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INTEGRAL-OPERATOR ANALYSIS OF LAYERED MEDIA AND

INTEGRATED MICROSTRIP CONFIGURATIONS

By

Yi Yuan

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1991

To

the loving memory of my father

ACKNOWLEDGMENTS

Foremost thanks are due Dr. Dennis P. Nyquist, my academic advisor, for his
sincere guidance and generous support throughout the course of this work. I am also
very grateful to Dr. Kun-Mu Chen and Dr. Edward J. Rothwell for their thoughtful
advice and encouragement during the period of this study.

I would like to thank Dr. B. Drachman for his generous donation of time and
assistance. I would also like to thank my friends and colleagues for their support and
encouragement during my years in MSU.

I am deeply grateful to my uncle and aunt, Mr. and Mrs. Lu Fan Huang, for their
support and understanding. Finally, I must thank my wife, Xiaochun, and my mother,
Mrs. Qiu Ouyang Yuan, for their continued love, sacrifice and understanding throughout

my many years of graduate study.

TABLE OF CONTENTS

LIST OF TABLES ....................................... iii
LIST OF FIGURES ..................................... iv
CHAPTER 1 INTRODUCTION ........................... 1
CHAPTER 2 GREEN ’8 FUNCTIONS AND ELECTROMAGNETIC
FIELDS IN PLANARLY LAYERED STRUCTURES ..... 5
2.1 Hertzian Potential and its Boundary Conditions .......... 5
2.2 Solution for Tri-Iayer Structure .................... 10
2.3 Green’s Functions and Fields in the Cover Region ........ 13
2.4 Green’s Functions and Fields in the Film Region ......... 18
CHAPTER 3 INTEGRAL EQUATION DFSCRIPTION OF
MICROSTRIP CIRCUITS ...................... 24
3.1 General Integral Equation for Microstrip Circuits ......... 24
3.2 Green’s Dyad and Integral Equation for Microstrip Lines . . . . 26
3.3 Propagation Spectrum; Excitation and Coupling of Microstrip
Lines .................................... 29
CHAPTER 4 ANALYSIS OF A SINGLE MICROSTRIP
TRANSMISSION LINE ........................ 38
4.1 Formulation of the Problem ...................... 38
4.2 Method of Moments Solution ..................... 41
4.3 Computational Results .......................... 46
CHAPTER 5 ANALYSIS OF COUPLED MICROSTRIP
TRANSMISSION LINES ....................... 54
5.1 Formulation for Coupled Microstrip Lines ............. 54
5.2 Method of Moments Solution .......... » ........... 59
5.3 Computational Results .......................... 62

CHAPTER6

EFIE-BASED FULL-WAVE PERTURBATION THEORY
FOR COUPLED MICROSTRIP TRANSMISSION LINES . 75

6.1 Integral-Operator Based Perturbation Theory ............ 76
6.2 Perturbation Solution of Two Coupled Microstrip Lines ..... 80
6.3 Numerical Results ............................ 83
6.4 Experimental Validation ........................ 86
6.5 Discussion ................................. 88
CHAPTER 7 ANALYSIS OF THE CHARACTERISTIC IMPEDANCE
OF MICROSTRIP TRANSMISSION LINES .......... 99
7.1 Circuit Description of Microstrip Lines ............... 99
7.2 Formulation of the Problem ...................... 101
7.3 Fields and Power in the Cover Region ................ 104
7.4 Fields and Power in the Film Region ................ 109
7.5 Numerical Results ............................ 116
CHAPTER 8 MATERIAL LOSSES, LEAKAGE AND RADIATION
CHARACTERISTICS OF MICROSTRIP
TRANSMISSION LINES ....................... 118
8.1 Propagation Spectrum and Leaky Modes .............. 119
8.2 Complex Integral Equation Analysis of Leaky Modes ...... 122
8.2.1 Surface Wave Pole Singularities and Surface Wave leakage . . . 123
8.2.2 Branch Point Singularities and Radiation leakage ......... 130
8.2.3 Branch Point Singularities and Root Search on the Complex
{-Plane ................................... 136
8.3 Numerical Procedure and Results ................... 138
CHAPTER 9 CONCLUSION ............................. 155
BIBLIOGRAPHY ...................................... 159

ll. '1‘ r'|l IL [‘ [A I 1" Is. I Ill 1"! .1
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LIST OF TABLES
Table 4.1 Convergence of the propagation eigenvalues upon the numbers
of basis functions used in the current expansion ............ 53

Table 4.2 Expansion coefficients of the Chebyshev polynomial series for
the currents of the first two modes .................... 53

Table 8.1 Convergence of the propagation eigenvalues upon the numbers

of basis functions used in current expansion for the leaky EHO
mode at 10 GHz ............................... 153

iii

Figure 2.1
Figure 2.2

Figure 3.1
Figure 3.2
Figure 3.3
Figure 4.1

Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5

Figure 5.1

Figure 5.2
Figure 5.3

Figure 5.4

meiS

LIST OF FIGURES

Configuration of multilayer planar structure with current sources
immersed in the ith region ......................... 22

Configuration of tri-layered media with current sources in the

cover region ................................. 23
Configuration of general open microstrip integrated circuit ..... 35
Configuration of general microstrip transmission line structures . . 36
Singularities and integration paths in the complex f-plane ...... 37
Configuration of a single microstrip transmission line ........ 48
Dispersion characteristics of the principal mode and the first two
higher-order modes for the microstrip line ............... 49
Current distributions of the principal EH0 mode for the

microstrip line operating at two different frequencies ......... 50
Current distributions of the first higher-order EH, mode for the
microstrip line operating at two different frequencies ......... 51
Current distributions of the second higher-order EH2 mode for

the microstrip line operating at two different frequencies ...... 52

General configuration of N coupled microstrip transmission lines . 64

Configuration of two identical coupled microstrip transmission

lines ...................................... 65
Dependence of propagation phase constant 1' upon line spacing

for the EH0 system modes of two coupled microstrip lines ..... 66
Dispersion characteristics of the EH0 system modes for two

microstrip lines with a separation equal to the strip width,

compared with the isolated profile .................... 67

Dispersion characteristics of the EHO system modes for two
microstrip lines with a separation equal to 30% of the strip
width ...................................... 68

iv

Figure 5.6

Figure 5.7

Figure 5.8

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5
Figure 6.6

Figure 6.7

(a) Longitudinal current distributions of the E11,, system modes

for two microstrip lines with a separation equal to the strip

width ...................................... 69
(b) Transverse current distributions of the EHO system modes

for two microstrip lines with a separation equal to the strip

width ...................................... 70

(a) Longitudinal current distributions of the EH0 system modes

for two microstrip lines with a separation equal to 30% of the

strip width .................................. 71
(b) Transverse current distributions of the EH0 system modes

for two microstrip lines with a separation equal to 30% of the

strip width .................................. 72

(a) Longitudinal current distributions of the EH0 system modes

for two microstrip lines with a separation equal to 2% of the

strip width .................................. 73
(b) Transverse current distributions of the EH0 system modes

for two microstrip lines with a separation equal to 2% of the

strip width .................................. 74

Comparison of MoM and perturbation solutions for dependence
upon line spacing of propagation constant 1- for the EH0 system
modes of two coupled microstrip lines ................. 90

Comparison of MoM and perturbation solutions of dispersion
characteristics for two coupled microstrip lines with a separation
equal to the strip width ........................... 91

Comparison of MoM and perturbation solutions of dispersion
characteristics for two coupled microstrip lines with a separation

equal to 30% of the strip width ...................... 92
(a) Comparison of longitudinal current distributions for isolated

and coupled microstrip lines with three different separations . . . . 93
(b) Comparison of transverse current distributions for isolated

and coupled microstrip lines with three different separations . . . . 94
Dependence of coupling length X, upon line spacing ......... 95
Illustration of experimental setup ..................... 96

Comparison of perturbation-theory—predicted and experimentally
measured voltage distributions on driven line ............. 97

Figure 6.8

Figure 7.1

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 8.7

Figure 8.8

Figure 8.9

Figure 8.10

Figure 8.11

Comparison of perturbation-dreary-predicted and experimentally
measured voltage distributions on coupled line ............. 98

Frequency dependence of characteristic impedance for the
microstrip transmission line ........................ 117

Illustration of the migration of the surface-wave poles in the
complex E-plane as the microstrip modes make the transition
from purely bound to leaky regimes ................... 141

(a) Pole and inversion contour for f,. (b) Pole and inversion
contour for f: ................................. 142

(a) Common inversion contour for the case of 5,, < 0 (decaying
wave). (b) Common inversion contour for the case of 5,, > 0
(growing wave) ............................... 143

(a) Negative real-axis as branch out in the complex ifi-plane.
(b) The mapping of the branch out in the iii-plane into the
complex £b-plane ............................... 144

Singularities and deformation of integration paths for a nonleaky
mode in the complex 5 -p1ane ....................... 145

Illustration of the migration of the p, branch points and their
associated hyperbolic cuts in the complex f-plane as the
microstrip modes make the transition from purely bound to

leaky regimes ................................. 146
Singularities, inversion contour and deformation of integration

paths for a leaky mode in the complex £-p1ane ............ 147
Branch points and the associated branch cuts for $1. in the

complex {-plane ............................... 148
Deformed integration path for numerieal computation ........ 149

(a) Dispersion curves of the principal mode and the first three
higher-order modes in both bound and leaky regimes for the

microstrip line ................................ 150
(b) Attenuation constants of higher-order EH, and EH2 modes
for the microstrip line in the leaky regime ............... 151

(a) Enlarged dispersion curve of the EH, mode in the transition
region between bound and leaky regimes for the microstrip line . . 152

vi

(b) Migration trajectory of the TM0 surface-wave pole E,
(c) Migration trajectory of the branch point E, for the EH0
microstrip mode ............................... 153

vii

CHAPTER ONE

INTRODUCTION

Microstrip configurations in a planarly layered environment are essential
structures in contemporary microwave and millimeter-wave integrated circuits (MMIC’s).
Recent advances in monolithic MMIC technology increasingly require rigorous analysis
and accurate modeling of the dynamic behavior for a single device as well as circuit
systems, since quasi-static analyses begin to break down for large systems operating at
very high frequencies. Also, in CAD simulators for MMIC design, accurate circuit
descriptions of individual components such as microstrip patches or lines, are necessary
so they can be used together to analyze more complex circuit systems.

In this dissertation, a rigorous full-wave integral operator formulation for
electromagnetic interactions among microstrip configurations in the layered environment
is developed. This integral equation theory is then applied to analyze one of the most
essential elements, the microstrip transmission line. It is intended to provide a complete,
rigorous analysis and an accurate modeling of the EM phenomena associated with
microstrip transmission lines, including their propagation spectrum, dispersion
characteristics, coupling effects, current and field distributions, power propagation, losses
and leakage and/or radiation properties.

Because of the importance of the subject, abundant research efforts have been
devoted to this area. The study of waves and fields in planarly layered media is quite
a classic problem. Many books have been written on this and related subjects [1]-[11].

The development of dyadic Green’s functions for layered media is relatively recent and

a good review of it can be found in [12]-[20]. For the analysis of rrricrostrip lines,
various methods, from quasi-TEM approximations to spectral-domain approaches, have
been developed. A comprehensive collection and review of papers up to 1986 can be
found in [21]-[22], and we refer to [23]-[33] for a review of the various approaches.
More recently, a rigorous integral equation formulation for the microstrip structure has
been introduced [18], [34]. Similar approaches have been developed and used by other
researchers for the study of various rrricrostrip devices such as microstrip patch antennas
and transmission lines. The integral equation approaches have gained increasing favor
and a large number of papers have been published. Reference is made here only to
representative papers [351-[45]. More papers are referenced in our discussion of relevant
subjects in later chapters. Most of the studies have been concentrated on the dispersion
characteristics of discrete, bound propagation modes which are now well understood,
while the knowledge of the leakage and radiation characteristics and other properties of
microstrip lines remain incomplete and require continued research efforts.

This dissertation is divided into nine chapters. Chapter 2 presents the
development of the dyadic Green’s function for layered media within which arbitrary
current sources are immersed. The Hertzian potential, and associated electric and
magnetic fields are obtained in terms of the Green’s function and current sources; they
enable us to develop integral equations for various problems. This work follows exactly
that of Bagby and Nyquist [18], except that here magnetic contrast among the media are
introduced in addition to dielectric contrast, and therefore the study of layered magnetic
media is enabled.

In Chapter 3, an electric field integral equation (EFIE) description for general

microstrip circuits is developed, and then applied to microstrip transmission lines. In a
Fourier transform-domain analysis, the complete propagation-mode spectrum of
microstrip lines is identified from a singularity expansion of its current in the complex
spatial frequency plane. A singularity expansion method (SEM) in the spatial frequency
domain is consequently identified. Arr excitation and coupling theory for the
electromagnetic response of microstrip lines to impressed radiation is developed.

Propagation modes on a single lossless microstrip transmission line are analyzed
in Chapter 4. Numerical solution to the homogeneous EFIE is implemented by
Galerkin’s method of moments. Chebyshev polynomials with square-root edge factors
are utilized as basis functions. Results of dispersion characteristics and current
distributions for the principal and higher-order modes are presented. Particularly, the
currents are obtained in a convenient quasi-closed form of rapidly convergent Chebyshev
polynomial series.

In Chapter 5, a coupled microstrip transmission line system is studied by an
approach similar to that used for a single line. Rigorous full-wave solutions to the
coupled integral equations are pursued again through the Galerkin’s method of moments.
The dependence of coupling effects upon line spacing, the behavior of the coupled-mode
waves, the dispersion characteristics and the current distributions of the coupled system
are analyzed. Numerical implementations are developed and results are obtained.

Chapter 6 presents a full-wave perturbation theory for N coupled microstrip
transmission lines based upon the rigorous EFIE formulation. In solving the coupled
EFIE’s, the eigenmode currents of the isolated line are used as first order approximation

for the nearly degenerate eigenmode currents of the loosely coupled system, and a simple

perturbation matrix eigenvalue equation is obtained. The perturbation theory provides
good physical insight regarding the coupling effects and greatly simplifies the
computational procedure. An experimental verification is carried out and the theoretical
and experimental results are in a good agreement.

Chapter 7 describes an accurate circuit modeling of the non-TEM microstrip line
structure. To obtain a complete equivalent circuit for the structure, not only the
dispersion characteristics, but also the characteristic impedance must be determined. A
full wave analysis for the characteristic impedance of microstrip lines is developed based
on the same EFIE formulation. Discussions are included on the definition of
characteristic impedance for microstrip transmission lines, the current, field and power
calculations and their applications.

In Chapter 8, leakage and radiation characteristics and material losses for
microstrip transmission lines are analyzed in a consistent manner using the integral
equation formulation. The full-wave EFIE formulation for general lossy microstrip lines
is implemented by extending the real analysis to a unified complex analysis. In
discussions of the bound waveguiding modes and the non-spectral radiative leaky modes,
the propagation spectrum is reviewed and clarified. The migration behaviors of the pole
and branch-point singularities in the complex spectral planes are carefully examined. The
branch cuts and the inversion contour for the Sommerfeld-type inverse integrals are then
determined and justified. From the complex EFIE analysis, the leaky-mode and radiation
properties of microstrip transmission lines are deduced naturally and consistently.

Finally, we conclude this dissertation in Chapter 9 with some general discussion

and comments on this work and some recommendations for future research efforts.

CHAPTER TWO
GREEN’S FUNCTIONS AND ELECTROMAGNETIC FIELDS

IN PLANARLY LAYERED STRUCTURES

In this chapter, general dyadic Green’s functions for electromagnetic fields in
planarly layered structures are developed. First Hertzian potentials in layered regions
are introduced and their boundary conditions are deduced. The Helmholtz equations for
Hertzian potentials are solved by a Fourier transform technique, and consequently
Hertzian potential Green’s functions are found in a form of Sommerfeld’s inverse
integral. The electromagnetic fields are then obtained in terms of associated Hertzian
potentials and the dyadic Green’s functions for electric - fields are identified
correspondingly. The properties of the Green’s function are studied. The fields
produced by an arbitrary volume source are then presented in a volume integral form,

which enable us to develop integral equations for various problems.

2.1 Hertzian Potential and its Boundary Conditions

Consider the multi-layer planar structure depicted in Figure 2.1. Each layer has
constitutive parameters (6,, 11,) and thickness t,. An arbitrarily directed electric current] (F )

is immersed in the ith region and produces electric fields in each layer. Assuming time-

harrnonic excitation with exponential factor aprt), the electromagnetic fields in each
region can be described in terms of an associated electric Hertzian potential I10") [47]

which satisfies the Helmholtz equation

Wm?) + k’fim = -.7(F)/jme. (I)

The electric and magnetic fields are given by

Em = (k1 + worm)
17m = free v x film.

(2)
To solve the Helmholtz equation for Hertzian potential in each layer as a

boundary value problem, we need the boundary conditions on that potential between the

interfaces of different layers. Let 31.: and ”1.: be the fields in two adjacent layers

having constitutive parameters (51.2 , p 1.2): respectively. The boundary conditions at a
point at the interface require tangential components of both electric and magnetic fields
to be continuous. Adopting the coordinate system shown in Figure 2.2, where i and i

are tangent to the interface, we have

51.42. tin-H... <3)

The field components are given in terms of Hertzian potential by

Ea ' kink * %(V°fl) Hz: ””549; " %]

 

2 a _ . an: an!
Eb . kin!) + 5(V‘fi) Hg, #05; az ’ fa;- 1: 1,2. (4)

E,z = kin, + Fez-(VII) Ha mural? - 33%]

The direct implementation of boundary conditions (3) to field components in (4) in terms

of Hertzian potential appears difficult since all components of II and its derivatives are

involved. To convert the boundary conditions on E and H into simpler ones for 11,
various cases with different orthogonal components of exciting current J and the
associated Hertzian potential are considered independently. In each case, the
enforcement of the continuity of tangential E and H fields always leads to the same
restriction on 11. By the property of linear superposition of the fields and potential, the

following general boundary conditions for Hertzian potential are deduced

 

 

 

II“ = szerzrnz. 111, = N22111:,
an . an (5)
a; ‘ 221%.— (a—gy-zz ’ 3‘1) =(N221M221 ' ”[63: + 6:2!)

where a =x,z denotes the tangential components; N3, = 62/6, and M22, - 112/"; are

dielectric and magnetic media contrasts. The boundary conditions show clearly that the
normal component of Hertzian potential is coupled into tangential components at the
interface.

The total Hertzian potential in each layer is the sum of a primary part and a
scattered part. The primary potential is produced by the exciting current in that layer
and satisfies the inhomogeneous Helmholtz equation, while the scattered potentials are
those reflected or transmitted at the interfaces and satisfy the homogeneous Helmholtz

equation, consequently

Vim?) + kffifm = -.7(i")/jue,,

2 (6)
Vin?) + mix?) = o.

Fourier transform techniques are used to solve the Helmholtz equations. Define a 2—D

transformation on the spatial variables tangential to the layer interfaces, which are x and

z in our case, as follows

fi(X;y) = ”mm-I‘m:

 

'2 (7)
11(7) = 1 ii(5.°;y)e"°'d2).
(2%)2 .‘U

where X =£E +£C is the 2-D transform variable; 12 = £2 + {1, d2}. =d£ dc and

ds = dxdz . Taking the transform on the equation of scattered potential, we have

321370; y)

3y - pitta») = o (8)

where pf = 1.2 - k} = E” + C2 - k} is the y-direction wave number. This equation has a

plane-wave type solution in the transform domain

filmy) = mm” + Won”. (9)

and inverse transformation gives the space domain result

1 . "’ r "‘ 'r Io
ilk?) = Ear—)2 U(W,me” + W,(}.)e ”)d '42). . (10)

The scattered potential usually has two parts: transmitted and reflected components.
Physically, the scattered potential can be viewed as superpositions of plane waves

traveling among layers and transmitted and reflected at interfaces according to the

boundary conditions.

The solution for the primary potential due to the exciting current is

fl7(7) = [G’Wlfl'zfldv’ (11)
v 1061
where G'(i’|?’) = exp(-jk,R)/4rrR is free space Green’s function which can also be

expressed in a spectral integral repesentation, and leads to the primary potential

1.0-1") -m|r-r’l
1'1" 7 dV’Z—L—f ’ ‘1 ‘ d2).
'( )= f [:f 2(2fl)2pc

 

 

3f]. f](fl) e-Ix'F’e’PIIP'IIIdV’ ejx'7d21
we 2am». (12)

 

(2 )2 ff #0 y)e1"d=r
it

where

 

j(i'.’) e-jxrf’e"tl"”ldyl
rue. 2p.

