELECTRIC CONSTANT = 1.74
**************************
Problem 7.6. Using a symmetric TFE section design an attenuator pad that will
produce 20 dB attenuation. Assume prefered resistor values for R 1 and R2..
Calculate the input impedance and attenuation of a TEE pad constructed from the
prefered components selected.
Solution:
][ FORMATTED LISTING
FILE: PROGRAMME 4.8 ATTN
PAGE-1
10 REM
20 REM
--- ATTN---
368
Design of microwave network modelling & computer aided design
30 REM
40 REM THIS PROGRAMME COMPUTES
50 REM THE RESISTANCE VALUES
60 REM NECESSARY TO CONSTRUCT
70 REM SYMMETRICAL TEE AND PI
80 REM ATTENUATOR CIRCUITS
90 REM THE POWER DISSIPATED BY
100 REM EACH COMPONENT IS ALSO
110 REM CALCULATED.
120 REM THE PROGRAMME CAN ALSO
130 REM FIND THE ATTENUATION
140 REM AND I/P IMP. OF SYMM.
150 REM PI AND TEE CCTS.
160 REM WHEN THEIR COMPONENT
170 REM VALUES ARE KNOWN.
180 REM
190 REM RO = CHARAC. LINE RES.
200 REM R1, R2 = RESISTIVE ELEMENTS
210 REM ALPHA = ATTN. IN DB
220 REM POWER = I/P POWER (WATTS)
230 REM
240 REM INPUT DATA
250 REM
260 HOME
270 PRINT
280 PRINT “DO YOU KNOW THE CCT. PARAMETERS”
290 PRINT “IF YES ENTER 1 ELSE 0”
300 INPUT A = 0 THEN 340
310 IF A = 1 OR A = 0 THEN
320 ELSE 235
320 IF A = 0 THEN 340
330 IF A = 1 THEN
770
340
PRINT “I/P CHARAC. LINE RES. (OHMS)”
350
INPUT RO
360
PRINT “ ENTER DESIRED ATTN. IN DB”
370
INPUT ALPHA
380
PRINT “ENTER I/P POWER IN WATTS”
390
INPUT POWER
400
PRINT
410
PRINT “*****************************”
420
PRINT
430
LET N = 10 ^ (ALPHA / 20)
440
REM TEE SECTION
450
LET K = 0
460
470
480
490
500
510
520
530
540
550
560
570
580
590
600
610
620
630
640
LET R1 = R0 * (N – 1)/(N + 1)
LET R2 = R0 * 2 * N/(N * N – 1)
LET V1 = SQR (R0 * POWER)
LET I1 = V1/(R1 + ((R1 + R0) * R2/(R1 + R2 + R0)))
LET P1 = I1 * I1 * R1
LET P2 = (V1 – I1 * R1) * ((V1 – I1 * R1)/R2
LET P3 = (V1 – I1 * R1)/(R1 + R0)) ^ 2 * R1
PRINT “A SYMMETRICAL TEE SECTION”
PRINT “WITH ATTENUATION OF “ALPHA” DB”
PRINT “GIVES R1 = “INT (R1 * 1000 + .5)/1000” OHMS”
PRINT “AND R2 = “INT (R2 * 1000 + .5)/1000” OHMS”
PRINT “POWER DISSIPATION IN R1 a = “ INT (P1 * 10000 + .5)/ 10000” WATTS
PRINT “POWER DISSIPATION IN R2 = “ INT (P2 * 10000 + .5)/ 10000” WATTS
PRINT “POWER DISSIPATION IN R1 b = “ INT (P3 * 10000 + .5)/ 10000” WATTS”
PRINT “FOR AN INPUT POWER LEVEL OF “POWER” WATTS”
PRINT “AND CHARAC. RESISTANCE “RO” OHMS”
PRINT
PRINT “******************************”
PRINT
369
Microwave Circuits & Components
650
660
670
680
690
700
710
720
730
740
750
760
770
780
790
800
810
820
830
840
850
860
870
880
890
900
910
920
930
940
950
960
970
980
990
1000
1010
1020
1030
1040
1050
1060
1070
1080
1090
1100
1110
1120
1130
1140
1150
IF K = 1 THEN
1110
REM PI SECTION
LET K = 1
LET R1 = R0 * (N + 1)/ (N – 1)
LET R2 = R0 * (N + N – 1 )/(N – 1)
LET R3 = R1 * R0/(R1 + R0)
LET P1 = V1 * V1/R1
LET P2 = (V1/(R2 + R0 * R1/(R0 + R1))) ^ 2 * R2
LET P3 = (((V1 * R3)/(R2 + R3)) /R1 ) ^ 2 * R1
PRINT
PRINT “A SYMMETRICAL PI SECTION”
GO TO 540
PRINT
REM
PRINT “DO YOU REQUIRE TO ANALYSE”
PRINT “TEE OR PI SECTION”
PRINT “IF TEE SECTION ENTER 1 ELSE 0”
INPUT A
IF A = 1 OR A = 0 THEN
840 ELSE 620
PRINT “INPUT RESISTORS R1, R2”
INPUT R1, R2
IF A = 0 THEN 990
LET R0 = SQR (R1 * R1 + 2 * R2)
LET ALPHA = 20 LOG (1 + R1/R2 + R0/R2)/LOG (10)
PRINT
PRINT “************************”
PRINT
“PRINT CHARAC. RES. R0 = “INT (R0 * 100 + 0.5)/100” OHMS”
“ TEE PAD ATTENUATION = “INT (ALPHA * 100 + .5)/100” DB”
“FOR R1 = “R1” AND R2 = “R2” OHMS”
PRINT
PRINT “************************”
PRINT
GO TO 1110
LET ALPHA = LOG ((1 + R2/R1) + SQR ((1 + R2/R1) ^ 2 - 1))
LET ALPHA = ALPHA * 8.686
LET N = 10 ^ (ALPHA/20)
LET R0 = 2 * R2 * N/(N ^ 2 - 1)
PRINT
PRINT “************************”
PRINT
“PRINT CHARAC. RES. R0 = “INT (R0 * 100 + .5)/100” OHMS”
“PI PAD ATTENUATION = “INT (ALPHA * 100 + .5)/100”
“FOR R1 = “R1” AND R2 = “R2” OHMS”
PRINT
PRINT “************************”
PRINT
PRINT “DO YOU WANT ANOTHER GO ?”
ENTER 1 IF YES; ELSE 0
PRINT L
IF L = 1 THEN
260 ELSE 950
PRINT
PRINT “*** END OF PROGRAMME ***”
END
1160
1170
1180
LISTING
RUN
THE CCT. PARAMETERS
ENTER 1 IF YES; ELSE 0
LINE RES. (OHMS)
ATTN IN DB
370
Design of microwave network modelling & computer aided design
?10
ENTER I/P POWER IN WATTS
**********************
PRINT A SYMMETRICAL TEE SECTION
PRINT WITH ATTENUATION OF 10 DB”
PRINT “GIVES R1 = 25.975 OHMS”
PRINT AND R2 = 35.136 OHMS”
PRINT POWER DISSIPATION IN R1 a = .5195 WATTS
PRINT POWER DISSIPATION IN R2 = .3286 WATTS
PRINT POWER DISSIPATION IN R1 b = .0519 WATTS”
PRINT FOR AN INPUT POWER LEVEL OF 1 WATTS”
PRINT AND CHARAC. RESISTANCE 50 OHMS”
PRINT
PRINT “******************************”
PRINT
PRINT A SYMMETRICAL PI SECTION
PRINT WITH ATTENUATION OF 10 DB”
PRINT “GIVES R1 = 96.248 OHMS”
PRINT AND R2 = 71.151 OHMS”
PRINT POWER DISSIPATION IN R1 a = .5195 WATTS
PRINT POWER DISSIPATION IN R2 = .3286 WATTS
PRINT POWER DISSIPATION IN R1 b = .0519 WATTS”
PRINT FOR AN INPUT POWER LEVEL OF 1 WATTS”
PRINT AND CHARAC. RESISTANCE 50 OHMS”
PRINT
PRINT “******************************”
PRINT
PRINT “DO YOU WANT ANOTHER GO ?”
ENTER 1 IF YES; ELSE 0
PRINT L
PRINT
PRINT
DO YOU KNOW THE CCT. PARAMETERS
ENTER 1 IF YES; ELSE 0
PRINT DO YOU REQUIRE TO ANALYSE
PRINT TEE OR PI SECTION”
PRINT IF TEE SECTION ENTER 1 ELSE ENTER 0
PRINT INPUT RESISTORS R1, R2
?100
??67
****************************
CHARAC. RES. R0 = 50.09 OHMS
TEE PAD ATTENUATION = 9.56 DB
R1 = 100 AND R2 = 67 OHMS
**********************
PRINT
PRINT “DO YOU WANT ANOTHER GO ?
ENTER 1 IF YES; ELSE 0
END OF PROGRAMME ***
Problem 7.7. Design an 31.3 series elements of microstrip matching circuits
shown in Fig.7.27.
The other specifications that may be used are = g /4, f = 800 MHz, r =
10.0 thickness h = 0.5 mm.
Solution: For frequencies lower than 2 GHz, static - TEM methods are sufficiently
accurate (within 1per cent accuracy). Firstly, we carry out a rapid approximate
determination using Presser’s graphical technique.
371
Microwave Circuits & Components
l
Figure 7.27. Series elements for microstrip matching
further
10 31.3 98.979
z01
(7.180)
From presser's graph, we obtain
q = 0.68
(7.181)
eff 1 q (r 1)
(7.182)
eff 7.12
Therefore the improved value of z01 is
7.12 31.3 83.51
z01
Again referring to the graph, we get
q = 0.7; w/h = 2
and
eff = 7.3
Repeating the above process, we again get w/h = 4, q = 0.76 finally we get
w/h = 2.225; eff = 7.09
Graphical synthesis doesn’t give accurate value of eff as such but gives
fairly accurate results for w/h. This is because q cannot be accurately determined
from the graph.
Since h = 0.5 mm with w/h = 2.2
w = 1.1 mm
The length is now required is
300
g
f eff mm
372
Design of microwave network modelling & computer aided design
f is in GHz
l
g
4
75
0.8 7.09
= 13.22 mm
Problem 7.9. Design a four finger large coupler to provide a 10 dB coupling in a
50 system operating at 5 GHz. The substrate to be used is of thickness of
0.234 mm and permittivity 2.2.
Solution: Since coupling C = 10–(10/20), C = 0.3162.
q 2 [c 2 (1 c 2 ) (k 1) 2 ]
(7.183)
k is the specified number of lines within the coupler.
Therefore
q 2 = [0.3162 2 + [1 - 0.3162 2 (4 - 1) 2 ]
(7.184)
Z oo 80
cq
(k 1) (1 c)
124
z oe z oo
z oe
Applying coupled line synthesis as discussed in the previous problems,
we get w/h = 1.103 and s/h = 0.39
W = 0.258 mm and S = 0.091 mm.
Using Akhta Zad’s technique
z
Z ose oe 34.675
2
z oo
Z oso
18.01
2
Using Presser's graph technique
(w / h ) se 2.0 and (w / h ) so 5.0
Using Akhtar Zad’s curve, we select the broken (even-mode) curves
which has parameters (w/h)se = 2. Next select the solid (odd-mode) curve which
has parameter (w/h)se = 5.
The point where these two intersects is the design point and its
coordinates gives the results.
w/h = 0.85 and s/h = 0.25
Because
h = 0.5 mm, w = 0.425 mm, and s = 0.125 mm
373
Microwave Circuits & Components
7.10
CONCLUSION
In the present chapter, the steps followed in various calculations for
microwave circuits is given. The different methods for estimating parameters of
microstrip and other planar structures have also been discussed. Many problems
using these methods are also included in the Chapter. However, many numerical
techniques have not been discussed. Let us take the case of numerical integration.
Numerical integration is used in science and engineering whenever any function is
in closed form and it cannot be integrated. It can also be used when the function is
described in the form of data. Several integration rules have been developed. The
common ones are:
(a)
(b)
(c)
(d)
(e)
Eulers rule
Trapezoidal rule
Simpsons rule
Newton-Cotes rules and
Gaussians (quadrature rules)
Numerical modellings of guided-wave passive components have been
important area in which lot of contribution is being made. The planar integrated
circuits do not have closed-form analytical expressions. The only possibility is to
use numerical techniques for calculations. At the moment circuit designers use
CAD packages. The use of computer facilitates utilisation of numerical techniques
such as the finite difference method (FDM), finite-element method (FEM) and
method of moments (MOM) to evaluate the field distribution in practical but
complicated geometries. The finite difference method divides the solution domain
into some finite discrete points and replaces the partial differential equations with
a set of difference equations. The solution is not exact. The mesh size of discreted
solution domain is a measure of accuracy of solution. The smaller the mesh size the
better the accuracy. An iterative technique known as successive over relaxation
method is very useful way of solving the difference equations of FDM.
The finite element method has also been exploited to large extent in
approximate solution of many complicated problems. It is a method well developed
and has perhaps the widest scope or versatality in dealing with vast ranges of
component geometries and material distributions. It is an optimisation method that
basically minimises the total energy stored in system subject to some constraints
dictated by boundary condition. One of the most important advantages of the FEM
is that it treats complicated boundary conditions with minimum difficulty. Another
important advantage is that it can handle the analysis of fields and multimaterials
quite easily. It introduces the boundaries of weighted residuals. This leads to
variational method upon which we choose to base finite element method.
The method of moments appear to be best choice to determine electric and
magnetic fields in problems with open boundaries. This method uses general
integral method, the retarded potential equations. We do have not to define the
solution with finite boundaries. This method employs step functions as basic
functions. However, choices for basis and testing of functions that can be much
more flexible. The basis and testing functions are identical in the Galarkins method,
and the resulting solutions are known to be variational. Normally, this technique
374
Design of microwave network modelling & computer aided design
requires knowledge of the change or current distributions on existing boundaries.
Sometimes this information is not available. However, when the potential is given
on the boundary the distributions of the change or current can be numerically made
available by dividing the boundary into number of elements. The field distribution
can then be determined everywhere in the system.
In addition to above there are other methods are used in the design of
microwave circuits. Some of these are
(a)
(b)
(c)
(d)
(e)
Integral equation method
Mode matching method
Generalised scattering of matrix method
Spectral domain method
Use of planar circuits model
Each method has its advantages and disadvantages. As an example the
finite elements method takes considerable time for calculation but is versatile.
The spectral domain method is efficient but cannot be applied in all cases. Table 7.2
gives various aspects of numerical method. The evaluation is generally qualitative.
The steady improvement of personal computers can be used for numerical
calculations for complicated cases. In this chapter a few of these techniques have
been discussed. However, it is not possible to discuss all of them in detail. This has
not been attempted.
Table 7.2. Comparison of numerical methods
S.No.
Method
1.
Finite
2.
Finite
3.
Boundary
4.
Transmission
5.
Integral
6.
Mode
7.
Transverse
8.
Method of
9.
Spectral
Storage
Requirement
CPO
Time
Generality
Large
difference
Large
element
Moderate
element
Moderate
line method
(TLM)
Small/
equation
Small/
matching
Large
Very Good
Nil
Moderate
Very Good
Large
Very Good
Small
Moderate
Small/
resonance
Small/
lines
Small
375
Preprocessing
Small
Moderate
Very Good
Large
Small
Small/
Moderate
Small/
Moderate
Good
Moderate
Good
Moderate
Moderate
Small/
Moderate
Small/
Marginal
Moderate
Good
Moderate
Small
Marginal
Large
Moderate
Large
Microwave Circuits & Components
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Strong, G. Linear algebra and its applications. Academic Press, 1980.
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Thom, A., & Apelt, C.J. Field computation in engineering and physics.
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Nostrand and Reinhold, New York, 1987.
Hoole, S.R. Computer-aided analysis and design of electromagnetic
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378
CHAPTER 8
ELECTRONIC WARFARE
8.1
INTRODUCTION
Electronics plays an important part for armament, communication,
surveillance, standby or support in defence systems. EW refers to a set of radioelectrical techniques which in themselves do not correspond to any specific
military application but used to gain control of radio waves to guarantee the use
of radio spectrum to good advantage but prohibits its use by enemy. The knowledge
of extent of threat is an essential element in defining and designing new equipment.
The stakes are so heavy that it justifies specific hardware and concept programmes
which are nontraditional and are grouped together with the name of Electronic
Warfare (EW). What exactly is the electronic warfare? It can be defined as a
form of military action aimed at
(a)
Drawing information from enemy emissions in order to take counter
action.
(b)
Reducing or preventing the use of RF, microwaves by enemy or
modifying them to our use.
Thus EW is a technique which leads to satisfactory defence of military
mission and are incorporated into the systems in which they are associated with
conventional methods.
What is the utility of EW? The first application is in communication
which includes (a) Morse code (b) Voice communication and (c) Digital data
communications which are protected by coding, microwave and satellite links.
The techniques to intercept or exploit enemy communication is an
important part of EW. COMINT (Communication Intelligence) is an important
step in this direction. It involves wide variety of techniques to collect and analyse
signals, direction finding and transmitter locating, the analysis of signal or signal
processing and lastly the exploitation of information thus collected.
The other major use of radio frequency is radar which was introduced in
1940. The aerial surveillance or early warning radar, ship radar, airborne radar, fire
control radar and missile seeker radar are different varieties of radars which are
used in EW and also the techniques developed to stop their use. First radar signal
interception techniques were developed, then locating radars, signal processing
Microwave Circuits & Components
and analysis of signals received. The non-conversational area meaning without
any conversational content is known as Electronic Intelligence (ELINT). ELINT is
identified with Electronic Support Measure (ESM). It is possible to build
broadband transmitters which prevent the use of radio spectrum for
communication. The jammers have become more sophisticated keeping with the
progress made in frequency management. It makes use of frequency hopping and
spreading the spectrum psuedo-random encoding which makes use of modulation
(white noise). Recently jamming techniques have also been extended to
non-communication systems which make use of high speed circuits, programmable
logic and very advanced technology microwave component. Jammers make use of
deception devices such as intrusion, range or velocity pull-off. These are referred
to as Electronic Countermeasures (ECM). It includes:
(a)
(b)
Radio navigation in which EW is used to jam or modify the signal.
