Microwave planar passive circuits and filters 9780471940562, 0471940569

Table of contents :
PREFACE
1 TEM AND QUASI-TEM TRANSMISSION LINES
1.1 Introduction
1.2 Propagation in coaxial line
1.3 Telegraphist equations
1.4 Parallel-wire transmission line
1.5 The stripline line
1.6 The microstrip line
1.7 The coplanar waveguide
1.8 The slot-line
1.9 Suspended stripline
Further reading
2 SUBSTRATE MATERIALS AND PLANAR FABRICATION
2.1 Introduction
2.2 Dielectric constant
2.3 Dielectric loss tangent
2.4 Anisotropy
2.5 Tensor permeability in infinite space
2.6 Initial or demagnetized permeability
2.7 Low field losses in unsaturated gyromagnetic medium
2.8 Magnetic loss tangent
2.9 Fabrication of planar circuits
2.10 Thick-film fabrication
Problems
Further reading
3 IMMITTANCE AND SCATTERING PARAMETERS
3.1 Introduction
3.2 Impedance matrix
3.3 The admittance matrix
3.4 The scattering matrix
3.5 Relationships between S, Z and Y matrices
3.6 The unitary condition
3.7 Admittance and scattering relationships in lossless network
Problems
Further reading
4 TRANSMISSION PARAMETERS
4.1 Introduction
4.2 ABCD parameters
4.3 ABCD relationships
4.4 ABCD parameters of series, shunt, tee and pye networks
4.5 Ideal transformer
4.6 Uniform transmission line
4.7 ABCD parameters of radial transmission lines
4.8 Circuit relationships
4.9 Kuroda's identities
4.10 Scattering transfer parameters
4.11 Frequency variable
Problems
Further reading
5 SYNTHESIS OF COMMENSURATE LINE NETWORKS
5.1 Introduction
5.2 Richards' variable
5.3 Inductors and capacitors in the t-plane
5.4 Positive real commensurate immittance functions
5.5 Synthesis of t-plane inductors and capacitors
5.6 Richards' theorem (the unit element)
5.7 Synthesis of t-plane reactance circuits using unit elements
5.8 Partial extraction of unit elements
5.9 Kuroda's identities
Problems
Further reading
6 QUASI-TEM WAVEGUIDE MODEL OF MICROSTRIP
6.1 Introduction
6.2 Characteristic impedance and phase velocity
6.3 Equivalent waveguide model of microstrip line
6.4 Frequency effects
6.5 Planar capacitors
6.6 Mitres, steps, tapers, tee-junctions
6.7 STM solution of planar triangular resonator
References and further reading
7 PROPAGATION ON MAGNETIC SUBSTRATES
7.1 Introduction
7.2 Birefringence and effective pemeability in infinite space
7.3 Equivalent waveguide model of microstrip on a magnetic substrate
7.4 Frequency-dependent parameters
7.5 Synthesis of microstrip on garnet
7.6 The edge mode effect
Problem
References
8 ELECTRIC MAGNETIC AND CONDUCTOR LOSSES ON MICROSTRIP LINE
8.1 Introduction
8.2 Dielectric loss in quasi-TEM lines
8.3 Magnetic loss in quasi-TEM lines
8.4 Effective dielectric loss tangent
8.5 Conductor loss
8.6 Effective magnetic loss tangent
8.7 Measurements of magnetic and electric quantities
Problems
References and further reading
9 PLANAR RESONATORS
9.1 Mode patterns in planar circular disc
9.2 Modes in planar ring resonator
9.3 TM field patterns of triangular planar resonator
9.4 Equivalent waveguide model of planar resonators
9.5 Planar radiators
Problems
References and further reading
10 PLANAR 90° COUPLERS AND HYBRIDS
10.1 Scattering matrix of directional coupler
10.2 Even and odd modes theory of directional couplers
10.3 Operation of microstrip directional coupler
10.4 Synthesis of branch coupler
10.5 Frequency response of branch-line coupler
10.6 Odd- and even-mode parameters of parallel lines
10.7 Balanced microwave mixer
Problems
Further reading
11 180° HYBRID CIRCUITS
11.1 Introduction
11.2 180° waveguide hybrid network
11.3 Synthesis of ring hybrid
11.4 Even- and odd-mode description of ring hybrid
Problems
References and further reading
12 PLANAR POWER DIVIDERS
12.1 Introduction
12.2 Wilkinson power divider
12.3 Even- and odd-mode adjustment of Wilkinson divider
12.4 Frequency behaviour of Wilkinson power divider
12.5 Variable power dividers
12.6 Variable Butler dividers
Problems
References and further reading
13 PHYSICAL PARAMETERS OF PARALLEL LINES
13.1 Introduction
13.2 Odd and even impedances and phase constraints
13.3 Parallel striplines
13.4 Symmetrical parallel bars
13.5 Unsymmetrical parallel lines
13.6 Adjustment of arrays of parallel lines
13.7 Coupled microstrip lines
13.8 Broadside parallel couplers
13.9 Suspended substrate coupled lines