£70.; y) = f— (13)

is the transformed primary potential. The total potential in any layer is then obtained as
fi,(?) = 13(7) + mm. (14)
The unknown spectral amplitude coefficients 17:0.) implicated in I170") are

determined by application of boundary conditions at the interfaces for a specific

9

configuration; they depend on the exciting current through the primary term. Combining
(10) and (12) in (14), the total Hertzian potential can be expressed as a volume integral

in terms of a spatial Green’s function and source current

rim = [cirrus-flaw (15)
v 1‘“:

Where the dyadic Green’s function is in a form of 2-D inverse integral, and can be
identified in each region for that specific structure. Once the Hertzian potential is
determined, the E and H fields can be found by equations (2), and an electric field dyadic

Green’s function can then be defined. Enforcement of boundary conditions at interfaces

results in linear equations for the TWO.) which, in principle, can always be solved but

could be very tedious to do for large number of layers. A chain matrix approach to
evaluate the reflection and transmission coefficients has been developed by Nyquist [48].
In the following sections, the case of a tri-layered structure will be studied in detail for

our later use in the analysis of microstrip circuits.

2.2 Solution for Tri-Layer Structure

Consider the tri-layered structure depicted in Figure 2.2. A film of thickness d
having constitutive parameters (6, , 11,) is deposited over a substrate (y < ~d) described by
(e. , A). The region (y > O) is the cover with parameters (6, , #9- An arbitrarily directed
electric current I is immersed in the cover region and produces electromagnetic fields in
each layer. The transform domain Hertzian potential in each region is given by (9), (13)

and (14). In the cover region, the total potential includes a principal part and a reflected

10

part as
many) = may) + may”. (16)

In the film region, the total potential is composed of a transmitted wave and a reflected

wave with

ii,(1;y) .. Wyn)!" + 57(1)?" (17)

while in the substrate region, only a transmitted part exists

mm) = icon”. (18)

Applying boundary conditions at the y = 0 and y = -d interfaces, the spectral
amplitudes can be determined. To facilitate the use of boundary conditions, they are

transformed into the spectral domain as

2 2 _ 2
“r. " Nerzr‘nz. “r, ’ "2132,

an , 61:2. (19)

8y 21-6).—

 

33 31:
(Ta)?! - 51!) = (N22151:! - l)(j£fik +1111”)

with a = x, z denoting the tangential components. Notice that the boundary condition for

tangential components is independent of that for normal components; we solve for the

tangential components first. At y = O, we have

mam - Nzufitwm) + W,:(z)1 + no) = o
p.W.'.(A) + Nipxwgtm - W,:(x)1 - p.V.(A) = o

(20)

where

11

(21)

 

J 7’ 4: 1’ 1.”
v.0.) = nf,();0) = f f( )‘ ‘ dV’
V Jwel 2pc
is the principal potential produced by source current at the y = 0 interface, which is

considered a known function here. At y = —d, we have

Wink)!“ + "920)!” - NiMrWztmfl" = o (22)

p,W,:0)e"”’ - p,W,:0)e'I" - Wigwam?" = 0
Relations (20) and (22) form a set of linear equations. Omitting the algebra, we have

the following solution in terms of Va

W.2.0) = 12,0) V.0) W,:.0) = 1,0)V.0)

, , a =x,z (23)
nga) = 19.0)".(1) Wu“) = T.0) v.0)

where R,()r) and T1,“) are reflection and transmission coefficients for tangential
component potential waves in each region, respectively, and are listed later when needed.
Now we solve for the normal components. The boundary conditions at y = 0

in combination with (23) give

W.;0) - Nyil W50) + W30» + v,0) = o
p,W.',0) + p,t W50) - W,;0)1 - p,V,0) - A0)Li£V,0) + 1020)] = o

(24)

where .40) - mm: - 1)[R,,0) + 30)] isan auxiliary function. Similarly, boundary
conditions at y = -d give
W50) + W,;(x)e't" - N3W,‘,(A)e"" - o (25)

Mime” - Mime" - W40)?" + B0) 05130) + 1040)] = o

12

where BO.) .. (N514; - 1) T.(A)e "-‘ is also an auxiliary function. Expressions (24)

and (25) are a set of linear equations which have the following solutions:

W50) - 12,0) V,0) + C,0-)U£V,(l) + 1030)]
W50) = 19.0) V,0) + c,0)UEV,0) + imam
W50) = r50)v,0) + c,0)UtV,0) + 10:0»
W50) = 1.0) V,0) + C.(A)UtV,0) + jCV,0)l

(26)

where Rhos) and Th0) and are reflection and transmission coefficients for normal
potential components in each region; C50) and C,,()\) are coupling coefficients for
reflected and transmitted waves, respectively. Results (23) and (26) in combination with
(l6), (l7) and (18) completely determine the transform-domain Hertzian potentials in
each of the three regions. By inverse transformation, the spatial domain potentials, and
therefore the Green’s function, can be obtained correspondingly. 'Ihe Green’s functions
in the cover and the film regions are of primary interest to our problems and are

explicitly studied below.

2.3 Green’s Functions and Fields in the Cover Region
In the cover region where the source current is immersed, we have both the
primary potential and the scattered potential. Correspondingly, the Hertzian potential

Green’s dyad is found to decompose into a principal and a reflected part as
G(?|i") = G'(r|r’) + 6501:”). (27)

From (11) and (12), the principal Green’s dyad

13

from = 10%|?) (28)
with

G'(?|?’) = If

[Mr-f) ~r.|r-r’l
‘ ‘ r121 . (29)
2(2 rr)2pc

 

From (23) and (26), the tangential and normal components of the reflected

spectral-domain Hertzian potential in the cover region are

«2.0.; y) = Ra(x)v,(x)e"¢’ a =x,z
«2,039 = (rum/,0) + mammal) +J’CV,(A)1 1e” .

(30)

Inverse Fourier transformation of (30) gives the potentials in space domain. Exploiting
(21) in the inverse integral, and noticing that it ~a/ax, jC ~a/6z for the inverse

transform, we have

,1! -(r-r') e 1.0%

2(2 )2 d2}. a =x,z (31)
n C

 

1.0) "
11:.0) = [WE L f 12.0)

and

 

(J 7 " jf-(r-r’) 1.000
1130*) - de’ 1L2 f 650. 3 ‘ ‘ d3).

- , I
cit-(r-r’) e M! y)
20102:),

a eff-(1'4") embrr')
a .._..
31 2(2x)’p.

 

d1). (32)

d2).

 

 

l4

Examination of (31) and (32) reveals the reflected Green’s function component as
5:00) = (as + 22mm) + 90% 6.:(rl 7) + yam?) + 2% 6.20m] (33)

where

r ' ’1

6.01?) . 12,0) (1,,(,_,,)¢,,‘(,,,,, 34
lG;(rI?')> = ff R50) 421 < )
,G.2(r|r')1 " Cam 2‘2“”:

 

 

 

with the reflection and coupling coefficients

 

 

 

R¢(A) = AXA) r Rail) = AJA) a Cer(1) = 2p6F(A) (35)
2‘0) 2'0) z'mzvx)
where
11,0.) = (MfiMfipy, - pp.) + (Mime. - Mfrp} )tanhtpfi)
Z ‘0) ‘ (Militia?) + m) + (Mime. + Mir} )th)
44.0.) = (NfiNépcpf - plp‘) + (sz‘p‘ - Nfip} )tanh(p’t) (36)

Z ‘00 = ("/1 Nine, - pp.) +(N13M. - "39} MW!)
F0.) = (”1:”): ‘ ”(N391 * Prmipr‘))(M:rpr + p,tanh(p,t))

+ (NjMf, — r)p,’/oosh¢(p,r).

Summing up the above results, the spatial Hertzian potential dyadic Green’s function for

the cover region can be expressed as

6,(7|r') = (22 +z‘i‘)(G' + 65) + yy(or + 65,) + weir}; + 23826;) (37)

15

and the Hertzian potential is

fi.(?) = f G.(rlr’)-Z.@dV’ (38)
V «’ch

Having obtained the Hertzian potential, both the E and H fields can be found by

(2). The electric field in the cover region is thus

5.0) = 03 + vvof cyrplyflavl. (39)
V Juec

Exchange of the W- operator with the spatial integration is legitimate under the

condition that the integrand is continuously differentiable. The reflected part of the
Hertzian potential Green’s function is a well behaved function, but the principal part is

singular at the source point. So the term

W-f IGPG’ Ir’) -.7(i-”)dV’
V

produces the source point singularity [l6], and needs special consideration. Appropriate
application of Leibnitz’s rule and excluding a ”slice” principal volume at y = y' leads

to

W! IG’(?|7’)-.7(i-") cl!” = P.V.{W'UG'G|7’)'J(?’)]dV’ + 94(7) (40)

where P.V. indicates that the spatial integration must be performed in the principal value

(P.V.) sense [19]. With that, the electric field can be written as

16

34?) = f 63? Wrfldv’ (41)
v yore

C

where the electric field dyadic Green’s function is defined by

6;(r|r’) = P.V.(Irf + wo)<’},(r|r’) + liar-7’) (42)

with the depolarizing dyad identified as f. = -yy . This Green’s function satisfies the

general reciprocity theorem for spatial domain Green’s functions, as is confirmed directly
for this integral representation. The reciprocity property of the Green’s function will be
discussed further in later chapters.

The magnetic field in the cover region is found by

In?) 81106ch f Gymr’yfldv’. (43)
V Jwec

Since only a single derivative is involved, interchange of the curl operation with the
integration is allowed in the above equation; therefore there is no source point singularity

for the magnetic field. Define magnetic dyadic Green’s function

6¢‘(r|r’) = v x GJFIF’), (44)

the magnetic field is then obtained as

110’) = f 637 |r')-.7(r’)dv’. (45)
V

17

Pole singularities within integral representations of the Green’s function, as
implicated in the reflection and coupling coefficients R50), Raps) and Caa), lead to the
surface wave characteristics of layered dielectric slab waveguiding structures. In fact,
2'0) = 0 and 2"(h) = 0 are the eigenvalue equations for TE and TM surface-wave
modes, respectively, supported by the layered structure, and can be written in

conventional form as

m}x(m,25 + mfy)

 

 

tan(lct) = 2 2 2 ‘ TB modes
mcmpt - mfyb
(46)
2 2 2
6 +
tan(ttt) = ""0“ n”) TM modes

222

n,n,x - nfyb

where x =jp,=,/k} - 1’; o =pc =,/)’ 4:3; 7 =p, =,/i’ -k}. Equations (46) are the

most general eigenvalue equations for guided waves of the tri—layered environment.

2.4 Green’s Functions and Fields in the Film Region

The Green’s function and fields in the film region can be deduced in a similar
manner as for the cover region. Since no current sources exist in this region, the only
potential is the scattered potential, which has both reflected and transmitted parts. From
(17), (23) and (26), the spectral-domain Hertzian potential has the following tangential

and normal components

19.0.; y) - 1,0) V.(}.)e”’ + 19(1) V.(1)e"” a =x,z (47)

18

1:50;» a (1‘50) V,(A) + Cfi(1.)[j£Vx()\) . KEG-)1)?”

(43)
+ {125(k) yr“) + C5001)? V1.0) + ICI’,(1)lle"” .

Inverse Fourier transformation of (47) and (48) gives the potential in the spatial domain.
Exploiting (21) in the inverse integral, and noticing that it » a/ax , jC ~ 6/62 for the

inverse transform, we have

'P,’ ejx°("") e 1”,

 

 

(“’)"
J“ [r T(A)e"’ + (De (1’).
6a U R” 201011), (49)

n,.(r)= de’j

u. a = x’z
and

jx-(r-r’) 19”
‘ ‘ an

 

r = dV’ (:1 "’+CA "1’
n5()f 7—:(::)£fl,()e ,e() l; 20102:).

_ r
er-(r-r’) e m 2

 

f f [150)!" + may”)

 

 

I“)e 2(2:r)2pc (50)
J ( F") - aefl-(r-r’uwr’

+ A P]! 1 P], 2 .

, “We-[flame +C()e 1; 2(2”): 4),}

The space domain dyadic Green’s function for the Hertzian potential in the film region

is therefore identified as
(3,01?) = (a +22)G,(7|7’) + y(x§ofi(r|r) +yafl(r|r) +£§Gfi(flr’)) (51)

where the scalar components are

19

_ I
eJX'U-I’) e Pa,

2(2 102p,

d2; (52)

v

6500:) ff ( you” + 19(2):”
6501? ) "' £90)!” + chap”!

 

{0,006} . '90)!” + 190W!"

 

 

with the transmission, reflection and coupling coefficients

p,(M§p, + p,)e”‘ = p,(M§p, - p,)e""'

 

 

 

 

 

 

10) = (r) (53)
” N5cosh(p,d)z*(x) R” N5oosh(p,d)z*(r)
2 + r,4 1 ._ 7,4
7'50) = p,(N4p, p,)e ’ R!" A) = p,(N¢p, p,)e ’ (54)
eosh(p,d)z'(l) oosh(p,d)z'(r)
r A r A.
9.0) = 2p; ,( ). C50) g 2:»: ,( )‘ (55)
N52 (r)z (1) N52 an (r)
where 2*(1) and Z‘O.) are given in (36) and
F0) = (N541; - 1)(N,.p, + p,)(M3p, + p,)mnh(p,d))e""lcosh(p,d)
- (”314;— 1)p,(N5p. - p,)/cosh’(n,d) 56
( )
£0) = (N544; - 1)(N,p, - ppmip, + p,)tanh(p,d))e"‘/cosh(p,d)
- (N341: - 1)p,(N,ip. + p,)/cosh’(p,d)
With this Green’s function, the Hertzian potential in the film region is
10")
g I ,_ I 5
fig?) £65m?) 1,“ch (7)

20

Since no source current is present in the film region, there is no source point singularity

involved in the Green’s dyad or the fields. The electric field in the film region is then

Eye) = (Ir,2 + vv-)fc$c(r|r’)-@.dv’
y 1(06

C

(58)
= f G;(r|r’)~——]:(FI) dV’
1’ 106‘
where the electric Green’s dyad is defined as

G;(i°|r’) = (Ir,2 + vv-)G,(r|?’). (59)

Similarly, defining the magnetic Green’s dyad
c?,"(r|r’) = v x é,(r|r’), (60)

the magnetic field in the film region is then given by
12(7) = n15] 050 Ira-106W. (61)

v

Notice that the coefficients in (53), (54) and (55) have factors 2(1) and 2(5) in
their denominators. Therefore, the Green’s function in the film region has the same pole
singularities as the Green’s function in the cover region leading to surface-wave
excitation in the tri-layered structure.

The potential, fields and Green’s function in the substrate region can be obtained

in the same way. For the sake of brevity, the details are omitted here.

21

 

Region

i-E

 

Reglon

l-l

 

Region

(' source

 

Region

1+1

 

Region

3+8

 

-ol(l-8)

-d(i-l)

"d(|)

-o|(i+1)

-d(l+8)

—o|(i+3)

Figure 2.1 Configuration of multilayer planar structure with current sources immersed

in the ith region.

22

"3.... p {H 3'
cover (cc, uc) a E K

 

‘<
II
0

film (cf, uf) ‘qx 35“,!

 

substrate (es, us)

25 a

Figure 2.2 Configuration of tri-layered media with current sources in the cover region.

23

CHAPTER THREE

INTEGRAL EQUATION DESCRIPTION OF MICROSTRIP CIRCUITS

The Green’s function and Integral equation obtained in Chapter 2 are applied in
this chapter to the analysis of microstrip configurations, especially microstrip
transmission lines. A general electric field integral equation (EFIE) description of
microstrip circuits is developed and then applied to microstrip transmission lines. In a
Fourier transform-domain analysis, the complete propagation-mode spectrum of
microstrip lines is identified from a singularity expansion of its current in the complex
spatial frequency plane. An excitation and coupling theory for the electromagnetic

response of microstrip lines to impressed radiation is developed.

3.1 General Integral Equation for Microstrip Circuits

The general configuration of an open microstrip circuit is depicted in Figure 3.1.
A dielectric and/or magnetic film is deposited over a perfectly conducting substrate.
Conducting circuit devices (e.g. , patches or lines) are embedded in the cover layer
adjacent to the film-cover interface of the tli-layered conductor-film-cover environment.
The coordinates are chosen so that the y-axis is normal and the x, z axes are tangential
to the film-cover interface.

The electric dyadic Green’s functions for the tri-layered conductor-film-cover
structure are special eases of the general ones given in Chapter 2. By letting the
conductivity of the substrate become infinite in (2.35), the reflection and coupling

coefficients in the Green’s dyad for sources immersed in the cover region reduce to the

24

following

4,0)
Z ”(1)

_ 4.0)

2 2
. 450) - . = “New ‘1”; (I)
2‘0)

zt0)z'0)

 

 

 

Raul) 0.0)

where

.40) = 1145pc - p,eolh(p,r) 4.0) = N5);c - p,tanh(p,r)
z*0) - 415)), + p,eolh(p,r) 2'0) = Nfip, + P,tanh(p,r) .

(2)

The corresponding electric dyadic Green’s function G '(F | F’) is obtained when the above

coefficients are exploited in (2.36). Since only the Green’s dyad in the cover region will

be used in this chapter, the subscript C of the dyad is omitted for the sake of simplicity.

If excitation is provided by an incident electric field E ‘( i’) maintained by an
impressed current, a surface current I?( F) is induced on the surface of conducting

devices. This induced current in turn maintains a scattered field 3'0") in the system.

The scattered field can be expressed in term of induced current and the Green’s dyad as

2’0) = [Girlie-flash (3)
3 Joe

The boundary condition requires that the total tangential electric field vanish at the

conducting surfaces of the devices, that is

f-(E‘m + E'm) = o res (4)

25

where S is the total device surface composed of the that for each element Sn and i is a

unit tangent vector at any point on the surface. This leads to the following electric field

integral equation (EFIE) for the induced current on the system

i-j 6‘01 r’)-1?(r’)ds’ = - % £~E‘(r) i-‘eS (5)
S c

where kc=0\/-€,—ll: = rtcm‘Jc0 and nc = JECI—ec = menolnc with n‘, m, the
dielectric and magnetic indices, and ko, no the free space wavenumber and impedance,
respectively.

The EFIE obtained is the fundamental integral equation for general microstrip
circuits. It could be applied to specific microstrip devices such as microstrip patch
antennas or microstrip transmission lines. A more broad and detailed analysis of

microstrip lines, based on the above formulation, will be given in the next section.

3.2 Green’s Dyad and Integral Equation for Microstrip Lines

As a specialization of the general microstrip environment, the configuration of
microstrip transmission lines is shown in Figure 3.2, where the conducting strips extend
infinitely along the waveguiding z-axis. Since the Green’s function is z—axis invariant,
as indicated in (2.29) and (2.34), the axial integral with respect to z in the EFIE is
convolutional. Consequently, Fourier transformation on the axial variable 2 is prompted.
As the transformation pair is defined by (2.7), transform domain quantities are denoted

by lower case letters as

26

mime) = 7(5. C). (6)

Subsequent to application of the convolution theorem, EFIE (5) is transformed into the

spectral domain as

‘ .k A
t'fimiI:>".C)-IF(i>".C)dz' = lair-aim) rise (7)
C

C

where C is the boundary contour about the microstrip conductors in the transverse plane
and therefore axially invariant, ii = ix + yy is the 2-D transverse position vector and 5'

is the transform variable corresponding to z.

The transform domain electric Green’s dyad is given by

.. 8
§‘('P' IE'J) = I’VUCa2 + W°)§(3 I5“) + [5(5 " 5’) 0

where 5 = V, + fit with V, the transverse del operator. The transform domain Hertzian

potential Green’s dyad is obtained by

smear) = zléwlficzn. (9)

Again the dyad decomposes into a principal and a reflected part. Corresponding to

equations (2.27), (2.29), (2.33) and (2.34), we have the spectrial domain components as

m Iran = ig'ta lam + m ”5.6) (10)
where

27

 

" mx-x’) -r.lr-r’|
§'(5|6.C) =f‘ 4‘ 4;, (11)
.. We
and
2'05 lair) = (a + mails I 5’4) + 90,545 I 'p".€) + 935.0 I am) (12)
with

efiflf’x)‘ .. Ra0) , _ + ,
8;(?| 7!, c); = f R“(A) gnu-x )e PA, I ) d5 , (13)
g.’.(rl r’. C) "' Cam 4””;-

 

 

where 61- = .t 613:: + ijc; the reflection and transmission coefficients are the same as

those given in (l) and are implied functions of ( through A.
As stated before, the space domain Green’s function satisfies the general

reciprocity theorem, and therefore

6:.(3 I 5’.z-z’) = G;.(5’I 5.z’-z) (1,0 =x,y,z. (14)

Since

8:,(6I5’.c) = «9',{G:p(5|b”.z)}, (15)

a Fourier transform theorem gives

3:.(6 Ira’. C) -= 35.045. -c) (1,0 . am. (16)

This is the reciprocity property of the transform domain Green’s function, and will be

28

used extensively in the following analysis of microstrip line.