Optics for imaging, guidance, designation range finding using infrared which
are being used for EW. This involves location and detection of sources for
jamming, neutralizing or destruction of this source.
Protection of electronic equipment and its efficient use is also important.
The protection includes techniques for modifying the electronic camouflaging. It
also includes pulse coding and designated the phrase Electronic CounterCounterMeasures (ECCM). These are described in detail here.
8.2
ELECTRONIC SUPPORT MEASURES
In ESM, there are many aspects of radar receivers that needs to be
considered which involve techniques to detect signals below noise level. It also
involves sensitivity of radar receiver. The waveforms transmitted by these radars
have to be analysed in terms of pulse width, spectrum, the frequency, pulse
repetition frequency and the illumination period. This is to be utilized for
recognizing the signature of radar. The electronic facilities should have certain
performance capabilities like broad frequency surveillance radars at low
frequencies (L band), tracking radars with K band, sensitivity of detecting radars in
presence of noise and signal processing is also needed.
8.2.1
Noise, Probability & Information Recovery
Noise exists in all electronic systems, the general problem will be
discussed first and then the intentional introduction of noise like hostile noisejamming against radio or radar transmitter. The presence of noise in a system
introduces uncertainty as to information content and thereby degrades system
performance as measured in terms of accuracy, reliability, information handling
capacity or some similar criterion. The desired signal may be required to compete
with one or more information bearing (but undesired) signals of a nature similar to
its own. For purpose of semantic clarity it is conventional in discussion of
information theory to refer to an emanation whose reception is desired for purpose
of information extraction as a signal regardless of physical composition.
Conversely, any other emanations which signal must compete for recognition at
reception are classified as noise.
380
Electronic warfare
The fundamental problem of information recovery from a signal
immersed in noise is essentially independent of exact value of either signal or noise
at least within the framework of consideration of EW. The problem of information
recovery is basically the determination, at certain intervals of time, for both CW
and digital communication, the presence or absence of a specific signal or signals.
The signal must be distinguished in terms of signal parameters such as frequency,
time of occurrence, duration, amplitude, etc. The problem of recovering desired
information in noise environment is also similar. Simple pulse radar systems
transmit an interrupted RF carrier, sometimes with the echo. This provides
information regarding reflector targets, if any, within the range of the radar. The
recovery of information may be considered as imposing additional restriction or
group of restrictions on definition of signal. Suppose it is desirable to determine
radial velocity of an aircraft target in relation to radar side, the recovery of
information problem is for a series of Doppler frequency shifts fi (change in
received carrier frequency due to reflection from a radially moving target,
corresponding to series of radial velocities Vi). Does a received signal with Doppler
shift fi exist at any given instant? Therefore the problem is reduced to determination
of presence or absence of carefully defined frequency in a signal at an given instant.
Also the other thing to be determined is that whether a signal processing specified
characteristics are present at time T1, T2 ……, etc.
Very often statistical probability is a very useful concept in extraction
of signal from the noise.
8.2.1.1 Elementary probability theory & statistics
Let us try to understand some ideas of Probability theory. Suppose that a
number of balls numbered consequently from 1 to m are placed in a container
thoroughly scrambled. The container is large enough to scramble m balls in a single
push. Each ball is settled in small depression. The probability of one ball falling in
any of the depressions is one. Mathematically, the probability of any event
occurring is equal to number of different ways in which event probability may
occur divided by total number of equally likely outcome of the situation. The joint
probability of two independent outcomes A and B is equal to
(8.1)
P A,B P A P B
If outcomes are independent, i.e., the occurrence of A does not affect
probability of occurrence B PA B P B . Thus, the joint probability of
independent events A, B …..r is P(A, B, …….. r) is
P A,B,..........r P A P B .......P r
r
A P i
(8.2)
If there exists several mutually exclusive possible outcomes of an act,
then the cumulative probability that either outcome A or outcome B will occur is
P A , B P A P B
(8.3)
It is now desirable to apply the probability distribution to physical noise
problems. Let us take the example of free electrons emitted from thermionic
381
Microwave Circuits & Components
cathode; it is known that these particles may possess linear velocities of any
value within a wide range. They are not limited to any finite number of possible
discrete velocities. Hence, the velocity probability distribution function, P(v) is
continuous i.e., v is said to be a continuous variable. By definition the probability
of v lying in the incremental region between vi and Vi Vi is P Vi . v . It
should be remembered that
s
PVi v 1
(8.4)
i 1
It can be written in the integration form as
P v dv 1
(8.5)
P(v) is a continuous function of v, such that the probability of a single
electron having a linear velocity between v1 and v2 is
P v1v2
v2
P v dv
(8.6)
v1
form
It has been experimentally determined that actual distribution of the
P v k1e k 2 v 2
(8.7)
This is called normal or Gaussian distribution function and is plotted for
one pair of values, k1 and k2 as shown in Fig. 8.1. The Gaussian distribution is
symmetrical about its mean value and describes distribution of continuous
stochastic (random) processes. The exact shape depends, of course on constants, k1
and k2 as well as upon . It is often convenient to write the normalized Gaussian
distribution function as
K1
K1
K
P(V)
V
1
2
v
e
2
/ 2 2
1
2 2
V
Figure 8.1. Normal gaussian probability distribution function
382
Electronic warfare
Pv
1
2
e 2 v
2
(8.8)
2 2
The parameter is the standard deviation and determines its relative
width and height. 2 is the mean squared deviation of the distribution about the
mean and is useful measure of noise power or energy in many applications. P(v)
does not go to zero for any finite value of v indicating that the finite probability of
finding a small number electrons with very positive velocities towards anode.
8.2.2
Recovery of Signal from Noise
If noise is superimposed on any pulse signal and if noise is strong to
completely superimpose the signal, it may be difficult to extract signal from the
receiver. One possible way is to find the average. For noise only, average will be
zero whereas for signal alongwith noise may be different from zero.
The averaging process in reality is a filtering (low pass) process as the
averaging time is increased progressively, lower frequency components of
incident waveform is averaged out until limiting case of infinite integration is
reached, corresponding to low-pass filter of zero bandwidth. It is apparent that
the noise energy in a given bandwidth of width f is directly proportional to
value of f , as f goes to zero so does the noise power..
AMPLITUDE
The desired signal, in practice, is never of zero bandwidth. Information
can only be conveyed via some part of modulation of the carrier, resulting in a set
of modulation products or side bands of non-zero width. One of the most important
results from the information theory is establishment of a direct relationship
between system information-transmission capacity and bandwidth. It is obvious
that band-pass just sufficient to enclose entire signal will produce greater signal-tonoise ratio than a wider band pass filter. Many cycles of carrier frequency
ordinarily correspond to the length of one single pulse or one noise peak as
indicated in Fig. 8.2. Hence, the average value of the RF waveform is zero. The
problems of detection or extraction of signal then arises.
TIME
Figure 8.2. Modulated carrier noise with signal wave before deletion
383
Microwave Circuits & Components
8.2.3
Detection & Correlation
Probably, the simplest method consists of rectification of the RF or
Intermediate Frequency (IF) signal, followed by averaging (low-pass filtering) over
a sufficient time to effectively smooth out carrier-frequency ripple. This system
displays two characteristics that makes its use undesirable. Such a system suffers
from incoherent detection, and has a undesirable effect on signal-to-noise ratio.
The degradation is so small as to be insignificant in cases where the detection input
S/N is considerably larger than unity, it becomes increasingly severe for input S/N
ratios mean of unity, as may happen in a severe noise-jamming environment. As
the detector input S/N drops below unity, the incoherent detector further
deteriorates the signal-to-noise-ratio, so that a detector input S/N of –10 dB yields
an output S/N of –13 to –14 dB.
The alternative to incoherent detection is known as coherent detection.
To analyze the relative significance and the difference between coherent and
incoherent detection, it is necessary to return to statistics and information theory.
What is coherent detection? It can be defined as the descriptive parameters (i.e.,
frequency and phase) or two or more signals are functionally related in a specified
manner, then those are said to be coherent. The existence or absence of such
relationship is investigated in terms of statistical concept of correlation.
The time functions, f1(t) and f2(t), might be defined by the correlation
factor 12 , thus
12 lim
T
1
2T
T
f1 t f 2 t dt
(8.9)
T
Thus, the correlation factor of two identical signals will be some positive
number that for two same waves of the same frequency, but separated by 90 degrees
in phase, will be zero, since the product f1( f ) f2 ( f ) will be positive as often as
negative, averaging to zero over large T. Similarly, the average value of product of
any two functions arising independently (a random noise and any of other time
functions, or two same waves of different frequencies, etc.) will be zero over a long
time T. If one of the two signals is delayed by a time T relative to the other, and the
correlation factor is then determined as a function of delay time, the yield is crossrelation function 12 T
1
T 2T
12 T lim
T
f1 t f 2 t c dt
T
(8.10)
Two unrelated signals will still yield 12 T 0 for all T since the output
impedance Z merely introduces a varying phase relationship, and the concept of
phase for independent signals. For the case of two signals with periodic
components of same frequency, however, 12 T will be periodic in , yielding
maxima for such that signals are in 0° or 180° phase relationship, and zero as
passes through 90° and 270°.
384
Electronic warfare
The cross-correlation technique may be applied to detection of a
received signal in the presence of noise. Consider first the hypothetical
arrangement of Fig. 8.3. An oscillator is maintained at the exact frequency and
phase of the expected received signal. When no signal is present, the oscillator
signal is cross-correlated with received noise; the correlator output is zero
because only signal arising from a common source can, in practice, be correlated
over many cycles. When the received signal is present, the correlator output
becomes a steady DC voltage, thus accomplishing both signal detection and
separation of signal from noise by a process of multiplication and averaging using
a priori knowledge of the received signal.
Full information regarding parameters of received signal is seldom
available. The multiplication-averaging process of coherent detection does not
introduce the S/N ratio degradation inherent in incoherent detection (rectificationaveraging). However, in many cases the type of detection system first derivative
will provide a significant advantage in S/N over an incoherent detector, despite
degradation result from imperfect synchronization of the derivative. The coherent
reference signal is useful in many applications.
Another statistical process close to cross-correlation is autocorrelation.
The autocorrelation function of f1(t) is 11 .
1
T 0 2T
11 lim
T
f1 t f1 t dt
(8.11)
T
Auto-correlaion is thus seen to be equivalent to the cross-correlation of a
function with itself (i.e.,, of two identical functions). It is clear that auto-correlation
function of an periodic component is periodic ; also 11 0 is a positive for any
ANTENNA
SIGNAL AND
NOISE
CRO SS-CO RRELA TOR
(MULTIPLIER AND
AVERAGING CIRCUIT)
SIGNAL OUTPUT
MAINTAINED AT EXACT
FREQUENCY AND PHASE OF
EXPECTED SIGNAL
COHERENT
OSCILLATOR
Figure 8.3. Ideal cross-correlator reciever
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Microwave Circuits & Components
0
0
(A)
(B)
(C)
Figure 8.4.
Auto-correlation function, 11 for (a) 11 sinosoidal signal (b)
11 random noise (c) bandwidth noise.
f1(t). A random noise is, by definition, independent in amplitude at any instant of
its amplitude at any other instant thus 11 for any non zero is zero. Practical
noise is never truly random, because it must pass through finite pass bands in the
antenna and recover before reaching the detector, its high frequency components
are attenuated, resulting in some degree of short term autocorrelation and
autocorrelation function that decreases rapidly but not instantaneous by . These
are shown in Fig. 8.4. The auto-correlation receiver is shown in Fig. 8.5. Unlike the
cross-correlator the autocorrelator requires accurate prior knowledge of the
received signal. The correlation concept provides an extremely powerful and
useful mathematical tool for dealing with many noise and signal problems. The
mathematical theory provides rigorous justification of a statement that has been
f1(t)
tn2(t)
CROSSCORRELATOR
11 2Q12 22
BOTH= 0 FOR
f1(t)= SIGNAL
tn2(t)= NOISE
DELAY
Figure 8.5. Simplified auto-correlation receiver
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= 0
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made on essentially intuitive basis. The autocorrelation function of a given time
function is the Fourier Transform of that functions' spectral distribution. Thus, a
truly white noise with spectral density constant over all frequencies has a spike
auto-correlation, functions, possessing a finite value only for E = 1 and a zero
value elsewhere.
8.2.4
Characteristics of Microwave Receivers
In this section some common terms representing the characteristics and
performance of EW receivers will be discussed. These terms include sensitivity,
dynamic range, Probability of Intercept (POI).
In most microwave receivers there is a video detector that converts RF
energy into video signals. The sensitivity of receivers is limited either by
characteristics of video detector or by internally generated noise of the receiver. If
the gain in front of detector is high enough then sensitivity level thermal noise is
generated by thermal motion of electrons. Noise generated by a resistor R can be
represented by a noise generator in series. The available thermal noise power at
input of a receiver can be expressed as
N i kTB
(8.12)
k is Boltzmann’s constant 1.38 10 23 j k , T is the temperature of resistor
and B is the bandwidth in the hertz, Ni is the noise power in Watts. Another
expression for power is in dBm,
w
h
e
r
e
,
P
P(dBm) 10 log
1mW
(8.13)
In this expression, power on right hand side is in milliwatts. For 1 mW, P(dBm) =
0. When P > 1mW P(dBm) is positive when P < 1mW P(dBm) is negative. The
thermal noise at room temperature (where T = 290k) can be represented in dBm as
P(dBm) = -174 (dBm)/Hz or P(dBm) = -114 (dBm)/MHz
(8.14a)
Equation 8.14 is derived by substituting the values of R and T in
Eqn. 8.12. These two values are used in determining receiver sensitivity. Noise
figure is defined as
F
Si N i
Signal to noise ratio at the input of receiver
So N i Signal to noise ratio at the output of receiver
(8.14b)
This expression is always greater than unity. Also remember that
(8.15)
F dB 10 log F
If there are N amplifiers connected in cascade, the noise figure can be
expressed as
F 1
FN 1
F2 1 F3 1
.......
G1
G1G 2
G1G 2 ........G N 1
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(8.16)
Microwave Circuits & Components
where, Gs are gains of various amplifiers and Fs are the noise figure. The
derivation of Eqn. 8.16 can be illustrated with two amplifiers connected in
cascade as shown in Fig. 8.6. The noise generated at the output of amplifier 2 is
Figure 8.6. Two cascaded amplifiers
No = noise at amplifier 1 at the output of the amplifier 2 + noise
introduced in amplifier 2.
The noise contribution from amplifier 1 is kTB F1 amplified by G1 and G2.
the noise generated by amplifier 2 is G2kTBF2 which contains the noise G2kTB
generated by amplifier 1. Thus, the noise generated by amplifier 2 alone is
G2kTB F2 - G2kTB
Therefore,
N0 = kTBF 1 G1 G2 + G2kTB F 2 - G2kTB
(8.17)
From the definition of the noise, figure given in Eqn. 8.14
F
RTB F1 G1 G 2 G 2 RTBB2 G 2 RTB
F 1
F1 2
G
RTBG1G 2
(8.18)
Thus, we get the Eqn. 8.16
8.2.5
Tangential Sensitivity
The tangential sensitivity of a receiver is measured through visual display
on an oscilloscope monitoring the output of the diode detector or output of video
amplifier following the detector. The input must be a pulsed signal. Figure 8.7
shows a trace of oscilloscope. On the scope the minimum of the noise trace in the
pulse region is roughly tangential to the top of the noise trace between the pulses as
shown in Fig. 8.7 is tangential sensitivity (TSS). Frohmainer 1 experimentally
determined that at the TSS the signal is 8 dB above the noise level at the output of
the detector with standard deviation of 0.4 dB. Experiments carried out by
Williams2 indicate that the spread among a group of observers in setting of
tangential level is likely to be 1 dB about the mean. TSS is a very widely used
criterion because of extreme simplicity and its ability to give a convenient
comparison of sensitivities of widely different receivers.
The TSS depends on the RF bandwidth (BR), the video bandwidth (BV),
the noise figure receiver and characteristics of detector. In a receiver BR is almost
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Electronic warfare
Figure 8.7. Detector output at tangential sensitivity
always greater than BV. The discussion on TSS is given by Lucas3. The signal
strength TSS required at input of a detector to produce tangential sensitivity at
output is obtained by Lucas. Lucas’s results can be written in different format which
is
ABv
TSS 114 10 log FT 10 log 3.15B R 2.5 2 B R BV Bv2
2
G
T FT
for Bv B R 2 Bv
(8.19a)
ABv
TSS 114 10 log FT 10 log 6.31Bv 2.5 2 B R BV Bv2
2
G
T FT
for B R 2 Bv
(8.19b)
where, GT and FT are the overall gain and noise figure from the input of the
receiver to the detector. BR and BV are in MHz. The value –114 is from Eqn. 8.14
and is the thermal noise floor of a 1 MHz2 bandwidth system at room
temperature. The constant A is related to the diode characteristics and the noise
figure of video amplifier following the diode. It can be expressed as
4 FV
(8.20)
10 6
RTM 2
where, FV is the noise figure of the video amplifier expressed in power ratio and
M is the figure of merit of diode detector and can be expressed 4 as
A
M
(8.21)
R
where, is the detector sensitivity in volts per watts and R is the dynamic
impedance of the diode in .
