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This is to certify that the

dissertation entitled

THE SINGULARITY EXPANSION METHOD FOR
INTEGRATED ELECTRONICS

presented by
George Warren Hanson

has been accepted towards fulfillment
of the requirements for

 

 

PhoDo degree in Electrical
Engineering

§ \

4'

Major profess

DMW

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

If

 

Ir

 

 

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LEBMRY
Michigan State

University L
Le

 

 

—.____.

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity institution
c:\citc\dahedue.pm3-p.1

 

 

 

——_

 

THE SING

 

THE SINGULARITY EXPANSION METHOD FOR INTEGRATED ELECTRONICS

BY

George Warren Hanson

 

 

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

 

i
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In thi
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representati

 

 

 

 

ABSTRACT

THE SINGULARITY EXPANSION METHOD FOR INTEGRATED
ELECTRONICS

By

George Warren Hanson

In this dissertation, an approximate theory for the analysis of systems of microstrip
devices in the resonant frequency regime is presented. Standard integral—operator
techniques applied to this type of problem are often computationally inefficient due to the
presence of Sommerfeld integrals associated with the Greens functions which describe
the layered environment. When the near-resonant frequency regime is considered, the
unknown current on the microstrip device may be represented by a series of pole—
singularities in the complex frequency plane, leading to an efficient formulation of the
problem.

An electric field integral equation (EFIE) is developed for conducting devices
embedded in the tri-layercd conductor/ film/ cover environment typical of microstrip
circuits. This EFIE is conceptually exact, and forms the basis for most rigorous
investigations of the electromagnetic (EM) properties of such systems.-

It is well known that isolated and loosely coupled systems of microstrip components
exhibit highly resonant behavior. This motivates expanding the unknown current on the
device in a series of pole-singularities in the complex temporal-frequency plane. This

representation for the device current leads to an efficient technique for the relatively

 

 

 

general I
microstri
effective;

The
from a 0
leading [t
is found 1
moments

presented

 

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general deduction of EM prOperties of microstrip circuits. The specific example of a
microstrip dipole excited by a nearby transmission line is studied to demonstrate the
effectiveness of this method.

The EM properties of systems of coupled, nearly-identical devices are investigated
from a coupled set of EFIE’s. The singularity expansion technique is again invoked,
leading to an approximate perturbation solution for the system—mode resonances. This
is found to be an accurate and efficient method when compared to the direct method of
moments (MOM) solution to the same problem. Numerical and experimental results are

presented for a two-dipole system to support the validity of this approximate solution.

 

 

 

TO MY BEST FRIEND, SHEILA

 

 

 

iv

 

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SUpport,
his willin
of my st]
Byron Dr

Furth
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Finall
their cons
would als

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ACKNOWLEDGEMENTS

I would like to express sincere thanks to Dennis P. Nyquist for his guidance,
support, and inspiration throughout the course of this research. I especially appreciate
his willingness to accommodate my rather difficult schedule during the last few months
of my studies. Special thanks are also due Kun-Mu Chen, Edward J. Rothwell, and
Byron Drachman for their assistance in completing this research.

Furthermore, I would like to thank James Kallis for the outstanding job he did in
performing the experimental component of this work. I would also like to acknowledge
my fellow graduate students, in particular Jack Ross for the generous use of his computer
programs which enabled efficient data collection for this and other projects.

Finally, I am grateful to my wife, Sheila Hanson, and my mother, Ruth Hanson, for

their constant encouragement and support during the course of my graduate study. I
would also like to thank Dave and Carol Hyster for their generous provision of

computational resources.

 

 

TABLE OF CONTENTS

 

INTRODUCTION .................................. 1
ELECTROMAGNETICS OF PLANARLY LAYERED MEDIA ...... 6
2.1 Introduction .................................. 6 l
2.2 Electric Hertzian potential dyadic Green’s function .......... 7
2.2.1 Primary Green’s component .................... 11
2.2.2 Reflected Green’s dyad for sources in the cover ....... 13
2.3 Electric dyadic Green’s function ..................... 17
2.3.1 Source-point singularity of the electric Green’s dyad ..... 18
2.4 Analytical and numerical considerations in the evaluation of the
Green’s dyad ................................. 19
2.4.1 Spectral singularities of the Green’s dyad ........... 20
2.4.2 Integration techniques for the efficient numerical
evaluation of the Green’s dyad .................. 26
2.5 Summary ................................... 31
THE SINGULARITY EXPANSION METHOD FOR INTEGRATED
ELECTRONICS ................................... 33
3.1 Introduction .................................. 33
3.2 Formation of the electric field integral equation ............ 36
3.3 Singularity expansion of device currents ................ 38
3.3.1 Motivation of current expansion: the transient singularity
expansion method .......................... 39
3.3.2 Determination of natural modes ................. 41
3.3.3 Determination of excitation amplitudes ............. 42
3.4 SEM Analysis of the microstrip dipole ................. 46
3.4.1 Normalization constant ...................... 50
3.4.2 Coupling coefficient ........................ 56
3.5 Summary ................................... 61

vi

 

 

 

 

 