References and further reading
14 LUMPED ELEMENT MICROWAVE CIRCUITS
14.1 Transmission-line identities
14.2 UHF lines as circuit elements
14.3 Planar lumped element series inductors and shunt capacitors
14.4 Discrete inductors and capacitors using monolithic techniques
14.5 Distributed lines as series and shunt resonators
14.6 Dissipation
14.7 Equivalence between shunt resonator and half-wave long line
14.8 Lumped element resistors
Problems
Further reading
15 SYNTHESIS OF LOW- AND HIGH-PASS SEMI-LUMPED FILTER CIRCUITS
15.1 Feldtkeller condition
15.2 The approximation problem
15.3 The Chebyshev (or equal ripple) low-pass filter approximation
15.4 Element values of low-pass prototypes
15.5 Low-pass to high-pass frequency transformation
15.6 Element values of high-pass prototypes
15.7 Low-pass filter circuits
15.8 High-pass filter circuits
15.9 Frequency response of microwave filters
Problems
Further reading
16 COUPLED PARALLEL LINE CIRCUITS
16.1 Introduction
16.2 lmmittance matrices of parallel coupled lines
16.3 lmmittance matrices of parallel coupled lines in t-plane
16.4 Parallel line bandstop filter circuit
16.5 Parallel line bandpass filter circuits
16.6 Equivalence between coupled parallel line circuits
16.7 Self and mutual capacitances of parallel lines
16.8 Open-circuit parameters of parallel lines in inhomogeneous medium
Problems
Further reading
17 PARALLEL LINES BANDSTOP FILTERS
17.1 Introduction
17.2 Exact synthesis of distributed bandstop filters
17.3 Exact synthesis of bandstop filters using parallel lines
17.4 Bandstop filters using open-circuited stubs
17.5 Frequency response of bandstop filter
Problems
References and further reading
18 DIRECTLY COUPLED BANDPASS FILTERS
18.1 Introduction
18.2 t-plane synthesis using t-plane inductors and capacitors
18.3 Ladder network using t-plane inductors and ideal t-plane inverters
18.4 Immittance inverters using lumped element transformations
18.5 Practical immittance inverters
18.6 Directly coupled microwave filters
18.7 Capacitively coupled microstrip bandpass filter
18.8 Dual mode ring filters
Problems
References and further reading
19 BANDPASS FILTERS USING PARALLEL LINES
19.1 Introduction
19.2 Semi-ideal impedance inverters using UEs
19.3 Ladder network using t-plane inductors and semi-ideal t-plane inverters
19.4 Exact parallel line t-plane networks
19.5 Synthesis of parallel line filters using short-circuited lines
19.6 Synthesis of parallel line filters using open-circuited lines
Problems
Further reading
20 THE INTERDIGITAL FILTER
20.1 Capacitive matrices of parallel line arrays
20.2 Parallel lines identity (Grayzel theorem)
20.3 Synthesis of parallel lines using unequal strips
20.4 Equivalent circuit of interdigital circuits
20.5 Degree of interdigital filter
20.6 Capacitance matrix of interdigital filter
20.7 Design of interdigital filter
20.8 Bandpass resonators using Y-resonators
20.9 Bandpass resonators using ring resonators
20.10 Tapped interdigital structure
Problems
Further reading
21 COMBLINE FILTER
21.1 Introduction
21.2 Exact equivalent circuit of combline filter
21.3 t-plane bandpass prototype
21.4 t-plane LC resonators
21.5 Adjustment of combline filter
21.6 Mixed lumped/distributed combline filter
21.7 Equivalent circuit of UE using ideal admittance inverter
21.8 Degree-4 combline filter
Further reading
22 OPEN DIELECTRIC RESONATORS
22.1 Introduction
22.2 Frequency of dielectric resonator
22.3 Maxwell's equations in cylindrical coordinates
22.4 Characteristic equation of dielectric waveguide
22.5 Open dielectric resonator
22.6 Dielectric resonator circuits
22.7 External Q-factor
Problems
Further reading
23 IMPEDANCE MATRIX OF JUNCTION CIRCULATOR
23.1 Introduction
23.2 Network definition of junction circulator
23.3 Eigenvalue adjustment of three-port circulator
23.4 Impedance matrix of junction circulator
23.5 Complex gyrator immittance of three-port circulator
23.6 Synthesis of junction circulators using resonant in-phase eigen-network
23.7 Equivalent circuit
23.8 Junction circulators with Chebyshev characteristics
Further reading
24 THE STRIPLINE JUNCTION CIRCULATOR
24.1 Mode chart of gyromagnetic resonator
24.2 Impedance matrix
24.3 Circulation solution
24.4 Frequency response of weakly magnetized circulator
Further reading
INDEX