3.3 Propagation Spectrum; Excitation and Coupling of Microstrip Lines
The singularity expansion method (SEM) [49] in combination with the EFIE is
utilized to study the propagation spectrum of microstrip lines. Singularities in the

integral representation of Green’s functions lead to similar singularities in the spectral
domain microstrip current 5G5, C). These singularities in the complex {-plane consist of

two types: simple pole singularities which correspond to discrete propagation modes and
square-root branch-point singularities which lead to the radiation field with a continuous
spectrum.

Pole singularities of the current lead to discrete propagation modes. For the sake
of simplicity, we assume a single isolated strip line here in our analysis. For (- near to
a discrete propagation-mode pole eigenvalue 5,, the transform domain current can be

approximated as

.. a‘i‘fii)
k (6") - LL— 1
’ (tag) (a

where 12(6) is the eigenmode current of the corresponding pth discrete mode

propagating in the 1!. direction. It will be demonstrated later that I = 1, so the poles
are simple. Substituting (17) in EFIE (7) results in

4» ~ .. .. .. -,_,, I (kw-u»
t' ‘ ,c 'k )dl ='—t° (sC)r
“19), {Help ) ,(p '7. e p (18)

"56C.

29

Since the impressed field 3‘( f), C) is regular at f= 3F (3,, the integral in the above

equation must vanish at {= 3F I, to provide an indeterminate form. Thus the eigenmode

current satisfies the following homogeneous EFIE

f-jr'o'la’xrfiwidl’ = 0 a 60 (19)

with non—trivial solution only for r= :F 5;. This EFIE consequently defines the discrete
propagation modes and associated propagation constants (eigenvalues).
Inverse Fourier transformation of the spectral current gives the spatial domain

current

£0) = i f Raceway (20)
21: .-

Subsequent to the application of Cauchy’s integral theorem, the real-axis inversion
integral is deformed into the upper (lower) half {-plane for z > 0 (z < 0), respectively.
The integration path in the complex {-plane is shown in Figure 3.3 along with the pole

and branch point singularities. The space domain current then becomes

120:2) = Z aria: 12:0)8’9’ - 5‘; f i.(b'.0e"‘dc - (21)
P c:

The sum of discrete modes in (21) is the residue contributions of pole singularities while
the continuous-spectrum contribution arises from deforming the integration path about
the branch cuts along C} in the upper (lower) half plane. The continuous-spectrum

current is therefore identified as the forced solution to EFIE (7) for points 3' along C5.

30

Expression (21) therefore identifies the complete propagation-mode spectrum.

Having expressed the space domain current in (21), we here establish the pole
order I in (17) and evaluate the amplitude coefficients a; of discrete modes excited by
impressed currents. To facilitate appropriate use of Green’s dyad reciprocity, apply the

following testing operator
fame-(m) (22)
c
to replace the 31-») operator in EFIE (7). This is legitimate since the surface current
{’1 'p') is everywhere tangential to the conducting surface; the resulting equation is thus
equivalent to the original equation (7). Exploiting reciprocity of spectral domain Green’s

function (16) gives

1:
fdl’fdlflb’.C)°§‘(B’lb’.-C)°i:(b‘) - -Jn—‘fd1£',‘(a)-e‘(a,c). (23)
C C r: C

Expanding the Green’s dyad in Taylor’s series about 3F g; and using (17) for the current

in the above, the EFIE becomes

a;

(CtC,c)'

 

[dz T50") fdt{ §‘(b”|p.tC,°) E50)

+_ .-./-._ (24)
a(Nola.

     

(c a c,) + }-E,‘(i>')
3‘,

'k
= J7? ff;(a)oz‘(5.c)dz.

31

Invoking defining EFIE (19) for kfl‘p‘) , the contribution from the first term of the

expansion vanishes. The leading non-vanishing term is consequently the first-order
contribution proportional to ((3: r, , which can annul at most a first-order pole under the
limit {-9313 (3,. It is thus established that I = l and the poles are simple.

If reciprocity of the Green’s dyad is invoked again, and only the leading terms
are retained, coupling coefficients for the excitation amplitude of discrete

propagation-mode current are then obtained as

'k
a; = r ’ ‘. fk:(5)-z‘(5.oa. (25)
”cc? C

 

A convenient choice of normalization constraint for the current is

a ”3.. 6 3.. .T't-o g
c, = {dildl’kfim'azfi(plfi’.C)L'k,(p’) :1. (26)

Coupling coefficient (25) is an overlap integral of the impressed field with the
eigenmode current. If the impressed field is maintained by an impressed currentf(i5,€)
in the cover region, then

aim = -’—£’—‘ f§‘(5|5.0'f'(5’.049’. (27)

C 8

Substituting (27) into (25), and subsequently invoking reciprocity of the Green’s dyad,

results in

32

.k ‘
a; = 4’04 fi‘(a'.C)-e';(a)4s’ (28)

where

25(5) = 3,1 f §'(6Ia’.rc,)-£;(a’)dt’ (29)
c C

is the eigenfield associated with the propagation eigenmodes. Expressing the spectral

domain current f(p’, C) in (28) as the forward transform of spatial current .7‘(i’) , we get

0k ‘
a; ‘ ‘1?ch ‘0") -é‘:(a’)e"“’dv'. (30)
e V

This expresses the excitation amplitude of discrete propagation modes travelling in the
21:2 direction as an overlap integral of impressed current with the corresponding

eigenfields. With this, the space domain discrete-mode current is found, by (21), as

k - -
125(7) = 4&6) [106-250) U[t(z -z')le "r" “at” (31)
"c V

where 00:) is the unit step function. This expression clearly identifies the spatial
eigenmode current that is excited by an impressed current and propagates along :2

direction.

The above analysis also applies to the coupled multi-strip situation. For N
coupled microstrip lines, the discrete system eigenmode currents satisfy the coupled
system of homogeneous EFIE’s

33

N

5.2 [2'0 I5’.o-k.;(b°)dz’ = o (32)

3-1 C.
-- ii €C., m = 1,2, ...,N

with non-trivial solution for g-= T I”. The eigenmodes again correspond to simple pole
singularities in f-plane, and have spectrial properties similar to the isolated case, but are
complicated by the cross coupling effects. In the following chapters, The EFIE’s
developed in this chapter will be solved numerically or analytically and additional

properties of the microstrip transmission line will be studied in detail.

34

 

 

é resonator 2
A r 1

 

 

 

DIelectrIc Film at, u,

 

Conducting Ground

Figure 3.1 Configuration of general open microstrip integrated circuit.

35

 

 

y
Sm cover (cc, pc) Sn
I\\\\\\\\\\\l z l—A x |\\\\\\\\\\\]

film (of, pf)

Y = 0

 

 

y"’///////////cm®am ]////7/////

Figure 3.2 Configuration of general microstrip transmission line structures.

36

‘Imm

 

 

Im { § b} =0
poles
(9Q
-ke
D
kc
® (9
poles

 

RelC]

C;

Figure 3.3 Singularities and integration paths in the complex f—plane.

~37

CHAPTER FOUR

ANALYSIS OF A SINGLE MICROSTRIP TRANSMISSION LINE

Having established the fundamental integral equation theory for microstrip
transmission lines, we are in a position to actually solve those equations, numerically or
approximately, to analyze them and quantify practical parameters such as propagation
constant and characteristic impedance. In this chapter, numerical solution of the EFIE
for a single microstrip line is implemented by Galerkin’s method of moments with
Chebyshev polynomial basis functions. Principal and higher-order propagation modes

are studied. Results of dispersion characteristics and current distributions are presented.

4.1 Formulation of the Problem

As a specialization of the general configuration of microstrip circuits, the structure
of a single microstrip line is shown in Figure 4.1. For our consideration, the strip is
assumed perfectly conducting, with infinitesimal thickness and a width of 2w. In
general, the film may be a lossy dielectric/ magnetic material; the cover is air.

From the discussion in chapter 3, the eigenmodes propagating along the microstrip
line satisfy the homogeneous integral equation (3.19), which, for the present case,
reduces to

ff; i- £150) Ix’.y’=o.c)-ic;(x’)4x’ = o (1)

for -wsxsw

where the Green’s dyad is given by expressions (3.8) to (3.12). Exchange of the limit
38

y -» 0 with the integration over x' is permissible under the condition that the integral
remains convergent. When the source point and the observation point coincide, special
consideration may be needed. Convergence of the scalar integral components will be
examined later.

The above integral equation defines an eigenvalue problem. Solutions of (1) give
the eigenvalues (propagation constants) and eigencurrents of corresponding discrete
eigenmodes (propagation modes). For a single discrete mode, the spectral domain
current is given by (2.17). Inverse transformation of that result gives the space-domain

eigenmode current as

1?;(7) = E;(x)e”‘r‘ (2)

Expression (2) clearly demonstrates that this is a wave propagating in 1:2 direction along
the microstrip line with propagation constant 1;, and current distribution {’(x) . The
electromagnetic fields excited by the eigenmode current has the same 2 dependence

through the common phase factor exp( 3F j (z).

The surface current on the infinitely thin strip has only tangential components

P0) = i k’(x) :t i kz(x) . (3)

The current distributions for the 1:2 directed waves are essentially the same. In later
discussions, we assume the case for +z traveling waves only. Due to the symmetry of
the problem, there are two kinds of modes for the microstrip, even modes and odd

modes, defined according to the symmetry of the 1: component current as

39

k,( -x) = - lcx(x), Icz( -x) = kz(x) even made, (4)
k‘( -x) = kx(x), k,( -x) - - k,(x) odd mode.

Integral equation (1) decomposes into two scalar equations after current (3) is

substituted into it. After intensive algebra, we have

{g;.(x.y Ix’.0;C)k,(x’) + agony (neuronal = o

i":

 

 

w (5)
f {8;(x.y Ix’.0;C)k,(x’) + g;(x.y Ix’.0;C)k,(x’)}dx’ = 0
---for —wsxsw;y~0.
The scalar components of the Green’s function are given as
r I = i . Jw-x’) w
g..(x.y Ix .0) 2” [exam e d: (6)
a, B = x’z
where the coefficients are function of f and 3' as
2 2 _ 2 2 1 _ 2
0,02) = WWI“ 1” p‘ + ”‘0‘“ 5’. (7)
Z I'(C)Z ‘(E) Z "(C)
(Mimi-map, _ Mic: (8)

 

Cg 5.0 = Cg(E.C) = .
< Z “(5)2 ‘(0 Z “(5)

40

 

 

2 2 2 2
(Nfiufi-mz c + 111,0, —c’). (9)

9,0,0 = , h
z (020) z (r)

Exchanging the spatial integration with the spectral one in (5) results in the following

equivalent equations

f area” f (Cn(E)kx(x’) + Cn(5)kz(x’))e"""°dx’ = o

" “' 10
I die?" I (Cu(£)kx(x’) + Cu(£)kz(x’))e"°‘""dx’ = 0 ( )

---for -wsxsw; y-oO.

Equations (10) describe the behavior of the single microstrip line. A moment method

solution of them is pursued in the next section.

4.2 Method of Moments Solution

The integral equation (1) is solved by Galerkin’s method of moments combined
with entire-domain basis functions [50]. The surface current, as well as the fields on the
strip, has the well known edge singularities. Entire-domain basis functions are preferred
since the edge singularities can be explicitly accommodated by including proper edge
condition factors, and the current is represented in a compact form with a relatively small
number of unknown expansion coefficients [51]. In our solution, Chebyshev polynomials
with square-root edge factors are utilized as basis functions. The transverse and

longitudinal current components are then expanded in Chebyshev series as

41

N N
19.0) = 2 a..e..(x) k,(x) = 2: a..e,(x) (11)
“'0 n-O

with
e,(x) = T.(x/W)I/1 — (xIW)2
e,(x) = T.(xIW) I J1 — (rt/w)2

-w s x s w (12)

where 1;(x/w) is a Chebyshev polynomial of order n of the first kind and a, and an are

unknown expansion coefficients. Exploiting the current expansions in (10) gives

" N N
f drive“; {onto {3345(5) + 0.0) agar} = o

' N u (13)
f dEe'mgib {Cum 2513(5) + 02(5) anu(E)} = o
for-wsxsw; y-oO
where
4.“) = f e,,(x’)e"“’dx’ p = 1,1. (14)

'19

Following Galerkin’s method, the same basis functions are utilized as testing

functions. Define testing operator

Id; e..(x){ } a ..-. x, z

42

then applying to (13), we have

,, .
1:323 fdte""e..(t)Ia..C,(t)f..(E) + a,C,(t)f,.(t)) = o

(15)
N . -
13323 [are "’g,.(E)(a..C,(£)f,.(E) +a,.C,,(E)f,.(t)l = o
where
8...“) = fe"(x)e"’dx a = x,z. (15)
Let
4:; - ling f e""C.,(E)e...(C)f,,(E)dC «.9 = 1.2. (17)
I" ..
then equations (15) become
N
2 (43% + .45'au) = o m = 0,1,---,N,
u-o (18)
N
2: (Area. + Agra”) = 0 m = 0,1,---,N.

The above is a set of linear equations for unknown expansion coefficients am. It can be

written in a convenient matrix form as

43

 

 

A" a
a R IN a 0 m," = 0,1,...’N. (19)
A: A: “so
am.

Here A is a 2(N + 1)x2(N +1) matrix. To obtain a nontrivial solution of the
homogeneous matrix equation, the determinant of A muSh vanish. Since the elements
of A are functions of 1’, this requirement gives the propagation eigenvalue 3;. The
corresponding expansion coefficients are then evaluated and the eigenmode currents are
evaluated from (11) in a quasi-closed form Chebyshev polynomial series.

One advantage of using Chebyshev polynomials as basis functions is that the
spatial integrals in (14) and (16) can be evaluated analytically in closed form. Noticing
the fact that Chebyshev polynomials of even order are even functions and odd order are
odd functions, the spatial integrals reduce to the following four generic types and are

evaluated as [52]

" Thu/w)

0 J1 - (xlw)2

 

cos(€x)dx = (-1)'-"5”-'J,,(tw) (20)

" T...,(x/w)

o ([1 - (x/w)2

 

sin(Ex)dx .. (-l)" EEC-Imam (21)

f r2,(x/w),/1 - (x/w)2 cos(€x)dx
0

(22)
=(-1)'-"7“’- 1,.(tw) + %Jz,..,,(tw) + fi-J,,-,.(ew)
f T2..,(x/w)I/1 - (xIW)’sin(Ex)dx
0
(23)

= (-1>'14‘3[I..a<£w> + 514mm) + glance

where J,(x) is the Bessel function of first kind. As discussed before, propagation modes
for a single microstrip are either even or odd as defined in (4). Therefore, in the current
expansion for the even modes, odd order Chebyshev polynomials should be used in the
x-component current expansion and even order ones in z—component expansion; vice versa
for odd modes.

The matrix elements given by (17) are inverse Fourier transform integrals
involving coefficient C048) and Bessel functions. For bound propagation modes with or
without material loss but no leakage or radiation involved, the real axis is taken as the
inverse integration path in the complex f-plane. The surface wave poles produced by
2(5) and 2"(9 in Ca,(£) in that case are on or near the imaginary axis in the proper half
plane, and are not implicated in the integrals. Coefficients 0,5(8), and therefore the
integrands, are well behaved functions. By examining the asymptotic behavior of the
integrands for £ approaching infinity, it can be shown that all the integrals converge

regardless the value of the y factor. Thus interchanging the limit y - 0 with the

45

integration is justified and the value of y is set to zero in all the integrands. Integrals

(17) are evaluated numerically.

4.3 Computational Results

Numerical implementation of the Chebyshev Galerkin’s MoM solution is
developed in Fortran programs. Results are obtained for the typical configuration shown
in Figure 4.1 with various relevant parameters. Dispersion characteristics, current
distributions, the effects of material losses and other properties are discussed and
compared with results found in prior research publications.

Considering a microstrip line with parameters 2w = 3.0 mm, t = 0.635 mm,
c, = 9.8, and p, = 1.0, Figure 4.2 shows the dispersion curves for the principal mode
and the first two higher-order modes. The results are in good agreement with published
ones. The dashed line is the dispersion curve of the TM, background surfacewave
mode, that is the solution of 2'0) = 0 for h = hr For bound modes, the propagation
constant satisfies relation A, s g- s k}, and x, provides a cutoff condition with
corresponding cutoff frequency. As shown, the principal EH0 mode has no cutoff, but
the higher-order modes do. When the frequency becomes lower than cutoff for a higher-
order mode, it becomes a leaky mode, and background surface wave and/or space
radiation are excited. The leakage and radiation properties, as well as the transition
between bound and leaky modes, will be studied in detail in a later chapter.

Figure 4.3 shows the current distributions of the principal E11,, mode for the same
microstrip line operating at two different frequencies. Solid lines and dashed lines show

the normalized longitudinal and transverse currents on the strip, respectively. The

46

principal mode is an even mode. The magnitude of transverse current is quite small
compared with that of the longitudinal one. It increases significantly with increasing
frequency, while the change in longitudinal current is relatively small as a function of
frequency. There is a 1r/2 phase difference between those two current components. The
currents here were obtained in a four term Chebyshev polynomial series expansion.

Figures 4.4 and 4.5 show the current distributions of the first higher-order EH,
and the second higher-order EH2 modes, respectively, for two different frequencies.
Unlike the principal mode, the magnitude of the transverse current decreases as the
operating frequency is increased. For higher-order modes, the relative amplitude of
transverse components are significant compared with the longitudinal ones. These results
were also obtained using a four-term Chebyshev series.

Table 4.1 gives the propagation constants of the first three modes at 30 GHz when
different numbers of Chebyshev basis functions are used in the current expansions.
Table 4.2 lists the corresponding current expansion coefficients for the first two modes
when four terms are used in the Chebyshev series. It is clear that the propagation
constants converge quickly for the principal mode as well as for higher-order modes.
From the numerical results, we see that, for this case considered, three or four terms are
sufficient to accurately represent the currents of the first several modes and give
convergent eigenvalues. The currents, obtained in compact, quasi-closed-form
Chebyshev series representation, are very convenient for later use in analytic and

numerical applications.

47

w= 1.500“ 8 =9.8
t=0.635lun uf-1.0

 

(-—r+-—)

Figure 4.1 Configuration of a single microstrip transmission line.

48

 

 

 

 

 

 

 

j E k, I
b 3.0 E 53.0
- f .—
*’ 2 EH0 Z
a : I-
‘6’: - i
a : =
o 2.0 1 :20
o I C
or .. —
cl - -
£1 - -
0.. : I
8 1.0 l ——————— :10
N g I
ifi : cutoff :
5 : :
h
o - ..
Z 1 I
0.0 I I I I I I I I I l I I I I I I 1 I I I I I I I I I I I I I I I I I I I I I I 0.0
0.0 10.0 20.0 30.0 40.0

Frequency (GI-I2)
Figure 4.2 Dispersion characteristics of the principal mode and the first two higher-

order modes for the microstrip line.

49

 

 

 

 

 

 

AI 1.00 5 0.03 .3?
o E 1: f=1OGHz I,” 2 ‘\ .s
5 5 2 f=4OGHz / \ g
‘- 0.80 1 ,’ \
g : ’ \t 5
U I ’ U
- / \
«a E z t 0.02 0
a 0.60 T: I, “ a
g I II ‘ g
a 2 I \
so a x i
=1 0.40 -_ , g;
.3 - , .
: r ‘ 0.01 re
'd ' ’ 9 ‘ o
o I — 1 N
N Z I
:5 0 20 -_ 4 , ...... ~. 3
a E I, ,IT’I ‘\\ E
‘4 -' I av, \ o
O I I ’ ” \ z
z 0.00 :ILI’ I r l l I l I 1 I l I Ifi I I r I l l I fl 1 l 0.00
0.0 0.2 0.4 0.6 0.8 1.0

Position x/w
Figure 4.3 Current distributions of the principal EH0 mode for the microstrip line

operating at two different frequencies.

50

 

p.
O
0
9
IF
0

.0
on
o
I
l
I
I
I
I
c
on
o

.0
Q
0
I
I

 

.0
I:
o

.0
N
o

 

~

Normalized longitudinal Current k.
I
/
. .° .
to
0
Normalized Transverse Current in,

 

.°
0
o

lllllrl]1llllilllllr‘l‘IIoooo
0.0 0.2 0.4 0.6 0.8 1.0
Position x/Iv

Figure 4.4 Current distributions of the first higher-order EH, mode for the microstrip

line operating at two different frequencies.