FULL-WAVE SOLUTIONS OF THE FUNDAMENTAL EFIE AND
EXPERIMENTAL AND THEORETICAL VALIDATION OF THE

SEM THEORY .................................... 64
4.1 Introduction .................................. 64
4.2 Hallen-form solution with sub-domain basis functions ........ 67
4.2.1 MOM Solution of the general HFIE ............... 71
4.2.2 Separation Of the Hallen equation for even/Odd mode
symmetry ............................... 77
4.3 Entire-domain basis function solution of the EFIE for microstrip
dipoles ..................................... 79
4.4 Comparison Of MOM solutions ...................... 84
4.5 Results ..................................... 96
4.5.1 Current distribution ......................... 97
4.5.2 Frequency response ........................ 100
4.5.3 Variation of dipole current as a function Of
dipole/transmission-line separation ............... 103
4.5.4 Loss considerations ......................... 105
4.6 Summary ................................... 110
COUPLED MICROSTRIP DEVICES ...................... 112
5.1 Introduction .................................. 112
5.2 Approximate perturbation theory for coupled devices ........ 113
5.2.1 Natural system-modes ....................... 114
5.2.2 Coupled-mode perturbation equations .............. 116
5.3 MOM Solution for coupled dipoles with EBF’S ............ 119
5.4 Numerical and experimental results for coupled dipoles ....... 126
5.5 Summary ................................... 134
EXPERIMENTAL METHODS .......................... 136
6.1 Introduction .................................. 136
6.2 Isolated dipole resonant characteristics ................. 141
6.3 Transmission line fed dipoles ....................... 148
6.3.1 Negligence of secondary coupling effects ........... 149
6.3.2 Forced current distribution .................... 151
6.4 Coupled dipoles .......... _ ..................... 155
6.5 Summary ................................... 158
CONCLUSIONS AND RECOMMENDATIONS ............... 160

vii

 

 

 

Table 4.

Table 4..

Table 7..

 

 

LIST OF TABLES

Comparison Of resonant wavenumbers Obtained by different

Table 4.1:
singleoterm current distributions ..................... 84
Table 4.2: Effect of dielectric and ohmic loss on the complex resonant
wavenumber ................................. 108
Table 7.1: Theoretical and measured Quality factors ................ 148

 

viii

 

 

 

Figure 1

Figure 2
Figure 2

Figure 2

Figure 2.
Figum 2.

Figlue 2.

Figure 2.

Figure 3.

Figure 3.:
Figure 3.:

Figure 3.4

Figure 3,5

 

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LIST OF FIGURES

 

 

 

Figure 1.1: Typical microstrip system consisting of transmission line and

dipole elements. ‘ .............................. 2
Figure 2.1: Tri-layered background environment for integrated electronics. 9
Figure 2.2: Principal and scattered electric Hertzian potential components. . . . 10
Figure 2.3: Complex lambda-plane singularities of the Green’s dyad

components. ................................ 22
Figure 2.4: Complex lambda-plane with integration contour. .......... 23
Figure 2.5: Branch cuts in the complex lambda-plane. .............. 27
Figure 2.6: Proper branch cuts and the associated integration contour for

studying resonant phenomena ....................... 28
Figure 2.7: Alternative branch cuts for investigation of resonant phenomena. 29
Figure 3.1: General conducting device embedded in a tri-layered

conductor/film/cover environment. .................... 34
Figure 3.2: Microstrip device excited by an impressed source J. ......... 37
Figure 3.3: Microstrip dipole excited by a nearby transmission line ........ 47
Figure 3.4: Measured real resonant frequency and Q—factor of a dipole

excited by a microstrip transmission line as a function of

dipole/line separation. ........................... 49
Figure 3.5: Microstrip dipole eigenmodes and their associated current

distributions Obtained by pulse-function MOM solution, 40

pulses. ..................................... 51

ix

Figure I

Figure 3

Figure 3
Figure 3
Figure 4
Figure 4.
Figure 4.
Figure 4.
Figure 4..

Figure 4.1
1:igure 4.“

Figlue 4.5
Figure 4.9
Figure 4.1

Figllre 4,1

 

 

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.' M . ‘ -' ‘ .. ' _ _ ‘-_ . - - _
mamwe_mhw flan—tees . 3.;h .

Figure 3.6:

Figure 3.7:

Figure 3.8:

Figure 3.9:

Figure 4.1:

Figure 4.2:

Figure 4.3:

Figure 4.4:

Figure 4.5:

Figure 4.6:
Figure 4.7:

Figure 4.8:

Figure 4.9:

Figure 4.10:

Figure 4.11:

“‘3.“

Comparison Of nullspace current distribution (pulse function
MOM) and approximate current distribution (eq’s. 12,13) for the

first even/Odd modes ............................ 53
Comparison Of nullspace current distribution (pulse function

MOM) and approximate current distribution (eq’s 12,13) for the

second even/Odd modes. .......................... 54
Electric field distribution of a microstrip transmission line,

principal even propagation modes, x-component of current ...... 59
Local and global coordinate system used for field component

evaluation. .................................. 60
Microstrip dipole subdivided into segments for pulse function

expansion (a). Pulse function distribution (b) .............. 72
Convergence of pulse-function MOM solution for real resonant
wavenumber. ................................. 88
Convergence Of pulse-function MOM solution for imaginary

resonant wavenumber. ........................... 89
Convergence Of entire-domain basis function MOM solution for

real resonant wavenumber. ........................ 90
Convergence Of entire-domain basis function MOM solution for
imaginary resonant wavenumber. ..................... 91
Real resonant wavenumber versus film permittivity. ......... 93
Imaginary resonant wavenumber versus film permittivity. ...... 94
Comparison Of free-space, coupled dipole natural modes and

microstrip modes. .............................. 95
Current distribution near first even resonance for a parallel-

coupled dipole. ................................ 98
Current distribution near first even mode for a perpendicular-

coupled dipole. ................................ 99
Comparison between SEM theory and PF MOM solution for

current amplitude vs. wavenumber. ................... 101

 

 

 

Figure 4
Figure 4
Figure 4
Figure 4

Figure 5.
Figure 5.
Figure 5.

Figure 5.

Figure 5 .