Citation preview

MICROWAVE

PLANAR PASSIVE CIRCUITS AND FILTERS J. HELSZAJN

MICROWAVE PLANAR PASSIVE CIRCUITS AND FILTERS

MICROWAVE PLANAR PASSIVE CIRCUITS AND FILTERS

J. Helszajn Heriot-Watt University UK

JOHN WILEY & SONS Chichester ∙ New York ∙ Brisbane ∙ Toronto ∙ Singapore

Copyright©

1994 by John Wiley & Sons Ltd. Baffins Lane, Chichester, West Sussex PO19 1UD, England Telephone ( + 44) (243) 779777

All rights reserved. No part of this book may be reproduced by any means, or transmitted, or translated into a machine language without the written permission of the publisher. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA Jacaranda Wiley Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W 1L1, Canada John Wiley & Sons (SEA) Pte Ltd, 37 Jalan Pemimpin #05-04, Block B, Union Industrial Building, Singapore 2057

Library of Congress Cataloging-in-Publication Data Helszajn, J. (Joseph) Microwave planar passive circuits and fibers I J. Helszajn. p. cm. Includes bibliographical references and index. ISBN 0 471 94056 9 1. Microwave circuits. 2. Microwave fiters. 3. Electric fiters, Passive. I. Title. TK7876.H423 1994 93-3∞30 621.381'32— dc20 CIP

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 471 94056 9 Typeset in 10∣⁄12 Palatino by Thomson Press (India) Ltd, New Delhi and Printed and bound in Great Britain by Bookcraft (Bath) Ltd.

CONTENTS

PREFACE

1

TEM AND QUASI-TEM 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

2

TRANSMISSION

LINES

Introduction Propagation in coaxial line Telegraphist equations Parallel-wire transmission line The stripline line The microstrip line The coplanar waveguide The slot-line Suspended stripline Further reading

SUBSTRATE MATERIALS AND PLANAR FABRICATION 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

3

xi

Introduction Dielectric constant Dielectric loss tangent Anisotropy Tensor permeability in infinite space Initial or demagnetized permeability Low field losses in unsaturated gyromagnetic medium Magnetic loss tangent Fabrication of planar circuits Thick-film fabrication Problems Further reading

IMMITTANCE 3.1 3.2 3.3 3.4 3.5 3.6 3.7

AND SCATTERING PARAMETERS

Introduction Impedance matrix The admittance matrix The scattering matrix Relationships between S, Z and Y matrices The unitary condition Admittance and scattering relationships in lossless network Problems Further reading

1 1 1 5 6 7 9 12 14 16 18

19 19 19 21 22 26 29 30 33 35 36 37 38

41 41 41 44 46 50 52 53 54 55

vi

4

Contents

TRANSMISSION PARAMETERS 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

5

SYNTHESIS OF COMMENSURATE 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

6

LINE NETWORKS

Introduction Richards' variable Inductors and capacitors in the f-plane Positive real commensurate immittance functions Synthesis of f-plane inductors and capacitors Richards' theorem (the unit element) Synthesis of f-plane reactance circuits using unit elements Partial extraction of unit elements Kuroda's identities Problems Further reading