51

E"
o
o

 

.°
4:-
o

P
to
0

Normalized Longitudinal Current k.

.0
o
o

-R

 

.°
0

11111!gjllLlljlllllLJllllLllllll[11111111111111]
\

   

 

1 f=3OGHz
2: i=4OGHz
0.80 1
, .. - _ - 0.30
/ I l \
0.60 x x
I
,’ 2 0.20

 

 

I‘lllllillll

0.2

0.4

0.40

Normalized Transverse Current k,

[IWIIIIIII]0.00

0.6

0.8

1.0

Position x/w
Figure 4.5 Current distributions of the second higher-order EH2 mode for the microstrip

line operating at two different frequencies.

52

functions used in the current expansion.

 

Table 4.1 Convergence of the propagation eigenvalues upon the numbers of basis

 

 

 

 

 

 

 

 

 

 

 

of the first two modes.

 

 

1 fl
Number of EH0 node EH. mode EH, mode
terms N

1 3.247828 2.526470 0.531384

2 3.026762 2.577995 1.410514

3 3.026577 2.578243 1.399051

" 4 3.026571 2.578239 1.398955
ll 5 3.026583 2.578244 1.398961

Table 4.2 Expansion coefficients of the Chebyshev polynomial series for the currents

 

 

 

 

 

 

 

 

 

53

 

n EH0 an EH0 a, EH, an EH, a.
1 0.457x1o2 0.174x103 -o.421x103 0.763x103
2 0.867x10° -0.731x102 0.204x1o3 -o.232x103
3 -o.354x1o° 0.503x10‘ -o.931x10l 0.212x102
4 -0.557x10° 0.100x10‘ 0.144x10° 0.100x10‘

 

CHAPTER FIVE

ANALYSIS OF COUPLED MICROSTRIP TRANSMISSION LINES

Coupled microstrip transmission lines are a configuration of considerable practical
interest due to their extensive use in microwave and millimeter-wave integrated-circuit
components and devices, such as directional couplers. At high frequencies, accurate
dispersion and coupling characteristics are essential to the analysis of wave propagation
along the lines. But this is a difficult problem due to the complicated geometry of the
structure and the interactions among adjacent lines and different modes. Based on the
integral equation formulation, we study the coupled microstrip lines in this chapter by
an approach similar to that used for a single line. Rigorous full-wave solutions to the
coupled integral equations are pursued again through the Galerkin’s method of moments.
The dependence of coupling effects upon line spacing, the behavior of coupled mode
system, and dispersion characteristics and current distributions of the coupled modes are

analyzed. Numerical implementations are developed and results are obtained.

5.1 Formulation for Coupled Microstrip Lines
The general configuration of N coupled microstrip transmission lines is depicted
in Figure 5.1. From the discussion in chapter 2, the discrete system eigenmodes
propagating along the lines satisfy the following coupled integral equations
A N at
as}: [m I a';0-k.,<aodz’ = 0
III] e. (1)
.. .9. E C.’ m = l,2,...,N

54

with non-trivial solution only for eigenvalue r = 3F i}. 5;,(5) is the current distribution

of the pth propagation mode on the nth strip and 3F g; is the associated propagation

constant. The space-domain eigenmode currents are then

1?;(5’2) = £11165) e ”('29 n = 19 29 "'9 N- (2)

This is a current wave propagating in i2 direction along the nth strip. Since the current
distributions for :2 directed waves are essentially same, we restrict our discussion to the
+2 case.

Consider the structure of two identical coupled microstrip lines as shown in
Figure 5.2. The strips (1 and 2) are assumed perfectly conducting and infinitely thin,

with width of 2w, and separated from center to center by 2b. For this case, (1) reduces

to
~(b-w) 1””
limé' f £‘(x.y|x’.0;C)-Ez(x’)dx’ + f§‘(x.ylx’.0;C)-Fl(x’)dx’ = 0
"0 -(b+w) b—w
for -(b+w) xx 5 -(b-w),
(3)
-(b-w) bw -.
limfl' f §‘(x.ylx’.0;l)'k;(x’)dX’ + f§‘(x.y|x’.0;0°k1(x’)dX’ = 0
1‘0 '(WW) b-w
ufor b-w s x s b+w
where

55

1210:) = sicko» + £ku(x)

Ea) = £15.00 + 215,00

(4)

are the surface currents on strips 1 and 2, respectively. Due to the symmetry of the
problem, the two currents have the following relations, and two kinds of system modes,
symmetric and antisymmetric, are defined according to the symmetry between the 2-

components on the two strips as

15.0!) = - 1x( -x), €11) = ku(x) symmetric mode
k7,(x) = ku( -x), that) = -ku(x) antisymmetric mode.

(5)

With the current relation in (5), only one of the equations in (3) is independent. We
solve the second one for the current on strip 1. Using (5) and changing variables, the

first integral in the equation is modified as

45-")
f £‘(x.y Ix’.0. 0125,06 + 215,05]de
-(b+w)
b (6)
= f §‘(x.yl-x’.(nC)°[=F£k,,(x’)tzkuu’ndx’
b-w

where the upper and lower :1: sign corresponds to symmetric and antisymmetric modes,

respectively. With this, we obtain the inteng equation for current 21(1) as

56

bflv

lim 5" f §'(x,y| -x’,0;C) -[z£kk(x’) t £ku(x’)]dx’
r0

b-III

(7)
bflv
+ f §‘(x,ylx’,O;C)-[£kh(x’)+£ku(x’)]dx’ =o

b-w

for b-w s x s b+w

01'

MW

1:13? ° f {[§'(x.y|x'.0.0 : 2‘tx.yI-x'.0.0]-xk,,(x’)
b-w '

(8)
+ [§‘(x.ylx’.0;€) : mm -x',o;c)]-2I,,(xo}dxl = 0

for b-w s x s b+w.

This equation decomposes into two scalar equations after the vector operations are carried

out. Omitting the algebra, we have

hm
f {[gflm lx’.0) a: g;(x.yl -x’,0)]kk(x’) +
b-w

[8.3m Ix’.0) 1g;(x,y|-x’,0)]k,,(x')}dx’ = o

btw (9)
f {[8500 Ix’.°> . 3.3x.” -x'.0)]k1.<x') +
b-Iv

[8.1m Ix’.0) : g;(x.y | -x',o)]k,,(x’)}dxl = o

for b-wsxsb+w; y-°0.

57

The scalar components of the Green’s dyad are obtained from (4.6) as

. I = _JL_.. Itcxsxi '1»,
8.,(x.yltx.0) anC..<£>e 6 d5 (10)

“’B = x’z

where the coefficients are given by (2.7), (2.8) and (2.9). Exchanging the spatial

integral with the spectral one after (10) is substituted into (9) leads to

bow

f are?" f {Cn(E)kk(x’)[e""'=re1‘”] + CR(E)ku(x’)[e""’¢em’]}dx’ = 0
- b-w

(11)
. bw
f dEe"" f {Ca(E)kn(x’)[e"“lxe""] + Cu(£)kn(x’)[e"“':e""]}dx’ = 0
-. b-Iv

forb-wsstHw; y-oO.

These equations completely describe the behavior of the coupled system. However, the
spacing parameter b is not explicitly in the integrand functions, but implied in the
integration limits. To get a more convenient form, we shift the origin of coordinates
from the center of the system to the center of strip 1 (local coordinates). This coordinate

shifting actually corresponds to the following variable change

x’-ox’+b, x~x+b. (12)

With the variable changes, after some manipulation, (l l) is rendered into (in the local

coordinates)

58

fdte""ei‘<"‘> f {-jcncokucosinax’w) + c,<:)k,,(x'>cost(x’+b)}dx’ = o

(13)
dee""e""“” f {-jc,<t>k,,<x'>sint(x’+b) + c,<£)k,,(x0cost<x’+b>}dx' = o

for -wsxsw; y~0

for the symmetric modes, and

dee”"e’“‘“’ f {Cn(E)ku(x’)cosE(x’+b) - jCz(£)ku(x’)sin£(x’+b)}dx’ = 0
.. ,, (l4)
fd:a""a“‘”" f {C,<£)k,,<x')cos£<x’+b) - jC,,(£)k,,(x’)sinE(x’+b)}dx’ = o

for -wsxsw; y-o0

for antisymmetric modes. Equations (13) and (14) are ready to be solved numerically.

5.2 Method of Moments Solution
Following the same approach used for the single microstrip, equations (13) and
(14) are solved by Galerkin’s method of moments. Chebyshev polynomials with square-

root edge factors are utilized as basis functions, and the transverse and longitudinal

currents are expanded as

59

N
k1,,(x) = 2 a... T.(x/w) 1— (ac/w)2
II-O (15)

N
knot) - E 0,, T.(x/W)/\/1 -(1cIW)2
n-O

where 7;(x/w) is a Chebyshev polynomial of order n of the first kind and a, and an are
unknown expansion coefficients.

According to Galerkin’s method, the same basis functions are utilized to test the
EFIE after the above current expansions are exploited in (13) and (14). Omitting the

mathematical details, the EFIE is finally rendered into the following matrix equation:

= o m, n g 0,1,-~-,N (15)

 

 

where the matrix elements are given by

Ar; = um [ e""C.,(0u..(0v,,(0dc «.0 = m (17)

r0 -.

with the spatial integrals defined as

uel(() ‘ fa..(xl)ej£(x’tb)dxl a = 1:1 (18)

W

v..(0 = f—ie,.(x’)sinc<x’+b)dx’

v,,<0 = f eu(x’)cosC(x’+b)dx’

”W

for symmetric mode

(19)

v..(o = f encxocosaxhwdx’
"" for antisymmetric mode.

v,.(c> = f—je,(x’)sin«x’+b)dx’

'49

To obtain a nontrivial solution of the matrix equation, the determinant of A must
vanish. Solution of det(A) = 0 gives the propagation eigenvalue 5, and the
corresponding expansion coefficients give the eigenmode currents as compact Chebyshev
polynomial series.

Due to the cross coupling, the current distributions on the strips are neither even
nor odd about the center of the strips. Therefore, in the current expansions (15), for any
modes, both even and odd order polynomials are needed, and more terms are required
in the series to accommodate the complicated current distributions caused by coupling
effects. The spatial integrals in (18) and (19) can still be evaluated analytically in closed
form. Even though it is cumbersome, these integrals can always be decomposed into a
combination of the four generic types given and evaluated in (2.20) - (2.23), and
therefore be calculated accordingly.

The matrix elements given by (17) are inverse Fourier transform integrals similar
to those for a single strip. All the discussion about integration in the complex £-plane

in section 4.2 applies here. Again, examination of the asymptotic behavior of the

61

integrands permits interchange of the limit y - 0 with the integration in (17). However,
the coupling effects introduce some fast oscillating factors into the integrands and make
the inverse integrals numerically less stable, requiring more computational effort.
Moreover, relatively large numbers of basis functions make the A matrix become even
larger, and thus very computer-time consuming to evaluate and iterate its determinant for
3),. Therefore, approximate methods are desired. In later chapter, a full-wave

perturbation theory based on the same EFIE’s will be developed.

5.3 Computational Results

Numerical results are obtained for the Chebyshev Galerkin’s MoM solution for
the structure shown in Figure 5.2, with parameters 2w = 3.0 mm, t = 0.635 mm, c,
=9.8, and It: = 1.0. Figure 5.3 shows the dependence of the normalized propagation
phase constant g- for the EH0 system mode of the two identical coupled microstrip lines
upon the line spacing. The operating frequency is 10 GHz. The results show that when
the two strips are wildly separated, the propagation eigenvalues become those of the
corresponding isolated line limit. As the two strips become closer, the coupled system
modes, symmetric and antisymmetric, emerge and separate further from the isolated
mode. The propagation constant of the symmetric mode increases while that of
antisymmetric mode decreases as the coupling becomes stronger. When the two strips
contact each other at the edges, the symmetric mode reverts to the EH0 mode of an
isolated strip with double width, and the antisymmetric mode approaches the EH, mode
of the same double-width strip. The latter mode, however, might be cut off for certain

configurations and frequencies, and the antisymmetric mode therefore may become a

62

leaky mode rather than a bound mode in such cases. The properties of leaky modes have
been studied and will be discussed in a later chapter.

Figures 5.4 and 5.5 show the dispersion curves of the EH0 mode for the same two
coupled microstrip lines with two different separations, respectively, along with that for
the corresponding isolated strip. In Figure 5.4, the separation is equal to the strip width.
This is a loosely coupled system; the changes of propagation constants are relatively
small for both the symmetric and antisymmetric modes over the entire frequency range.
Also notice that the shifts of propagation constants from the isolated one for symmetric
and antisymmetric modes are almost symmetric (equal but opposite). A more strongly
coupled system is shown in Figure 5.4 where the separation is 30% of the strip width.
Similar behavior is observed. From both cases, it is shown that for fixed separation the
coupling effect becomes smaller as the frequency becomes higher, since the separation
normalized to wavelength (bl M, or the relative separation, becomes larger.

Figures 5.6 to 5 .8 show the normalized longitudinal and transverse current
distributions of the EH0 system modes on strip 1, respectively, for three different line
separations at 10 GHz. The isolated profiles are also included for comparison. It can
be seen that, due to the repulsion of the surface charge, the current for symmetric modes
becomes smaller near the inside edge but larger at the outside edge. For antisymmetric
modes, however, the current behaves in the opposite manner due to the attraction of
surface charge. The variations in current due to the coupling are relatively larger for the
transverse component than for the longitudinal one. For the calculation of currents on
this coupled system, six or more basis functions are required for accurate and convergent

results of the currents as well as the eigenvalues. It is time consuming computationally.

63

 

Conducting Ground

Figure 5.1 General configuration of N coupled microstrip transmission lines.

e—n—r

 

Figure 5.2 Configuration of two identical coupled microstrip transmission lines.

65

 

    
   
  

 

 

 

 

{ 2
”sm-
‘S: : EH, symmetric
2 3 EH, isolated
3 2.90 —
a, .
m ..
g .
o. 2.80 - EH, antisymmetric
U 1
0
.t‘ "
'6 _; r,=10 GHz
E 2‘70 w=1.500 mm
‘7 t=0. 6.35 mm
2
2.60 lllllIllllleIllIIIIIjIlllITUrIIIIIIFIII
0.00 1.00 2.00 3.00 4.00

Distance Between Coupled Strips (b-w)/w
Figure 5.3 Dependence of propagation phase constant 3' upon line spacing for the EH,

system modes of two coupled microstrip lines.

S"
o

 

 

 

 

s I _ =

b _ (b w)/w 1.0
3.00 - l,

"E _ ,/

o q

.5; .

S J

0 2.90 . 1

3 1 o

E ..

a. 2.80 4

U I

0)

.2 u

E 2.70 - 0: isolated EH, mode

3 - l: coupled symmetric mode

2 j 2: coupled antrsymmetric mode
2.60-firIITTIIITIIITIIIIIIIIIIIIIllirililllr

0.0 10.0 20.0 30.0 40.0

Frequency (GHz)
Figure 5 .4 Dispersion characteristics of the EH, system modes for two microstrip lines

with a separation equal to the strip width, compared with the isolated profile.

67

 

(b—w)/w=0.3

    
     

N
co
0

2.80

2.70 0: isolated EH, mode
1: coupled symmetric mode

2: coupled antisymmetric mode

Normalized Phase Constant {/ko

lLllIlllJlllllllllllJJJ

 

 

 

2.60 [IlllIIIIIIITIIllleIlTllllIr1lllTrllIl
0.0 10.0 20.0 30.0 40.0
Frequency (GHz)
Figure 5.5 Dispersion characteristics of the EH, system modes for two coupled

microstrip lines with a separation equal to 30% of the strip width.

68

1 .00

 

l I ! !

isolated mode
..... symmetric mode
.. _ _ antisymmetric mode

 

.0
\l
or

 

 

Normalized Current kz
o
01
o

.0
M
(It

 

IllllLlllJlllll

 

 

 

0000 Tlljlrjljlill1IIIIIIWIIIIIIIIIIIIIIITT

-1 .00 -0.50 0.00 0.50 1 .00
. Position (x-b)/w

Figure 5.6(a) Longitudinal current distributions of the EH, system modes for two

coupled microstrip lines with a separation equal to the strip width.

69

Normalized Current k,

 

0.02

.0
9

411111111

.°
0
o

 

I
.0
9.
11111111

  

isolated mode
..... symmetric mode
.. _ _ antisymmetric mode

 

 

 

-0.02 IIIIIIIII

—1.00

-0.50

IIITIFIT1I1'IIIjTIIIII'UUTIIII

 

0.00 0.50 1 .00
Position (x—b)/w

Figure 5.6(b) Transverse current distributions of the EH, system modes for two coupled

microstrip lines with a separation equal to the strip width.

70

1 .00

 

isolated mode

 

.0
\l
U"

..... symmetric mode I
- _ _ antisymmetric mode

‘—

 

 

.0
M
(II

 

 

Normalized Current kz
0
0|
0
11111111141111

 

 

 

0.00 IUIIIIIIIIIIIIIIIUIIIIIUIIllIllFllelll

-1 .00 -0.50 0.00 0.50 1.00
Position (x-b)/w

Figure 5.7(a) Longitudinal current distributions of the EH, system modes for two

coupled microstrip lines with a separation equal to 30% of the strip width.

71

0.02

 

llJlllll

 

 

 

 

 

 

:3 0.01
4.1
8
t
3
o c-
.0 0.00
O -
J! -
a -
-g _
z -0.01 - \ /
‘ \ / / isolated mode
" \ / ..... symmetric mode
3 \ \ fl , z ’ ._ _ _ antisymmetric made
-0002 IUIIIIIIIIIIIIrUFfiIIIIIIIUITIIU'TUIIII
-1.00 -0.50 0.00 0.50 1.00

Position (x-b)/w
Figure 5.7(b) Transverse current distributions of the EH, system modes for two coupled

microstrip lines with a separation equal to 30% of the strip width.

72

1 .00

 

gag.

 

 

 

 

 

 

 

 

isolated .mode
£0.75 2 ..... symmetnc mode
a q - _ _ antisymmetric mode
C
32’ I
D
o -I
o 0.50 -
.8 ‘
5 I
g - .
z 0.25 - 1 _ _ ,
2 I
’ /
1 ................
0.00 IllIlllllllfllllllllllllllIIIIIIIIIIIII
—1 .00 -0.50 0.00 0.50 1.00

Position (x-b)/w
Figure 5.8(a) Longitudinal current distributions of the EH, system modes for two

coupled microstrip lines with a separation equal to 2% of the strip width.

73

0.02

 

.0
0
111111111

 

 

Normalized Current k,‘
c
o
o

 

 

 

 

I
-0.01 - ..
- \ I .. ’ ’ isolated mode
‘ \ \ _ _, 5 ’ ..... symmetric mode
‘ .. _ _ antisymmetric made
-0002 IleII—UTTTIIFIIIIIIIITjTTIIIIIIIUIltllI
-1.00 -0.50 0.00 0.50 1.00

Position (x-b)/w
Figure 5.8(b) Transverse current distributions of the EH, system modes for two coupled

microstrip lines with a separation equal to 2% of the strip width.

74

CHAPTER SIX
EFIE-BASED FULL-WAVE PERTURBATION THEORY FOR COUPLED

MICROSTRIP TRANSMISSION LINES

Coupled microstrip lines have been studied in Chapter 5 by a rigorous integral
equation analysis. As shown there, direct solution of the coupled EFIE’s is very
involved both analytically and numerically, especially for coupled multi-strip systems and
those with non-identical configurations. In this chapter, a full wave perturbation theory
for the system of N coupled microstrip lines is developed based on the full-wave EFIE
formulation [44]. In solving the coupled EFIE’s, the eigenmode currents of the isolated
line are used as a first order approximation for nearly degenerate eigenmode currents of
the loosely coupled system, and a simple perturbation matrix eigenvalue equation is
obtained. This perturbation method is easy to apply to nearly degenerate multistrip
coupling, and greatly simplifies the computational procedure. Rather than being merely
a mathematical approximation, the perturbation theory provides good physical insight
regarding the coupling effects. A physical justification of the perturbation approximation
is based on the strip current distributions. Numerical results obtained by the perturbation
theory are compared with those obtained by method of moments solutions of the coupled
EFIE’s. To verify the perturbation theory, an experimental implementation of two
coupled microstrips is set up in a PC-board configuration, and the electric field
distributions along the strips are measured. The theoretical and experimental results are

in good agreement.