Figure 5.1
Figure 5 .‘
Figure 6.l

Figure 6.2

Figure 63
Fi’o’llre 64
Figure 65

Figure 6.6:

 

Figure 4.12:

Figure 4.13:

Figure 4.14:

Figure 4.15:

Figure 5.1:
Figure 5 .2:
Figure 5.3:

Figure 5 .4:

Figure 5.5:

Figure 5.6:

Figure 5.7:

Figure 6.1:

Figure 6.2:

Figure 6.3:

Figure 6.4:

Figure 6.5:

Figure 6.6:

-0.“ .. ' .. -. '..—. 4.... *..‘,,7' .- "4-4; 3.: -.-_..---

Comparison between SEM theory and EBF MOM solution for
current amplitude vs. wavenumber. ................... 102

Comparison between SEM theory with and without finite
conductor impedance accounted for, and measured Q-curve. . . . . 104

Experimental and theoretical current amplitude vs. separation

for a parallel-coupled dipole, with measured Q—factor. ....... 106
Experimental and theoretical current amplitude vs. separation

for a perpendicular-coupled dipole. ................... 107
A system of two coupled dipoles. .................... 115
Global and local coordinate systems Of a two—dipole system. . . . . 121
System-modes for two identical, parallel coupled dipoles. ...... 127
Measured parallel-coupled dipole response vs. frequency and

separation. .................................. 129
Resonant wavenumber vs. longitudinal separation dsz. ........ 130

Resonant system-modes for non—identical, parallel-coupled

dipoles. .................................... 132
Resonant system-modes as the relative angle between two

dipoles is varied ................................ 133
E—field probe structure used in measuring microstrip device
characteristics. ................................ 139
T—line probe structure. ........................... 140

Investigation of isolated-dipole resonant frequency and quality factor
using T-line and E-field probes. ..................... 143

Typical results for transmission (Sn) measurements made on a
isolated microstrip dipole. ......................... 144

Theoretical current and charge distribution (magnitudes) for an
isolated microstrip dipole. ......................... 146

Measurement system for the investigation Of transmission-

xi

 

 

Figure .

Figure t

Figure l

 

Figure 6.7:

Figure 6.8:

Figure 6.9:

 

line/ dipole interactions. ........................... 150

Experimental set-up for measuring microstrip dipole current
distribution. .................................. 152

Experimental configuration for the investigation of coupled-
dipole characteristics. ............................ 156

System-modes of two coupled, 5 cm dipoles separated by .281
cm. ....................................... 157

xii

 

 

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CHAPTER ONE

INTRODUCTION

This dissertation presents an approximate theory for the analysis of systems of
microstrip devices in the resonant frequency regime. A typical microstrip system is
depicted in Figure 1.1, consisting Of a transmission line feed for a two dipole array.
This dissertation is intended to provide an efficient method of analysis for the
investigation Of electromagnetic phenomena associated with these systems.

Early work on the analysis of microstrip radiator characteristics centered on

approximate modeling techniques, such as applying transmission line analogies to

 

 

rectangular patches fed at the center of a radiating wall [1]. A more sophisticated
technique, the modal-expansion method, was latter applied to study a variety of radiator
shapes [2]. A thorough survey of microstrip antenna element technology from its
inception until 1981 is given by Carver and Mink [3], while a similar survey of
microstrip array technology is found in Mailloux et al. [4].

Most of the early methods are approximate, and do not account for all phenomena

associated with the radiator itself, and the background environment in which it resides.

 

A rigorous study Of microstrip dipole elements was presented by Rana et al. [5], using

 

 

 

 

 

 

Figure 1. 1

 

 

Transmission line

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Dipoles

 

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Figure 1.1: Typical microstrip system consisting of transmission line and dipole
elements.

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integral equations involving the conceptually exact Green’s functions for the layered
microstrip media. Recent efforts have concentrated on this approach [6-11], and
considerable attention has been given to evaluation of the slowly convergent Sommerfeld
integrals associated with the Green’s functions [12-16]. This technique accounts for
space and surface-wave radiation, dielectric loss, and mutual coupling among system
elements. A disadvantage Of this method is the long computation times needed to
evaluate the Sommerfeld integrals, even with relatively efficient integration routines.

As an alternative to the above method, an integral-Operator approach is presented
here which involves the rigorous Green’s functions in an efficient manner. This
efficiency is due not to the specific integration scheme employed, but rather to the
utilization Of known characteristics Of microstrip dipoles near resonance, which is
generally the frequency regime of interest. Thus, the theory is built on an exact model,
and approximations are made at a later stage in the problem. This contrasts with other
approximate theories, which are not based on exact models.

The text is divided into seven chapters. Chapter 2 presents a derivation of the
electric dyadic Green’s function associated with the layered-background microstrip
environment. This work was originally performed by Bagby and Nyquist [17], and is
‘ included here for completeness. The Green’s dyad is in the form Of a two-dimensional,
inverse Fourier transform integral in the spectral plane. Singularities Of the spectral
integrand include branch-points and surface-wave poles (swp’s), and the physical
significance Of these singularities is discussed, along with their implication to numerical

evaluation of the Green’s functions.

 

 

 

 

 

 

 

 

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In Chapter 3, the steady-state singularity expansion method (SEM) for integrated

electronics is presented. This method, based on the SEM for transient scattering [18-20],
is an integral-Operator description of currents induced on conducting integrated devices.
The SEM evolves from the fundamental electric field integral equation (EFIE) for
integrated electronics, thus it inherently includes all loss mechanism associated with both
the device and the layered surround. It is found to be a computationally efficient method
for the analysis of integrated devices, yielding results which agree with experiment and
other full-wave methods. These other methods are presented in Chapter 4, along with
theoretical and experimental results for the example Of a dipole fed by a nearby
transmission line. Two different method of moments (MOM) solutions to the
fundamental EFIE are developed, which are used in part to validate the approximate
SEM.

Coupled systems of devices are considered in Chapter 5. An approximate
perturbation theory for coupled devices is presented, and applied to the problem of
coupled microstrip dipoles. A full-wave MOM solution is also developed, which is
intended to provide a comparison to the approximate theory. Theoretical results were
found to agree with measurements made to identify the system—mode resonances of a
two-dipole system.