QUASI-TEM WAVEGUIDE MODEL OF MICROSTRIP 6.1 6.2 6.3 6.4 6.5 6.6 6.7

7

57

Introduction ABCD parameters ABCD relationships ABCD parameters of series, shunt, tee and pye networks Ideal transformer Uniform transmission line ABCD parameters of radial transmission lines Circuit relationships Kuroda's identities Scattering transfer parameters Frequency variable Problems Further reading

Introduction Characteristic impedance and phase velocity Equivalent waveguide model of microstrip line Frequency effects Planar capacitors Mitres, steps, tapers, tee-junctions STM solution of planar triangular resonator References and further reading

PROPAGATION ON MAGNETIC SUBSTRATES 7.1 7.2 7.3 7.4 7.5 7.6

Introduction Birefringence and effective pemeability in infinite space Equivalent waveguide model of microstrip on a magnetic substrate Frequency-dependent parameters Synthesis of microstrip on garnet The edge mode effect Problem References

57 57 58 60 62 63 63 64 65 66 68 69 70

71 71 71 73 74 74 77 78 81 82 84 84

85 85 85 89 91 91 94 98 102

103 103 103 106 107 109 109 113 113

Contents

8

ELECTRIC MAGNETIC AND CONDUCTOR LOSSES ON MICROSTRIP LINE 8.1 8.2 8.3 8.4 8.5 8.6 8.7

9

Introduction Dielectric loss in quasi-TEM lines Magnetic loss in quasi-TEM lines Effective dielectric loss tangent Conductor loss Effective magnetic loss tangent Measurements of magnetic and electric quantities Problems References and further reading

PLANAR RESONATORS 9.1 9.2 9.3 9.4 9.5

10

Mode patterns in planar circular disc Modes in planar ring resonator TM field patterns of triangular planar resonator Equivalent waveguide model of planar resonators Planar radiators Problems References and further reading

PLANAR 90 o COUPLERS AND HYBRIDS 10.1 10.2 10.3 10.4 10.5 10.6 10.7

11

180 11.1 1 1.2 1 1.3 11.4

12

Scattering matrix of directional coupler Even and odd modes theory of directional couplers Operation of microstrip directional coupler Synthesis of branch coupler Frequency response of branch-line coupler Odd- and even-mode parameters of parallel lines Balanced microwave mixer Problems Further reading o

HYBRID CIRCUITS Introduction 180 o waveguide hybrid network Synthesis of ring hybrid Even- and odd-mode description of ring hybrid Problems References and further reading

PLANAR POWER DIVIDERS 12.1 12.2 12.3 12.4 12.5 12.6

Introduction Wilkinson power divider Even- and odd-mode adjustment of Wilkinson divider Frequency behaviour of Wilkinson power divider Variable power dividers Variable Butler dividers Problems References and further reading

vii

115 115 115 116 117 118 120 122 124 124

125 125 133 135 141 143 144 144

145 145 148 151

153 154 155 156 157 157

159 159 159 163 164 172 172

173 173 173 174 177 178 182 184 184

viii

13

Contents

PHYSICAL PARAMETERS OF PARALLEL LINES

185

Introduction Odd and even impedances and phase constraints Parallel striplines Symmetrical parallel bars Unsymmetrical parallel lines Adjustment of arrays of parallel lines Coupled microstrip lines Broadside parallel couplers Suspended substrate coupled lines References and further reading

185 185 186

13.1 13.2 13.3 13.4 13.5 13.6 13.7

13.8 13.9

14

LUMPED ELEMENT MICROWAVE CIRCUITS 14.1 14.2 14.3 14.4

14.5 14.6 14.7

14.8

15

SYNTHESIS OF LOW- AND HIGH-PASS SEMI-LUMPED FILTER CIRCUITS 15.1 15.2

15.3 15.4 15.5

15.6 15.7

15.8 15.9

16

Transmission-line identities UHF lines as circuit elements Planar lumped element series inductors and shunt capacitors Discrete inductors and capacitors using monolithic techniques Distributed lines as series and shunt resonators Dissipation Equivalence between shunt resonator and half-wave long line Lumped element resistors Problems Further reading