75

6.1 Integral-Operator Based Perturbation Theory
As indicated in section 5.1, the discrete system propagation modes supported by
N coupled microstrip lines satisfy the homogeneous integral equations (5.1), rewritten

here for convenience as

N
f... 23 f are Ii .0 ~k,f,(i5’)dz’ = o (I)
III C.
" 60., m = 1,2, ...,N.
Applying the testing operator

mm
C-

to replace the i, -( ...) operator in the above EFIE’s leads to the following modified

integral equations:

(2)
* .56., m = 1.2, N

Here 2;“)(5) is the current of the pth propagation made along the mth microstrip with

propagation constant 5(9),; it satisfies the isolated EFIE

5,. f§‘(filfi’.tig)°k3°’(5’)dl’ = o. i5 eC.- (3)
CI

76

Nontrivial solutions of (2) at f = $ 3', give the coupled system propagation eigenvalues
and eigenmode currents.
Invoking the reciprocity property of the transform domain Green’s dyad (3.16)

in equation (2) gives

N
2: fdl'ic‘;(a’> ~ f §‘(b”|6.-C)-E;‘°’(fi)dl = o.
..I c. ,_ (4)

:5 5C“, In =1,2,...,N.

In the case of loose, nearly degenerate coupling, the isolated eigenvalues of different
linesarenearly equal, or (2, I (:2) for m, n = 1, 2, m, N. The coupled system-mode
eigenvalue is not greatly different from the isolated (unperturbated) ones, in other words,
the cross coupling effect is only a perturbation to the uncoupled one. A Taylor’s series

expansion of the Green’s dyad about C = I: (2) is therefore suggested in (4) and results

N
Ira-o. c-oI-o. (0) +1 c-oI-._
f2 idllgp’) (II-{£01 Ip.=tC.,) acflp In. C)“:

(5)

-(c r I?) + }-£;§°’(5)d1 = o. m = 1.2. N

Referring to definition (3) for 2:9(8) , the leading terms in the above equations vanish

for m = n. Dropping second-order small quantities and retaining only the leading

nonvanishing terms, the above equations become

77

[dram cf: —§‘<a ’Ia;

     

“:(c r (2;) {imam

(6)

N
+ 23 fdt’”;‘°’<a'> f it"(a’lpucfi’) 3;“)(6)dl= 0,
c, c.
m = 1’ 2’...’N.

Now we assume that for the case of loose, nearly-degenerate coupling, the current
distributions of the coupled system modes are approximately weighted replicas of the

corresponding isolated currents; that is

5,105) - a:F;‘°’<is) n = 1.2. ...,~ (7)

where the a, are unknown amplitude coefficients. This is the basic perturbation

approximation. Substituting approximation (7) into (6) leads to

etfdt'fimta'r j: —§‘(B’IB;-C) (C : C2)'F;(°’(6)dl

*lr

 

N ,. -, (8)
2 0:14” 31% ”9'1?" w’lfistc‘S’I-kf’cnduo.

m = 19 129 "'9 N0

Invoking the reciprocity of g ' again, we finally obtain the following coupled-mode
perturbation equation

78

am r (‘33) f d1 f dz’if’m .31“, mm ~i;‘°’(a’)
c. c. 3‘ «2

N * I"*(°)-o .. .. ~I (0) “am-0 (9)

+ E 0..dede k.., (0)100)II>;¥C.,,)-k.,P (p’) = 0.

”I- c_ c,
.. B e c_, m = 1.2, N
Let
C..*... = f d! f dt’Ez,‘°’(aI-61§'<alr’;0 ~E:,‘°’(i>") (10)
c. c. C .t:

0:. = CI d! J dt’f;‘°’(r)-r'(3mercfi’rif’cs’) (11)

which are normalization and coupling coefficients, respectively. Equation (9) then

assumes the following convenient form as

c;,,(c r (52,»; + 2 C,:,,a: = o, m -- 1,2, N

.‘l

which can be expressed in a matrix form as

cm e 0?. Ci, a:
. (13)

II
C

 

 

 

 

Cg, Catecfi ‘ _a,:‘

This is the basic matrix equation for the coupled-mode perturbation theory. Here 1' is
the unknown propagation constant, which is the value that leads to a nontrivial solution

79

for a, , and is determined by requiring the determinant of the coefficient matrix in (13)

to vanish. Notice that both C; and C; are g- independent, and are constants

determined only by the isolated eigenvalues ; (g and currents 53°) . As indicated in

Chapter 4, these are readily obtained, in a simple quasi-closed form, from the full-wave
moment-method solution for an isolated strip. The perturbation solution therefore
involves simpler analytical and numerical procedures than direct numerical solution of
the coupled EFIE’s. Another advantage of the perturbation theory is that it can easily
be applied to configurations of multiple nonidentical coupled lines, which are very

complicated and difficult to solve directly by the method of moments.

6.2 Perturbation Solution of Two Coupled Microstrip Lines

As an example, we consider the problem of two coupled microstrip lines. Let 1',
and I, be the isolated propagation eigenvalues of a certain mode of the two coupled
strips, respectively. For simplicity, assume the wave is propagating in the -z direction.

Then the coupled mode matrix equation (13) for that case becomes

c [41].. .1.)
“2

12
C21 C22( 5 ’ C2)

 

 

and al .1 0, (12 t 0 requires that

(15)

C"r1(C " (1) Cu
C21 C22“ ’ C2)

 

80

This is the eigenvalue equation for propagation constant I and is readily solved to obtain

C=Et6 (16)

where z = (C, +C2)12 is the average of the isolated propagation constants, and
b = A2 +1:2 is a shift, with A = (Cl - (2)/2 and x1 = CuCnlCuCn. The coupling

coefficients C,, and C,, are given by (10) and (11) as integrals involving isolated
currents; it can be shown that for large separations, both C,, and C,, approach zero and
1' reduces to its unperturbated isolated values. With the propagation constant obtained,
the relation between the relative amplitude coefficients of the currents on the two strips

are found correspondingly from (14) as

<3 1
a, = 23—11;“, -c,) =r o)a, (17)

For the special case of coupling between two identical strips, such as that shown
in Figure 5.2, since the isolated currents and propagation constants are same for the two

strips, it can be seen from (10) and (l 1) that C,, = Q, and C,, = C,,. This leads to
C = C0 1 0
(18)

“23*01

where 3', is the isolated phase constant and the shifi 6 = ICuICu | . The normalization

and coupling coefficients are given by

C11 " 51; f 45 [5%Cn(E.C)L s,(£)f,(£) + 5%Cu(£,()|‘ 3405(5)] (19)
.. . ’

81

C12 = 31; fdte‘m‘lcna.c,)g,(0f,(0 + C,,(t.c,)g,(t)f,(m (20)

where C,, and C,, are due to the Green’s dyad as given by (4.7) to (4.9) and

3.10 = f k£°’(x)e/‘*dx. a =x.z
"" (21)

f.(£) = f ki”<x)e"‘*dx. a = x.z

where kfio’(x) is the eigenmode current of the isolated strip and is obtained from the

moment method solution in the form of the rapidly convergent Chebyshev polynomial
series (4.11).

Equation (18) clearly shows that two system modes, symmetric and
antisymmetric, emerge and shift symmetrically from the isolated mode phase constant 1“,.
The propagation constant of the coupled system is that of the isolated one modified by
a small shift 6. The positive (negative) shift corresponds to a symmetric (antisymmetric)
mode in which the 2 component currents in the two lines are in same (opposite) direction.
Once the system propagation constants are determined, the current waveforms along the
coupled lines are readily obtained.

As a simple application, consider the situation of two coupled lines with matched

load: a microstrip coupler. Suppose there is only a single isolated mode 1', excited, then

system eigenvalues C“ = C, :1: 0 represent the two coupled modes. If line 1 is fed at

z = 0 with I,(0) = I, and line 2 with 12(0) = 0, the currents along the two lines are

82

I,(z) = I,cas(bz)e"‘°z
(22)

12(2) =j10sin(oz)e"‘°‘.

These equations show clearly that the input power on line 1 is coupled to line 2 and the

energy is exchanged continuously between the two lines. Define coupling length A, as

it = J!- (23)

which depends upon line spacing and is a quantity that describes the coupling strength.
Its physical meaning is clear; after a propagation distance 1,, the input power of the
driven line is completely coupled into the parasitic one. The coupling length is obtained
numerically from results of the perturbation solution; it is compared with experimental

measurement in the following sections.

6.3 Numerical Results
Numerical implementations of the perturbation theory were developed. The same
configuration of two identical coupled microstrip lines that has been studied in Chapter
5. by Galerkin’s moment method is solved by the perturbation approximation, based on
the MoM solution for the isolated currents. The results obtained by these two different
approaches are compared, the physically motivated perturbation approximation is justified
and its regime of validity is identified.
Figure 6.1 shows the dependence of propagation phase constant 3' for the EH, system
modes of two coupled microstrip lines upon the line spacing. The two microstrips,
shown in Figure 5.2, have the same parameters given there. The solid line is the MoM

83

result and dashed line is that of the perturbation theory. The results show that when the
two strips are wildly separated, the propagation eigenvalues become that of the
corresponding isolated-line limit. As the two strips become closer, the coupled system
modes, symmetric and antisymmetric, emerge and shift symmetrically away from the
isolated mode. Comparing the results obtained by the two different approaches, we see
that the perturbation approximation is in very good agreement with the MoM solution for
loose coupling; it is also good for moderately strong coupling. In the particular case we
studied, the perturbation method is valid up to the point where the separation distance
is approximately 30% of the strip width.

Figures 6.2 and 6.3 show the dispersion curves of the EH modes for the same two
coupled microstrip lines with two different separations, respectively. In Figure 6.2, the
separation is equal to the strip width. For this loosely coupled ease, the perturbation
results are in good agreement with those of MoM solution and the propagation constant
of the coupled modes shift symmetrically from the isolated one over the entire frequency
region. For the moderately-strong coupled case shown in Figure 6.3, where the
separation is 30% of the strip width, the perturbation results give a reasonable
approximation and the shifts are roughly symmetrieal over the entire frequency range.

To acquire a clear physical picture of the perturbation approximation, we compare
the current distributions of the two coupled strips when they are separated by three
0 different distances. Figures 6.4(a) and 6.4(b) show the normalized longitudinal and
transverse current distributions, respectively. Solid lines are the corresponding isolated
currents which are used to quantify the perturbation theory. Dashed lines are the

rigorous coupled currents obtained by MOM solutions to the coupled EFIE’s, where short

84

and long dashes correspond to symmetric and antisymmetric modes, respectively. In the
case where the separation distance is equal to the strip width, the longitudinal
components of both isolated and coupled currents are almost identical. The transverse
components of coupled strip currents are slightly different from the isolated current.
However, since the transverse components in this case are less than 1% of the
longitudinal component, it has negligible contribution to the coupling. Correspondingly,
the perturbation approximation is good for this loosely coupled case. When the two
coupled strips are separated by 30% of the strip width, the coupled currents do exhibit
some change relative to the isolated ones. Consequently, there are errors in the
perturbation results. Referring to Figure 3, this is a case of moderately strong coupling,
and the perturbation results give a reasonable approximation. When the two strips are
closely spaced, with a separation equal to 2% of the strip width ((b-w)/w=0.02), the
coupled currents are significantly different from the isolated ones, especially for the
antisymmetric mode. This is a very strong coupled system. At this separation, the
perturbation result fails, since the perturbation assumption is no longer valid.

Figure 6.5 shows the dependence of normalized coupling length Mk, upon line
spacing for the same two coupled strips, where A, is the free space wavelength. This
result is obtained from the perturbation approximation, and is only valid for separations
greater than 30% of the strip width, since after this point the eigenvalue shift could be
considered as symmetric. The results of such curves provide a valuable reference for
engineering design of microstrip couplers.

Due to the simpler matrix equation involved, the numerical procedure of the

perturbation method is much more efficient than that of the MoM. For the same

85

eigenvalue computation, the perturbation method requires only about 5 % Of the computer
time needed by the MOM method. Also, the MOM approach (especially in the coupled
case) is very sensitive tO the initial eigenvalue guess, so the perturbation approximation

provides a good initial guess for the case where a more accurate MOM result is needed.

6.4 Experimental Validation

Consider two coupled microstrip lines with arbitrary loads which support only the
EH, mode while all other modes below cutoff. From the results Of the perturbation
theory, the coupling effect results in two system modes which have symmetric shifts from

the isolated propagation phase constant. The coupled EH, symmetric and antisymmetric

modes have propagation constant B * = 13, 1; 6 , respectively. The currents along the two

lines therefore are

I,(z) = 1:51” + Hem" + 1:61“ + I; e’“ (24)

12(2) = 13""! + 1,1!” - [go-m - gel”.

Assume the microstrip transmission lines have characteristic impedance 2,. Then the

associated voltage along the lines can be written as

V,(z) = 200:5”? - 1,114: + Ige'm - 1,11“) 25
( )

V2(z) '3 200:5”.z " Keir" - IJe-Ifi'z + 1:61P?)

or, equivalently

86

V1(z) = COS(0Z)(VuC(B(BoZ) + VuSin(poz))

-jsin(bz)(V,,cOs(8,z) + V14Sin(floz))
(25)

V,(z) = c08(5z)(V,,c08(B,z) + Vasin(B,z))

- jsin(Oz)(V23cos(B,z) + Vusm(B,z)).

The amplitude coefficients V” in the above equations are determined by the terminal
conditions Of the transmission lines.

Equations (26) clearly show that the voltages (currents as well) along the two lines
are standing waves composed with two parts: a rapidly changing component characterized
by the isolated phase constant 3,, and a slowly-varying component characterized by
coupling shift 5 which modulates the rapidly-varying component. By experimentally
measuring the voltage pattern along the coupled lines, we can verify the phase-canstant
shift 5 as well as 8, obtained numerically from the theoretical analysis.

As shown in Figure 6.6, an experimental implementation Of two coupled
microstrip is set up on a common PC-board. A monopole microcoax probe is used tO
measure the electric field amplitude axially along the coupled system. The probe is
inserted through the conducting ground screen into the film and the cover layers to
measure at that point the normal electric field, which is proportional to the voltage there.
A line Of small holes is drilled through the PC-board closely along the edges Of both
strips tO implement probe insertion. A harmonic heterodyne detector assures
measurement linearity, and the normalized voltage amplitude is read from a SWR-meter.
Strip 1 (driven line) is fed at one end by a microwave generator and shorted at the Other

end. Strip 2 (coupled line) is shorted at both ends.

87

The PC-board has a dielectric film layer with relative permittivity 6,, = 2.20 and
thickness Of 1.22 mm. The two strips have an equal width Of 4.0 mm, length Of 183 mm
and separation Of 1.0 mm. The working frequency is 10 GHz (X, = 30 mm). With
these experimental parameters, we Obtain the following numerical data by using moment
method for isolated calculation and perturbation method for coupled calculation:
Propagation constant Of isolated EH, mode: 6, = 1.4040k,
Coupling phase-constant shift: 6 = 0.0301k,
Coupling length: A, = 8.3005),

where k, = 21/), is the wavenumber in free space.

Exploiting this propagation constant and phase constant shift in (26), in
combination with the terminal conditions, we Obtain the theoretical voltage distributions
on the microstrip lines. The actual voltage distributions along the lines are measured and
the two results are compared. Figure 6.7 shows both the analytically predicted and
experimentally measured voltage distributions along the driven line. Figure 6.8 shows
the voltage distributions along the coupled line. We see that the theoretical results are
generally in a gOOd agreement with the experimental ones; the theoretical predicted B,
and 6 are consequently verified. The errors are due mainly from end effects and the

inaccurate knowledge Of the probe hOle positions.

6.5 Discussion
The above results show that the full-wave EFIE based perturbation theory is a
simple but powerful method tO approximate the coupling among adjacent microstrip lines.

The perturbation approximation is valid over a broad range Of loose, nearly degenerate

88

coupling, with good physical justification. The numerical results of the perturbation
dreary are in gOOd agreement with those Of Galerkin’s moment method solutions over a
broad range Of weak tO moderately strong coupling, indicating that the perturbation
theory yields a gOOd approximation. Numerically, the perturbation theory greatly
simplifies the computational procedure and therefore greatly reduces computation time.
Also, the perturbation method is easy tO apply to nonidentical multistrip coupling. TO
verify the perturbation theory, an experimental implementation Of two coupled
microstrips is set up on a PC-board configuration, and the voltage distributions along the
strips are measured. The experimental results are in a gOOd agreement with theoretical

predictions Of the perturbation theory.

89

 

3.10

 

    

 

 

 

 

1
g I ..... Perturbation
b 3-‘ Galerkin's MOM
3.00 - ~
‘3' 2 EH, symmetric
2 1 EH, isolated
8 2.90 -I
q) -1
0‘1 .1
E .
O. 2.80 -
'0 I
o
.E '
'6 _: f,=10 GHz
g 2-70 - w=1.500 mm
0 - t= 0.635 mm
2 ..
2.60 q IIIIIIIII'IIIIIIUTI'IIIIIITIIIrIIIIIUII
0.00 1 .00 2.00 3.00 4.00

Distance Between Coupled Strips (b—w)/w
Figure 6.1 Comparison Of MOM and perturbation solutions for dependence upon line
spacing Of propagation constant 3' for the EH, system modes Of two coupled

microstrip lines.

90

9"
o

 

 

 

 

.I
A? . 1 (b-w)/w=1.0 I
b i
3.00 - /,
"é ‘ /”
O ..
'3‘; ..
g u
o 2.90 T. 1
0 ‘ . .
3 Z t I: I! 1- Perturbation
0:. 280 _ ----- - GOIGI’kih'S MOM
v I
0
a: :
E 2.70 - o: isolated EH, mode
'5 r 1: coupled symmetric mode
2 I 2: coupled antlsymmetnc mode
2.60qlllllllTllrrrIllllllIIIIIIIII—ITIIIIIIII
0.0 10.0 20.0 550.0 40.0

Frequency (GHz)
Figure 6.2 Comparison Of MOM and perturbation solutions Of dispersion characteristics

for two coupled microstrip lines with a separation equal to strip width.

91

 

9"
o

   
 

 

 

 

'1 t
f 1 (b-w)/w=0.3 I
\ ..l
V 3.00 -
g - 1
3 I
g 2 90 1
0 ° . 0
3 ..
O I Perturbation
0.:- 2.80 - ----- Galerkln's MOM
'0 I
d.)
.2 -I
E 2.70 - , o: isolated EH, mode
3 ' 1: coupled symmetric mode
2 I 2: coupled antlsymmetnc mode
2.60ulllll[Ill]IIIIIIIIIIIIIIIIIIIIIWTIIIII
0.0 10.0 20.0 30.0 40.0

Frequency (GI-l2)
Figure 6.3 Comparison Of MOM and perturbation solutions Of dispersion characteristics
for two coupled microstrip lines with a separation equal to 30% Of the strip

width.

92

1 .00

 

  
   

 
   

 

   

 

 

 

— — — antisymmetric 1: b—w /w=1.00
.. ..... symmetric 2: b-w /w=0.30
52‘ 0.75 isolated 3: b—w /w=0.02
is”, I
t -
3
0 -
D 0.50 -
o ..
.t‘ _
a d
E 1. 1
o I \
z 0.25 - \...3-»~--s ~——-—-L------— ’ , l
' \ " “ - - -2 -------- I
.. \ I /
: ~ ----- 3— --------- r
0000 llIlIl1j11j1lllllljllllllliiilijlilliri
-1 .00 -0.50 0.00 0.50 1.00

 

Position (x-b)/w
Figure 6.4(a) Comparison Of longitudinal current distributions for isolated and coupled

microstrip lines with three different separations.

93

0.02

 

 

 

 

 

 

 

'1 - - - antisymmetric
a ------ symmetric g-.. —————— -.
- isolated xv" _, ...... 5‘
a? 0.01
4.0
E
g
3
0
.0 0.00
o
.t‘
'6
g
E —0.01 - \\
: \\ 3 ’a’d'x/ 1: b—w /w=1.00
- \\ - -- ’ / ’ 2: b-w /w=0.50
. \ a Z. .9 ’ 3: b-w /w-0.02
-0002 tiYUIIUIfiIIIIIIIIIIUITIIIIIIIIIIITIrTj
-1.00 -0.50 0.00 0.50 1.00

Position (x-b)/w
Figure 6.4(b) Comparison Of transverse current distributions for isolated and coupled

microstrip lines with three different separations.

94

 

250

fo=10 GHZ
w=1.500 mm
t=0.6.'55 mm

_L ..a N

O U" 0

O O O
IIIIIIIIIIJIIIIIIIILLIII

01
0

Normalized Coupling Length kc/h,

 

 

 

 

 

O llll IIIIIIIII IITI llll

l I I
0.0 0.5 1.0 1.5 2.0 2.5
Distance Between Coupled Strips (b-w)/w

Figure 6.5 Dependence Of coupling length A, upon line spacing.