The description of the experimental methods used in the course of this research is
presented in Chapter 6. Measurements were made to quantify the EM properties of both
isolated and coupled microstrip dipoles. Some experimental results are presented in this

chapter, although most are dispersed throughout the text where appropriate. Finally,

 

 

some gr

7.

dyads v

lead to 1

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although

examples

 

some general conclusions and recommendations for future work are provided in Chapter
7.

Throughout this dissertation, vectors will appear overstruck with a single arrow,
dyads with a double arrow. The assumptions that:

(1) All media are linear, isotropic, and non—magnetic
(2) The time dependance is harmonic (e’""’) and is suppressed

lead to Maxwell’s equations in M-K-S units as

V‘e'E = p (la)
W? = 0 (lb)
VxE = jeep}? (1C)
VxI? = j+jooe‘E. (1d)

Lastly, the term "device" is used throughout this dissertation, and refers to an
arbitrarily shaped conductor embedded in the layered surround. The techniques
presented here are sufficiently general to be applicable to a wide variety of shapes,
although the specific class of narrow, conducting dipoles or resonators are considered as

examples.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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CHAPTER TWO

ELECTROMAGNETICS OF PLANARLY LAYERED MEDIA

2. 1 INTRODUCTION

In this chapter, the electromagnetics of planme layered media are investigated.
Fields in the layered environment are obtained as Fourier transform integrals, and are
expressed in dyadic notation. This formulation provides a conceptually exact description
of electromagnetic interactions in layered media, and will form the basis of all subsequent
analysis.

The rigorous study of layered media problems began in 1909 with Sommerfeld [21],
who addressed the problem of the lossy half-space. His intention was to study wave
propagation along the earth’s surface, using integral-transform techniques to Obtain the
fields due to radiating elements above the earth-air interface. The resultant integrals
were highly oscillatory and slowly convergent, and have formed the generic basis for a
class of integrals known as "Sommerfeld integrals". Efficient evaluation of these
integrals remains an active research area today [12-16]. A good historical overview is

found in Bar‘ios [22].

In Section 2.2 the field equations are formulated by expressing the electric and

it

 

magneti

volume
are deve
bottom
circuit e

The
dyad. l
Green’s
this matt
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[23]. ll
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2.2 a

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magnetic fields in terms of a Hertz potential, II. This potential is in the form of a

volume integral Of a dyadic Green’s function and a current density. The field equations
are developed for a general tri-layered environment, as shown in Figure 2.1. Later, the
bottom layer will become conducting, forming the typical microstrip\millimeter—wave
circuit environment.

The next section address the problem of the source-point singularity of the Green’s
dyad. It is well-known that great care must be exercised when forming the electric
Green’s dyad in regions where the source and observation points coincide. In the past,
this matter caused some confusion since the principal value of the integral in question
depends critically on the shape of the infinitesimal singularity excluding volume used
[23]. The Green’s function is, Of course, unique, which is required by the uniqueness
theorem, and is derived in this section.

The last section address the various singularities encountered in the complex spectral
Fourier transform space, knowledge of which are necessary to compute the inverse
transform integrals. These consist of surface~wave poles (swp’s) and branch-point
singularities of the spectral integrand. The physical significance of these singularities is

discussed, as well as their implication to numerical evaluation Of the field quantities.

2.2 ELECTRIC HERTZIAN PQTENTIAL DYADIC GREEN’S FUNCTION

The Hertzian potential dyadic Green’s function is formulated in this section. The

derivation is based on the classical development Of Sommerfeld [24], utilizing Fourier

 

 

 

 

 

 

 

 

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fl . . - ., _ . . .... . . _. ..__ . J— . . . . ' '2!-
l

transform techniques. This Green’s function was originally developed by Bagby and
Nyquist [l7], and is valid for arbitrary tri-layered media. Subsequent to development
of the general Green’s function, media for y< -t will become conducting, forming the

typical microstrip\millimeter-wave environment.

fl

Consider the layered environment shown in Figure 2.1. Electric current density J

is immersed in the cover region Of a tri-layered substrate/film/cover background
environment. The film layer Of thickness t is embedded between unbounded substrate
and cover layers. The origin of coordinates is chosen at the film/cover interface, with
y normal and x,z tangential to that interface. Each layer is assumed to be linear,
isotropic, and homogeneous, with dielectric and magnetic properties ei=n2i60 and pi=p0
for i=s,f,c, where ni is the electric refractive index. The electric contrast between layer
i and j is given by Nji=nj/n,. The wavenumber and intrinsic impedance of each layer are

k,-=n,k0 and ni=n0/n,, where (komo) are their free-space counterparts.

The impressed current .7 (or an impressed polarization I3=.7/jro) radiates into the

 

 

cover region of the multilayered structure, generating electric Hertzian potential in each

 

layer, as shown in Figure 2.2. The primary potential propagates directly from the source
to a field point in the cover layer, and the scattered potential (reflected or transmitted)

arrives at a field point after being scattered from interfaces between adjacent layers. The

total potential in the cover layer is the sum of a primary potential II" and a scattered

potential II". In the i¢ 0 layer there is just a scattered potential. All components of

potential satisfy the Helmholtz equations (A.7)

 

 

y=o

 

Flgure 2| .

 

 

 

 

 

 

 

 

 

a 3
AN RR R
Cover Layer QRRRECR
(8 ) “R's" RR
CJILLC A:&RR
il yer/
(5t [1213; /
y}, //fl///////// %

 

 

 

 

 

 

Substrate layer
(SSHLLS)

 

Figure 2.1: Tri-layered background environment for integrated electronics.

9

 

 

 

Figure 2.:

 

 

 

 

 

Figure 2.2: Principal and scattered electric Hertzian potential components.

10

 

 

as derive

asasup<

will be s

2.2.1 P]

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unbound:

point. T

Where

is the fan
diStance f

makes in

”__3: i “' -' . . '. .._‘ ' .__. .-___
“ ’kn’r “an. s..-_.,.---. - .2._:.-. ' -‘.“.‘.“"‘ __ _.,_. "‘-."."“" 3.... . . . '.