The value of A can be determined from the Eqns. 8.20 and 8.21 if the
figure of merit M(or and R) and noise figure F1 are known.
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Microwave Circuits & Components
It can also be measured experimentally. If video detector and video
amplifier under test are considered as the only component in a receiver and there is
no gain or loss in front of detector then GT = FT = 1 from Eqns. 8.19a and 8.20. In
this case, A is the only dominant term in these equations and these two equations
can be approximated by
(8.22)
TSS 110 10 log ABV
By measuring TSS experimentally, A can be determined by Eqn. 8.23.
The error introduced in this approximation is usually less than measurement error.
8.2.6
False Alarm Rate & Probability of Detection
In order to operate a receiver in a satisfactory manner a certain threshold
must be set up to keep the false alarm rate below the desired value.
Let the threshold be set at VT as shown in the Fig. 8.8. The average time
interval between crossings of threshold by noise is defined as the false alarm time
Tfa in seconds.
T fa ln
1 N
TR
4 R 1
(8.23)
where, TR is the time between crossing of threshold voltage VT by the noise
envelope as shown in Fig. 8.8. The probability of false alarm Pfa may be defined as
duration of time if the envelope is actually above the threshold to the total time of
observation. It can be expressed as
Pfa
1
T fa BR
(8.24)
ENVELOPE OF THE OUTPUT OF THE RECEIVER
VOLTAGE
where, BR is the RF bandwidth in Hertz.
TR 1
TKS1
TKS1
TKS2
TK1
VT
FO1/2
Figure 8.8.
Envelope of the output of reciever producing false alarm due to noise
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Electronic warfare
If gaussian noise were passed through the narrow-band IF filter, the
probability density of the envelope of the noise voltage output is
P E
E2
E
exp
20
0
(8.25)
where, E is the amplitude of the filter output. The probability that the noise voltage
envelope will exceed the threshold voltage VT is defined as the probability of false
alarm.
Tfa P EdE
VT
E2
exp
dE
20
VT
VT 2
exp
0
E
0
(8.26)
Remember that noise voltage V entering in the RF filter is assumed
Gaussian with a variation 0 ; the mean value of V is equal to zero. If a same wave
signal of amplitude A is present along the noise in the receiver, the probability
density becomes
P E
E 2 A2
E
exp
20
0
EA
I0
0
(8.27)
where, I0(Z) is the modified Bessel function of zero order and augmented z is
defined as
I0 z
Z 2n
n 0
2n n n
(8.28)
The probability of detection is
Pd
P E dE
(8.29)
VT
The threshold VT in Eqn. 8.29 is the same value of VT in Eqn. 8.26 but the
probability density function is defined in Eqn. 8.26 rather than Eqn. 8.25. To
illustrate the process of threshold detection the probability density function of
Eqns. 8.25 and 8.27 are plotted in Fig. 8.9. In this figure an arbitrary threshold VT
is selected. In the figure, the shaded area represents the probability of false alarm
and the area under the curve S/W=1/ Right of VT represents the probability of
391
Microwave Circuits & Components
detection. It is possible to increase the threshold voltage VT to reduce the
occurrence of false alarm but it also reduces the probability of detection. Emerson5,
Robertson6 and Abreheim7 have worked on this subject.
8.2.7
Introduction to Dynamic Range
Dynamic range is commonly used for input signal amplitude range that
receiver can process properly. The lower limit of dynamic range is the sensitivity
of receiver. The standards selected are minimum detectable signal (MDS). This
means S/N=1(0 dB). The other standard concerns tangential sensitivity and
P(E)
S/N=0
S/N=1
E
VT
Figure 8.9. Probability functions for noise alone and for signal plus noise
operational sensitivity. It must be remembered that there is no standard definition.
Someone measures it to be 50 dB and the other 30 dB. It depends on definition of
dynamic range. When the receiver provides only the frequency information and
does not measure the amplitude, the dynamic range is usually defined as weakest
signal level where the measured frequency error is within certain predetermined
range. The upper limit is the strongest signal level where the measured frequency
error is within the same predetermined range.
For Electronic Support System (ESM) these studies are essential.
Sensitivity and dynamic range should be emphasized because they are the most
important factors in a receiver. Tangential sensitivity and operational sensitivity
are commonly used terms as they are experimentally measurable. The noise figure
studies are useful in receiver design. False alarm rates measurement is needed.
This is closely related to the sensitivity calculations. There are many definitions of
dynamic range but the commonly used ones are single signal and two-tone spur
free dynamic ranges. POI is a useful term in EW receiver and therefore has been
briefly discussed. The graphic method of determining receiver sensitivity and
dynamic range are important in estimating receiver performance.
There are some aspects of the Electronic Support Measure (ESM) which
is briefly discussed. These are
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Electronic warfare
(a)
(b)
(c)
(d)
Direct detection receivers and superheterodyne receiver.
Spectrum analyser
Direction finding techniques
Location measurements
Now essential feature would only be described here due to constraint of
scope and time. The details would be referred to the standard text and original
papers.
8.2.8
Direct Detection Receiver & Superheterodyne Receiver
Some of the features of these systems are already discussed in the
preceding sections but mention should be made for simplest receiver system which
can be used. A desirable feature is that it consists of direct detection receivers
which is actually crystal video detector and is described below. This type of
receiver was used for the first time to detect electronic intelligence. This is a
broadband receiver covering one or several octaves using very broadband antenna.
It carries out video detection of microwave signal.
The advantage of this type of receiver is that it has good probability of
intercepting the signal at low cost but the sensitivity is very low. It may have high
detector noise ratio. It is non-selective in terms of frequency. To improve signal
following facility is added.
(a) Broadband amplifier can be used to compensate for antenna detector link
losses thereby reducing the noise figure and also improving selectivity.
(b) Microwave filter banks may be used for improving the sensitivity.
Measurements centered on such a detector concern the envelope of
signals received i.e., the levels pulse widths, pulse reception period, illumination
period, etc. The frequency, a major parameter of radar signature is now measured,
leading to development of devices such as IFM (Instantaneous Frequency
Measurement) receivers. This technique is based on principle of instantaneous
phase measurement between two channels, one delayed by time from the other..
For a measurement of the phase difference , the frequency F of the incident
signal is then given by
F
2 C
(8.30)
We must remember that phase measurement is ambiguous measurement
(measurement within 2 ) that several measurement channels needs to be used,
presenting different delays to clean up any ambiguity leading to a relatively high
cost. Thus, the IFM receiver and a direct detection receiver makes it possible to
measure the received level and instantaneous frequency in a wide frequency band.
8.2.8.1 Superheterodyne detection receivers
As evident from above discussion, the limited sensitivity and lack of
frequency selectivity of direct detection receivers have lead to superheterodyne
receivers. The frequency transposition of received microwave signal with local
393
Microwave Circuits & Components
oscillator signal and amplifying and filtering in a limited band around the lower
intermediate frequency leads to good frequency selectivity and sensitivity.
The main drawback of this is a low probability of intercepting radar
signals spread out over a wide frequency band whereas the receiver frequency
band only extends from a few MHz to hundreds of MHz. Therefore, small
frequency scan is possible to improve detection several times by connecting
several superheterodyne in parallel (multichannel receiver) to increase
bandwidth and thereby the probability of intercepting radar signals.
8.2.9
Spectrum Analysers
To improve the instantaneous broadband high sensitivity, frequency
selectivity, high intercept probability, instantaneous spectrum analysis have
developed in two main areas. First, the use of frequency dispersive electroacoustic-mode in compressive receivers or Surface Acoustic Wave (SAW)
receivers. If the incident microwave signal is modulated by a frequency saw tooth
signal generated by the dispersive line and feeding of the intermediate frequency
obtained into a dispersive line whose slope, delay, versus frequency is inverse of
that of the saw tooth, then we obtain a time/frequency tranposition of signal
received during the saw tooth signal.
An instantaneous spectrum analyser can separate two or several
simultaneous signals at different frequencies in compressive receiver pass band.
The pass band is around 500 MHz and analysis is carried out in 1 s . It means that
signal processing should be at every 2 ns to obtain 1 MHz resolution.
Simpler receivers like delay line discrimators directly using the
properties of dispersive lines; signal delay at the line output proportional to the
incident frequency, the use of acoustic-optical processor is based on principle of
Bragg cells. A coherent beam is directed into a medium which can be electrically
excited in order to generate acoustic waves. The laser beam is then deviated
which is proportional to the excitation beam frequency.
An optoelectronic interface comprising photodetectors measures the
deviation laser beam in order to work back the frequency of electromagnetic signal.
This type of receiver requires the development of integrated optical circuits, which
makes it possible to widen the instantaneous frequency band of frequency around
2 GHz but leads to an instantaneous dynamic and lower sensitivity than those of a
comprehensive receiver.
8.3
DIRECTION FINDING TECHNIQUES
Direction finding is based on (a) amplitude (b) phase or (c) time of arrival
measurements.
8.3.1
Direction Finding by Amplitude Measurement
A high-gain rotating antenna can be used in direction finding. The method
is very simple. The maximum signal gives the direction from which signal is
received. This method has the capability of finding out good angular selectivity. But
the instantaneous coverage is small. The size of antenna is large and slow due to
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Electronic warfare
inertia of the rotating system. This definitely means that it cannot be used or
installed in aircraft. Perhaps in such system phase array antenna can be used but it
cannot cover 360° without using more than one antenna system. Several antennas
with angular aiming off by small phase angle is used. Directional measurements
are made from differences in the received signal of each antenna. In this way
there can be solutions comprising twin-lobe antennas comparable to monopulse
technique used in different radars. This system can be used with detection
receivers with each antenna having its associated reception channel leading to an
instantaneous direction measurement.
Rotmann lens microwave antenna can measure directions accurately.
Phase difference between two received signals on two different antennas can be
converted into amplitude modulation by arrival compensation techniques. It is
possible to measure the direction by locating a maximum or minimum level.
8.3.2
Direction Finding by Phase Measurement
The difference in phase is measured at the output of the antennas by using
a plane interferometric network. The direction of arrival is reconstituted by
calculation from these phase measurements. The phase is only measured to
within 2°. Direction ambiguities may appear as a function of wavelength of the
incident signal relative to the dimensions of the measurement base. Butler matrix
of antenna array of microwave circuits make it possible to carry out directional
measurements by relative phase measurements and provide omnidirectional
coverage. However, this cannot be used in aircraft.
8.3.3
Directions Finding by Time of Arrival Measurement
It is the output of two antennas receiving the same incident signal. What is
measured here is a time of arrival of two signals. In this case large bases are needed
to ensure good angular accuracy.
8.4
LOCATION MEASUREMENTS
It is information detected by monitoring receivers to determine the
location of radar transmitters. Some of the following methods can be used:
(a)
(b)
Some of the techniques are based on measurement of the received level,
either by directly estimating the transmitted range as a function of
received level or by dynamic estimation of range on the basis of a
variation of this level with time. However, these are only approximate
methods because the received level depends upon parameters which are
not controlled such as attenuation of radiations during their propagation,
fluctuation of antenna pattern gains and above all, the possibility of radar
transmitters being able to modify their transmission level.
Techniques based on geometrical calculations such as
• Direction finding bearing lines received from a singular carrier
moving with respect to the radar can be utilized for finding the exact
location.
• Triangular method can also be utilized for this purpose. It may be
mobile or may be on other carriers.
395
Microwave Circuits & Components
(c)
Analysis of difference of time of arrival by hyperbolic location from
two or more monitoring sites.
These techniques can be used together according to requirement. For
carrying out angular measurements from mobile carriers the direction finding
bearing methods during movement of carriers which supply acceptable accuracy if
the angular movement of radar to be located with respect to carrier vehicle is
sufficient with direction finder having accuracy of 1°, range location of 5 per cent
requires angular movement between carrier vehicle and radar to be located to
approximately 30°.
The precision location is linked directly to the direction of
measurement base formed by several sites used for monitoring angular
measurement within 1° from 1 km base results in relative accuracy of above 50
per cent only for a radar transmitter situated at 50 km perpendicular to the
measurement base.
Different approaches have been developed to minimize the locating
error as a function of measurement made. However, it should be remembered
that radar transmitters to be located are totally separated from rest of the
environment. In the case of dense environment where radar needs to be located
very close to other, general sorting, touching methods have been developed.
These take account of all the measured parameters whether they are radioelectrical, they relate to direction or time of arrival. The data may be used for
radar, transmitter, sorting and location control then these have to be grouped
together and classified in multidimensional system.
8.4.1
Evolution of the System
The evolution of signals transmitted by radars needs monitoring in many
directions.
(a)
Expansion of frequency band covered in order to take account of modern
radar.
(b)
The use of following detector system to optimize sensitivity dynamics,
selectivities and intercept probabilities.
• Direct detection receiver
• Hetrodyne receiver
• Spectrum analyser
(c)
(d)
(e)
The new technologies should be used, like MMIC, VHSIC, VLSI, VHDL
etc., for enabling higher integration within receiver units
To develop techniques for improved filters for
• Spatial
• Frequency
• Tune related signals
These would certainly help in signal processing and
The signal processing facilities should be further improved for sorting
and locating of radar signals in dense environments. For example,
clustering methods and for signal identification using artificial
intelligence techniques.
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Electronic warfare
8.5
ELECTRONIC COUNTERMEASURES (ECM)
The term ECM generally applies to all facilities used for implementing
jamming or decoy application procedures against radars of enemies. To trigger
countermeasure activities, ECMs include receivers for analysing the
environment and working the best suited average merits against enemy
preparations, which involve neutralizing their radars.
Historically, EW has been principally concerned with techniques for
seeking the enemy targets in either normal or countermeasure environments or in
preventing the enemy from detecting our targets and other electronic areas, i.e.,
communications or navigation targets have received less emphasis. Radar and radar
countermeasures received large attention during Second World War. It is expected
that any developments as useful as radar is bound to receive much attention. To
nullify vulnerablility to countermeasures, Radar electronic warfare becomes
even more important.
The countermeasures require undertaking of the various types of radar
systems and their principles of operation. There are different classes of radars
like pulse, Continuous Wave (CW) and Pulse Doppler (PD). Generally, an ECM
system is designed to operate against a particular type of radar. As new types of
radar are being developed, new ECM systems must be devised to handle them.
8.5.1
Pulse Radars
A schematic block diagram of typical pulse radar is shown in Fig. 8.10.
Radar energy is generated by a magnetron oscillator powered with high voltage
pulses supplied by the modulator. Pulse modulator contains capacitors and
inductors which store energy and release them in pulses. The pulse is of the order
of a few microseconds. Though magnetron is most commonly used pulse-radar but
very often TWT is used as amplifier, the source may be low power frequency
synthesizer. The pulse is transmitted to antenna through a cable or waveguide. The
angular resolution of a radar is obtained with directional antenna that focuses
radar energy into a narrow beam. As an example, at 10 GHz an antenna 6 ft. in
diameter has a power gain of 20,000 and 1° beamwidth. During the search mode,
antenna systematically scans through the solid angle in which targets are
expected. The recovery system in the radar is protected from the high power
transmitter with a series of TR and anti TR (transmit and receive) duplexing
devices (gas suitables) that operated by ionization during transmission thus
protecting the receiver from burnout.
When radar echo is received it is delivered to microwave mixer, where it
is heterodyned or mixed with local oscillator signed to generate IF by crystal diode
inserted in waveguides or coaxial line. Noise places a fundamental limitation upon
the sensitivity of radar receiver. Even if the receiver and mixer do not generate
noise, the effective resistance of the antenna will provide a source of thermal
noise. The amount of noise power that is coupled into a receiver from antenna
that will pass through IF amplifier pass band is kT f , where k is Boltzmann's
397
Microwave Circuits & Components
MIXER
PULSE
MODULATOR
IF AMPLIFIER
AMPLITUDE
D ET EC TO R
MAGNETRON
AF C
VIDEO
AMPLIFIER
LOCAL
OSCILLATOR
SWEEP
G E N ER A T O R
Figure 8.10. Non coherent radar
constant, T is temperature in degree kelvin and f is the bandwidth in Hz (kT is
about 4.2 x 10-21 W-sec). This ideal situation is never met in practice, since the
mixer and IF amplifier always generate some noise, so the noise power
intercepted by IF amplifier is greater than kTf by a factor known as noise
figure NF. This is assumed that receiver is ideal and that the noise from the
antenna is
(8.31)
Pn kT f NF
The IF amplifier must be sufficiently wide to pass the narrow signal
pulses, yet if it is made too wide then it will also pass excessive noise. A
rectangular radar pulse is shown in Fig. 8.11. It is a sinusoidal voltage sin 2f c t
is made into narrow voltage pulses of duration 0 and at pulse rate of fr. The
resulting frequency spectrum is also shown. The spectral components spaced
apart in frequency by repetition rate is utilized which is centered about carrier
frequency fc is fr to EC volts peak. Each component by itself is a distinct
sinusoidal signal. A narrow band receiver tuned to any frequency would find it a
continuous wave. The central components rotates fc times per second, the next
one to the right rotates (fc+fr), the next (fc+2fr) and so on. Thus the relative phases
1
between all the vectors are continuously changing. Once during every period
fr
the vectors align themselves for the internal 0 so they are all in a straight line
(the positive one adding, the negative ones subtracting), which adds upto voltage
EC. During the remaining position of each period, however, the vectors spiral
around in such a manner that the result is exactly zero.