Feldtkeller condition The approximation problem The Chebyshev (or equal ripple) low-pass filter approximation Element values of low-pass prototypes Low-pass to high-pass frequency transformation Element values of high-pass prototypes Low-pass filter circuits High-pass filter circuits Frequency response of microwave filters Problems Further reading

COUPLED PARALLEL LINE CIRCUITS 16.1 16.2 16.3

16.4 16.5 16.6 16.7 16.8

Introduction lmmittance matrices of parallel coupled lines lmmittance matrices of parallel coupled lines in I-plane Parallel line bandstop filter circuit Parallel line bandpass filter circuits Equivalence between coupled parallel line circuits Self and mutual capacitances of parallel lines Open-circuit parameters of parallel lines in inhomogeneous medium Problems Further reading

190

195 197 199 204 207 208

211 211

212 214 217 218 221 223

225 226 226

227 227

228 231

233 235 236 238 241 242 243 243

245 245 245 247

249 251

255 258 259 261

261

17

PARALLEL LINES BANDSTOP 17.1 17.2 17.3 17.4 17.5

18

Contents

ix

FILTERS

263

Introduction Exact synthesis of distributed bandstop filters Exact synthesis of bandstop filters using parallel lines Bandstop filters using open-circuited stubs Frequency response of bandstop filter Problems References and further reading

DIRECTLY COUPLED BANDPASS 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8

FILTERS

Introduction f-plane synthesis using f-plane inductors and capacitors Ladder network using f-plane inductors and ideal f-plane inverters Immittance inverters using lumped element transformations Practical immittance inverters Directly coupled microwave filters Capacitively coupled microstrip bandpass filter Dual mode ring filters Problems References and further reading

19

BANDPASS

19.1 19.2 19.3 19.4 19.5 19.6

Introduction Semi-ideal impedance inverters using UEs Ladder network using f-plane inductors and semi-ideal f-plane inverters Exact parallel line f-plane networks Synthesis of parallel line filters using short-circuited lines Synthesis of parallel line filters using open-circuited lines Problems Further reading

20

THE INTERDIGITAL FILTER 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10

21

FILTERS USING PARALLEL LINES

Capacitive matrices of parallel line arrays Parallel lines identity (Grayzel theorem) Synthesis of parallel lines using unequal strips Equivalent circuit of interdigital circuits Degree of interdigital filter Capacitance matrix of interdigital filter Design of interdigital filter Bandpass resonators using Y-resonators Bandpass resonators using ring resonators Tapped interdigital structure Problems Further reading

COMBLINE FILTER 21.1 21.2

Introduction Exact equivalent circuit of combline filter

263 263 267 271 274 274 275

277 277 278 281 285 286 287 290 293 294 295

2 97 297 297 301 304 306 309 311 312

313 313 315 318 322 322 325 327 329 331 331 332 332

335 335 335

Contents

x

21.3 21.4 21.5 21.6 21.7 21.8

22

OPEN DIELECTRIC RESONATORS 22.1 22.2 22.3 22.4 22.5 22.6 22.7

23

Introduction Frequency of dielectric resonator Maxwell's equations in cylindrical coordinates Characteristic equation of dielectric waveguide Open dielectric resonator Dielectric resonator circuits External Q-factor Problems Further reading

IMPEDANCE MATRIX OF JUNCTION CIRCULATOR 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8

24

f-plane bandpass prototype f-plane LC resonators Adjustment of combline filter Mixed lumped/distributed combline filter Equivalent circuit of UE using ideal admittance inverter Degree-4 combline filter Further reading

Introduction Network definition of junction circulator Eigenvalue adjustment of three-port circulator Impedance matrix of junction circulator Complex gyrator immittance of three-port circulator Synthesis of junction circulators using resonant in-phase eigen-network Equivalent circuit Junction circulators with Chebyshev characteristics Further reading

THE STRIPLINE JUNCTION CIRCULATOR 24.1 24.2 24.3 24.4

Mode chart of gyromagnetic resonator Impedance matrix Circulation solution Frequency response of weakly magnetized circulator Further reading