-.7/jtoei i=C (1)
0 all 1'.

ft’.’
V2+k.2 ‘ =
( .){fi.}

as derived in Appendix A. A solution for the total potential in any region can be written

as a superposition of Hertzian potentials. A solution for the total potential of the form
_, .. -' -°/
n(r)= [Gay’s—J? )dV’ (2)
V free
will be sought, where C(FIF’) is the dyadic Green’s function to be determined.
2.2.1 PRIMARY GREEN’S COMPONENT
The primary wave of potential corresponds to the potential generated by a source in an

unbounded homogeneous medium, which propagates directly from source point to field

point. This potential can be written for the cover region as [25]

I ~/
EEG) = f G P(r|r’)-{(—’—ldv’ (3)
V J (DEC
where
-jk,|r-7’|
GP(?|F’) = _‘——— (4)
41: |?- i" |

is the familiar free-space Green’s function in spatial form. The quantity I11? ’ I is the
distance from a source point at F’ to a field point F . The presence of this quantity

makes it difficult or impossible to analytically integrate G’ into other functions, which

11

lllllllll
iiiiiiiiiii

 

 

 

 

will be

for G”

variable.

The

where .‘

Wavenur

between

is well-k

Uniquene

It Shl

' Spatial re

slowly cc

bot also t

will be required to perform a numerical solution. An alternate spectral representation

for G” is developed in Appendix B, which has a simple dependence on the spatial

variables x,y,z, thus facilitating the numerical solution for II.
The spectral representation for the principal Green’s dyad is found in Appendix B
as

ejio— r) e-pcly- yl

GP(F F’)= Eda»- (5)
I '[fe 2(2702P
where X=JEE +z‘C is a 2-D spatial frequency with AZ=EZ+C2 and dzl=dEdC.

Wavenumber parameters are p, =‘Mz —k,.2 with Re{p,} >0 for i=s,f,c. The equivalence

between the spectral and spatial forms,

-jk.lr‘- F’l °' jI-(r—r’) -p. Iy-y’ I
e = ff e e d2).
41: IF—F’l __ 2(21t)2pc

is well-known as the Weyl identity [26], and can be confirmed by direct integration or
uniqueness arguments.

It should be noted that the source point singularity at F =F ’ , which is obvious in the

~ spatial representation, is still contained in (5). As F-F ,equation (5) becomes very
slowly convergent. This is due not only to the loss of the exponential decay as y—>y’,

but also to the loss of the oscillatory nature of the integrand as x,z—»x’,z’. For F =F’ ,

12

 

 

 

 

 

which is

present 5

2.2.2 R

Equat

variables
y, to allc

Fourier 1

where I

Equation

Where thi

 

GP ~=~I ~_1 ”51:1
(r r) 2(21r)2 if A

which is a divergent integral. Therefore, the source-point singularity of 0(1/ IF -F’ l)

present in the spatial form corresponds to non-convergence of the spectral integral.
2.2.2 REFLECTED GREEN ’S DYAD FOR SOURCES IN THE COVER

Equation (1) is solved for the scattered potential by Fourier transforming on spatial
variables tangential to the layer interfaces. This will preserve the normal spatial variable
y, to allow implementation of the appropriate boundary conditions. A two-dimensional

Fourier transform pair is defined as

 

.. _. 1 .. .. _. g” 2
II = A a, I d a. (6)
(r) (2“), f f ( y)e

K(X,y)= f] fi(r)e-ii"‘dxdz

where I = )25 +26 . Operation of the Fourier transform on equation (1) results in

(3:; more.» = 0. (7)
Equation (7) has solutions
Aida) = WW” (8)

where the coefficient W.’ is determined by application of the appropriate boundary

l3

£.___~._3x;r-r, y; ,

 

 

 

 

conditio

The con

bounded
Desi
the total

potential

as showr

:11

where t]
implemer

Thel

I“Elohim

The total

 

conditions, derived in Appendix C. Substitution of (8) into (6) results in

- ”W.’ x ,.
Him = if (2.1:)? e!“ e man. (9)

 

The correct branch of p,(k) must be chosen to yield spectral components which remain
bounded and propagate outward as y»: on. This will be discussed in Section (2.4.1).
Designating the cover, film, and substrate layers as regions 1, 2, and 3, respectively,
the total potential in region (1) is found as the sum of a principal potential and a reflected
potential,
fig?) = fif(r)+fij(r) (10)
as shown in Figure (2.2). Using (5) and (9), equation (10) may be written as

_, .. .-_ - .1174 -pl(k)|y-y’| _, _. _
mm = 1 [fen-r [.1 L_e__dvl+W.’o)e""” 4’4 (1“
(2n)2 _.. ijel 2‘01“)

 

 

where the spatial and spectral integrations have been interchanged to facilitate
implementation of the boundary conditions.

The total potential in region (2) is the sum of a transmitted and a reflected potential,

_. _.

112(7) = 1150*) +fl§(r‘)- (12)

In a manner similar to (10), equation (12) may be written as

 

fizfr‘) = 1 2 ffeii" [ W2'(X)e"2“”+W2'(X)e "”1“” ] dzi. (13)
(2n) ..

The total potential in region (3) consists of only a transmitted wave,

14

 

 

 

 

 

where

Apr
is quite
speciali:

environi

where tl

The prin

where

 

 

fin?) = fix?) <14)

where

fir?) = 72:35 [fem mocha” ] an. (15)

Application Of the appropriate boundary conditions to determine the various W,"s

is quite tedious, and is summarized in Appendix C. Also in Appendix C, the
specialization Im{n,}—>oo is implemented, resulting in the desired cover/film/conductor

environment. The resulting potential in the cover region is given by
" ~ jnc “ — —-/ " ~/ /
II,(r) = -—fG(r r )~J(r )dV (16)
kc V
where the Green’s dyad is
time) = (Vain) + O'(F|F/).