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Electronic warfare
0
EC
0
T
1
fr
FR
0 E
1
C
0
F
TC
Figure 8.11. Radar pulses and the resulting spectrum
An infinite number of components is needed to represent a perfectly
rectangular pulse. Supposing that only the components within a rectangular
frequency band f wide, centered about fc are selected and the rest rejected, the
resulting pulse amplitude (at the centre of the pulse) can be obtained by summing
their spectral components as
E pr
where
fo
f fx
sin x
Ec 0 f r
Ec 0 f r
x
x f fc T0
fb
f fa
sin x
x
(8.32)
f b f a f
f
fb f c 2
When the number of the components to be summed is large, the
summation can be approximated by an integral noting that fr is the spacing
behaviour components
E pR Ec 0 f r
1
fr
fb
fa
Ec 0 2 f
Sinx
x
0 f 2
0 r
where
Ec
Si u
2
df
0
sin x
x
dx
Si 0 f 2
0
S in x
x
dx
399
(8.33)
Microwave Circuits & Components
For example, E pR Ec if 0f
wide bandwidth
Ec E pck 0.99E if 0f 1.2
corresponding to an infinitely
for optimum bandwidth
Figure 8.12 shows a plot of peak pulse voltage as a function of 0 f .
Equation 8.31 shows that noise power is proportional to f . Since power is
proportional to voltage square, noise voltage increases as f . This proportion is
shown as line in the figure. Note that either side of approximately 0 f 1.2 ,
the noise voltage increases faster than the pulse voltage. Thus 0 f should be
about 1.2 for maximum signal-to-noise ratio out of the IF amplifier. When
0 f 0.5 or 3, the ratio of signal-to-noise is reduced by 3 dB from the optimum
value which has the same effect on detection range as reducing the transmitted
power to one half.
The wave shapes of the output of the amplifier for several values of
0 f is shown in Fig. 8.13. It can be noted that bandwidth decreases if the pulses
get smaller; they also get wider so that although the peak pulse voltage decreases
proportionally with bandwidth 0 f 0.5 the pulse power does not decrease
proportionally with square of bandwidth but directly with bandwidth.
Although the ratio of peak signal to average noise decreases when
bandwidth is narrowed, the ratio signal to average noise does not. This suggests
that the IF bandwidth can be made very narrow and the resulting S/N from the IF
amplifier will still be just as satisfactory for target detection (of course the sigma
will no longer consist of pulses). This conclusion can be verified when we are
dealing with pulse Doppler radars.
The average power contained in spectral components within f can be
estimated in spectral adding powers of each spectral lines
PEAK PULSE VOLTAGE
F
5
PROPORTIONAL
f
1
0.5
0.1
0.1
0.5
0 f
10.0
50.0
100
Figure 8.12. Plot of integral of some function of x for different values of x
400
Electronic warfare
0F =2
1.5
EC
1
1/2
0
1
Figure 8.13. Plot of pulse distribution for different bandwidth of an amplifier
Pow
1
2R
fb
2
f fx
Es 0 f r Sinx
x
Es2
0 f r 2 2
2R
0 f r
Pav
Es2
0 f r 2
2R
where
Pav
0 f 2
2
Sinx x dx
2
0
(8.34)
Sin 2 2 0 f
Si 0f
f
2 0
ES2
f r if 0f
2R
Pav 0.84
ES2
0 f r if 0f 1.2
2R
The variation of average pulse power with 0 f is plotted in Fig. 8.11.
Pulse detection and integration is not discussed here.
8.5.1.1 Pulse radar range
If radar transmits a pulse at some target at range R, the peak power in the
received pulse is
PRR
(8.35)
Ae
4R
4R 2
PPR is the peak power of the transmitted pulse and PPR/4R2 is the peak
power of density at the target if the power were radiated isotropically. Gt is the
transmitting antenna gain over an isotropic radiator, so the product of the first two
terms represents the power density at the target. The radar backscatter cross-section
is defined as an area normal to the direction of wave propagation that captures
PR
2
GE
401
Microwave Circuits & Components
all the power incident upon it and reradiates this power isotropically. This area is
purely arbitrary. There is no physical area.
The power density at the receiving antenna is given by the product of the
first three terms, and the power collected by receiving antenna is Ae times the
incident power density. Receiver noise is given by Eqn. 8.31. Assuming the
transmitting antennas are the same it can be shown that signal-to-noise ratio in the
output of IF amplifier is
Pr
PPRGt22L
S
(8.36)
N
IF Pn 4 3 RT NF fR 4
This is the conventional pulse radar range equation. It can be modified to
a more basic formulas. The peak and average powers are related by
(8.37)
Pav PPR 0 f r
where, 0 and fr as shown in Fig. 8.11. The optimum receiver bandwidth is
0 f 1.2
The antenna gain can be expressed as
G
4
(8.38)
b
where, b is the solid angle (in steradians) of the antenna beam. The effective
signal-to-noise ratio coherent integration of N pulses is
S
S
N NN
eff
if
The number of pulses revealed is
(8.39)
(8.40)
N Ti f r
where, Ti is the integration time, the time required for antenna beam to traverse the
target, fr is the pulse repetition rate. If the antenna beam is required to search a
solid angle s in a time TS assuming that it spends almost an equal time searching
in all directions within the solid angle.
Ti b
Ts s
It can than be shown that
(8.41)
Ae PavTS L
S
N
eff 1 24RT NF R 4 s
(8.42)
It can be seen that this range depends upon Pav Ts S the energy per
solid angle, and Ae, effective antenna area. It is independent of peak transmitted
402
Electronic warfare
power, pulse length, pulse rate and wavelength. Equation 8.42 indicates how
various factors are interrelated and how variation in them will change the range. It
is difficult to estimate radar cross-section of most of the targets. Radar crosssection varies with frequency. It also varies with target aspect angle if the target is
many wavelength in dimension.
In deriving Eqn. 8.38 it is assumed that antenna gain is constant between
half power points and zero elsewhere. This assumption is usually correlated by a
beam-loss factor of 1 to 2.5 dB.
In practice, the range of radar is calculated as accurately as possible,
taking into account as many corrections as necessary and then accounting for any
discrepancies between calculations and flight test by adjusting the operator or field
degradation factors. It is possible to use Eqn. 8.43 to estimate the effects of any
changes in radar parameters.
8.5.2
CW Radars
The pulse radars discussed in the preceeding section are often unable to
detect small targets because of echoes emanating from the land or sea
surrounding the target. The land or sea clutter can give echoes that are sufficiently
strong and irregular so that they completely obscure the target echoes. A radar
with two degree antenna beamwidth transmitting pulses will receive an echo from
low flying aircraft 30 miles away but with the echo clutter added from an area of
ground 6000 ft. wide and 500 ft. long. The effective backscatter cross-section of
this clutter would easily be 15,000 ft; 150 times as large as any fighter aircraft.
The principal difference between many aircrafts, truck and tank is velocity of the
target which results in a shift in the echo frequency due to Doppler effect.
The continuous wave radars are able to use this frequency shift to resolve
the moving targets. A simplified diagram of continuous wave radar (CW radar) is
shown in Fig. 8.14. The transmitter consists of microwave, source like Klystron
and Magnetrons.
It may be noted that in this case we cannot have same antenna for
reception and transmission. Separate antennas are to be used and they are oriented
in such a way that there is minimum interference between them. The received
echoes are mixed with a local oscillator signal to produce an IF signal. The IF
amplifier requires enough bandwidth to pass the expected Doppler spread of
frequencies. The amount of Doppler shift fd is equal to rate of change of round trip
radar path length expressed in wavelengths per second.
2v c
(8.43)
where, vc is the radial or closing velocity and is wavelength. At 10 GHz a
2000 fps closing velocity would shift the frequency by 40 KHz. A narrow band
filter selects the Doppler frequency component corresponding to velocity to be
detected for more than one frequency simultaneously. Each of the filters is timed to
progressively increasing but overlapping frequency bands. Amplitude detectors
are used to demodulate the output from each filter and produce a continuous
fd
403
Microwave Circuits & Components
MIXER
RF
GENERATOR
IF
AMPLIFIER
NARROW
BAND FILTER
AMPLITUDE
DETECTOR
COMMUTATOR
LO
GENERATOR
ANTENNA SCAN SIGNAL
INDICATOR
Figure 8.14. CW radar
voltage that indicates the presence of a target at velocity corresponding to filter.
A fast commutator is used for rapid examination of each filter detector in term.
The principal quantities measured by CW radars are angles and velocity.
Since velocity is not a dimension of space, display of type PPI are possible.
Therefore it is a problem. One of the indicator which can be used is shown in
Fig.8.15. The indicator type spot is swept in the velocity dimension in
synchronization with commutator and is moved back and forth in azimuth direction
corresponding to the antenna motion. The commutated output from the detectors
cause the spot to be brought up and the target is displayed. The band in the centre
of the indicator is at zero velocity due to a ground clutter and signal leakage from
the transmitter.
8.5.2.1 Doppler filter bandwidth limitations
The sensitivity of the receiver is limited by amount of noise, that passes
through the narrow-band filters, the noise being proportional to the bandwidth. It is
not possible to make bandwidth arbitrarily small due to several reasons. First, the
Figure 8.15. Velocity indicator for stationary radar. The centre band is due to
ground clutter and spurious modulations.
404
Electronic warfare
antenna beams are scanning past the target, causing the echo-signal amplitude to
vary as two-way gain of antennas. In other words, the continuous received signal is
changed into a pulse of duration, equal to the time antenna beam spent traversing
the target. The spectrum of this pulse is best explained by considering an antenna
made of an array of dipole elements instead of a reflector and feed (Fig. 8.16). As
the antenna rotates, the dipoles at one end move toward the incident radar wave,
while those at the other end move away. This motion results in a Doppler frequency
along the surface of the antenna. If the antenna has a diameter, and an angular
velocity, WT (radius) see the Doppler variation across its surface as
V
W d
fd 2 C r
(8.44)
The antenna beamwidth can be expressed as
Q
70 radius
.
37 d
(8.45)
For an antenna with tapered illumination across its aperture and the time
the beam spends on the target is
Ti
(8.46)
Wr
Combining Eqns. 8.44 and 8.46 gives
fd
1.2
Ti
(8.47)
It can be seen that spectral spread is approximately the reciprocal of pulse
duration. Thus, if antenna dimension is d , the spectrum is not spread beyond
2
Wr d .
+VE
INCIDENT RADAR
W
WAVES
R
D/2
+NE
Figure 8.16. Doppler shift due to rotating antenna
405
Microwave Circuits & Components
Normally a band of Doppler filters are needed for CW radar. A single
filter could be tuned through the frequency range to be monitored. However,
sluggishness of the filter should be considered. If the signal operating at its
resonant frequency is suddenly connected across a simple RLC filter (having
3 dB bandwidth of f , its response builds up as 1 exp t , reaching 63 per
cent of its final value in 1 f sec. Assuming that a filter should not be tuned
through any single frequency in less than this response time, the maximum tuning
range is
df
f
(8.48)
f 2
dt 1 f
It can be noticed that it would take the 36 cps filter 8.4 seconds to
examine 15 fps velocity range which would require the antenna to scan 0.27
degree/sec. This scanning is slow and wastes lot of radar power. In those cases
where maximum sensitivity is required, a single swept filter is much simpler than
a bank of many fixed filters. Many other factors would also give some type of
spectrum like target motion, antenna motion, yawing of aeroplane, propeller or
jet-turbine rotation. Amplitude or frequency modulation takes place in such a
case, the spectre of ground and sea returns are also spread. The system noise
enters the receiver by direct leakage from the transmitter or as modulation on
clutter return. Frequency modulation is sometimes, used in CW radars to obtain
target range. The echo signal is amplitude-limited so it will be independent of
target size, then it is compared in phase with the transmission modulation
generating a voltage proportional to Sin m 2 . This determines target range,
since is proportional to range.
If the spurious modulation of transmitter is reduced so that its spectral
lines are confined to a narrower bandwidth, the clutter becomes less of a problem.
Targets with positive-Doppler frequencies can be slower and still be sorted from
the clutter. Narrower Doppler filters can be used. If the filters are narrower than the
Doppler spread due to variations of ground velocity over illuminated ground area,
they will pass only a portion of the total power received within the beamwidth.
This is analogous to the pulse radar situation, where a narrower pulse reduces
ground clutter because only a portion of ground area covered by the antenna beam
is resolved at any instant. Then the velocity resolution of a moving airborne CW
radar can be used effectively to increase the radar ground mapping resolution,
beyond that provided by the antenna beam, just as narrow pulse increases radar
resolution in range.
8.5.2.2 CW power range
The basic detection range of a CW radar is quite easily calculated, using
the same reasoning that was developed earlier except that average power is used
instead of peak power.
P G A
Pr av t v
4 2 R 4
(8.49)
406
Electronic warfare
The receiver noise Pn kTf NF , and the signal-to-noise ratio is then
Pav Gt Av L
S
N 4 2 RT NF fR 4
(8.50)
The factor L is introduced to account for any system losses.
This is the signal-to-noise ratio after the narrowband Doppler filter
which has a bandwidth of f . The optimum value of f is 1.2 Ti . Where Ti is
same that an antenna spends scanning past the target.
Using Eqns. 8.31, 8.38 and 8.41, (these relation apply to all radar), gives
Ae PavTs L
S
N
eff 1.24 RT NF R 4 s
(8.51)
S
is the same as obtained for
Thus it can be seen that relation for
N eff
pulse radars Eqn. 8.42.
8.5.3
Pulse Doppler Radars
Pulse radars provide a nice, map like presentation of the surrounding
terrain and targets but it has been found that very often the targets are observed by
the ground return. Targets like, trucks, tanks and fighter aircraft do not have radar
crosssections which are large enough to compete with the echo from ground.
CW radars take advantage of Doppler shift from moving targets to discriminate
them from the clutter. However, these do not provide range resolution. They
provide velocity resolution and can use FM techniques to measure range on a
single target if is resolved by velocity but if objects are moving with same
velocity they cannot be distinguished or resolved.
It would be better if the functions of pulse and CW radars characteristics
are combined. If this is possible then both range and velocities are resolved. A
block diagram of such radar is shown in Fig. 8.17. The transmitter generates
narrow pulses for transmission like a pulse radar. It must also develop a coherent
local oscillator signal, and in this respect it is like CW radar. This coherence is
generally acquired by amplifying the signal from a stable source and then pulsing
the signal supplied to the RF power amplifiers.
The received echo pulse is mixed with local oscillator and is passed
through a range gate that is triggered open at the time of pulse execution. It is
closed between pulses and thus shuts out. Figure 8.18a shows the pulse
waveforms, and spectra in Fig. 8.18(b) shows the removal of noise between
pulses present in Fig. 8.18(c). The spectrum of waveform in Fig. 8.18b is shown
in Fig. 8.18c. Figure (d) and (e) are expansions of the frequency scale, which
illustrate the individual spectral components of the pulse. Since the pulse
envelope is purely periodic and is imposed upon CW carrier (no spurious carrier
AM or FM), all components of spectrum are pure CW. If the target is moving
towards the radar, the frequencies of all spectral components are shifted by
407
Microwave Circuits & Components
MIXER
RF
AMPS
RF
SOURCE
RANGE
G AT E
IF AMP
N NORMAL
BAND
FILTERS
TO OTHER RANGE
CHANNELS R1 R3
LO
GENERATOR
N AMP
DIL
TO LARGEST
SENSING
EQUIPMENT
RANGE DELAYED NOSE
PULSE
GENERATOR
GATE PULSES FOR OTHER CHANNELS
Figure 8.17. Block diagram of a pulse Doppler radar
Doppler effect, as shown in Fig. 8.18e. One of the central spectral components
can be regarded as the signal in CW radar and can be processed as such. Figure
18.8 (f) shows array of narrow band Doppler filters used to sort out the target
component, which happens to pass through filter no. 6. The Fig. 8.18d, e and f
show that the pulse-repetition frequency period must exceed the Doppler range
of possible target. This ensures that not more than one spectral time can pass
through filters and produce velocity ambiguities.
Though it is not apparent at first glance that filtering out one spectral
line gives satisfactory sensitivity. If EC is the peak voltage of the pulse, the voltage
(B)
(A)
T0
1/FR
PULSE BEFORE
RANGE GATE
SPECTRUM
'C'
EXPANDED
(C)
F
PULSE AFTER
RANGE GATE
(D)
FR
(E)
FR
1/T0
SPECTRUM 'D'
EXPANDED
SPECTRUM OF
PULSE 'B'
FIF
F
(F)
F
RESPONSE IF
DOPPLER
FILTER BANK
12345678910111213
F
Figure 8.18. Characteristics of pulse Doppler filter
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Electronic warfare
amplitude of central line is 0 f r E c . The power of this line is only 0 f r times
the average pulse power. However, the range gating has also reduced the noise
power density by a factor 0 f r so that signal-to-noise ratio into narrow-band
Doppler filter is in the ratio of average signal to average noise as for a CW radar.
Noise out of the range gate can be computed, the gating action can be
regarded as multiplying the IF amplifier output by a function that is unity during
pulse interval 0 and zero between pulses. Each of its spectral lines can be
regarded as individual local oscillator and the multiplication is the equivalent of
mixing each of the “local oscillators” with noise and summing the results.
The amplitudes of local oscillator signals are
Sinx
x
0 fr
(8.52)
where, x f fi 0
and the mixer output voltages (RMS value per cycle bandwidth) and
1
Sinx
(8.53)
Dn 0 f r
2
x
where Dn is the noise voltages per cycle out of the IF amplifier. The IF band pass
will be assumed as constant between fa and fb and zero elsewhere. The noise
voltage have random relative phases and thus cannot be coherently added but their
powers can be. The resultant noise voltage density Dr is
fb
Dr2 12 Dn 0 f r
ffa
1
2
Dn 0 f r
2
Sinx
x
2
1 b Sin2 x
2 dx
f r 0 a x
2 b Sin2 x
2 dx
a x
b a 2 0 fb f a
1
2
Dn
2
0 fr
(8.54)
(8.55)
If IF bandwidth is large
Dr2
The
1
2
1
2
Dn
2
(8.56)
0 fr
value results from mixing.