95

 

OOOOOOOOOOOOOOO

 

 

 

 

 

 

 

 

 

 

 

 

shorted [ /(——— shorted
shorted- L j
o o o o o o o o o o o o o o o;
L
probe
SWR METER DETECTOR GENERATOR

 

 

 

 

 

 

 

 

 

Figure 6.6 Illustration Of experimental setup.

96

 

 

 

 

 

 

 

 

 

 

 

 

1.20 *
° 1 00 1 Dash: experiment
B -. Solid: theory
T: I
8080:'
2 ' .. l
o - 1
0’ q 1
«g 0 60 q | J
o J ,
> . 1 ,
E040 - l f“
E - l f I
a " ’ f f
o 0.20 - I ,, ‘ ‘
Z ' . I- ‘

: ' 1 w l l 1
0.00 lilllllIIIIIIlllllll'IlI—TIIIIIIIIIIIIIII
0.0 50.0 100.0 150.0 200.0

Position Z(mm)
Figure 6.7 Comparison Of perturbation-theory—predicted and experimentally measured

voltage distributions on driven line.

97

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

llIIIIIIIIIIIIIIIIIUIWT1IIIIlIrIIIIITIII

 

.m .mln l l l I ..lllltl...
om \.alal|al‘lluu.l| “v
eon f
we 11......
0000 .5 III.
Ill“
00 £111!
DS ill
a d u — q u u — u - u — u - - — - u q
0 O 0 0 0 0 O
2 O. on 6. A 2 0.
1 4| 0 o 0 o 0

0632003. oooto> 63:08.62

100.0 150.0 200.0
Position Z(mm)

50.0

0.0

Figure 6.8 Comparison Of perturbation-theory-predicted and experimentally measured

voltage distributions on coupled line.

98

CHAPTER SEVEN
ANALYSIS OF THE CHARACTERISTIC IMPEDANCE OF

MICROSTRIP TRANSMISSION LINES

In previous chapters, the dispersion characteristics, current distributions and
leakage and radiation properties were analyzed rigorously for nricrostrip line structures.
In order tO accurately analyze the dynamic behavior Of those structures in their increasing
presence in the high-speed interconnections and micro/millimeter-wave integrated
circuits, an accurate circuit description Of them is necessary. The circuit models Of
microstrip lines, in combination with Other device models, can then be used together in
larger simulators tO analyze more complex circuit systems. TO Obtain a complete circuit
equivalent for the structures, not only the dispersion characteristics, but also the
characteristic impedances must be determined. In this chapter, we present a full-wave
analysis for the characteristic impedance Of microstrip lines based on the same integral-
equation formulation. Discussions are included on the definition Of characteristic
impedance for microstrip lines, the current, field and power calculations and their
applications. Results Of previous chapters are utilized in the evaluation Of various

parameters. Numerical implementation is developed and results are Obtained.

7.1 Circuit Description Of Microstrip Lines
A microstrip transmission line, as a waveguiding structure, is a non-TEM system,
and therefore a non-ideal transmission line. Unlike a pure TEM system, non-TEM

structures, such as the microstrip line, have no unique definition for characteristic

99

impedance. TO Obtain an equivalent transmission line (circuit) model, a suitable
definition for characteristic impedance, in the circuit sense, is needed so that the
microstrip structure can be analyzed along with other TEM structures, such as loads and
sources, that are connected tO it. For a pure TEM system, the voltage and current are
uniquely defined in terms Of path integrals Of the fields, and have the conventional circuit
meanings; the complex power flowing along the system is defined by the integral Of

Poynting’s vector across the waveguiding cross section and satisfies the relation
P = I ‘ V/ 2 . The characteristic impedance Z, Of the TEM system is therefore uniquely
defined by any Of the three equivalent definitions, i.e., the voltage—current definition

Zo = VII, the power-voltage definition Z0 = |V|2/(2P‘) , and the power-current
definition Z0 = 2P] II I”.

Previous discussion shows that the propagation modes along microstrip line are
non—TEM hybrid modes. Due tO the longitudinal field components Of the non-TEM
modes, current and voltage cannot be uniquely defined by the usual TEM path integrals
and do not always have the usual circuit interpretations. All the three definitions Of
characteristic impedance have been used for microstrip lines in the research literature,
but lead tO different results due tO the ambiguity in the meaning Of current and voltage.
However, it has been shown by Brews [53], [54] that it is possible to Obtain a circuit
model as a transmission line by satisfying certain requirements and by suitably choosing
either current or voltage as the circuit variable. The equivalent transmission line is
required to have the same propagation constant and propagate the same complex power

as the microstrip line. As in the TEM system, the propagated power remains

100

P = 1‘ V12 . By choosing either current or voltage as the dependent variable with the

circuit interpretation and satisfying the above relations, the other one is determined
(though without circuit interpretation), and the three definitions for characteristic
impedance give the same result and are therefore equivalent.

For the microstrip line, the power propagating along the waveguiding axis is
found by integrating Poynting’s vector over the transverse cross section. If the current
is chosen as the dependent variable, it is the total longitudinal current flowing along the

strip. The corresponding characteristic impedance is given by power-current relation

2,, = 21"] II Iz, and the voltage is given by V = Z”! which is a weighted average over

the cross section and has nO circuit meaning. If the voltage is selected as the dependent
variable, it is chosen as the strip center voltage and another equivalent model results.
There has been much discussion about these different models. It is generally agrwd that
the model based on the total longitudinal current is the most accurate one [55]-[6l]. It
has the most TEM-like character. Also, the total longitudinal current on a microstrip is
a well defined physical quantity, as can be seen from the results Of previous chapters,
and is conserved when the microstrip is connected tO other devices. Therefore, the
power—current model is considered as the most appropriate one for describing the

behavior Of microstrip line among other TEM structures, and is adopted in our analysis.

7.2 Formulation of The Problem
Consider a single microstrip line with only the principal mode propagating along

the +2 direction. The characteristic impedance is defined by

101

2P

2. = — (1)
I1012
The total longitudinal current I, is given by
10 = I fix) 2112: = f k,(x)dx (2)
c '19

where the current density k,(x) is Obtained in (4.11) and (4.12) from the MOM solution
tO the integral equation as a rapidly convergent Chebyshev series. Utilizing the
symmetry Of the current distribution and the orthogonality Of Chebyshev polynomials,
integral (2) can be done easily to Obtain

I0 = Itwaw (3)

where a,, is the first expansion coefficient Of the z-component current.
The power propagating along the microstrip line is defined by the integral Of

Poynting’s vector over the transverse cross section, as

P

%£I(Exlr)°idS
(4)

g 1' 1 (5,11; - £,H,’)dxdy

where E, and H, are the spatial domain fields excited by the eigenmode current

(eigenfields). For the single principal mode, from the discussion in Chapter 3, we have

it?) -- and“. (5)
Hm = Sore“.

102

with 5(5) and 5(5) the spectral-domain transverse field distributions and I, the

propagation constant. The power then becomes

P = %e”"’°’ff(e,h; - eyh;)dxdy. (6)
3?

Combining this with (2), and noticing that the exponential decay factor for lossy lines is
canceled with the similar one in the total current, we Obtain the characteristic impedance

2 12 2 ”wt; - e,h;)dxdy. (7)

“Wazox’

zc=

 

The spectral domain electric and magnetic fields can be Obtained via Hertzian
potentials, as demonstrated previously, through the Green’s function and known

eigenmode current, as

2.05) = (k? + Waste).

_, _. _ 11,1 _ -. (3)
h((p) " — VXfit(p)9
"I
and
.. - in, .. _. _,, _, ,
70(9) - -7- [3,0 Ip.-c,)-E<p')dt (9)

C C

where l = c, f denotes the cover and film regions, respectively. Consequently, the
power flow in each Of the regions can be evaluated to Obtain the characteristic

impedance. In later sections, the power in each region will be calculated in detail.

103

7.3 Fields and Power in the Cover Region

In the cover region, the eigenmode electric field is given by

3.0:.» "11,1 f I.‘(x.ylx’.o;-c,>-£(xrdx'

where the Green’s dyad has the following scalar components

‘ l . ‘ x-x ’.
§.,(x.ylx’.0.0 = 5; fC.,(E.C)e’“ It ' ’4:

mmfl=nna

(10)

(11)

Some Of the coefficients have been given in chapter 4. For our interest here, those

related tO the x and y components Of the field are listed below

2 2_ 2 2 2_ 2
= (~er I): , + Mfitk. t)

 

 

 

 

CWK) .
’4 Z‘(E)Z‘(E) Z "(0
2 2_ 2
c300 = (~er 1m” ‘ - -M"—“.
25‘(£)13‘(E) EZ‘(E)
. N2M2 '1 1+ .MZ
0,315.0 = 1(1, 1: )(E (“M _ 1 1,619,,
1! (5)ZZ‘(E) i! (E)
. N2”: -1 2+ 1 .M2
0,;(£,() = K; ,t )(E ()6 _ I ,an,

 

2*(E)Z‘(£) 2"(0 '

(12)

(13)

(14)

(15)

Substituting the above Green’s function into (10), and exchanging the spatial integration

with the spectral one, the scalar field components are Obtained. Omitting the algebraic

104

details, we have

 

 

J'rl, , - .
gm) = — Zak. [Ere-mat e ”dt.
(16)
e (x y) = - jn‘ 11"(5 -C )e’“e""dE
’ 2 2nkc -. y 9 o 9
where
F,(£.-C,) = C;(E.-C,)f,(£) + C;(E.-C,)f,(E). (17)
F,(£.-C,) = C;(E.-C,)L(E) + C;(E.-C,)f,(5).
with the spatial integral over current being
f,(£) = [hearsay p =x, z. (18)

Remembering that the current in the above expressions is the eigenmode current Obtained
from the MOM solution Of the integral equations, and is in the convenient form Of a
Chebyshev polynomial series given by (4.11) and (4.12), integral (18) is readily

evaluated in closed form. Referring tO the discussion in Section 4.2,

N
f3“) " Z a,,.f,.(£) I) = x, z (19)

where f,,(£) is given by (4.14) and eventually evaluated in terms Of Bessel functions.
The eigenmode magnetic field in the cover region is found though Hertzian

potentials from (8) and (9), as

105

5.0:.» = Vx f r.(x.ylx’.o;-c,)-E<x’ldx’
"’ (20)

= f §,"(x.ylx’.(k -c,)-E<x')dx’.

Since no source point singularity is involved with magnetic field, interchange Of the curl
Operator with the integration in (20) is permissible. Consequently the magnetic Green’s

dyad is defined as

§‘(x.y lx’.y’;C) = Vx§(x.y lx’.y’;C). (21)

The transform domain Hertzian potential Green’s dyad is given by (3.9) to (3.13).
Substituting it in (20), after intensive vector Operations and many manipulations, the
scalar components Of magnetic field are Obtained as

W

h.<x.y) = f {3.105 Ix’.0;-C,)k,(x’) + stray Ix'.0.-c,)k,<x'>}dx'

., (22)
a = x, y, z.
The scalar components Of the magnetic Green’s dyad are given by
k I - 1 . h not-2’) ‘PJ
9 9O; - — C 9 d
2.,(x y Ix 0 2” j. ..,(t 0: e a (23)

a, p = x, y, z,
with the coefficients

(N121; - 1) CE
2*(02'00

= (Nina-1)? _ M21». (24)

0.3.0.0 = ,
2*(02'10 2*(0

 

 

. C,,,(E. C )

106

mic

2*(0 ’

mi:

, (25)
2'10

 

 

0,115.0 = 6,350 = -

_ (~04; - nez , Mir.

_ (”liuri'l)“ (26)
210210 2"(0’ '

c350 = ,
z (02*(0

0,105.0 =

 

 

Exchanging the spatial integration in (22) with the spectral one in (23) after the Green’s
dyad (23) is substituted into (22), the field components can be further simplified. The

transverse components needed in power calculations are then given as

h,(x.y) = 3‘; f G,(E.-C,)e"’e""d£.

(27)
h (x y) = $160 -c )e""e""d£
’ ’ 2n __ ’ ’ ° ’
where
G,(£.-C,) = cat-(01,10 + C£(E.-C,)f,(t). (23)

G,(£.-C,) = C,§(E.-C,)f,(t) + age-C0140.

and 120E) and 12(5) are given by (18).

With the transverse fields Obtained, the power in the cover region is found from

(6) by

PC = I'dx‘idflexh; - eyhx'). (29)

N1—

Notice that the inverse integrations for the e, and It, fields in (16) and (27), respectively,
are independent, and therefore should be distinguished. Substituting (16) and (27) into

107

(29), we get

in, ' ° . -, . . -’x':
p,=- 2*. (2”) [dz fdy «1inF,<£)e’“e"d£}{£6,(£5el‘e”de’1

 

 

(30)
i a.

2* (27:).f {dy {Lamina-Md“{£G;m""’¢""d£’l.

 

 

The integration order is exchangeable under the condition that the integrand is sufficiently

well behaved that the integral converges. This condition is satisfied here and (30)

becomes

_in.
2k (211')2

C

 

 

fdedE’ F<£>G ,'(E’)fdye 0" ””14“!" w:

-.--

(31)
in. 4». r2» 1(:-¢’)x
2k: (ziyffdzdz’ F406 (tofdye fdxe

 

 

By the completeness condition for exponential functions and the concept of distribution
5(5)

f em’w‘dx = 21:5(5-5'). (32)
Also, letting
32(5) = fe'mwb’dy = _L_:, (33)
o P, + P.
we get

108

in. ' . .
. - m f {F,(E)G,(€) - F,(£)G.(e)}r,(e)de (34)

C-.

 

with g’ = {0 for the principal propagation mode. Examination of the integrand in (34)
shows that it is an even function of :3, so the power in the cover region is finally obtained
as

in.
‘ 21rkc

 

0"”

{F.(£)G;(E) - F,(£)G.‘(£)}r,(t)de. (35)
This integral is evaluated numerically.

7.4 Fields and Power in the Film Region

The spatial domain Green’s function in the film region has been obtained in
Chapter 2 for a general three-layered structure. For the conductor-film-cover structure
of a microstrip environment, the corresponding transmission, reflection and coupling

coefficients are obtained by letting conductivity of the substrate become infinite in (2.50)

 

 

to (2.52), as
9,4 _ 7,4
em = p“ . 1w) = ..,. , <36)
Nfisinh(pfd)Z"(A) Nfisinh(p,d)z*(x)
PI" 7,4
NA) = p“ Rm) = p“ (37)

 

 

”' cosh(p,d)z'(x) ’ cosh(p,d)z'(x) ’

109

(NfiM; - 1)(l + tanh(pfd))pc
N;2‘().)Z'(A) ’

 

0,0) =
(38)
(NfiM; - mg - tanh(p,d)&

c (A) =
"' Ngz’mz‘at)

 

The corresponding spatial-domain Hertzian potential Green’s dyad is obtained when the
above coefficients are exploited in (2.51) and (2.52); the electric and magnetic dyadic
Green’s functions are consequently defined by (2.59) and (2.60), respectively. The
spatial domain fields in the film region are given by (2.58) and (2.61).

For the problem of microstrip lines, as shown before, it is more convenient to
work in the spectral domain. By taking Fourier transform of (2.51) and (2.52), spectral

domain Hertzian potential Green’s dyad is obtained

§f(6|fi’:C) = (it + £68}; + 9(273819 + 98‘”. + 2118;) (39)

where the scalar components are

81;“? I islet) .. rTfiOJe”, + 19(1):?” l
8,..(5l5’.€) = [[4 5(1):” + 19.0)!”
8,,(5'5’,C) "' [Cfl(l)e"’ + Cfi(1)¢vflJ

ejz(8-x’) e ’3’, de . (40)

V

 

41rpc

 

 

Correspondingly,

£,‘(i>’|i>”.() = 0:} + Wows Iran.

§,"(5|i>”.€) = vxmlaco.

(41)

and

110

’5‘ f§;(b°ls’.C)-Fmdz’.

c C

axis) = -

 

13(5) = Nfifrfia I ran-RWI'.
C

(42)

When the propagation constant and eigenmode currents are substituted, equations

(42) give the eigenfields of the corresponding propagation mode in the film region. The

scalar components of electric field are given by

W

e.(x.y> = f {g:.(x.y|x’.o;-c,)k,(x0 + g;(x.y|x'.m-c.)k,<x'>}dx’

'W

 

 

a = x, y, z.
The components of electric Green’s dyad are
c l . c x-x’
880(x’y'xl’mc) ’3 .2—1; £B¢p(e’c’y)eju )dE as B 3 xsyszs
where
2 . 2 2
page,» = if: - E” WAN) _ s’ixem- n mun/(y +00,
Nfiz'm smhp,d NfiZ"(A)Z‘(}.) coshp,d
. 2 2
3;“.30’) = ——k’2 ' ’2 m” ”D - Czpflnfufi'” “my ”0.

 

 

Ngzm) sinhp,d N},z*(2)z'(2) coshp,d

111

(43)

(44)

(45)

(46)

ftp coshp,(y+ d) jwc} +p})(N;M;-1)sinhp,(y+d)

 

 

 

 

 

 

8‘ C.£.y) = (4 7)
’4 N;Z"(A) sinhp,d Nfiz‘(2)z'(2) coshp,d
. 2 2 2_ . +

3’2“”) = up, cosrap,o+d)+iC(k,+p,’)(N,.M,. 1) smhpgy d), (48)
Nfiz'n) smhp,d Nfiz‘mz'ot) coshp,d

B;(€.C.y) = 830w).

(49)
k’- 2 ' +d 1 N’M2-r +d
33“”): , E smhp,(y )_ €p,( 2 fl )coshp,(y ).

N,’,z*(2) sinhpfd N;Z*(A)Z'(A) “8“de

With the above Green’s function, the transverse electric field components are obtained

as inverse spectral integrals, after exchanging the order of integration in (44), as

 

 

 

abs» = - 2’3; 1 (F,(:)sinhp,(y+d) + a,(e)¢oshp,(y+d)).wdg,
.. (50)
e,(x.y = - ’ '7‘ hf (F, (ecoshpxwd) + G ,(e>sinhp,(y+d))e“*de.
where
2_ 2 _
..,.... <2 ”:30 2:134“,
n 1’1" (51)

(NfiMfi' 1)£p,[£f(5) "' {8(5)}
N; coshp,dZ"(l)Z‘(l)

 

0,,(0 = -

112

ip,[£f.(£) + we]

F (E) = .
’ 1v; sinhp’dZ’Tl)

 

i(N,.M,. -1)<k, +12?) rem) + we]

G,(E)=
N, coshp)dZ “(10212)

 

and IKE) and 5(5) are those given by (18).

(52)

Similarly, the magnetic fields are evaluated. The magnetic Green’s dyad has the

following scalar components

It , ___ 1 . I. ’,-,I _
8¢p(xs}’|x’aosC) 2—7; £B¢p(E’C9y)e1£( )dE ... asp "' x.y.1.

where the coefficients are

(1),;sz - l)£C sinhpf(y +d)
N;z*(2)z'(2) coshp,d ’

 

335.6.» =

P, COShP/(y + d) _(Nfchc ' 1):: SinhP/(y "’ d)

 

 

Elm.» = ,
Ngz'xx) smhp,d N Jfiz’*(2t)z'(zt) coshp,d

B ,’,‘,.(E by) = imfl’m.
Nfiz'm sinhpfl

° sinhp,(y + d)
s’<£.c.y = - ’ ‘ , .
’2 Nfizhm mnhp,d

 

113

(53)

(54)

(55)

(56)

(57)

 

 

B.f“..y<£t)=- -_L°°‘hpr(y*d>_ (NfiMfi-I)E’sinhp,(y+d)

(58)
Nfiz z‘m sinhp,d N;z*(r)2'(x) coshp,d

(Nina-1):: sinhpxy+d>

(59)
N;z*(x)z'(2) coshp,d

8.2m.» = -

 

Again, after exchanging the integration order and many manipulations, the transverse

magnetic field components are obtained as

h ,(x.y) = N,.— f (U (osinhpxyw) + V,(E)coshp,(y+d))e’“dz

 

 

(60)
l . O
’5‘“) = ”23; f v,(£)smhp,(y +d)e"*d:,
where
v (a) = (”11”; ' DC [We + we]
1 2 h c ,

Nfi: “Shad Z (1)2 (A) (61)

V2“) = 2 P,f,(5) ’

”I: Sinhpfl Z ‘0.)

U,<£) = yum) - we]. (62)

 

N; sinhpIdZ'Kl)

As before, 1,;(5) and 3(2) are those given by (18).