The principal and reflected components may be decomposed as

 

 

’

3G
CxA+Gry+ 02+2Gr£
6x " dz '

where

15

 

 

The
maintain
poteno'al

of potent

C(i) arr

where

P016 Sing

ofzham

 

e11(r—7)e-pcly-yl

GP(F|F’)= H 20 )2p ————d2A
TC

GKFIF’) .. 1m)

G,{(F|F’)= ff

Gc'(F F’) ‘°' C((A) 2(2n)2p,

e jI-(r-F’) e -P.(y+y’)

(17)

The reflected Green’s component G,’ yields tangential components of potential

maintained by tangential components of current, while G; yields normal components Of

potential maintained by normal components of current and G: gives normal components

of potential coupled to tangential components of current. Coefficients R,(l), Rn(l), and

C(A) are given in Appendix C as

 

A
so) = ”if”. ,(A) = —N’( )
Z (A) Z ‘0»)
2 N2 —1
C (it) = ___: f” )p‘
z (nzra)

where

N10) = pg TprOthpft)
N20) = min—dentin,»
Z‘o) = Nip.+p,tanh<p,r)
Z ”(1) = pc +pfcoth(pft).

(18)

(19)

Pole singularities of the reflection and coupling coefficients, associated with the vanishing

of Z " and Z ‘, lead to surface waves excited along the layered background environment.

16

 

 

 

 

 

 

 

 

These w:

2.3.11;

The 6
quantity i

is given i

It is desir

potential 1

Where @‘
as it; 1)

manipulati
3 numeric;

integral re

These will be discussed further in Section (2.4.1).

2.3 ELECTRIC DYADIC GREEN’S FUNCTION

The electric field in the cover region, rather then the Hertzian potential, is the needed
quantity for analyzing the electromagnetics of the layered environment. This relationship
is given in Appendix A, and with (16) becomes

-1'n,
kc

 

5(a) = (kc2 + W.) [C(FIF’) -.7(F’)dV’. (20)
V

It is desired to pass spatial derivatives under the spatial integral found in the Hertzian

potential to obtain

_knc f6 ‘(FlF’)-.7(F’)dV’ (21)

C V

13(7) =

 

where G"(F|F’) is an electric dyadic Green’s function. Representation (21) is desirable

as it; 1) provides a compact notation for the electric field amenable to algebraic
manipulations, and 2) allows implementation of the spatial derivatives analytically before
a numerical solution is undertaken. Passage of these spatial derivatives under the spatial

integral requires special care, which is the subject of this section.

 

 

 

 

 

2.3.1 5

under th
be contir
Green’s
of the al
arising f
can be h
incorpor.

The

given by

The third

invoking

 

 

 

 

2.3.1 SOURCE-POINT SINGULARITY OF THE ELECTRIC GREEN ’3 DYAD

The electric field in the cover is recovered from the Hertzian potential as (20) where
C(FIF’) is given by (17). It is desired to pass the spatial differential operator W-
under the volume integral in (20). This requires that the integrand of the volume integral
be continuous with continuous second derivatives [27]. The reflected components of the
Green’s dyad posses this property, but the principal component does not. The presence
of the absolute value function ly-y’ I in (4,5) gives rise to a source-point singularity
arising from derivatives with respect to the normal coordinate variable. This situation
can be handled by defining the spatial integral in a principal value (P.V.) sense, and

incorporating an appropriate correction term [23].

The electric field in the cover region, which contains electric current sources, is
given by (20), rewritten as

-1'n.
k

C

Em = (k,2 +vv-) [G'-T(F’)dV’
V
-jnck, TGP-flF’) dV’ (22)
V

flvvffcria’) dV’.
k V

C

The third term in (22) demands careful attention. It can be properly evaluated by

invoking Leibnitz’s rule, and excluding a specific principal volume [28], ie.;
W‘IGP-KF’MV’ = P.V.fVV'GP~.7(F’)dV’ + foe-7’)
v v

18

 

 

 

 

 

where P

and RV.

dyad, wt
for the It

above, ti

where

As

tangentizu

2.4 A

E
The t
integrals l
oscillatoq
is used to

undertake

 

 

where P.V. designates evaluating the integration in a principal value sense,

lim
/ _ /
P.V.f{---}dV —"s“0 f {---}dV
V V4,

and RV. is a shape dependant principal value. The term EMF— F’) is a depolarizing

dyad, which is dependent on the specific principal volume used. It should be noted that
for the topic Of this work, a slice principal volume is naturally assumed [28]. With the

above, the electric field may be written as

15(7) = 5% f6‘(r‘°|r’)-J‘(F/)dv’ (23)
C V
where
6‘6“er = P.V.(kcz+VV-)G(F|F’)+f5(F-F/). (24)

As will be seen later, the depolarizing dyad i=9)? is never required, as only

tangential current components are needed to describe the desired interactions.

2.4 ANALYTICAL AND NUMERICAL CONSIDERATIONS IN THE
W
The Green’s dyad components given by (17) are generally known as Sommerfeld
integrals [24], which are notoriously difficult to compute. In general, they exhibit very
oscillatory, slowly convergent behavior. Typically, the Method of Moments (MOM) [29]
is used to solve the equations associated with these dyads. The type of MOM solution

undertaken, such as the use of Galerkin’s method or the type of expansion or testing

19

 

 

 

 

 

function
topic is i
the folk
Sommer
branch p
with ph

significa

2.4.1 8]

Spec
h-plane.
allows fc
transfom

Pole
Physicall
layering,
is the solr

Unde
P016 singi
aXiS- Re;

 

 

 

function chosen, will greatly effect the accuracy and speed of obtaining a solution. This
topic is discussed in Chapter 4, after the relevant equations are formulated. In this and
the following sections, the singularities inherently present in the integrands of the
Sommerfeld integrals are discussed. These singularities are in the form of poles and
branch points in the complex X-plane, as shown in Figure 2.3, and each is associated
with physical phenomena. Knowledge Of these singularities, and their physical

significance, is paramount to accurate evaluation of these integrals.