The effect of the range gate upon signal pulse can be computed in a
similar manner. If Ec is the peak pulse voltage, the peak pulse amplitude of
voltage, the voltage amplitude of the central spectrum component is
409
Microwave Circuits & Components
fb
12 0 f r Sinxx 0 f r Sinxx
ES
f fa
12 EC 0 f r
ES
1
2
2
EC 0 f r 2
b
1
(8.57)
Sin x
dx
0 fr x
2
2
a
b
2
Sinx x dx
2
0
The signal-to-noise density power ratio after gating is
2
2
E
E
S
2
s c 0 f r
N d Dr
D
n
The function
2
b
b
2
Sinx x dx
2
(8.58)
0
2
Sinx2 x dx
0
is plotted in Fig. 8.12 showing that it increases as IF bandwidth increases. Thus IF
bandwidth can be wider than 0 f 1.2 , the optimum value derived for pulse
radars. The Doppler filter bandwidth will restrict the noise. In this case
2
S
E
Dc 0 f r
Nd n
(8.59)
It should be noted that
Ec
Dn
2
ratio of peak pulse power
R M S noise power density
(8.60a)
Thus 0 f r reduces peak power to average power. Thus
ratio of average signal power
Output S after gating
N
noise density before gate
a
The average signal power before gate
Pr
Pav Gt Ae
4 2 R 4
The noise power density after gating is RT NF , therefore after gating
410
Electronic warfare
S
PavGtAe
N d 4 2 R 4 RT NF
ratio is
(8.60b)
The Doppler-filter bandwidth is f cycles and signal-to-noise power
S
S 1
PavGtAe
2
N N d f 4 RT NF f R 4
(8.61)
The optimum filter bandwidth is
f 1T.2
i
where, Ti is the time required for antenna to scan past the target. Substituting the
T
4
relation c b in the relation G
, one obtains
s
Ts s
Ae PavTs L
S
N
eff 1.24 RT NF R 4 s
8.62a)
PavAeTs L
S
N
1.2 4RT NF R 4s
8.62b)
Thus,
Equations 5.62a and 5.62b show that
equations developed for pulse and CW radars.
S
N
is the same as the radar-range
The range gate shown in Fig. 8.17 shows that the bank of N Doppler
filters can be allowed to quantric one range interval into N-possible velocity
values. If M range intervals are to be examined simultaneously, M range gates and
Doppler filter banks are required. The total number of Doppler filters is M times
N, which is concurrently large. The number of filters can be reduced as for CW
radars, by making them wider than optimum and inserting low-pass networks after
amplitude detectors to narrow the effective bandwidth.
Since pulse repetition frequency must be greater than maximum
Doppler shift, number of range gates required is more than that would be needed
in a conventional pulse radar. To accommodate a 4000 fps target closing velocity
10 GHz radar would need a repetition frequency of more than 80 KHz. Assuming
1 ms pulse's maximum possible range gates would be about 11 an 80 KHz
repetition rate results in range ambiguities every 62.50 ft. Such ambiguities can
be resolved by changing the repetition rate among several different values and
noting the shift in position of the echo pulse in relation to transmitted pulse or by
coding the transmissions and measuring the round trip propagation delay.
411
Microwave Circuits & Components
Pulsing the RF amplifiers introduces additional complications for purity
of spectrum. Any variations in the shape of pulse envelope from pulse to pulse, or
any phase modulation of the amplifiers that is not the same for each pulse will
spread the spectral lines. In short, unless all pulses are exactly same and
transmitter output is strictly periodic the pulses will not be represented by
individual spectral lines spaced far apart.
There are many indicators used in radar, but they are not discussed here
because of nature of book which does not only concern radar. Some of the
associated topics are
(a)
(b)
(c)
(d)
(e)
Moving target in action
Target tracking systems
Angle tracking
Range tracking
Velocity tracking
8.6
GENERAL RADAR CONCEPTS
It is possible to consider the effect of various types of electronic
countermeasures (ECM) on each type of radar discussed above. Specific radar
systems have been discussed. The types of ECM would be equally specific like
jammers for pulse radars, jammers for CW radars and so on. The disadvantage of
this approach is that it does not allow one to treat ECM on future radars or classified
radars. It misses several very fundamental concepts pertaining to radars and their
ECM in vulnerability. A better approach to this problem is to develop these
fundamentals, relate them to radar systems already present and then consider their
implications in EW. The detection range fundamentals are more or less three types
of radars viz., pulse, CW and pulse Doppler. The detection range equations for all
three radars is the same. Considering the difference between pulse and CW
concepts, they seem to be opposite approaches to radar problem and it is not
intuitively obvious that one equation could describe the range of both the
systems. The range equation is
R4
Ae R Ts L
1.24RT NF
(8.63)
S
N
The expression can be written in this form because the factors can be
obtained from actual system specifications, simplifying numerical calculations.
When the radar is searching the solid angle s for the target which is at
range R, the power can be regarded as uniformly distributed over an area equal to
that portion of spherical surface of radius R intersected by antenna beam. The
average power density at the target is
Pav
(8.64)
bR2
by
To obtain average echo power density at the radar Eqn. 8.64 is multiplied
412
Electronic warfare
4R 2
This is the definition of radar cross-section.
(8.65)
To obtain the receiving antenna waveguide power, the power density in
front of antenna is multiplied by Ae. A factor L is used to account for any losses.
The echo energy, E, received from the target during one scan interval as Ti times
the average power or
P A L T
E av e i
(8.66)
b R 4 4
Let us take noise also in consideration which affects the range. The
receiver noise is given by
(8.67)
Nd kT NF
where
k = Boltzmann constant
T = Temperature
NF = Receiver noise figure
Nd is the power density in watts per cycle per sec. Using the Eqns. 8.66
and 8.67, the ratio of signal power to noise power in front of the receiver is then
S
E
(8.68)
N 1 .2 N d
Remembering that the probability of target detection is directly related to
S/N which is equal to the ratio of received target echo energy to noise power
density. Theoretically it can be said that the detection is independent of type of
radar modulation used and pulse, CW and pulse Doppler radar are equally good. In
practice such as clutter, target size, resolution and accuracy requirements and case
of implementation can make one type of radar much more preferable but all of
them do need sufficient echo energy from the target. Radar will be used to make
successive measurement, its transmission will be assumed to be periodic, all
periods identical, each period equal to Tp. Then the spectrum of radar
transmission will consist of a series of discrete lines. Each line is sinusoidal of
constant amplitude and frequency having some relative phase relationship. Each
is spaced in frequency 1 T p . The sum of these spectral components produce
waveforms of specified radar transmission.
2 Es a1Cs W1 1 a2 Cs W2 t 2 ......... a x cos Wn t n
n
2 Es g R t
(8.69)
R 1
The highest voltage that can be obtained is simply the sum of all
components added in phases. This summation can be accomplished by first
heterodyning all components to a common frequency so that the phases between
components are independent of line. This is obtained by multiplying the signals.
413
Microwave Circuits & Components
E s a1Cu W1t 1 E m b1Cu W1t 1 Wi f t
(8.70)
Filtering out the resulting frequency difference gives
1
2
E s a1 E m b1 C u Wi f t
(8.71)
The phase shift, 1 , is eliminated and Wif is the desired common
frequency. The amplitude of sum of all components is
S
1
2
E s E m a1b1 a 2 b2 .......... . a n bn
(8.72)
The amplitude of noise voltage accompanying each signal component is
1
2
E m b1 N d ,
1
2
E m b2 N d ,......... ..
(8.73)
where, Nd is the noise voltage (RMS) per cycle per second accompanying the
signal, since the noise is added on power basis, the resulting noise combined with
the signal is
N
1
2
Em N d
b12 b22 ....... bn2
(8.74)
Taking signal-to-noise ratio, voltage ratio and maximizing S/N as a
function of b1, b2, ……. gives the result that aR = bR. Thus, the echo and reference
signal frequency component should both have same relative amplitudes for
maximum signal-to-noise ratio. It can be easily found that whatever form of
modulation a radar uses, the echo signal is processed by multiplying it by a
reference signal that has the same frequency components and thus the same
waveform as echo itself.
8.7
EFFECT OF ECM ON RADARS
The generalized radar transmission has the spectrum consisting of
n spectral lines spaced at uniform frequency by interval I T p . It may be possible
for enemy to readily deduce the spectral arrangement of radar transmission by
simply observing its waveform. A plain pulse transmission is easily determined
inspite of its extensive spectral composition.
Suppose the enemy does attempt to jam the generalized radar but fails to
arrange the phase of n spectral components in the same manner as the radar
transmission, then on reception, the radar fails to arrange the phase of n spectral
components in the same manner as the radar transmission. On reception, the radar
receiver will process the jammer signal, as it will be the real target echo by mixing
it with proper reference signal. Since the jammer signal spectral components are
incorrect the process will not result in summation of n in phase voltages but of n
voltages having random phases. If the amplitudes of the individual voltages are a1,
a2,…..,an for the desired echo, they add n phases as
a 1+ a 2+a 3+……a n
(8.75)
414
Electronic warfare
For the random phases of the ECM signal, they add as:
(8.76)
a12 a 22 a 32 ...............a n2
If the relative amplitudes of the individual spectral components are all
approximately equal, then the ratio of signal voltage to ECM voltage is
na1
na12
n
(8.77)
and the ratio of echo to ECM powers is n. Thus, the ECM is degraded in favour of
the echo. [Relation of type 1.2 0 f r spectral line within hard bandwidth
f (assuming 0 f r =1.2)]. A jammer that attempted to produce pulses with
required spectrum but was unable to get the proper phase arrangement resulting in
random phases which would be degraded by a factor which is nearly 0 f r , which
is the ratio of pulse peak to average power. It should be remembered that a CW
radar, having a single spectral line, cannot code its spectrum so as to reduce the
effects of CW jammer. A pulse radar is not immense either, unless the phase
arrangement of its spectrum is altered in some way so that it not obvious to enemy.
As an example of radar with some ECM immunity, consider the MTI
system shown in Fig. 8.19 where conventional pulse radar can be modified to form
pulse Doppler that provides enhancement of moving targets, or Moving Target
Indication (MTI). In addition to usual IF channel, the local oscillator is mixed with
magnetron pulses and the resulting IF pulses are used to synchronize the phase of
an IF reference oscillator. When the reference is mixed with IF amplifier output,
the resulting difference frequency pulse is used to synchronize the phase of IF
oscillator. When the reference is mixed with the IF amplifier output the resulting
difference frequency is modulated by the Doppler. If IF pulse can be represented
by
x cos i
(8.78)
f di t
where, i f r and d are the IF and Doppler frequencies. Multiplying this
composite waveform by reference signal, cos( i f )t, and taking the difference
frequencies gives
x 1 cos di t
2
(8.79)
Thus, IF pulses are changed into video pulses that vary in amplitude as
cos( t ). Since adjacent pulses differ in amplitude, if each pulse is subtracted
from the pulse preceeding it by means of delay tunes followed by a compensator, a
difference or residue is obtained. If target is stationary no Doppler is obtained, as
all pulses will be of same amplitude and the residue will be zero. Thus, pulses
from moving target are enhanced while stationary clutter is rejected.
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Microwave Circuits & Components
MODULATOR
MAGNE TRON
M
IF
AMPLIFIER
M
IF REF
OSCILLATOR
M
DELAY AND
HIND
LOC AL
OSCILLATOR
VIDEO
AMPLIFIER
PULSE
GENER ATOR
SWEEP
GENER ATOR
INDICATOR
Figure 8.19. Moving target indicator in a radar
In this system if the delay line unit is removed and substituted by a range
and band of narrow band Doppler filters, such as those used in pulse Doppler
system shown in Fig. 8.18. If IF amplifier and reference oscillator outputs are
mixed to produce signals that are gated in filters as shown in Fig. 8.18. The
Magnetron phase is different for each transmitted pulse; also the IF reference
oscillator is adjusted to be in phase with each transmitted pulse, so the IF echoes
will be coherent. A jammer that generates radar like pulses would also have to
adjust the phase of each pulse in keeping with Magnetron pulse-to-pulse phase
changes.
The use of various types of coded radars such as matched filter or pulsecompression systems can reduce the effectiveness of a jammer that attempts do
duplicate the radar transmission. As the radars are designed to make them less
vulnerable, it becomes more difficult to defeat them by duplication and more
attractive to simply drown the radar echoes by transmitting wideband random
noise.
A jamming situation as shown in Fig. 8.20 in which radar is attempting to
locate target aircraft. In the absence of any jamming, the echo signal must be
detected in the presence of receiver noise. It can be recalled that Eqn. 8.63 can also
be written as
Pav Ts LAe
S
N 1.2 4 RT NF s R 4
The target is detected at a range Ro.
(8.80a)
When the barrage noise jammer is operating at a distance of Rj from the
radar, the receiver noise-power density (W/unit frequency) is given by Dj of an
jammer antenna in the radar direction Gr and the gain of the radar direction is Gr
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Electronic warfare
D j G j G 2
r
kT NF
4 R 2 4
j
(8.80b)
instead of only kTNF.
The gain of radar antenna in the direction of jammer is Gr. Under these
conditions the signal-to-jamming signal is given as
S
N
Pav Ts LAe
D G G 2
1.2 4 R 4j j j2 r 2 RT NF
4 R j
(8.81)
where, the new distance from radar to target is Rj.
Thus, it is quite clear that the radar has to have same signal-to-noise ratio
for the target in the presence of jamming signal as
S
S
(8.82)
N J
Then the fractional reduction of radar detection range due to noisebarrage jamming Rj /Ro is
4 2 kT NF R j 2
Rj
R
2 D j G j Gr 4 2 kT NF R j 2
0
4
(8.83)
In most cases of interest the jammer noise is considerably greater than
the recent noise. The large value of jammer noise-to-receiver noise allows
simplification of Eqn. 8.83 to
4
2
2
R j 4 RT NF R j
R
2 D j G j Gr
0
(8.84)
JAMMER
TARGET
RS
Rr
RADAR
Figure 8.20. Radar operation in presence of both jammer and target
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Microwave Circuits & Components
The ratio of S/N with and without jamming is also given by Eqn. 8.84
S N j 4 2 RT NF R 2j
S N 0
2 D j G j Gr
(8.85)
Consider an radar operating of at 10 GHz having a side lobe gain in the
direction of jammer of 10dB, and receiver noise figure of 10dB. A jammer at a
range of 20 nautical miles radiates a noise power density of 1W/MHz with antenna
gain of 10 dB. Then
S N j
S N 0
0.1
and
Rj
R0
0.56
Thus, jamming reduces the signal-to-noise-ratio by 10 dB and reduces
the single-scan range to 56 per cent of the non-jamming range. If the jammer is in
target, Rj= Rr and Gr becomes the gain of the radar antenna main lobe which
simplifies to Eqn. 8.84 still further, it can be noticed that if the radar is expected to
have increased detection range because of a lower noise figure. The result given in
Eqn. 8.84 is independent of the type of radar. The detection range in any radar is
limited by noise in whose presence the echo signal has to compete with it. The
noise jammer simply increases this noise, or, in other words it increases the radar
noise.
8.8
SOME JAMMING TECHNIQUES
There are many jamming techniques, a few of these will be discussed
here. The purpose of jamming is to interfere with the enemy’s effective use of the
electromagnetic spectrum. So far, only radar jamming was discussed briefly. We
will discuss the jamming in other parts of electromagnetic spectrum like
communication command signals to remote control by located instruments or
assets. In many cases the term electronic attack (EA) is also used for electronic
countermeasure (ECM). EA includes the use of high level radar power or directed
energy to physically damage enemy assets. As already mentioned, the basic
technique of jamming is to place an interfering signal into an enemy receiver along
with the desired signal. Jamming becomes effective when interfering signal is
strong enough to prevent the enemy from recovering the required information from
the desired signal. The information content may be overwhelmed by power of
jamming or the desired and jamming signal have such characteristics that the
processor is prevented from extracting useful information.
Table 8.1 defines several ways in which various the various classes of
jamming are differentiated.
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Electronic warfare
Table 8.1. Types of Jamming
Types of Jamming
Purpose
1. Communication jamming
Interferes with enemy's ‘ability’ to pass
information over a communication link
2. Radar jamming
Causes radar to acquire target to stop
tracking target or to output false alarm
3. Cover jamming
Reduces quality of the dense signal so that it
cannot be properly processed and
information it carries cannot be recovered
4. Deceptive jamming
Causes a radar to improperly process its
return signal to indicate an incorrect range or
angle to the target
5. Decoy
Looks more like a target. Causes a
guided weapon to attack the decoy rather than
its intended target.
8.8.1
Communications in Comparison with Radar Jamming
COMJAM or communication jamming is the jamming of
communication signals. In a way, it is jamming of tactical HF, VHF and UHF
signals using noise modulated cover jamming of point-to-point microwave
communications links or command and date links to and fro from remote assets.
Figure 8.21 shows the enemy communication link, which carries the signal from
a transmitter (XMTR) to a receiver (RCVR). The jammer also transmits into
receiver antenna, but it has enough power to overcome the disadvantage of
antenna gain and to be received and output to the receiver’s operation or
processor with adequate power to reduce the quality of the desired information
to an unusable level.
COMMUNICATION LINK
RCVR
XMTR
JMR
Figure 8.21. Communication jamming
As we have already seen, a radar jammer provides either a cover or
deceptive signal to prevent radar from locating or tracking its target.