INDEX

339 339 341 345 346 348 351

353 353 354 355 357 364 365 367 367 368

369 369 370 372 374 375 378 379 382 382

385 385 389 392 394 395

397

PREFACE

Planar microwave circuits represent an important class of devices in microwave engineering. The purpose of this book is to deal with some of the important parallel bandpass and bandstop passive arrangements such as planar resonators, directional couplers, power dividers, filters, gyromagnetic phase shifters, junction circulators and planar radiators. While the open dielectric resonator is not, strictly speaking, a planar circuit it is compatible with this sort of circuit and is therefore included. The text is intended to introduce the young engineer in industry and the taught postgraduate university student to this important area of microwave engineering. A planar circuit is defined for the purpose of this text as one which can be fabricated using either thin or thick film techniques and one for which propagation may be assumed to be TEM or quasi TEM. While the latter assumption breaks down at very high frequencies it has nevertheless served the engineering community well at both the conceptual and practical levels. The text includes design equations which are suitable for both simulation and implementation of a number of typical networks. Although not all the relationships met in the text have been derived from first principles the mathematical level adopted is still sufficient to provide an adequate intellectual insight into the operation of this class of circuits.

To my mother Dyna

1

TEM AND QUASI-TEM TRANSMISSION LINES 1.1

INTRODUCTION

At low frequencies, where electric potential and current are perhaps more meaningful concepts than electric and magnetic fields, electrical energy is transported by current carrying wires; the coaxial and the two wire lines being examples. This class of line supports a ΊΈΜ solution with no low-frequency cut-off condition. Two widely used semiplanar forms of these two lines are the microstrip and stripline geometries. Other structures which have some merit at still higher frequencies are the slot line, the coplanar line, the fin-line and the inverted microstrip. While strictly speaking, each of these geometries supports a quasi-TEM rather than a TEM solution in the microwave region, this has not prevented the wide use of such lines in modem assemblies and equipment. This is in no small part due to the ease with which it is possible to mount series and shunt elements along such structures without the need of drilling holes in the substrate. Quartz and alumina are two substrate materials employed in practice.

1.2

PROPAGATION IN COAXIAL LINE

A widely used transmission line that is readily amenable to a closed-form solution is the coaxial line illustrated in figure 1.1 having an inner radius b and an outer one a. Such a structure can support a purely transverse solution (TEM wave) for which Ez = H z = O

(la)

The simplest permissible solution in this type of transmission line is that for which the boundary conditions at r = a and b are Eθ = H r = Q

(lb)

and Er ≠ 0,

Hθ ≠ 0

(lc)

The description of the two components H θ and Er may be deduced by having recourse to Maxwell's equations. These equations are given in differential form for a charge-free region

2

ΊΈΜ and quasi-ΊΈΜ

transmission lines

Inner conductor Outer conductor

Fig.1.1 Schematic diagram of coaxial transmission line

(p = 0) by

V × E = - μ0 —

of

V × H = ε0

∂E of

(2a)

(2b)

V D = 0

(2c)

V∙B = 0

(2d)

In transmission theory it is usual to assume that all the fields vary in the positive z-direction as exp ( - γz)

(3a)

so that it is only necessary to determine the variations of the fields in the transverse plane. y is a complex quantity known as the propagation constant which satisfies the wave equation to be deduced 7 = α÷jβ

(3b)

a is the attenuation constant of the line per unit length (dB⁄m)z β is the phase constant per unit length (rad⁄m). It is also understood that all the field quantities vary with time as exp(jωf) where ω is the radian frequency (rad⁄s). The constitutive parameters μ 0 and ε0 are given by μ 0 = 4π × 10

7

ε0 = - - - - × 10 36π

H/m 9

F/m

(3c)

Propagation

in coaxial line

The two curl equations in circular variables with the boundary conditions in (1) give SEr — = - }ωμ0 H β OZ

(4)

and ∂H β — = - j ω ε 0 Er oz ∂(rH β ) ~ ∂r

(5)

o

(6)

respectively. The two divergence equations yield ∂(rEr ) _ ∂r

0

δθ

(7)

(8) equation for either

quantity by making use of (4) and (5). The results are ~ + ω 2 μ 0 ε 0 ∖ Er = 0 ∂z2 ⁄ or

∂z

->

Ď

— + ω 2 μ 0 ε0 )H θ = 0 ∂z 2 J

(9)

(10)

One possible product solution for H θ in a semi-infinite line which is independent of θ and which satisfies (6) with rH θ

(11)

Hθ = - e x p ( - γ z ) 2πr

(12)

β 2 = ω 2 μ 0 ε0

(13)

as a constant is

provided where I is the total current in the conductors. Er is now obtained in terms of H θ by having recourse to (5) Er

— He

jωε0

(14)

Product solutions of pure functions of R and Z which are independent of θ therefore satisfy both the curl and divergence equations.