2.4.1 SPECTRAL SINGULARITIES OF THE GREEN’S DYAD

Spectral singularities of the Green’s dyad are poles and branch points in the complex
h—plane. Accurate determination of the location of these singularities is possible, and
allows for determination of the correct inversion contour when evaluating the inverse
transform (spectral) integrals.

Pole singularities occur in the reflection and coupling coefficients given by (18).
Physically, these poles correspond to surface waves launched in the background dielectric
layering. There is no pole associated with the principal Green’s component, since this
is the solution for a homogeneous medium.

Under the assumption that the background environment has limitingly small loss, the

pole singularities are located in quadrants two and four, infinitesimally close to the real-k
axis. Representative pole locations are shown in Figure 2.3. Ignoring branch points for

the moment, Figure 2.4 depicts the complex—x plane with pole singularities p0, p1, pN.

20

 

 

 

 

 

 

 

 

Implied
numeric:
at i 0° 1
that the]
inversior
Cauchy-l
have no

Tran

leading t

where K=
is also tl
waveguk

Tran.

and ther

Equation
Symmetn'

least one

 

 

Implied inversion contour Cr is shown slightly offset from the real-A axis, to avoid

numerical instabilities near the surface—wave poles. By closing the original contour Cr
at -l_- on by a semicircle of infinite radius, and invoking the residue theorem, it is seen
that the poles contribute a discrete field component from each residue. Equivalently, the
inversion contour may be deformed around the pole singularities to allow use of the
Cauchy-Goursat theorem [30]. The integration along the infinite semicircle is found to

have no contribution, by proper choice of the branch cuts, as discussed later.
Transverse electric (TE) surface waves are associated with the vanishing of Z ”(A) ,

leading to the eigenvalue equation

tanoct) = i (25)
P

C

where K = ~ij is a commonly defined wave parameter for the film region. Equation (25)

is also the eigenvalue equation for TE—odd surface-wave modes of the symmetric slab

waveguide [31], which exhibit a low—frequency cutoff.
Transverse magnetic (TM) surface waves are associated with the vanishing of Z ‘(A),

and the resulting eigenvalue equation

mm) = wifl. (26)
K

Equation (26) is the eigenvalue equation for even TM surface—wave modes of the

symmetric slab waveguide. These modes do not have a low-frequency cutoff, and so at

least one pole singularity will always be present in the complex k-plane. The cutoff

21

 

 

 

 

 

 

Figure ‘

 

Figure 2.3:

 

 

Complex h-Plone

3*: Branch Point
><= SurFoce-Vove Pole

XZU

XX
Q'DX
F'OX
zUX

 

Complex lambda-plane singularities of the Green’s dyad components.

22

 

 

 

 

 

 

 

Figure 2

 

/ R—wo \

 

 

 

\
/ \
/ \

/ \

/ are 'k. \
\

l{ X---XX 9K 7>\Y‘
I \ / 316 XX X
I c k. at: P.

 

 

 

 

Figure 2.4: Complex lambda-plane with integration contour.

23

 

 

 

conditio

where t
will exi:
and cov
suppress
conduct
usually
techniqu
The
loss, as
nearby c

to the s;

Point sin
StrllCture

eXaminai

Only the

 

 

condition for the 11"“ TM or TE, even/odd surface-wave mode is given by

2
—‘ =——"— (27)

A0 2,5,7— "‘3

where t is the film thickness. Note that whether or not a particular surface-wave mode
will exist depends on the frequency, film thickness, and indices of refraction of the film
and cover. With the exception of the TM0 mode, these parameters may be chosen to
suppress or initiate a particular surface-wave mode. For the present discussion of
conductor-based microstrip or millimeter wave circuits, these surface-wave modes are
usually viewed as undesirable, although they form the basis of dielectric waveguiding
techniques.

The main physical consequence of surface-waves is that they are a source of power
loss, as they carry energy away from the circuit. These waves may couple to other
nearby circuits, complicating circuit/ system analysis. Numerically, they contribute poles
to the spectral integrals, complicating their evaluation.

The second type of singularities inherent in the Sommerfeld integrals are branch

points. Branch points arise from the multivalued nature of wavenumber parameters

p00.) and pr.) , resulting in a sign ambiguity. It can be shown in general that branch

point singularities are only associated with the outer layers of a multilayered dielectric
structure [25, p. 112]. For the specific example of the tri-layered structure studied here,

examination of the spectral integrands revel that they are even functions of pf. Hence,

only the branch points at A = :kc are of consequence.

\

24

 

 

 

 

 

 

Gen
problem

points 8.]

consider
decay a

Im{P }I

C

the stan
Riemann
integral
complex

Whe
positive
dependar
axis [32]
branch Cl
2-6. It
Alternate
Continuot
implicit

i“Version

 

 

Generally, physical constraints indicate which branch of the function to use. For

problems involving real frequencies and lossy or limitingly low-loss materials, the branch

points are below the positive real—)x axis, as shown in Figure 2.3. This can be seen by

considering the cover wavenumber to be kc =kc/ —jkc” where kc”>0. Requiring waves to
decay and propagate outward from a source point necessitates Re{pc} >0 and
Im{Pc} >0, to be consistent with exponential factors of the form e7"y . This leads to

the standard hyperbolic branch cuts [31], which separate the proper and improper
Riemann sheets, and are depicted in Figure 2.5. Also shown in Figure 2.5 is the implied
integral inversion contour , which is along the real-h axis or may be deformed into the
complex—x plane.