8.8.2
Cover & Deceptive Jamming
In this system high power signals are transmitted to the enemy
transmitter. The noise modulation makes it more difficult for the enemy to know
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Microwave Circuits & Components
that jamming is taking place. This reduces the enemy’s SNR (signal-to-noise)
ratio to the point where the desired signal cannot be received with adequate
quality. The Fig. 8.22 shows a radar plan position indicator with a return signal
and noise cover jamming strong enough to hide the return. The Deceptive
jamming causes a radar to draw the wrong conclusion from the combination of its
desired signal and jamming signal as is shown in Fig. 8.23. In this jamming, the
radar is reduced from target in the range, angle or velocity. To a radar operator it
appears as if he is receiving a jamming signal and thinks that radar is tracking a
valid target.
Figure 8.22. Cover jamming. It hides the radar's return signal
ACTUAL TARGET LOCATION
TARGET LOCATION
DETERMINED BY JAMMED RADAR
RADAR
Figure 8.23. Deception jamming in which target position is incorrectly detected
Cover jamming injects additional noise into receiver which is above
thermal noise already present (kTB in dBm + the receiver system noise figure in
dB). The cover jamming has the same effect as increasing the transmission path
length of or decreasing the RCS of a radar’s target. When the jamming noise is
significantly higher than receiver’s thermal noise, we speak of the jamming to
signal(J/S) ratio rather than SNR (signal-to-noise ratio). If the cover jamming is
increased gradually, the operator or the automatic processor of circuitry
following the receiver may never become aware that jamming is present except
that SNR is becoming extremely low.
As shown in Fig. 8.24, a receiving system discriminates to some extent
against all signals except the one it is designed and controlled to receive. If it has a
functional antenna pointing at the source of desired signal, all signals from other
directions are reduced. Any type of filtering reduces out-of-band signal. In pulsed
radar, the processor following the receiver knows approximately when to expect
a return pulse and will ignore signals that are not near the expected return time.
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Electronic warfare
CONTROL
JAMMER
E W
PROCESSOR
SETS
JAMMER
ON IN
FREQUENCY
TIME
DIRECTIONAL JAMMER
ANTENNA ARRAY
RECEIVER
DF SYSTEM
Figure 8.24. Power management system
The problem for jammer is that to be effective it must spread its
available power to cover over the entire frequency that receiver might be
receiving over all angular space that might contain the receiving antenna during
all time that the receiver might be accepting signal energy. The jammer output is
increased till the jammer output power has enough power so that it becomes
effective.
8.8.2.1 Power management
The more the jammer knows about the operation of the receiver the
more narrowly it can focus its jamming power to which the receiver will notice
jammer energy-focussing is called power management and it can only be as good
as the information available about the jammed receiver. This information
normally comes from a supporting receiver (either a jammer receiver or an
electronic support system), which receives, qualifies and measures the
parameters of signals that are thought to be received by the receiver being
jammed. Sometimes this is easy (as in a radar that is tracking the platform
carrying the jammer) and sometimes this is harder (for example communications
links or bistatic radars). The integrated EW system shown in Fig. 8.24 will
provide its jammer with information on direction of arrival frequency and timing
appropriate for managing its power.
8.8.3
Deceptive Jamming Techniques
Deceptive jamming is almost entirely a concept applicable to radars. In
this case, signal-to-noise ratio in the receiver is not touched but these techniques
operate on the radar processing to cause it to lose its ability to track a target.
Some cause the radar track to move from the target in range and some in angle.
Some of these do not work against monopulse radars, i.e., radars in which each
pulse contains all necessary tracking information.
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Microwave Circuits & Components
8.8.3.1 Range gate pull-off techniques
This is a self detection technique which requires knowledge of the time
of arrival of pulses at the target being tracked by radar. This technique may not
work against monopulse radars (radars in which each pulse contains all necessary
tracking information).
The jammer enters a false return pulse, which is delayed, from the
reflected radar pulse by gradually increasing amount as shown in Fig. 8.25. Since
the radar determines the range of the target by the time of arrival of reflected
pulses, this makes the radar to interpret that the target is further away than it
actually is. In this way the accurate range information is derived. This technique
requires 0 – 6 dB J/S ratio.
RADAR SIGNAL
TA RG ET
SIGNAL
JAMMER
Figure 8.25. The range gate pull-off jammer transmits a higher power return
signal and delays it by increasing amount.
An important consideration is how fast the jammer can pull the range
gate away from the target. Obviously, the faster the range gate is moved, the better
the protection. However, if it exceeds the radar's tracking rate the jamming would
fail.
Two counter-countermeasure techniques are effective against the range
gate pull-off jamming. One is simply to increase the radar power so that the true
skin return dominates the return signal tracking. This is in fact what happens at
burnthrough range. The second is to use leading edge tracking considering the
actual signal received by the radar during the range gate pull-off jamming. Inverse
gain jamming is one of the techniques used to cause a radar to lose angular track.
8.8.4
Inverse Gain Jamming Techniques
Inverse gain jamming is a self-protection technique that uses the radar's
antenna-scan gain pattern as seen by a receiver at target being illuminated.
Figure 8.26 shows a typical radar scan pattern. As the radar beam sweeps by the
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Electronic warfare
SKIN RETURN SIGNAL POWER TO RADAR
RECEIVER FROM THE RADAR ANTENNAE
POWER
JAMMING LEGEND WHICH WOULD
PRODUCE"INVERSE GAIN SIGNAL
POWER
Figure 8.26. Ideal inverse jammer
target, the time history of the power it applies to the target varies as shown in the
top part of the figure. This is called the threat radar scan. The large lobes occur as
the radar's main beam passes the target, and the smaller lobes occur as each radar
side lobes pass the target. The skin return from the illuminated target is reflected
back to the radar with the same pattern seen and the radar uses the same antenna to
receive the return signal. Basically, the radar determines the angle, the azimuth
elevation or both so the target to know where its main beam is pointed when it
receives the maximum skin return signal strength.
If a transmitter located at the target were to transmit a signal back
towards the radar with same modulation (i.e., the pulse parameters) as the radar
but with power versus time as shown in the bottom part of the figure, the received
signal power and radar's antenna would add to a constant. This means the receiver
in the radar receives the constant signal regardless of where its antenna beam is
pointed and would thus be unable to determine the angular information about the
target location.
Although this jamming burst-pattern will not consistently create 180°
tracking error (as it would if it were synchronised to scan the jammed radar) it
will cause erroneous tracking signals almost all the time.
8.8.4.1 AGC jamming
Automatic Gain Control (AGC) is an essential part of any receiver that
must handle signals over an extremely wide received-power range. The receiver’s
instantaneous dynamic range is the difference between the strongest and weakest
signals it can receive simultaneously. To accept a range of signals wider than this
instantaneous dynamic range it must incorporate either manual or automatic gain
control to reduce the level of all received signals enough to allow the strongest
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Microwave Circuits & Components
signal to be accepted. AGC is implemented by measuring power at some
appropriate point in the receiving system and automatically reducing the gain or
increasing an alteration enough to reduce the strongest in-band signal to a level
that can be handled by the receiver. Large variations in the target range and radar
crosssection (RCS) require use of AGC in radars. An AGC jammer transmits very
strong pulses at approximately the radar scan rate. These pulses capture the radar's
AGC. The resulting gain reduction causes all in-band signals to be greatly
reduced. The skin track signal is suppressed to such a low level that radar cannot
effectively track the target.
8.8.4.2 Velocity gate pull-off
Continuous Wave (CW) and Pulse Doppler (PD) radars separate signals
reflected by a moving object (for e.g., a low flying aircraft or a walking soldier)
from a signal reflected by the earth using of frequency discrimination. According
to the Doppler domicile, the reflected radar return signal from every object
within the radar's antenna beam will display changed frequency. The frequency
shift of the reflection from each object is proportional to the relative velocity of
the radar and the object causing the reflection. The return can be quite complex.
To track a particular target return in this mess, the radar needs to focus on a
narrow frequency range around the desired return signal. Since every frequency
in Doppler returns correspond to relative velocity, this frequency filter is called
a velocity gate and set to isolate the desired target return. During an engagement,
the relative velocity of radar and target may change rapidly and over a wide range,
for e.g., the relative velocity between two aircraft making 6 G turns can range
from Mach 3 to 6 and change at rates upto 400 kph per second. As the relative
velocity of target changes the radar velocity gate will move in frequency to keep
the desired return centered. The amplitude of the return signal can also change
rapidly because, the RCS of any object viewed from different angles can differ
significantly.
The velocity gate-pull-off (VGPO) jammer generates a much stronger
signal at same frequency at which the radar signal is received at the target. The skin
return will come back at radar with different frequency (Doppler Shift) but since
the target and jammer move together, the jammer signal will be identically
shifted and so will fall within the radar's velocity gate. When the jammer sweeps
the jamming signal away from the frequency of the skin return, (since the jammer
is much stronger), it captures the velocity gate away from the skin return. Then
the jammer causes the velocity gate to move far enough from the skin return so
that the skin return is outside the gate – breaking the radar's velocity track.
8.8.5
Deceptive Jamming Techniques used against Monopulse
Radars
A monopulse radar is tough to jam because it gets all the information
required to track a target (in azimuth and/or elevation) from each return pulse it
receives rather than by comparing the characteristics of a series of pulse returns.
Self-protection jamming of monopulse radar gets even trickier because the
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Electronic warfare
jammer is located on the target – a beacon that may make tracking even easier. If
a self protection jammer denies a monopulse jammer range information (with,
for e.g., cover pulses), the radar can usually track in angle, which may provide
enough information to guide a weapon to the target.
There are two basic approaches to deceive a monopulse radar. One is to
take advantage of some known shortcoming in the way the radar operates. The
second is to take advantage of the width of resolution cell that monopulse radars
develop for their angle tracking information within a single radar resolution. The
second approach is generally superior, and will be discussed first.
8.8.5.1 Radar resolution cell
The width of resolution cell is defined by the area that falls within the
antenna’s beam – which depends on the beamwidth and the distance from the radar
to the target. The beamwidth is usually considered to be the 3 dB bandwidth, so at
a range of n km the beam covers (2n x the sine of half the 3 dB beamwidth) km –
but that is not (on the whole) strong. The radar's ability to discriminate between
two targets in azimuth or elevation depends on the relative strength of the radar
returns from both the targets as the antenna beam is scanned across them. In case
the targets are far apart, so that both cannot be in the antenna beam at the same
time, the radar can discriminate between them (i.e., resolve them). Since the radar
can be assumed to have the same transmitting and receiving antenna patterns, the
return from a target located at the 3 dB angle from the antenna bore sight will be
received with 6 dB less power than a target at the bore sight. Now consider what
happened to the received signal power from two targets separated by one half
beamwidth as radar antenna moves from one to another. The power from the first
target will diminish more slowly as the power from the second target builds up –
so the radar will show continuous bump of return of power. At less than one-half
beamwidth separation, this is even more pronounced. When the two targets are
separated by more than a half beamwidth, the response has two bumps but they do
not become pronounced until the targets are about a full beamwidth apart. Thus
the resolution cell can be considered to be a full beamwidth wide, but considering
it to be one half beamwidth across is more appropriate.
The mechanism causing depth of resolution cell (i.e., the range
resolution limitation) is shown in Fig. 8.27. This figure shows a radar and two
targets (with distance to the targets obviously small relative to pulse width – PW)
when the two targets are separated in range by less than a pulse duration, the
illumination of the first target is complete. However, the arrival of return pulse
from the second target is delayed from the first return by twice the target
separation divided by speed of light – because the round trip time from the range
of the first target to the range of second target is added. Thus as separation of two
targets in the range is decreased, the return pulse do not start to overlap until the
range difference is reduced to half of pulse duration – limiting depth of
resolution cell to half a PW (in distance). From these discussions the
conservative definition of the radar resolution cell is the area enclosed by half
the beamwidth and the half the distance travelled by the radar signal during its
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Microwave Circuits & Components
TRANSMITTED PULSE
TARGET 1
RADAR
TARGET 2
TARGET 2
TARGET 1
RETURN
PULSE
Figure 8.27. Two targets separated in range by one pulse duration will produce
return pulse separated by pulse duration.
pulse duration8. The type of jamming used in this case is known as formation
jamming. It can be performed at greater target separation if cover pulse or noise
jamming is used to deny the radar range information. The required J/S for these
types of jamming is typically not high (0 to 10 dB).
8.8.5.2 Blinking jamming
Blinking jamming also involves two targets within a single radar
resolution cell. However, they carry jammers, which are used cooperatively. The
two jammers are activated at blinking rate that is close to radar’s guidance zero
bandwidth (typically 0.1 to 10 Hz).
8.8.6
Decoys: Applications & Strategies
Decoys can be classified according to the way they are placed in service,
the way they interact with threats or the types of platforms they protect; terms
abound for each category. The decoy type is defined in terms of the way it is
deployed. Table 8.2 shows decoy type along with the missions and platforms
typically associated with these decoys. Expendable decoys are ejected from pods
or launched as missile from aircraft and also launched from tubes or rocket
launchers from ships. These decoys typically operate for short periods of time
(seconds in air/ minutes in water).
Table 8.2 Missions and platforms typically associated with type of decoys
Decoy Type
Mission
Platform protected
Expandable
Seduction
Aircraft, ships
Saturation
Aircraft, ships
Towel
Seduction
Aircraft
Independent
manoeuvre
Detection
Aircraft, ships
A towed decoy is attached to the aircraft by a cable, with which it can be
controlled and/or retracted by the aircraft; towed decoys are associated with long
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Electronic warfare
duration operation. Towed barges, for ships use large corner reflectors and could
also be considered towed decoys (considered separately).
Independent manoeuvre decoys, are deployed on propelled, typically
airborne platforms. Examples are UAU decoy payload, used in ship protection
and decoys mounted on or below helicopters. When independent manoeuvre
decoys protect a platform, they have complete flexibility of relative motion (in
contrast to towed decoys, which must follow along, or expandable, decoys which
fall away or are fired forward).
Decoys have two basic missions to saturate enemy defences: to cause
an enemy to switch an attack from the intended target to the decoy and to cause an
enemy to express his offensive assets by preparing to attack an decoy. These
decoy missions are as old as history of human conflict, far preceding the age of
EW. The difference is that rather than directly deceiving the senses of human
warriors, modern EW decoys deceive the electronic sensors, which detect and
locate targets and guide weapon to them.
8.8.6.1 Saturation decoys
Any type of weapon has limitations in the number of targets it can
engage at a time. Since a finite amount of time is allotted for sensors and
processors to deal with each target it attacks, the limitation is more accurately
described as limit on the number of targets it can attack in a given amount of time.
The total time period during which a weapon can engage a target starts when the
target is first detected. It ends when either the target can no longer be detected or
the weapon has succeeded in performing its mission. The weapon will only
engage some maximum number of targets at once; if more targets are present,
some will escape attack, because weapon must operate above its saturation point.
8.8.6.2 Detection decoys
A new and particularly useful radar decoy causes a defensive system
such as air defence network, to turn on its radar – making is susceptible to
detection and attack. This typically requires independent manoeuvre decoys. If
decoys look and act enough like real targets, the acquisition radars or other
acquisition sensors will hand them off to tracking radars. Once tracking radars
turn on, they are vulnerable to be attacked by anti-radiation missiles fired from
aircraft outside the lethal range of the enemy weapon.
8.8.6.3 Seduction decoys
In seduction mission the decoy attracts the attention of a radar that has
established track on a target, causing the radar to change its track to the decoy.
8.9
RADAR COUNTER COUNTERMEASURE TECHNIQUES
The decision as to kind of ECM equipment to be developed depends
upon assessment of enemy’s technology, intelligence on his operating
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Microwave Circuits & Components
frequencies and types of radars. In absence of such data, it is often assumed that
enemy has achieved a state-of-art comparable to ours. Therefore, his radars
should be similar to ours. The problem is to devise ECM technique to become
sufficiently developed to show promise of eventually becoming operational and
more time to decide to devise countermeasures. As a result, ECM development
lags behind radar development. Likewise, radar counter countermeasures are
often prompted by new advances in ECM techniques.
Telecommunication frequencies cover a wide range extending from ELF
to millimetric. Within this wide range of frequencies there are many conceivable
EW threats, many conceivable countermeasures and many techniques to be
developed for countering the countermeasures. Most of the modern radars,
designed for military applications have built in ECCM (Electronic counter
countermeasures), which are designed in response to ECM specifications. First
it is proposed to concentrate on surveillance radar and the then briefly discuss
other methods.
The ability of a surveillance radar to form target tracks depends on
output, data rate and accuracy of each target measurements. Data rate is related to
scanning rate which typically is to the order of 5-6 r/min (10-12 s between data
samples) for long range surveillance radars, 15 r/min (14 s between data samples)
for tactical military radars, and 30-6 r/min (1-2 s between data samples) for short
range radars. The output of signal processesor can be a maximum of one target
report per radar resolution cell and minimum which corresponds to number of
targets detected by radar (400 targets is maximum in many cases). The range and
azimuth data are necessary to provide overall target position accuracy, while the
Doppler and amplitude data is necessary to identify those reports which are due
to clutter leakage and extraneous targets.
Some of the frequency trade-offs related to surveillance radars are
summarized in Table 8.3. The final frequency selection depends on application,
and operational surveillance radars can be found from VMF.
Table 8.3. Frequency selection for surveillance radar
Higher frequencies
•
•
•
•
•
Lower frequencies
Better resolution (angle and doppler
aperture production
Superior accuracy ECM resistant
• Better MTI performance larger power
Better low angle detection
Multipath resistant
Aurora clutter resistant
non-significant above L band
• Greater weight
• Limited absolute bandwidth
• Glactic nose limits sensitivity below UHF
8.9.1
• Precipitation, insensitivity, greater size and
often greater weight
Surveillance Radars – ECCM Considerations
The design of radars to counter ECM is a complex subject, which
depends on the type of ECM involved and the mission of particular radar under
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Electronic warfare
consideration. A truism in ECM-ECCM world is that any radar can be jammed
and any ECM can be countered depending upon resources, which either side is
willing to construct. Thus it becomes important to understand the basic principles
involved rather than to dwell on explicit examples which exploit weakness in
design implementations of either radar or ECM equipment.