With the transverse fields in (50) and (60), the power in the film region is readily

114

found by integration over the cross section as

Pf=

NI.—

jdx idyuxh; - eyhx'). (63)
-- -d

Similar to the case in the last section, since the fields are well behaved functions,
interchanging integration order in (63) is allowed after (50) and (60) are substituted into
it. Again, the integration with respect to 1: gives rise to a 5-distribution. The integrals
with respect to y fall into three categories

0

m) -- f sinhp,(y +d)sinhp;(y +d)dy. (64)
-d
0

ram = f coshpxy +d)coshp;<y +d)dy. (65)
.4
0

13(6) = f coshp,(y +d)sinhp,'(y NM» (66)
.4

The above integrals can be evaluated analytically. Also, the integrand in (63) is an even

function of £. Therefore the power in the film is finally obtained as
p — 1%"; - F(E)U'£)-G a 0‘ Y -F v‘ 1’
,- 3,7:- I. {I , ,( ,() .<m ,(t) ,(E) .(E) 2(a)
+ [G,(E)U,'(E) -F,(£) U;(E)]Y3(E) - G,(E) V;(E)Y;(E)}d£.

Inteng (67) is evaluated numerically.

115

7.5 Numerical Results
The characteristic impedance for the principal mode of a single microstrip line is

then obtained, from (7), (35) and (67), as

2
2
n’wza‘,

 

20 = (P: + Pf). (68)
Also, the associated electromagnetic field distributions are given by (16), (27), (50) and
(60), respectively, for each region. All those quantities have to be evaluated
numerically. Numerical implementation of those calculations has been developed in
Fortran 77.

Figure 7.1 shows the frequency dependence of the characteristic impedance for

the same single microstrip transmission line studied in Chapter 4.

116

 

Impedance (ohms)
N N
0 N
o 0

9°
0

llllllllllllllllllllllllllllllLLLljllll

 

 

 

16.0 IIIIIIIITIIIllllIIllllllIIllrllllllllll

0.0 10.0 20.0 30.0 40.0
Frequency (GHz)

Figure 7 .1 Frequency dependance of characteristic impedance for the microstrip

transmission line.

117

CHAPTER EIGHT
MATERIAL LOSSES, LEAKAGE AND RADIATION CHARACTERISTICS

OF MICROSTRIP TRANSMISSION LINES

Losses in microstrip transmission lines are of practical importance as well as
theoretical interest. The losses include conventional material losses (due to dielectric,
magnetic and conductor losses) in the microstrip lines, and the losses caused by leakage
of power due to the excitation of surface-wave modes on the layered background
structure and/or space wave radiation. The latter unconventional losses, usually
undesired and potentially dangerous to system performance, but sometime useful, are
novel and important especially for high frequency applications. In this chapter, the
material losses and the leakage/radiation characteristics for microstrip lines are analyzed
in a consistent manner using the integral equation formulation developed previously.
First, in discussions of the bound waveguiding modes and the radiative leaky modes, the
propagation spectrum is reviewed and clarified. Then the full-wave integral equation
formulation for general lossy microstrip lines is implemented by extending the real
analysis to a unified complex analysis. By studying the behavior of the pole and branch
point singularities in the complex spectral plane, the integration contour for the
Sommerfeld-type inverse integrals is determined and justified. From this complex EFIE
analysis, the leaky-mode and radiation properties of microstrip lines are deduced

naturally and consistently.

118

8.1 Propagation Spectrum and Leaky Modes

Losses and surface—wave/ radiation leakage exist in various multilayered microstrip
configurations. Our discussion here is restricted to a single open microstrip line, but the
physical mechanisms are similar and relevant for the other configurations. The
theoretical formulation is general, and is readily applied to other structures.

The conventional losses due to non-ideal dielectric materials, magnetic materials
or conductors have been studied by many researchers [621-[67]. For microstrip lines,
the material losses in the cover or film regions are relatively easy to handle. They are
incorporated in the integral equation analysis by introducing a negative imaginary part
in the permittivity and/ or permeability. The propagation constant then becomes complex,
acquiring an imaginary part which describes the attenuation effects. Power is dissipated
in lossy materials and the fields (or currents) of propagation modes decay axially.
Nevertheless, the modes with only material losses remain bound, no power is transferred
into and carried away by other type of waves, and the physical situation is essentially
unchanged. The analysis and results for the material losses is presented along with those
for the leakage and radiation losses, on which most of our attention is focused.

Radiation characteristics of microstrip lines were first presented by Errnert and
then studied by others [681-[79]. In the following paragraphs we give a brief review of
the previous research work (include the author’s) on this subject. It was revealed in that
work that a so called ”radiation region” exists in the propagation spectrum of microstrip
line when the higher order modes are below cutoff. The radiation region represents the
continuous spectrum of propagation and is characterized by leaky modes which reside

within this region. When a higher order propagation mode becomes cutoff, it enters the

119

radiation region and becomes a leaky mode, corresponding to a complex propagation
constant (eigenvalue) gr = 1', + j 5. Appropriate. solutions for leaky modes would
correctly represent the leakage and radiation characteristics of microstrip line. As an
application, a microstrip traveling wave antenna was proposed by Menzel, and was
studied experimentally and analytically [801-[82].

Consider a single open microstrip transmission line, for simplicity, assuming it
has little material losses and its dielectric film is thin enough so that only the lowest TMO

background surface wave mode is allowed. For a higher-order bound mode, its

propagation constant satisfies the relation kc < A, < C, < [C,, where A, is the propagation

constant of the TM surface-wave mode of the layered dielectric background environment.
The latter surface-wave mode satisfies Z'()\) = 0, and A, acts as the cutoff limit for the
bound microstrip mode. The fields of a bound mode are restricted in the vicinity of the
strip and decay exponentially in the transverse directions; power is only transported along
the waveguiding z-direction. As the operating frequency decreases, the mode may fall
below cutoff and become a leaky mode. Its propagation constant becomes complex,
acquiring an imaginary part that describes attenuation in the z-direction. The onset
condition for leakage of higher-order modes is therefore 3', < A” which defines the
boundary between bound and radiation regions. The principal mode, since it has no
cutoff, remains bound over the entire frequency range and has no corresponding leaky
region. .

There are two forms of leakage: surface wave leakage and space wave leakage

(or radiation leakage). Surface wave leakage arises when k" < C, < A, , which defines

120

the surface wave region as a subregion of the radiation regime. In this case, the

background surface wave is excited and propagates away from both sides of the strip at

approximately the angle 0 = cos'1 (Cr/1,) . Power leaks away transversely in the form

of a surface wave. As the frequency is decreased so that the phase constant becomes less

than that of the dielectric cover medium, that is O < C, < kg, space wave leakage occurs

in addition to that due to surface waves. In this case, a conical space wave is excited and

propagates away from the strip into the upper half space at a cone angle approximately
given by 6 = cos'1 (C,/k,,.) . Power is carried away in the normal y—direction by this

radiation wave.

In the leaky regime, the fields of the leaky modes seem to grow exponentially in
the transverse directions, i.e. , the x-direction for surface wave leakage and the y—direction
for space wave leakage. This is due to the assumed infiniteness of the ideal microstrip
line structure in the theoretical analysis, e. g. , infinitely long strip line and infinite planar
background environment. It is not (as it seems to be) a violation of the radiation
condition, which is based on the assumption of finiteness of radiated power in the far-
zone due to a source of finite size and power. Rather, it is a characteristic of leaky
modes. More discussion and physical explanation no this matter can be found in [79],
[83], [84].

The above description of leakage, presented mainly by Oliner and followed by
others, provides sound physical pictures and approximate numerical results. The theory,
however, lacks mathematical rigor and appears artificial, especially for the criteria

separating the leaky regimes, which have no sound explanation and become unclear when

121

material losses are included. In the next section, we present a rigorous and consistent

complex EFIE theory for the leakage problem on microstrip line structures.

8.2 Complex Integral Equation Analysis of Leaky Modes

The integral equation formulation, when defined over a complex domain,
completely and rigorously describes the behavior of microstrip lines, including the
radiation properties. Losses and leakage are therefore consistently incorporated into the
theory, and leaky modes are merely complex solutions to the EFIE on the complex
propagation constant (eigenvalue) plane. In the development of the relevant Green’s
functions as well as in solving the complex EFIE, the same Fourier transform technique
was applied. However, the Fourier transform is now defined over the complex spectral
domain, and therefore the inverse transform integrals that appear in the Green’s function
and the EFIE’s, such as (4.6) and (4.17), become complex contour integrals instead of
the conventional real-axis inverse integral. The inverse integration contour must be
properly determined to insure the existence and convergence of both the forward and
inverse transformation as well as to give the correct physical interpretation. To be able
to choose the proper path, properties and behavior of singularities in the Sommerfeld-
type inverse integrals on the complex plane are investigated thoroughly.

A typical inverse integral that appears in the EFIE solutions is rewritten here,

with some modification, as

1,,(x.y.c) = fc.,<:.c>r,,<oe""e“*d£ «.6 = m. (1)
C

122

 

where pc = J 52 + (2 -k3 and C,,,(fiJ) is given by (4.7) to (4.9) but with all the
variables being complex; C is the integration contour in the complex £-plane, yet to be
determined. Dielectric/ magnetic material losses are introduced by complex parameters

I .I I .I
51:51‘151 and I41: I’d-JP:-

Examination of (1) reveals that there are basically two types of singularities, pole
singularities and a square root branch point singularity, leading to the background

surface-wave and the radiation leakage, respectively, as the following discussion will
show. In our analysis, we assume propagation factor e *1“ for waves propagating in the

:tx-direction, respectively.

8.2.1 Surface Wave Pole Singularities and Surface Wave Leakage

Surface wave poles in the f-plane result from zeros of Z"(£) in the integrands.
As previously indicated, the poles characterize the surface waves supported by the
layered background structure. Considering only the principal TMO surface mode, its

propagation eigenvalue equation is rewritten here more specifically, as

 

 

 

Z‘(E) = N;J:’+c’-k.’ + J£’+c’-k} mn(\[e’+c’-r}:) = o. (2)

Since the above equation involves only 5’, its roots yield a pair of surface-wave poles
3F E, in the E—plane which, when accommodated, correspond to possible surface wave
propagation in the :tx-direction. Notice that (2) cannot be solved unambiguously unless
branch cuts for the square roots are appropriately specified. The determination of proper

branch cuts will be discussed in the next subsection in combination with radiation

123

leakage; here we assume the cuts are known and the poles are quantified correctly. As
shown in chapter 3, the spectral domain eigencurrent has only simple first order poles;
consequently the fields (including surface waves here) produced by the current should
also posses only such poles.

As the excitation frequency varies, the poles migrate on the f-plane. Figure 8.1
shows the migration of the poles as the mode make the transition from purely bound to
leaky regimes. First, suppose there is slight material loss but no leakage, then from (2),
1F E, lies in the second and fourth quadrants, respectively. As the material loss vanishes,
the poles approach the imaginary axis. For the bound case, of course, the inverse
integration path lies along the real axis. By deforming the real axis integral into a closed
contour integral on the upper (lower) half E-plane for x>0 (x<0), it can be shown that
the residue contribution from £=-£, corresponds to a evanescent ane propagating in the
x-direction for half space x>0, while the residue contribution from i=5, results in a
similar evanescent wave propagation in the -x direction for the half space of x< 0. As
the mode enters the leaky regime, the poles at $5, migrate across the real axis into the
first and third quadrants, respectively (see Figure 8.1). In this case, the real axis inverse
integration is no longer valid. Instead, in order to insure the existence and convergence
of the inverse, and the corresponding forward transform integrals, and to assure the
continuity of physical situation, a deformed integration path which maintains -£, on its
upper side and 5, on its lower side, as indicated in Figure 8.1, must be chosen. This
integration path can also be deformed into the closed contour path on the upper/lower
half £-plane, as for the bound mode, to capture the correct poles for propagation in the

ix direction; consequently, residue contributions from £= 1F .9, result in growing

124

propagating waves in the :tx direction for their > 0 (x < 0) half space, respectively, which
physically correspond to surface waves excited and propagating away from both sides of
the microstrip line.

In order to understand the problem better mathematically as well as physically
without involving the complicated expressions in the EFIE solution, we consider a
simplified example here. Let F(x) be a spatial domain function that describes a wave

originating at x=0 and propagating along the ix direction, that is

Ae'fl" x > 0,

F(x) -- Ae"""' = (3)

Aen" x < o,

where E, = E” + 1'5“ is complex propagation constant with s”) 0 but 5,; being positive

or negative, corresponding to a decaying or growing wave, respectively. This function

has no conventional Fourier transform. It can be decomposed into two parts as

HI) = 17.0!) + F.(x)

O x > 0’ (4)

Far) -

0

{Arm x 2 o,

x < 0,

P100 =
{Aem x s 0.

where function I"+ (x) and F,(x) are Fourier transformable, individually, on the complex

f-plane. Considering F, (x) first, its Fourier transform is

125

f¢(E) = yIF.(X)} = [flak-lbw
-. (5)

= [Aflflwarxetgnzode
0

This integral converges only when i, < -$,,. Thus, the Fourier transform of F,(x) exists
for all 5 in the lower half plane specified by 5, < -£,,, and is a uniquely defined analytic

function

A

f.“) = my

r e (5, < —£,,). (6)

As shown in Figure 8.2(a), function f,(£) clearly has a simple pole at £=-£p in the E-
plane which characterizes both the spatial and the spectral domain functions.

Inverse transform of (6) is a contour integral defined as

= .1. 1::
we 2“ cfxtee d5. (7)

Correct recovery of F, (x) depends on properly choosing integration path C, . To be
consistent with the forward transform, C, can and must be chosen to be any horizontal
line in the lower half plane ,5, < -£,,, and with this choice (7) recovers F,(x) correctly.
By exploiting Cauchy’s integral formula in combination with residue theory, integral (7)
can be evaluated as indicated below.

For x) O, the integral along the closed contour in the upper half plane, see Figure

8.2(a), gives

126

ff.(5)e"’d5 = ff.(E)e"‘dE + ff.(£)e"’d5 = zujrresim a: -E,l (3)
c, .

C.

By Jordan’s lemma, the integral along C: vanishes, and the residue of f,(£)e"‘ at ~51,

is evaluated as

residue = ' (E + Ep)f,(E)e"‘ = flied". (9)
£~-z,
Therefore
F (x) = _l_ ff(£)ejud5 = Ae-j‘f x > O. (10)
§ 2” c. O

For x< 0, the closed contour in the lower half plane is used. Since the integral along 0;

vanishes when x<0, and f,(£) is analytic in the lower half plane, then by Cauchy’s

integral theorem

1 1
mo = g cfxcoemde = -5 cff.(£)e"’di = o x < 0. (11)

Thus evaluating (7) along contour C, recovers F, (x) from f,(x)correctly.

When £,,<O, such that F,(x) represents a decaying wave, the pole at E =-£, is
located in the second quadrant. In this case, the real axis becomes a natural choice for
C,. When 5,, becomes positive, so F, (x) represents a growing wave, the pole moves
to the third quadrant and the real axis can no longer be chosen as C,. C, has to be a
line below the pole in the lower half plane. However, according to Cauchy’s theory, C,

can be deformed in various ways for our needs as long as it is not moved across a

127

singularity off, (5); that is the pole at 5=-5, should always be kept on the upper side of
contour C,.
The Fourier transform of F,(x) is defined in the same manner. In short summery,

= .1: 'lixd = -__A__, , , (12)
12(5) f -(x)e E j(,_,p) 896,:

= .1. It:
F.(x) 2n Cffme d5. (13)

Here 15(5) is an analytic function defined in the upper half plane 5, > 5", see Figure
8.2(b), and C, is any horizontal line above the pole at 59-5,. We can do the closed
contour integral in the upper/lower half plane for (13), and again the residue contribution
from pole at 5=5,, characterizes the -x directed wave F,(x).

If a common inversion contour C can be found so that it satisfies the requirement

for both C+ and C,, then letting C, = C, = C, we have

F (x)

F.(x) + F.(x) = 5‘; ff.(£)e"‘dE + 5‘; [1(5):de
C, C-

(14)

i It!
2” {fine as

where 1(5) = f,(5) + f_(5). The Fourier transform of F(x) is therefore uniquely defined.
When 5,,< 0, there is a common strip region around the real axis in the 5-plane in which
both f,(5) and f(5) are analytic, so any horizontal line in that strip can be used as

inversion contour C, and naturally the real axis is adopted. But when 5 A,,,>0, no straight

128

lines satisfy the requirement for both C, and C, to be a common contour. However, we
can deform both C, and C. to coincide with a curved contour C,, as shown in Figure 8.3,
which has pole 5=-5,, on its upper side while 5=5, resides on its lower side. C, satisfies
the condition for both C, and C, and is therefore qualified as a common inversion
contour.

The above example clearly demonstrates that for an inverse integral like (1), the
poles located in the left half 5-plane correspond to discrete surface waves in the x> 0 half
space, while the poles in the right half plane correspond to the same symmetric surface
waves in the x<0 half space. Consider a pole in the left half plane. When it is above
the real axis (second quadrant), it represents a evanescent wave in the x-direction and it
is said that the surface-wave is not excited; when it is below the real axis (third
quadrant), it corresponds to a growing (leaky) wave in the x—direction and the surface-
wave is said to be excited. Each pole corresponds to a single spatial frequency
component in the transform domain which characterizes a single surface wave
propagating away in a certain direction. Vice versa for a pole in the right half 5-plane.

In the above discussion, we assumed that only one single surface-wave is present.
For the case that more than one surface wave is excited (thick film structure), multiples
poles are implicated in the 5 -plane. These can be handled in a similar manner. For
some structures, such as shielded stripline, only surface wave leakage can exist. The
above discussion is sufficient for those problems. The problem of microstrip line,
however, is more complicated because, in addition to the pole singularities, there are also
branch point singularities present. In the determination of the proper integration contour,

not only the poles, but also the branch cuts, need to be considered.

129

8.2.2 Branch Point Singularities and Radiation Leakage

The p,(5) factors in the integrands of (I) produce square-root branch point
singularities with associated branch cuts. Since the integrands are even function of PKE)’
the p,(5) singularity is not implicated. Therefore the branch point singularity is

contributed only by the square-root factor p,(5) which is the wave number in the cover

 

region. Recalling that pc = 52 + (2 - k: , a pair of branch points can be defined as

(15)
£2 = k.’ - c’ = k3. - .2:- c3+c3+12(k.,k, - am.

so that

 

PC“) = t/E + 5‘ J5 - 5b. (16)

To unambiguously locate the positions of branch points if» in the 5-p1ane, the branch

cuts in the 5i-plane for the function (ft—i in (15) need to be specified. Also, appropriate

branch cuts in the 5-plane must be determined for the square root in (16) to specify the
value of p,(5). Expression (15) shows that 5, results branch cuts in {-plane too, which

will be studied in the next subsection.
First, remember that the space-domain propagation factore '1" ‘1‘” = e ‘1": *" ,

is assumed in the forward Fourier transform for waves propagating in the ix direction
for the x) 0 (x< 0) half space, respectively, and where B > 0 but a can be either positive

or negative, corresponding to a decaying or growing wave, respectively. To determine

130

the 5.2,-plane branch cuts, consider a general inversion integral similar to (l), as

F<x.y) = ff(E.C)e""e"‘dE (17)
cl

which requires that Re{5} > 0 (Re{5} < O) to correctly specify a wave in the 3Fx direction
for the x< 0 (x> 0) half space, respectively, as indicated previously by the discussion on
the residue contribution of poles, and as will be indicated below for branch cut
contribution. This requirement separates the 5 -plane into the right (5,> 0) half plane and
the left (5,< 0) half plane for the x< O (x>0) half space, respectively. It means that the
contributions (poles and/or branch cuts) from the right half 5-plane result in waves
propagating in the x< 0 half space, and those from the left half 5-plane give waves in the
x>0 half space. In the 5Z-plane the negative real axis maps into Re{5,} =0 in the 5,-
plane, and therefore is identified as the branch cut for 5,, as shown in Figure 8.4. This
branch cut insures that Re{5,,} > 0 while allowing Im{5,} to be either positive or negative
to accommodate the leaky or non-leaky regimes.

To see this more clearly, we consider the well-understood non-leaky situation here
first. For simplicity, we only describe branch point +5, and its associated cut in our
discussion; due to the symmetry between if», the behavior of -5, is obtained
correspondingly. For a slightly lossy, non-leaky mode, from (15), 5, lies in the fourth
quadrant with the negative-real-axis branch cut in the 52-plane. In this case, the most
frequently used branch cut for p, is the so called proper hyperbolic branch cut 3,, as
shown in Figure 8.5, which defines a top Riemann sheet on the complex 5-plane on

which Re{p,} > O, and therefore the factor exp(-p,y) satisfies the radiation condition.

131

Also, the real axis is used as the inversion contour for (17) as usual; we can evaluate the
inversion integral by closing the contour in the lower half 5-plane (see also Figure 8.5),

which results in

ff(£)e""e"’dfi =
c. (18)
21rj{residue at 5,} - ff(£)c-"’e"‘dE - ff(5)e"c’e1¢*dz

c; C.’