ΊΈΜ and quasi-ΊΈΜ transmission

lines

The wave impedance of the line is defined by equation (14). It corresponds to that of free space: η0 = ~ ~ / — = 1 2 0 π jωε 0 ∙√ ε0

Ω

(15)

The phase velocity of the line is defined by (13). It coincides with that of the velocity of light: υ = - = --- -- = 3 × 10 8 m/s p √μo¾

(16)

More generally, in the presence of a dielectric or magnetic filler with a complex relative constitutive parameter εr or μ r , ε0 →ε 0 (ε' — jε")

(17)

μQ→μ 0 (μ'r - ] μ " )

(18)

and both the impedance and propagation constants are complex quantities. Figure 1.2 illustrates the field patterns in this type of transmission line. At low frequencies it is customary to work in terms of voltage and current variables instead of electric and magnetic ones; it is also usual to work in terms of the characteristic impedance instead of the wave impedance. The voltage across the line is defined by rb

V =

Jα

E r dr

(19)

Writing Er in terms of (12) and (14) gives (20)

-------------- Electric field -------------- Magnetic field Fig. 1.2

Electric and magnetic fields in coaxial line

Telegraphist equations

5

and forming V gives the required result: v = ⅛∩ 2π

Ďλ ⁄

(21)

This last equation also defines the characteristic impedance of the line as (22)

and I without further ado as I= —

Zo

(23)

The characteristic impedance of the line may therefore be varied by adjusting the radii α and b.

1.3

TELEGRAPHIST EQUATIONS

It will now be demonstrated that the coupled equations between Er and H θ in (4) and (5) reduce to a standard form in terms of voltage and current variables V and I. Combining (12) and (21) gives V He=

,., η 0 r In (b⁄a)λ 1

(24)

Making use of (14) separately gives V Er = jio H β = - — — r In (b⁄a)

(25)

Substituting these relationships into (4) and (5) produces a pair of coupled equations in V and I instead of Er and H θ . ∂V ln(t⁄β) r — = - jωμ0 — — I dz 2π

(26)

∂I 2π „ — = — jωε00 -------- V dz ln(b⁄a)

(27)

The preceding equations are recognized as the standard coupled transmission line or telegraphist equations:

OZ

=-z'

(28)

TEM and quasi-TEM transmission

6

d L ∂z

lines

- YV

(29)

Z and Y are defined in terms of the inductance and capacitance per unit length of the line Z = jωL

(30)

Y=jωC

(31)

Scrutiny of the coupled equations indicates that the capacitance (F/m) and inductance (H/m) of the line are given respectively by 2πε0 C = -------In (b⁄α) t

≈⅛ln(⅛l

F/m

H/m

2π

1.4

(32)

(33)

PARALLEL-WIRE TRANSMISSION LINE

Another classic TEM transmission line is the twin or parallel wire arrangement. Although this geometry is amenable to a closed — form solution its derivation requires an appreciation

Fig. 1 . 3

Schematic diagram of parallel wire transmission line

------------- Electric field -------------- Magnetic field Fig. 1 . 4

Field patterns in parallel wire transmission line

The stripline line

7

of conformal mapping which is outside the remit of this work. The result will therefore be noted without further ado: Zo = 120 cosh

1

( — Vo

(34)

The schematic diagram of this arrangement is illustrated in figure 1.3 and the field patterns in figure 1.4.

1.5

THE STRIPLINE LINE

A semi-planar variation of the coaxial line that is widely used in the microwave region is the stripline geometry illustrated in figure 1.5. The derivation of its characteristic impedance starts by evaluating the magnetic field at the position y, in the configuration in figure 1.6, due to a current I in the centre conductor. h = --------- -------2 W + 2 t + 4y

(35)

The contribution from each ground plane cancels, since the magnetic field from a uniform infinite current sheet does not depend on the distance from it.