The major ECM threat to surveillance radar involves (a) Noise jamming
(b) Deception jamming (c) Chaff: strips of metal foils released in the air to
obstruct radar detection (d) decoys and expendables and (e) anti-radiation
missiles (ARM). These major threats against radars represent three possible
actions on part of enemy (a) Radiation of energy to confuse radar (b) Injection
of spurious target into radars surveillance volume, and (c) Destruction of the
radar. The first two actions are sometimes referred to soft kills while the last one
is hard kills.
The most common type of jamming is noise jamming. One option is to
increase average transmitted power. The next option is to reduce the data rate
requirement, thereby allowing a longer dwell time on the target. This is sometimes
referred to as a burnthrough mode and reflects the philosophy that it is desirable to
have some target data in a heavy ECM environment rather than no data at all. The
ability to vary data rate in an optimal scan rate and look back scanning are also
available in a limited number of surveillance radars 9.
The second principle of ECCM design for surveillance radars in a main
beam noise-jamming environment is to minimize jamming energy accepted by
radar over as wide a band as available while maintaining a radar bandwidth
consistent with radar range resolution requirement. If for example, a 150 to 300
MHz transmitting frequency range is available to S band, then the potential for a
150 to 300 dilution of jamming energy is possible. A jammer which works against
the 1 MHz radar bandwidth is referred to as a spot jammer, while a jammer which
works over full 150 to 300 MHz radar RF bandwidth is called barrage jammer. The
ECCM objective is to force the jammer into a barrage-jamming mode of operation.
Operation of radars over a wider bandwidth than that dictated by range
resolution requirements can be accomplished in several ways. Some radars
incorporate a spectrum analyzer, which provide an advance look at the
interference environment. This allows the radar frequency to be tuned to that part
of environment, which contains minimum jamming energy. This can be detected if
noise jammer has a look through mode and follow the radar frequency changes.
Frequency agility refers to the radar’s ability to change frequency after a
time period, which respond to the radar’s Doppler processing time. For an MTI
radar, this period may be as short as every two transmitted pulses. For pulsed
Doppler or MTI radars, a block processing interval may typically be 8–16
transmitted pulses. Frequency agility usually forces a noise jammer into a
barrage-jamming mode. Frequency diversity refers to use of several
complementary radar transmissions at different frequencies, either from a single
radar or several radars. The diversity is usually limited by practical considerations
429
Microwave Circuits & Components
to a finite number of frequencies5-10. Another method which is employed to
reduce the effect of main-beam noise jamming is to raise the transmitter
frequency to narrow the antenna bandwidth. This restricts what is blanked by main
lobe noise jamming and also provides a strobe in the direction of the jammer.
Strobes from two spatially separated radars pinpoint the jammers location.
However, with multiple jammers glistering can be a problem.
The ECCM design principles for main lobe noise jamming also apply to
side lobe noise jamming, with exception that the side lobe response in the
direction of jammer must be minimized. Ultra-low side lobes which are on the
order of 50 dB below the antenna's main lobe peak response are feasible using
currently available advanced technology 10-11. An adaptive antenna array has the
potential of placing antenna pattern nulls in the direction of side lobe jammers
while still maintaining the main-lobe pattern, thereby reducing the effects of the
jammer at the output port of the antenna. While adaptive arrays are applicable to
phased arrays, they are not appropriate for conventional single-element antennas.
However, by adding anxilliary antennas to conventional radar, an adaptive type of
action can be formed between main antenna and the added antennas. This
configuration is called a Side Lobe Canceller (SLC) 12,13.
Another class of radar ECCM techniques is aimed at controlling the
effects of noise jamming and other interference on radar system’s output. The
effect of additional noise on radar signal processing is large. This magnitude of
false-alarm increase would saturate the data processing capability of the radar
whether it is automatic or visual using a CRT display. Therefore, most radar signal
processors employ a constant false-alarm rate (CFAR) threshold control, which
maintains the design false alarm rate in presence of a variable noise or
interference background15.
Another countermeasure which frequently confronts radar consists of
clouds of electrically conducting dipoles called chaff which are injected into the
radar’s coverage volume. The chaff dipoles are approximately half-wavelength of
hostile radar frequency. Doppler processing in the form of MTI or pulse Doppler
processor is used by radar to extract targets from chaff. With MTI signal
processing a notch or null response is required at a different frequency than
required for ground or sea clutter because the notch must be adaptive to allow for
different chaff mean velocities, a difficult MTI design problems is generated.
8.9.2
Tracking Radar ECCM Consideration
The use of higher transmission frequencies and long target dwell times
for tracking radars generally make them less susceptible to noise jamming than
surveillance radars. Many radars make provision for angle tracking noise
jammers for target self-protection. Tracking a noise jammer in angle from two
spatially dispersed radars provide enough information to locate a target with
sufficient accuracy for use with a semi-active missile guidance weapon system.
These considerations lead to a general preferance for Deceptive ECM (DECM)
as a defensive system against tracking radars. Another factor which favours use
430
Electronic warfare
of DECM is that it generally requires considerably less energy than noise
jamming. Detection of the main beam radiation from a tracking radar associated
with SAM system or anti-aircraft system usually causes the activation of the
aircraft self protection ECM system (e.g., airborne self-protection jamming
system, ASPO). The function of this system is to cause the weapon-control
tracking radar to break lock.
The most common type of deception jammer is the range gate stealer
whose function is to pull the radar tracking gates from actual target through
introduction of a false target into radar’s range tracking circuits. Range deception
is accomplished by initially transmitting a sufficiently strong replica of radar signal
thereby applying radar's automatic gas control as mentioned earlier and then
progressively delaying the retransmission to cause range gate pull-off (RGPO).
The ECM system requires a microwave memory, which is usually implemented
using microwave acoustic memory or a digital RF memory (DRFM). After
capturing the radar’s tracking gates and accomplishing RGPO, the ECM system
stops retransmitting the false target which has the effect of causing the tracking
radar to loose track, thereby forcing the radar to reacquire the target.
A primary ECCM defence against RGPO is the use of leading edge range
tracker16. The assumption is that the deception jammer needs time to react and that
the leading edge of return pulse will not be covered by the jammer. Pulse Repetition
Interval (PRI) fitter and frequency agility both help to ensure that the jammer will
not be able to anticipate the radar pulse and lead the actual skin return.
Alternatively, the tracking radar might employ a multigate range tracking system in
conjunction with a wide dynamic range receiver to simultaneously track both the
skin and false target returns. This approach utilises the fact that both the jammer's
signal and actual radar return come from the same angular direction, so that the
radars angle tracking circuits are always locked on to the real target. Those
jammers which do not replicate linear response, scintillation or Doppler
characteristics of actual targets and hence, are easily defeated by modest ECCM
features. In other cases for tracking radars, it is usual to differentiate the output
from range tracks to obtain radial range rate or velocity data. If the measured
range rate exceeds that is expected from a real target, then this provides an early
warming of probable presence of a deception jammer. Coherent tracking radars
have less advantage of being able to compare radial range velocity derived from
Doppler measurements with those derived from differential range data. This
forces deception jammer to induce a realistic Doppler signature upon the
synthetic radar in order to meet the criterion set for a real target in radar. It is also
possible for coherent radars to measure other target signatures, such as those
induced by engine modulation17 to discriminate against the false target returns.
To be truly efficient that is to enable a radar to ensure its acceptable
level of operation in presence of an active passive jammer – ECCM functions
have to be brought into radar design. ECCM functions are integral to the choice
and definition of all sub assemblies antenna, transmitter, generation-reception,
signal processing, data processing, display as well as radar measurement.
431
Microwave Circuits & Components
8.9.3
Radar Range in Presence of Jamming
In a general case of a jammer transmitting continuous noise, the
jamming power density received by radar is defined by
J
ERP Gr 2
(8.86)
4 2 R 2j B j L j
where,
ERP = PjxGj = effective radiated power
Pj = jammer power
Gj
= jammer antenna gain
Bj
= jammer bandwidth spectrum
Gr
= Radar antenna gain in jammer direction
= wavelength of radiation
Rj
= jammer to radar range
Lj
= atmospheric loss (one way)
The total spectra noise density
(8.87)
N j J N0
where evidently
N0 = FRT0, F is the receiving system Noise factor; earlier we written
for the same expression NF RT meaning the same thing.
k = Boltzmann’s constant.
T0 = 290 K, the standard noise temperature, and
N = Radar noise power spectral density
If R0 represents range of the radar in clear conditions, its range R in the
presence of jamming is defined by the relation
R N j
R0 N 0
1 4
J
1
N0
1 4
J
N0
1 4
(8.88)
(The approximation is in valid if the jammer is effective i.e., J >> N0)
This expression shows that radar would be better protected at other
conditions being same if the jammer transmits throughout the broadband Bj and
antenna gain is lower during a reception Gr. For example, for surveillance radar in
S bandwidth 150 km range, facing a stand-off jammer with the following
parameters
ERP = 15kW
= 0.1 m
Dj = 100 km
432
Electronic warfare
R0 = 150 km
F = 4.5dB
Lj = 0.5d
J
15 10 3 10 1
1 . 58 10 2 10 5
or
J 8.5 1011
2
2
.
1 . 12 B j
Gr
Bj
(8.89)
Gr
. W H 2
Bj
N 0 2.82 1.38 1023 290
1.13 1020 W H 2
N j 7.5 100
Gr
Bj
There are two extreme cases. First the jammer is received in the main lobe
with Gr = 30 dB and concentrates its power narrow band where Bj=30 MHz
N j 7.5 10 9
10 3
6
3.75 10 5
20 10
The jamming power is approximately 56 dB above the thermal noise and
the radar range is reduced R = 6 km. Second, the jammer is received by side lobes
with Gr= –5 dB distributes its power in broadband, Bj = 400 MHz
N j 7.5 109
0.316
400 106
5.93
The jammer power is approximately 8 dB above the thermal noise and
radar range is still R = 92 km. This shows the advantage available for radar
manufacturer
(a) To reduce probability that an Suj will enter the main lobe j
(b) To construct an antenna with low side lobes.
(c) To prevent the jamming from concentrating its power in a narrow pencil
beam
A radar which has to operate in high jamming atmosphere must be
thought of according to the following three approaches:
(a) Minimizing the power of jamming, which enters the space which useful
signal can penetrate. This is achieved by spatial frequential or time- related
filtering.
(b) Not letting the jammer determine the directional characteristics or
frequency and time of signal transmission. These are the best ways to
increase its effectiveness.
(c) Prevention of the remaining false alarm to saturate the radar operating
system at the cost of sensitivity. This effect should be kept as low as possible.
433
Microwave Circuits & Components
The two systems which have been already discussed are
(a) Surveillance radars
(b) Tracking radars.
The ECCM techniques are discussed below according to conventional
breakdown of radars into subunits from the antenna to data processing units.
8.9.4
ECCM with Antenna
This has already been discussed earlier. Some features are repeated here
for understanding better. In those directions in which jammer exists the antenna
lobe during reception must have the smallest possible aperture (Fig. 8.28). It
shows the superiority of 3D surveillance radar compared with 2D surveillance
radar. It preserves the possibility of detecting a silent high altitude target at a
short range, protected by long range stand-off jammer at same azimuth. The side
lobes have angular scope proportional to the aperture of the beam angle of around
ten times the width of beam at 3 dB. This domain will be smaller in proportion to
the narrowness of aperture.
It is possible to choose transmission with a wide beam and to
simultaneously receive on several side lobes spread out or to transmit and receive
with narrow lobes. The first requires several receivers. The second requires a
complex design of antenna. The construction technology of antenna is governed
by both electrical requirement and mechanical design. It is not easy. Therefore,
today most radars use phased array antenna for anti-jamming. The jamming signal
picked up by auxiliary omnidirectional antenna is subtracted from the signal Sm
received on the main channel. Expressed mathematically
S m=Gm J(t)+Nm
(8.90a)
S a=Ga J(t)+N a
(8.90b)
FREE SPACE
JAMMER
FREE SPACE
JAMMER
Figure 8.28. Comparison of 2D (top) and 3D (bottom) surveillance radar
coverage against a long range stand-off jammer.
434
Electronic warfare
where, Gm, Ga are gain of the main channel and auxiliary channel on the jammer,
and Nm, Nn are the thermal noise of the receivers of the two channels where Nm
and Na are independent and have the same power Pn. J(t) is the jamming signal
with power Pj in the f radar signal band. Stating that J(t) is the same on both
channels expresses spatial coherence of the jammer.
To eliminate the jammer, attempt to carry out weighting W which
minimizes the jamming power on the main channel after opposition, W such that
2
E V m W V a
min
(8.91)
This is conventional least square calculation.
Wop
E VmVa
(8.92)
2
E Va
and
2
min E V m
Therefore,
E V mV a
2
2
min Pn
(8.93)
2
E Va
Gm Pj
2
. Pn Pn Gm Pj
(8.94)
Ga Pj Pn
If the jamming power on auxiliary channel is high compared with the
thermal noise then
min Pn
2
Gm
2
Ga
2
Pn
(8.95)
The first term expression corresponds to natural noise of the main
channel and second of the noise of auxiliary channel brought onto the main
channel. In this way the jammer is eliminated. However, the performance is
limited by antenna gain ratio. It may be noted that power after operation is, always
less than the input power whatever may be the gain.
8.9.5
Integrated SLC
Some antenna architecture are more suitable for layout of SLC function
without a need for auxiliary antenna. The most elaborate anti-jamming form
integrated in an antenna corresponds to beam forming system by computation.
The signal received by each eliminating feed is amplitude transposed into intermediate frequency then, coded in digital form on two channels in phase
quadration. The elementary signals are combined linearly to form several
simultaneous pencil beams with hollows in the direction of jammers.
435
Microwave Circuits & Components
Currently, the cost-effective way is to group the feeds conventionally in
high frequency to form subnetworks. The output of each subnetwork is
transposed into intermediate frequency then coded digitally. The main beam is
reconstituted by a linear combination of suitably weighted subnetworks.
8.9.6
Side Lobe Blanking
Side lobe blanking (SLB) eliminates pulse, transponder or relatively
slow scanning jammers penetrating through the side lobes. SLB works on
principle of comparing the signals received by the main channel with signal
received by auxiliary channel with overlaps of side lobes of the main channel
(Figs. 8.29 and 8.30). If the signal received by secondary channel is higher with
margin of X dB, the signal received by the main channel will be blocked. The
value of x is chosen to ensure that the main channel operates normally during the
periods when there is no jamming.
GAIN
MAIN BEAM
AUXILIARY BEAM
JAMMER DIRECTION
Figure 8.29. Side lobe blanking using main signals
8.9.7
ECCM with Transmitter
The development of very stable high power amplifier tubes has
considerably improved the resistance of radars to jamming.
The best means of jammer to improve efficiency is to
concentrate its
available power into radar signal band. The optimal way of preventing its to happen
MAIN
CHANNEL
CLEAR
RECEIVER
DELAY
SLB
W
DELAY
AUXILLARY
CHANNEL
Figure 8.30. SLB Implementation
436
CHOICE
SLC/SLB
Electronic warfare
is to transmit with frequency agility i.e., to change transmission frequencies from
one pulse to another so that the jamming spreads its energy out to the wider
frequency band.
Table 8.4 gives the reduction in jamming power reduction by frequency
agility using a reference of 20 MHz narrow band and a wide band which is 10 per
cent of transmitted frequency.
Table 8.4. Jamming reductions with frequency agility of 10 per cent
Radar band
L: 1 GHz
ERP reduction
(in dB)
8.9.8
S: 3 GHz
8
12
C: 5 GHz
X10 GHz
14
17
Pulse Compression
Pulse compression consists of transmitting a long frequency-or phasemodulated signal which occupies a broad spectrum. From the ECCM standpoint,
compression offers three advantages.
(a) For the same amount of emitted energy, the signal power is less. For the
jammer detection and measurements of received signal characteristics
becomes more difficult (quiet radar).
(b) While transmitting a long pulse, the radar preserves range resolution equal to
1/pulse duration
(c) The pulse transmitted by a jammer, or by another radar and which does not
have the transmitted signal modification law, and extended at the output of
the adapted filter and their peak power is attended by compression ratio
Tf . They are easily eliminated at outset by the range CFAT with a reduced
loss in terms of sensitivity.
8.9.9
ECCM with Receiver
To detect small signals in presence of very powerful clutter, modern
radars feature improvements which are extremely useful against jamming. It uses
double frequency change gain control. The input band of a gate radar is very wide
and the intermediate frequency (IF) preceding transposition into video, is
necessarily low around several tens of Mega Hertz with a single frequency
change. We obtain, at IF frequency at the mixer output.
(a) The useful signal corresponding to transmission frequency fij.
(b) Interference centered on the image frequency
fi fio IF , if ft f LO IF
(c) The spurious signals from a jammer using the non-linear feature of mixer by
emitting two frequencies, separated by intermediate frequency (IF).
The uses of a double frequency change with the first IF higher than the
radar agility and eliminates spurious signals by filter using the receiver output. A
high dynamic range associated with gain control is essential for avoiding, in most
437
Microwave Circuits & Components
cases, the saturation of the receiver and the ensuring the efficiency of linear
filters located at own stream and thus conserving the sensitivity of receiver.
Figure 8.31 shows the double frequency change ECCM technique.