By Jordan’s lemma, the integral along C; vanishes. So expression (18) clearly shows

that the integral has two contributions: a pole residue which corresponds to a single

surface wave, as demonstrated previously, and a contribution from the integral around

the branch cut contour C; . Notice that along the contour C,- , we always have Re{5} > 0

and Im{5} <0. Also with the hyperbolic cut 3,, Re{p,} >0 always. Therefore the

integral along G; represents a superposition of decaying wave components propagating

in the -x direction in the half space of x<0, which forms a continuous spectrum in the
spatial frequency domain.

Appropriate branch cuts for p,(5) are not unique, but must be selected under
certain constraints so that the factor axp(-p,y) properly describes the behavior of different
modes (leaky or non-leaky). The choice also needs to be consistent with other portions
of the integrand so that it is a well defined analytic function, and therefore a proper
common inversion contour can be found to insure the existence and convergence of both

forward and inverse transformations. In the above discussion of non-leaky mode, the

132

hyperbolic branch cut B, is used; but actually, any branch cut that extends from the
branch point downward to infinity, that is, within the fourth quadrant, is a valid cut for
this problem, based on Cauchy’s integral theorem. Particularly, a vertical cut B, is a
convenient choice. All the cuts can be distorted in arbitrary ways as long as they pass

downward in the fourth quadrant. It should be noticed that with the downward branch

cut, the integrand function is analytic in the E, > max{ Eu» é”) upper half 5-plane. This

condition is required by the pole singularity, as shown in the last subsection, for the
existence of the forward transform and its proper inversion. For a bound mode, both the
pole 5, and branch point 5, are located in the fourth quadrant. Therefore the real axis
is within the analytic region of the integrand function and becomes a obvious choice of
the inversion contour.

As the propagation eigenvalue r varies due to changes in frequency, the branch
points 5, migrate in the complex 5-p1ane. The migration of 5, is indicated in Figure 8.6.
This can be seen from (15) and the migration is similar to that of poles. As the mode

enters the leaky regime, the branch point 5, moves across the real axis up into the first

quadrant. In this case, the proper hyperbolic cut becomes B; which extends from 5,

upward along the imaginary axis to infinity. This cut is not valid for the leaky case for
the following two reasons. First, as stated before, the cut defines a top Riemann sheet
on which Re{p,.} > 0 so exp(—p,y) is always a decay factor and no growing wave is
allowed. If we choose an inversion contour on this sheet, no radiative leaky-wave
components will be contributed over the path of inverse integration. Secondly, and more

fundamentally, the branch cut goes upward to infinity and therefore breaks the analyticity

133

of the integrand function in the upper half plane which, as indicated previously, is

required by the pole singularity in that function. Thus, it is impossible to find a proper

inversion contour on the top Riemann sheet (equivalently under the 8,! cut) that retains

the analyticity of the integrand and, at the same time, includes the possibility for

radiation leakage. Some researchers [41], [76], in attempting analysis of the leakage

problem, adopted the 3,: cut as an extension of the non—leaky regime, but chose a

inversion path that crosses the branch cut and passes onto the bottom Riemann sheet, see
Figure 8.6, and therefore picked up radiation components. This is mathematically
incorrect for the reasons that: first, by crossing the branch cut, the function is no longer
single-valued and analytic over the path of integration; second, Cauchy’s theorem cannot
be applied to the integration contour since it is not on a single Riemann sheet, which is
required by the theorem; and third, since a portion of the inversion contour passes to the
bottom Riemann sheet, some poles on the bottom sheet should also be considered, in
particular leaky-wave—poles of the layered surround, and it is not clear how to decide
whether a pole on the bottom sheet should be included or not. Basically, any cuts that
pass upward to infinity are inconsistent with the condition for the existence of the
forward transformation and should be avoided; any proper choice of inversion contour
should lie on a single Riemann sheet so that Cauchy’s integral theorem is applicable and
contributing pole singularities can be identified.

Based on the above discussion, we can conclude that any branch cut that
originates at the branch point 5, and extends downward within the fourth quadrant to

infinity is an appropriate cut for the general (leaky and non-leaky) problem.

134

Correspondingly, by symmetry, the cut associated with branch point -5, extends upward
within the second quadrant to infinity in the same way. With the branch cuts chosen in
this way, the common inversion contour can be selected as any path that extends from -
co to +00 by passing below -5, and -5, but above 5,, and 5,. All the cuts and the
contours are equivalent and can be deformed according to convenience provided that the
above conditions are satisfied and the conditions for Cauchy’s theorem are not violated.
It is important to note here that, once the branch cut is selected, it should be used
consistently throughout the analysis, particularly in the search for pole locations by (2)
where it is evident that the pole locations depends on the choice of branch cut. The
author believes that some of the confusion in previous publications are caused by the
ignorance of this fact.

Figure 8.7 shows the‘ poles, branch points/cuts and inversion contour for a leaky
mode. In our analysis, the vertical branch cut B, and the inversion contour C, are
selected. This cut defines a top Riemann sheet on the 5-plane (which is the plane shown
in Figure 8.7 on which the real part of p, can be either positive or negative, allowing
both decaying and growing waves to exist. Since C, lies on the top sheet, Cauchy’s
theorem is applied, and integration along the closed contour in the lower half plane (for

x < 0) results in

ff(£)e""e"’d£ =
c. (19)

zuy'tresidue at 5,} - ff<£)e""e"’d£ - f f(E)e""e"‘d£
c,’ C:

Since B, is chosen to approach the negative imaginary axis, by Jordan’s lemma, the

135

integral along C; vanishes. Comparing the above with (17) and the previous discussion

of non-leaky modes, it is evident that the analysis and formulation for both cases are
consistent and unified. In fact, the non-leaky mode is just a special case of the leaky
mode. In the leaky case, expression (19) shows that the residue contribution of the pole

in the first quadrant corresponds to surface-wave leakage, as already known;

contributions from the integral around the branch cut contour C, represent the

continuous spectrum contributed by space waves which now include both decaying and
growing components. The integral along the portion of branch cut that passes into the
first quadrant gives a superposition of growing waves with different spatial frequencies,
which corresponds to radiation leakage, while the remainder just results in evanescent

waves.

8.2.3 Branch Point Singularity and Root Search on the Complex {-Plane

In solving the complex homogeneous integral equation for the leaky modes of a
microstrip line as an eigenvalue problem, the same Chebyshev Galerkin’s method of
moments that was utilized in Chapter 4 is applied, and a similar complex matrix
eigenvalue equation results. The matrix elements, which are now given as complex
contour inverse integrals for leaky modes, are evaluated according to the guideline

discussed in the last two subsections. Again, requiring the determinant of matrix A to
vanish gives the complex propagation eigenvalue 5,. The root search for det[A(C)] = 0 ,
however, is now performed on the complex {-plane. Therefore, the singularities on the

{-plane need to be considered. From expression (15), rewritten here as

136

 

a, =,/k,+c ,Ik,—c. (20)

it is evident that there are a pair of branch points ik, and associated branch cuts in the
{-plane. The branch cuts are determined by the required condition on 5,, which is
Re{5,} > 0. As previously demonstrated, this condition corresponds to the negative real
axis branch cut in the 5,2,-plane or equivalently the imaginary axis in the 5,-plane. From

(15), that is

muh=fi-é-é+é<0
(21)
Imrei} = 2(k,k,, - c,c,.> = o

Expressions (21) define two hyperbolas in the {-plane, and therefore the negative-real-
axis branch cut in 52-plane is mapped onto a pair of hyperbolic branch cuts in the
complex f-plane, as shown in Figure 8.8. It should be noticed that in the root search
process in the {-plane, it is important to implement this 5,-cut in the numerical
procedure, especially in the transition region, since when the root is close to the branch
point k,, it is very possible that the iterated values could cross the branch cut and make
the function discontinuous. There are pole singularities in the f—plane too. As revealed
in the discussion in Chapter 3, poles in the {-plane correspond to discrete propagation
modes which are just what we are searching for. Notice that the poles 5, now have
complex values, and both the k, branch cut and the g; poles reside in the fourth quadrant
while the -k, branch cut and the 4; poles reside in the second quadrant. Since the ('-
plane is the spectral-domain corresponds to the space variable 2, similar to the discussion

involving the 5-plane, it is clear that the waves propagating along the waveguiding i2-

137

direction are composed of two parts, the discrete eigenmodes and the radiation modes
with continuous spectrum components. The discrete eigenmodes are contributed by the
T g; poles, which could be either pure propagating (lossless bound) modes with the T g;
poles on the negative (positive) real axis, or decaying (lossy or leaky) modes with T g;
poles in the second (fourth) quadrant, respectively, propagating along the iz-direction
for z > 0 (z < O). No growing waves (modes) exist. Similarly, the continuous radiation
spectrum is from the contribution along the T k, branch cut, which is a superposition of

decaying wave components propagating in the iz-direction for half space of z > 0 (z < 0).

8.3 Numerical Procedure and Results

The losses and leakage properties of a single microstrip line structure are studied
numerically. The complex integral equation is solved by exactly the same Chebyshev
Galerkin’s method of moments that was outlined in Chapter 4, and results in the same
matrix eigenvalue equation as that given by (4.19). All the expressions in Chapter 4 are

valid when corresponding complex variables are substituted. The root searching for
det [A] = 0 is performed on the complex f-plane, but with same numerical technique.

The matrix elements are given by the same inverse integrals as (4.17), except now the
inversion contour C, is used for integration on the 5 -plane. Since evaluation of the
integrands involves computing Bessel functions, but the computation of complex Bessel
functions is time consuming, C, is deformed into C,, which is mostly along the real axis,
as shown in Figure 8.9, and therefore numerically more efficient.

Numerical results are obtained for a single open microstrip line that has the same

parameters as the one analyzed in Chapter 4, except some material losses are introduced

138

here. Figure 8.10(a) shows the dispersion curves of the principal mode and the first
three higher-order modes of the microstrip line. The phase constants are normalized by
the wave—number of free space k,. The dashed line is the dispersion curve of the TM,
background surface-wave mode. Approximately, when the phase constant 3', of a higher-
order microstrip mode crosses this line, this mode becomes leaky. Figure 8.10(b) shows
the attenuation constant j’, of the first three higher-order modes in the leaky regime. The
results are close to those obtained by Oliner using a different approach and Michalski
with a similar one [73], [76].

The transition behavior of microstrip modes between bound regime and leaky
regime is of interest and importance to the understanding of the leakage problem. Figure
8.11(a) shows an enlarged picture of the transition part of the dispersion curve for the
EH, mode. Figures 8.11(b) and 8.11(c) show the migration trajectories of the surface-
wave pole 5, and the branch point 5,, corresponding to the positions in Figure 8.11(a),
when the EH, mode enters into the leaky regime. The behaviors of the pole and the
branch point are in agreement with the theoretical prediction. Transition behaviors of
other higher order modes are potentially more complicated. Some recent reports have
indicated that more involved transition behaviors exist for microstrip lines and other
microstrip configurations [85], [86]. This is still a open area with many unanswered
questions. More study is needed and is presently ongoing.

Table 8.1 gives the propagation constants of the EH, mode at 10 GHz, where it
is in the leaky regime, when different current expansion terms are used. It is found that
in the leaky regime, both the phase constant and attenuation constants converge quickly,

as in the bound regime. However, in the leaky regime, complex computation is more

139

time consuming numerically.

140

 

 

 

 

 

 

 

Ci
#—
-* 5p leaky mode
+
Relél
A
-§,, *
’5
Ci
gr) )'—>_ * i,
bound mode (lossless) material loss

 

Figure 8.1 Illustration of the migration of the surface-wave poles in the complex 5
plane as the microstrip modes make the transition from purely bound to leaky

regime.

141

A 1mm

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)
Figure 8.2 (a) Pole and inversion contour for f,.

(b) Pole and inversion contour for f.

142

Imlé)

f- analytic

 

 

 

 

 

 

(a)

 

 

 

 

 

 

(b)

Figure 8.3 (a) Common inversion contour for the case of 5,,,(0 (decaying wave).

0)) Common inversion contour for the case of 5,,> 0 (growing wave).

143

‘Imagt

Re{§,}=0 . A

 

 

 

 

 

. »
B Retégl
(a)
A Imlébl
i'.-.33125's532323333352333233533332235233325332333223332233333323333
Re I i b1 <0 335133513331323:323:33}:{ififiitfiizgirfiié {$3 [#13331
’23:};23323:,23;31:.12:.:I;;I;;I;;I;:I:.;2:_;Z;;I:;Z;;I;;I >
:133:1}:33321332133i¥233§ri3:333:33}233321332133:I?Ezriiztiizzifzif 1%,}
(b)

Figure 8.4 (a) Negative real-axis as branch cut in the complex 5i—plane. (b) The

maping of the branch cut in the 52-plane into the complex 5,-plane.

144

 

 

 

 

 

Figure 8.5 Singularities and defamation of the integration paths for a nonleaky mode

in the complex 5-plane.

145

 

 

 

Bu bound mode

 

Figure 8.6 Illustration of the migration of the p, branch points and their associated
hyperbolic cuts in the complex 5 -plane as the microstrip modes make the

transition from purely bound to leaky regimes.

146

Alm[§}

 

 

 

 

 

 

 

 

 

Figure 8.7 Singularities, inversion contour and defamation of integration paths for a

leaky mode in the complex 5-plane.

147

‘Imtm

 

 

 

Figure 8.8 Branch points and the associated branch cuts for 5, in the complex {-plane.

148

 

 

 

 

 

Figure 8.9 Deformed integration path for numerical computation.

149

 

 

 

 

 

 

 

3 k I
g L——-——————: —————————— :
¢3°§##,,r, $3
H : Em 5
a z :
’3' : EH t
8ao§ ' ‘ an
o I I
3 a :
in? E c
-e 3 ..Af. _ .— 3
ng:-— -—- - ——————————— fifl
a a We 5
E) E leaky regime E
0.0 - I I I I I I I I I I I I I I I I I I I l I I I I I I I I I l I I I I I I I Ii '- 0.0
0.0 10.0 20.0 30.0 40.0

Frequency (GHz)
Figure 8.10(a) Dispersion curves of the principal mode and the first three higher-order

modes in both bound and leaky regimes for the microstrip line.

150

 

 

 

 

{ 10.0 1 _ 10.0
v : E
+3 E E
a : :
,3 8.0 ': :8.0
m "l .—
G I :
O : :
r: 6.0 E 56.0
.3 : EH1 :
«O-J -‘ .—
3 E E
G I :
4.0 : :4.0
g : EHa :
<3: : E
a 5 E
a 2.0 '3 "-32.0
g 0.0 : Hjj I I I I I I l I I I I I I I I I I I I I I H I I I : 0.0
0.0 10.0 20.0 30.0

Frequency (GHz)
Figure 8.10(b) Attenuation constants of the higher-order EH, and EH, modes for the

microstrip line in the leaky regime.

151

C)

 

IEFH nWCHje

'0
0|

.0
(O
(.n

 

2

Normalized Phase Constant {,/k,,
C3
C3
Ill]llllllIlllllllllllllllLllIlllllllJl

 

 

 

0.90 IIFIIllllIIllllIllllllllIIIII
1.3.60 15.72 13.84 13.96 14.08 14.20
Frequency (GI-12)
Figure 8.11(a) Enlarged dispersion curve of the EH, mode in the transition region

between bound and leaky regimes for the microstrip line.

152

0.3 -
f—Plone i

0.1 -‘

 

-003 -

, El/ko

2
fr/ko
mm
0.1 0.3 0.5

Figure 8.11(b) Migration trajectory of the TM0
surface-wave pole 5,,

 

Figure 8.11(c) Migration trajectory of the branch
point 5, for the EH0 microstrip mode.

153

Table 8.1 Convergence of the prOpagation eigenvalues upon the numbers of basis

functions used in the current expansion for the leaky EH0 mode at 10 GHz.

 

F

 

 

 

 

 

 

 

 

 

 

Number of terms Normalized phase Attenuation
N constant {,/k,, constant §',/k0
1 0.4752870 2.539088 I
2 0.4526595 2.481156
3 0.4525928 2.480961
4 0.4525926 2.480961
5 0.4525921 2.480959 I
u _

 

154

CHAPTERNINE

CONCLUSION

This dissertation presents a rigorous full-wave integral operator formulation for
electromagnetic interactions among microstrip configurations in the layered environment,
and a comprehensive study on one of the most essential elements, the microstrip
transmission lines. Broad electromagnetic phenomena associated with microstrip
transmission lines, including their propagation spectrum, dispersion characteristics,
coupling effects, current and field distributions, power propagation, losses and leakage
and/or radiation properties are investigated via the rigorous full-wave integral equation
approach.

First, electric and magnetic dyadic Green’s functions are developed for layered
media with both dielectric and magnetic contrast and within which arbitrary current
sources immersed. The Fourier transform technique is utilized in this development, and
consequently the Green’s functions are obtained in the form of Sommerfeld’s inverse
integral. The Green’s functions provide the foundation for the integral operator theory.
The Hertzian potential, and associated electric and magnetic fields are obtained in terms
of the Green’s functions and current sources; based on these, integral equations for
general microstrip structures are constructed.

This dissertation is mostly devoted to analysis of the microstrip line structure by
application of this integral-operator description. In Chapter 3, a Fourier transform-
domain electric field integral equation (EFIE) description for microstrip transmission

lines is formulated. The complete propagation-mode spectrum of microstrip line is

155

identified from a singularity expansion of its current in the complex spatial frequency
plane. A singularity expansion method (SEM) in the spatial frequency domain is
consequently identified. An excitation and coupling theory for the electromagnetic
response of microstrip lines to impressed radiation is developed.

Propagation modes on isolated and coupled lossless microstrip transmission lines
are analyzed in Chapter 4 and Chapter 5. A numerical solution to the homogeneous
EFIE’s is implemented by Galerkin’s method of moments. Chebyshev polynomials with
square-root edge factors are utilized as basis functions. Results of dispersion
characteristics and current distributions for the principal and higher-order modes are
presented. Particularly, the currents are obtained in a convenient quasi-closed form of
rapidly convergent Chebyshev polynomial series. For coupled lines, the dependence of
coupling effects upon line spacing, the behavior of the coupled-mode waves, the
dispersion characteristics and the current distributions of the coupled system are
analyzed. Numerical implementations are developed and good results are obtained.

A full-wave perturbation theory for N coupled microstrip transmission lines, based
upon the rigorous EFIE formulation, is presented in Chapter 6. In solving the coupled
EFIE’s, the eigenmode currents of the isolated line are used as first-order approximations
for the nearly degenerate eigenmode currents of the loosely coupled system, and a simple
perturbation matrix eigenvalue equation is obtained. The perturbation theory provides
good physical insight regarding the coupling effects and greatly simplifies the
computational procedure. An experimental verification is carried out and the theoretical
and experimental results are in a good agreement.

As an important application, an accurate circuit modeling of the non-TEM

156

microstrip line structure is described in Chapter 7. To obtain a complete equivalent
circuit for the structure, not only the dispersion characteristics, but also the characteristic
impedance must be determined. The proper definition of characteristic impedance for
microstrip line is discussed and the current-power definition is adopted. A full-wave
analysis for the characteristic impedance of microstrip line is developed based on the
same EFIE formulation.

Leakage and radiation characteristics and material losses are the relatively poorly
understood phenomena associated with microstrip transmission lines. In this dissertation,
much effort is made in developing a unified, consistent, complex integral equation-
formulation which correctly handles the problem of general lossy and leaky microstrip
lines. In discussions of the bound waveguiding modes and the non-spectral radiative
leaky modes, the propagation spectrum is reviewed and clarified. The migration
behaviors of the pole and branch-point singularities in the complex spectral planes are
carefully examined. The branch cuts and the inversion contour for the Sommerfeld-type
inverse integrals are then determined and justified with rigorous mathematical bases.
From the complex EFIE analysis, the leaky-mode and radiation properties of microstrip
transmission lines are deduced naturally and consistently.

The research in this dissertation has reflected some progress on the analysis and
understanding of electromagnetic phenomena associated with microstrip structures.
However, there remain many t0pics which deserve further research efforts in the future.
As indicated in Chapter 8, the transition behavior of microstrip modes between the bound
and leaky regimes is very involved, and remains as an unanswered question. Also, the

continuous spectrum radiation components have not been fully conceptualized and

157

quantified. The leakage and radiation properties of coupled multi-strip structures are an
important issue, and more quantitative study is nwded. Furthermore, for practical
MMIC applications, more complex structures and systems should be considered.
Directly solving the coupled integral equations for a large system is very complicated and
even impractical, therefore certain schemes like the perturbation approximation need to

be developed further.

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165

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