Ground plane Centre conductor

Fig. 1.5

Fig. 1.6

Schematic diagram of stripline transmission line

Definition of magnetic field in stripline transmission line

TEM and quasi-TEM transmission

3

lines

The average field is then given by A=

1 p Hj

0

Idy

(36)

2 W + 2 i + 4ι⁄

Integrating this quantity leads to , 1 1 ⁄ W + f + 2H h = — In ---------------4H Ď W + f

(37)

The voltage on the line is separately given by (38)

V = E d y = EH

The impedance of the line is now obtained by combining (37) and (38). The result is Z o = 3 0 π In

(39)

Electric field Magnetic field Fig. 1 . 7

200

- OHMS

140

t / b = 0.25 0.20 0.15 0 10 0.05 0

VFZ

160

0

180

Field patterns in stripline transmission line

t/b =0 — 0.05 — 0.10 — 0 15 — 0.20 r- 0 25

120 100

0.1

0.2

0.3

0.5 0.7

10

2.0

3.0 4.0

W /b

Fig. 1 . 8

Characteristic

impedance of stripline transmission line

9

The microstrip line

S/2

b/2

Fig. 1.9

Capacitance of stripline transmission line

W, H a n d t are the linear dimensions of the stripline in figure 1.5. Figure 1.7 shows the field patterns on this line; figure 1.8 depicts one graphical solution for Zo . The characteristic impedance of a TEM line may also be deduced from a knowledge of its total capacitance and its phase velocity. Figure 1.9 shows the parallel plate and fringing capacitances in this type of geometry. The parallel plate capacitance is readily given by (40) The capacitance at a single edge may be shown to be approximately fixed by Cf = ° - l n 2 H

F/m

(41)

The total capacitance of the line is then given by C t = Cp ÷ 2 C

f

(42)

The calculation of the impedance of the line based on this approach is left as an exercise for the reader.

1.6

THE MICROSTRIP LINE

A classic planar transmission line closely related to the two-wire geometry is the microstrip arrangement. Figure 1.10 illustrates its layout. The characteristic impedance of this line may be approximately deduced by assuming that the fields are quasi-TEM and that the metallization is infinitely thin. Making use of these assumptions indicates that it may be expressed in terms of the free-space phase velocity of the substrate and a knowledge of either the inductance (L) or the capacitance (C) per unit length of the line: zo = ⅛ vC

Ω

(43)

Z o = vL

Ω

(44)

or

10

ΊΈΜ and quasi-ΊΈΜ transmission lines

The capacitance of the geometry per unit length is given without further ado by ε ε w r = -------θ r C H

eF/m ⁄

(45)

and the phase velocity of the medium is c v = -----

m/s

(46)

where ε r is the relative dielectric constant of the substrate, H is its thickness (m) and W is the width of the centre conductor (m). The required result is now obtained by combining the preceding relationships:

Zo = -

-

Ω

(47)

This expression is valid for low-impedance lines; for high-impedance ones the effect of the fringing fields at the edges of the centre conductor must be accounted for in the manner mentioned in connection with (41). The field pattern in this type of line is indicated in figure 1.11. If the fringing is significant then the dielectric constant of the substrate is no longer appropriate to describe the phase constant of the line described by (46); it is then expedient to define an effective dielectric constant (εe ff) in order to accommodate this effect:

(48)

The effective dielectric constant may be derived by forming the ratio of the capacitance of

11

The microstrip line

Electric field Magnetic field Fig.1.11 Field patterns in microstrip

line

the line with and without the dielectric filler: C ε

(49)

eff —

where C is the capacitance of the line with the dielectric filler, while C' is that without it. This result may be readily demonstrated by investigating the parameters of the two arrangements. The three possible definitions of the characteristic impedance of free space line are given: ¾=y∣

(5o≡)

Z'0 = cL

(50b)

4 = -⅛

(50c)

and those of the loaded line are given by similar relationships with C' replaced by C and c replaced by υ. Combining the preceding equations and noting that the inductance is invariant under the two situations gives

c2 C'

v2 C

(51)

which satisfies (48) and (49) as asserted. For low-impedance lines with wide strips β

eff~÷

β

r

(52a)

ΊΈΜ and quasi-TEM transmission lines

12

and for high-impedance

lines with narrow strips (52b)

This interval is sometimes expressed in terms of a filling factor ⅛) βeff = l +

r

-l)

(52c)

which lies between 1 ≤