8.9.10 ECCM with Signal Processing
Doppler filtering and MTI conventional technique ensure the visibility
of target echoes hidden in gain or rain (clutter in the case of chaff). In intense
jamming environment characterized by simultaneous presence of stand-off
AGILITY BANDWIDTH
F= IFLO1 + IF
RF
AMPLIFIER
DUPLEXER
IFLO1
NARROW
BANDWIDTH
1ST IF
AMPLIFIER
MIXER
IF1
I
O
PHASE
DETECTOR
IF2
MIXER
2NDAMPLIFIER
FLOI = IF1 + IF2
R&D
BANDWIDTH
IF2
Figure 8.31. Double frequency change ECCM technique
jammers (SOJ), chaff and ground clutter, the most efficient processing for radar
consists of emitting burst, of ten or so pulses with a change of transmission
frequencies and bursts or repetition frequency to eliminate blind speed zones.
Pulses for same bursts go through Doppler filtering and processing is
carried out using narrow band filter bank. The output of each speed channel is
detected and followed by range of CFAR in order to eliminate mobile clutter and
continuous jammer (Fig. 8.32). This type of processing combines the advantages
TEMPORAL
CFAR
1
0
DOPPLER
BANK
TRANSVERSAL
FILTER
LOW VELOCITY
CHANNEL
RANGE
CFAR
M
RANGE
CFAR
A
MOVING TARGET
CHANNEL
X
RANGE
CFAR
Figure 8.32. Doppler signal processing
438
Electronic warfare
of Doppler Processing with those of frequency agility when confronted by a
jammer which measures the radar frequency on the first pulse. In this case one
has to use MTI type of adaptive processing with narrow pulses. Two filters are to
be used of three pulse canceller type. The system is shown in Fig. 8.33.
8.10
CONCLUDING REMARKS
It is such a wide subject that it is almost impossible to cover entire
subject in a chapter. There has been some development in ECM. It is theoretically
GROUND CLUTTER
MTI 3 PULSES
CANCELLER
GROUND L CLUTTER
MTI 3 PULSES
CANCELLER
RANGE
CFAR
RAIN CHAFF
VELOCITY
MEASUREMENT
Figure 8.33. Adaptive MTI processing
possible to go to ECCM. In EW advance measures have been made to use
electromagnetic spectrum for causing immense damage. Pentagon believes
newly developed weapon can inflict immeasurable damage. These are known as
E-bombs. American Army hopes to use E-bomb technology to explode artillery
shells in mid flight. The American Navy wants to use the E-bomb's high power
microwaves, microwave pulses to neutralize antiship missiles. The American Air
Force plans to equip its bombers, strike fighters, cruise missiles and unmanned
arrival vehicles with E-bomb capabilities. Basically, it is based on Compton
Effect18. Mention might be made of an article Listening for whispers. The
complex acoustic environment made of water poses new challenge for antisubmarine warfare19. Major Naval forces are focussing their efforts in
antisubmarine warfare (ASW) primarily on passive detection of nuclear powered
boats operating in the open ocean. The technologies to be developed include new
transducer materials, use of low frequencies to counter challenges posed by
littoral operations. Solid-state, phase-shifting aerodyne techniques are being
developed as high performance lightweight high reliability and low cost
alternative to medium power to Travelling Wave Tube Amplifier (TWTA)
generated deception jamming20.
High temperature super-conduction filter technology for improved EW
system performance is being studied.. Much progress has been made in
developing high quality microwave devices that possess lower insertion loss 21.
REFERENCES
1. Frohmainer, J.H. Noise performance of a three stage microwave receiver.
Electronic Tech, ,1960, 37, 245-46.
2. Williams, D. Visual measurement of receiver noise. Wireless Engineering,
1947, 24, 100-04.
439
Microwave Circuits & Components
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Lucas, W.J. Tangential sensitivity of a detector, video system with RF
preamplification. Proc IEE, 1966, 113, 1321-30.
Sareen, S. Threshold detector nanosecond fault or level detection RF
preamplification, Application Note p. 40, Artech Industries, summary,
CA,1975.
Emerson, R.C. First probability density for receiver with square law
detectors. J Appl Phy, 1953, 24, 1169-76.
Robertson, G.H. Operation of characteristics of linear detector of CW signal
in narrow bandwidth Gaussian noise. Bell System Tech, 1967, j (46), 755-74.
Abberhseum, W.J. A closed form approximations to Robertson's detection
characteristics. Proc IEEE, 1981, 69, 839.
Stuison. Airborne radar published by Hughes.
Bullock, C. Melotany surveillance radars. Interavial, 1982, 7.
Evans and Schrank, H. Low side lobe radar antennas. Microwave Journal,
1983, 7.
Shrank, H. Low-side lobe radar antennas. IEEE Antenna and Propagation
Society Newsletter, 1983, 4 .
Walsh, T. Military radar systems. Microwave Journal, 1978, 11.
Soccu, R. Aeg IS radar reciever. Microwave Journal,1978.
VanNrunt, L. Applie ECM. E.W. Engineering, 1978, 1, Va.
Schleher, D.C. Automatic detection and radar data processing. Artech House,
Dedham, MA, 1980.
Barton, D & Ward, H. Handbook of radar measurement. Artech House,
Dedham, MA, 1984.
Hynes, R. & Gardner, G. Doppler spectra of s-band and x-band signals. IEEE
Trans AES, EASCON supplement, 1967, 3.
Wilson, Jim. Popular mechanics, 2001, September, 51p.
Hewish, Mark. Listening for whispers. Janes International Defence Review,
2001, September.
Madani, A.M. & Endler, H.M. Solid-state multiple deception hamming
system for ECM applications. Aerospace conference, IEEE. 1998, 1,
331-43.
Ryan, Paul A. High temperature superconducting filter technology for
improved EW system performance. Aerospace and Electronic Conference,
NAWLON, 1997, Proc IEEE, 1997, 1, 392-95.
440
Appendix A
GRAPHICAL SYMBOLS UTILISED IN
MICROWAVES
Transmission lines
transmission line (general)
rectangular wavelength
ridge waveguide
coaxial line
stripline
microstrip
One-port devices
short-circuit
matched load
reflecting cavity
antenna
detector
oscillator
generator of unit step
Microwave Circuits & Components
Two-port devices
fixed attenuator
variable attenuator
fixed phase shifter
variable phase shifter
transmission cavity
low-pass filter
bandpass filter
isolator
TR cell
ATR cell
varactor
amplifier
Three-port devices
circulator
mixer
PIN-diode-switch
power divider
Four-port devices
switch
directional coupler
hybrid T
442
Appendix B
GLOSSARY
List of Symbols – A
Symbol
Units
Description
ai
W1/2
Complex normalized wave
A
A/m
Surface current density
Ae
m2
Effective reception area
W1/2
Complex normalised wave
B
T
Induction phasor vector
co
m/s
Velocity of light in vacuum
J/kg.K
Specific heat
D
As/m2
Displacement phasor vector
E
V/m
Electrical phasor vector
fB
Hz
Cyclotron frequency
Hz
Cutoff frequency
Hz
Doppler frequency
Hz
Transition frequency
F
1
Noise figure
Fm
N
Lorentz force
g
1
Lande factor
G
1
Power gain
H
A/m
Magnetic phasor vector
Hm
1
Hankel function
A
Equivalent line current
Ig
A
Waveguide current
J
A/m2
Current density phasor vector
Je
A/m2
Perturbing electrical current density
Jm
V/m2
Perturbing magnetic current density
Jm
1
Bessel function of first kind of order m
k
m-1
Wave number
kB
J/K
Boltzmann constant
kP
m-1
Resonance wave number
K
1
Distributed coupling factor
K(u)
1
Elliptical integral of first order
Km
1
Modified Bessel function of order m
bi
cp
fc
fD
ft
Ie
Microwave Circuits & Components
Symbol
Units
Description
LA
dB
Attenuation level
LC
dB
Coupling level
LD
dB
Directivity
m
Am2
Spin magnetic moment
M
A/m
Magnetization
N
W
Average noise power
NA
1
Numerical aperture
Nm
1
Bessel function of second kind of
order m
p
m-1
Transverse wave number
Pi
W
Pulse power
Q0
1
Unloaded quality factor
Q0m
1
Metallic quality factor
Q0e
1
Sample quality factor
Qc
1
Loaded quality factor
Qe
1
External quality factor
s
1
VSWR: Voltage Standing Wave
Ratio
(s)
1
Scattering matrix
Ta
K
Antenna noise temperature
Tr
K
Reciever noise temperature
Ue
V
Equivalent line voltage
vg
m/s
Group velocity
vj
m/s
Phase velocity
Yf
S
Beam equivalent admittance
Z0
Characteristic impedance of vacuum
Zc
Characteristic impedance of microstrip
Ze
Wave impedance
Zm
Metal characteristic impedance
ZUI
Wave guide impedance
(voltage-current)
ZPI
Wave guide impedance
(power-current)
ZPl
W
Np/m
Wave guide impedance
(power-voltage)
Attenuation per unit length
rad/m
Phase shift per unit length
c
1
Coupling factor
m-1
Propagation factor
Overvoltage factor
444
Appendices
Symbol
Units
Description
g
(sT)-1
m
Gyromagnetic factor
Skin depth
1
Kronecker delta symbol
As/Vm
Complex permitivity
0
As/Vm
Electrical constant
r
1
Relative permitivity
d
1
Mismatch efficiency
e
1
Electronic efficiency
p
1
Partial efficiency
pa
1
Power added efficiency
r
1
Coupling factor
s
1
Substitution efficiency
t
1
Total efficiency
c
m
Cut-off wavelength
g
m
Waveguide wavelength
g
m
Loaded waveguide wavelength
mnl
Am
Hertz Potential
Vs/m
Complex permeability
0
Vs/m
Magnetic constant
p
1
Carrier mobility
1
Relative permeability
Vs/m
Permeability tensor
mnl
Vm
Hertz potential
1
Reflection factor
C/m3
Charge density phasor
S/m
Conductivity
m2
Effective scattering cross-section
1
Transverse potential of TM mode
1
Transverse potential of TE mode
L
rad/s
Larmor angular frequency
M
rad/s
Magnetisation angular frequency
p
rad/s
Complex eigen-angular frequency
pr
rad/s
Eigen-angular frequency
r
445
Microwave Circuits & Components
List of Symbols – B
List of Symbols/Abbreviations
A
B
Bel
C
Ceq
dB
dBm
G
IL
Im
ISC
i(t)
j
K
KCL
KVL
L
L eq
N
Np
pF
Re
Req
Ri
v12(t)
Definitions
VOC
X
Y
YT
Z
ZL
ZT
Used to represent amplitude of a function
Susceptance
International unit for measuring attenuation
Capacitance
Equivalent lumped value capacitance
Decibel
Decibels referenced to 1 milliwatt
Conductance
Current through the load
The imaginary portion of a complex number
Short circuit current
Current with respect to time
An imaginary number where j = Ö-1
Overall voltage gain
Kirchhoff’s current
Kirchhoff’s voltage law
Inductance
Equivalent lumped value inductance
Used to represent the number of turns of wire in an inductor
Used to represent the unit Neper that defines attenuation
pico-Farad
The real portion of a complex number
Equivalent lumped value resistance
Used to represent a lumped element resistance
Notation used denote a difference in voltage between two design
at points, 1 and 2, in a circuit
Open circuit voltage
Reactance
Admittance
Thevenin admittance
Impedance
Load impedance
Thevenin impedance
Angle
Upright used to denote absolute value of the variable inside
Frequency given in radians/second
Magnetic flux
Phase of a function
Phase angle of a complex number
446
INDEX
A
D
Admittance 41
AGC. See Automatic gain
control
Amplifiers 221
Antenna 434
horn 5
Armament 379
Attenuation 29
Automatic gain control 423
DC biasing 197
Deceptive ECM 431
Decibels 234
DECM. See Deceptive ECM
Decoys
applications
strategies 426
detection 427
saturation 427
seduction 427
Detection 384
probability of 390
Dielectric resonator
oscillator 207
Digital RF memory 431
Digital signal processing 8
Direct analogue synthesis 220
Direction finding
by amplitude measurement 394
by phase measurement 395
techniques 394
by time of arrival measurement 395
Distortion
delay 31
frequency 30
waveform 30
Doppler filter
bandwidth limitations 404
Doppler shift 424
Double-stub matching 261
DRFM. See Digital RF memory
Drown 416
DSP. See Digital signal processing
Dynamic range 392
B
Bias decoupling 199
Bragg reflection and interference
transistor 196
Bump 425
Burnthrough 429
C
Chaff 429
Characteristic impedance 22
Circuit
model 17
COMINT. See Communication
intelligence
Communication
satellite 7
terrestrial 7
voice 379
intelligence 379
digital data 379
Continuous wave 424
power range 406
Correlation 384
CWo See Continuous wave
Microwave Circuits & Components
IMPATT. See Impact ionization
avalanche transit time
Impedance 41
measurement 5 I
matching 233, 248, 272
networks 276
Information recovery 380
Input impedance 43
Insertion loss 303
Integrated SLC 435
E
E plane Tee 136
EA. See Electronic attack
ECCM 428, 434, 437. See also
Electronic counter
countermeasures; Electronic
counter-countermeasures
ECM 428. See also Electronic
counter measures; Electronic
countermeasures
ECM-ECCM 429
Electromagnetic
wave 5
Electronic attack 418
Electronic camouflaging 380
Electronic countermeasures 8, 9,
380, 397,418
Electronic intelligence 380
Electronic support measures 8, 380
Electronic warfare 8, 9, 379
EUNT. See Electronic intelligence
ESM. See Electronic support meas ure
EW. See Electronic warfare
J
Jammer signal 414
Jamming
AGC 423
blinking 426
cover 419
deceptive 419
Radar 419
JFET 192
K
Klystron 5
L
F
Line
coaxial 33
analysis 35
typical 33
loss less 27
open two-wire 31
open circuited 43
quarter wave 45
resonant 43
slot 93
small losses 27
transmission 26
Linear lossless 133
Lines
fin 97
integrated fin 100
mono strip 100
Location measurements 395
Lossless line 27
Lumped elements 272
False alarm rate 390
FET. See Field effect transistors
Field effect transistors(FET) 167
Filters
HTS 9
Forbidden regions 267
Frequency
high 41
reference 21 7
Fuzzy
logic 9
processor 9
G
GaAs MESFET 167
Gain 201
H
H plane Tee 136
Hard kills 429
Hybrid T 139
M
Magic T 139
Magnetron 5
Matrix
connection 297
I
Ideal line 27
Impact ionization avalanche transit
time 210
448
Index
MESFET 192
Microwave
engineering
applications 7
spectroscopy 7
technology 5
devices 133
FET 184
tubes 209
transistor 200
Mixers 217
Modes
TE 65
Morse code 379
MOSFET 192
Q
Quality factor 43
Quarter-wave
transformer 250
R
Radar
concepts 412
cross-section 424
C W 403
ECCM 430
jamming 419
monopulse 424
pulse 397
pulse doppler 407
range 432
resolution cell 425
surveillance 429
Range gate pull-off 431
RCS. See Radar: cross-section
RCYR 419
Receiver
direct detection 393
superheterodyne 393
microwave 387
superheterodyne detection 393
Reflection coefficient 235
Representation
ABCD parameter 120
admittance 117
G parameter 119
hybrid 118
impedance ll6
network 115
Resonant transmitting diode 196
RGPO. See Range gate pull-off
N
Nepers 234
Network
microstrip matching 281
impedance matching 276
Noise 380
environment 383
performance 192
Normalised
impedance 271449
admittance 271
o
Open-circuited 41
Optimization 323
p
Parameter
conversion 121
scattering 124
inverse chain 121
PD. See Pulse doppler
Phase lock loop 215
PLL. See Phase lock loop
POI. See Probability of intercept
Power management 420
PRI. See Pulse repetition interval
Probability 380
of intercept 387
theory 381
factor 26
Propagation constant
equation 25
Pulse
doppler 424
radar range 401
repetition interval 431
S
SAW. See Surface acoustic wave
Scattering matrices 133
Short-circuited 41
Shunt susceptance 161
Side lobe
blanking 436
canceller 430
Signal
frequency shift keying 9
phase shifting key 9
information 383
model 168
449
Microwave Circuits & Components
Transistion
CPW 104
Transistor
mIcrowave 200
amplifier 20 I
bipolar 168
osci lIator 206
scattering parameters 176
silicon bipolar junction 168
Transition
CPS 105
Transmission line 17, 26
Transmitter 437
Travelling-wave tube am plifier 8
TSDF. See Timed synchronous
dataflow
TSS. See Tangential sensitivity
TWTA. See Travelling wave tube
amplifier; Travelling-wave tube
amplifier
Single-stub matching 256
SLB. See Side lobe blanking
SLC. See Side lobe canceller
Smith chart 233, 256
compressed 268
transmission line 236
Z-Y 271
Soft kills 429
Solid state source 210
Spatial 396
Spectrum analysers 394
Stability 201
Standing wave ratio 233
Stub matching 256
Surface acoustic wave 222
Surveillance 379
SWR. See Standing wave ratio
Synthesizer 215
architecture 220
double loop radar 229
microwave 224
piezoelectric 229
PLL 229
single reference PLL 229
System
evolution of 396
U
Unknown impedance 244
V
Velocity gate 424
pull-off 424
VGPO. See Velocity gate pull-off
T
Tangential sensitivity 388
Technique
deceptive jamming 421, 424
direction finding 394
Jamming 418
inverse gain jamming 422
radar counter countermeasure 428
range gate pull off 422
Temperature stability 199
Three stubs
matching 267
Tracking 430
W
Wave propagation 22
Waveform di stortions 30
Waveguide
coplanar 91
rectangular 67
ridge 100
White noise 221, 380
WLAN. See Wireless Ian applications
X
XMTR 419
450
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