When considering resonant phenomena, the frequency must become complex with
positive imaginary part to provide temporal decay consistent with the ej“ time
dependance. This leads to a migration of the branch points and poles across the real-k
axis [32], since the imaginary part of kc becomes positive for a low-loss cover. The
branch cuts to separate the proper from improper sheets now become as shown in Figure
2.6. It has been found [33] that the integration path must cross the branch cuts.
Altemately, physical reasoning would dictate that all quantities must change in a
continuous manner as the migrating singularities cross the real-)x axis (which is also the
implicit integral inversion contour for the non-resonant case). Since the original

inversion contour is above the singularities, it should remain above the singularities as

25

 

 

 

 

 

they m‘
manner
inversir

work, 2

2.4.2 1

The
are diff
topic [1

transfor

where

A possil
oscillate
feCtangu
function

Provides

they migrate across the real-)x axis, to keep all parameters changing in a continuous
manner Chew [32]. These branch cuts are shown in Figure 2.7, along with the new
inversion contour. The branch cuts shown in Figure 2.7 have been implemented in this

work, and yield good numerical results.

2.4.2 INTEGRATION TECHNIQUES FOR THE EFFICIENT NUMERICAL
EVALUATION OF THE GREEN ’8 DYAD

The Sommerfeld integrals associated with the Green’s function for the layered media
are difficult to compute, as evidenced by the large number of papers concerning this
topic [12-16]. Their evaluation involves a double infinite integration, which is often
transformed to a finite and an infinite integration by the transformation

~21:

f}{...}(123, .. ff {...};tded). (28)
—- o o

where

A possible problem with this formulation is that the finite integration becomes highly

oscillatory with increasing >\. Alternatively, the integration may be preformed in
rectangular coordinates [16]. This involves regarding the inner integral (over E) as some
function of C , and tabulating that function for different values of C . Interpolation then

provides the needed values when performing the outer integration, although numerically

26

 

 

 

 

 

 

 

 

Figure 2

 

 

 

x 2D

S< _u'

x 913'
n

\

 

 

 

Figure 2.5:

27

 

:50 X

JD X

Branch cuts in the complex lambda-plane.

X
j

 

 

 

 

 

Figure 2

>/

 

AR: x x ...x \
x...x x _k‘V 7 A

 

Figure 2.6: Proper branch cuts and the associated integration contour for studying
resonant phenomena.

28

 

 

 

 

 

 

 

Figure 2

7y

 

//"kc xx...>< >>\

x...>< x -kcar-i/

 

 

 

Figure 2.7: Alternative branch cuts for investigation of resonant phenomena.

29

 

 

 

it is mo

evaluati<

where

is the fr

found th

approxir
integral.

The
(1990) a
integram
requiring
Preferabl
this wor]

'tYPeofl

it is more accurate to do a function approximation rather then an interpolation, since

evaluation points may be chosen judiciously. This is illustrated by

1 = [dc 13(C)ff2(C,E)dE = ffl(C)fza(C)dC
C E C

where

f2.(C) = [anode
é

is the function to be approximated. This scheme proves to be efficient because it is ,

found that f2“ is a smooth function of C . As a result, once the function f2, has been

approximated, evaluating the integral I reduces to evaluation of a one—dimensional
integral.

The method of performing the spectral integration in rectangular coordinates is new
(1990) and is found to be very efficient and accurate. The oscillatory nature of the polar

integrand is avoided in rectangular coordinates, leading to greater accuracy while

 

requiring less evaluation time. There are, however, situations where the polar form is
preferable. Both polar and rectangular integral formulations have been implemented for
this work, and the question of which method to use has been found to depend on what

i type of MoM solution is being implemented. This is discussed further in Chapter 4.

a

30

 

 

 

 

cover/fl
Hertziar

as

where 1

where t
forming
mathem:

Which F

Source 1'

2.5 SUMMARY

The electric field induced by currents in the cover region of a tri—layered
cover/film/conductor environment is formulated in terms of Hertzian potentials. All
Hertzian potentials satisfy the vector Helmholtz equations (A.7), and can be expressed

as

fi a U-‘/
110’) = fG(F|F’)-—‘;:€)dV’

where GG‘IF’) is a Green’s dyad specific to the layered surround. Determination of

G(F|F’) requires matching the appropriate boundary conditions (C.3), for potential
components in each region.
Once the Hertzian potential is obtained, it is desired to form the electric field as

5(7) = %f@‘(r|r’)-i(r’)dv’ (29)
c V

where G’(F|F’) is an electric dyadic Green’s function. Care must be exercised when
forming (29), as spatial derivatives must be passed through a spatial integral in a
mathematically correct manner. This leads to a depolarizing dyad term, EMF— 1""),

which provides the field with the correct value when the observation point is in the

source region. The electric Green’s dyad may be written as

31

 

 

 

 

 

where I
sense.
Cor
present
As well
the forn
backgro
of the 3;
loss and
with rat

paramot

G‘WF’) = P~V<k3+vv~>é<rlroJae—w)

where P.V. indicates that the spatial integration must be preformed in a principal value
sense.

Components of the Green’s dyad are in the form of Sommerfeld integrals, which
present analytical and numerical difficulties to evaluation of the desired field quantities.
As well as being oscillatory and slowly convergent, they possess spectral singularities in
the form of poles and branch points. These singularities are associated with the layered
background environment, and important physical significance is attached to them. Poles
of the spectral integrand are associated with surface-waves which usually result in energy
loss and undesirable coupling between circuit elements. Branch points are associated
with radiation. Knowledge of the types and locations of these singularities is of

paramount importance in evaluating the Green’s dyad components.

32

 

 

 

 

 

In t
cover/f1
film lay
often cl
on the d

Invr
continue
teChnolo
Similar:
[5-11] I
compreh
microstr

A v

mentiom

 

 

 

 

THE SINGULARITY EXPANSION METHOD FOR INTEGRATED
ELECTRONICS

3. 1 INTROD TION

In this chapter, the analysis of a general conducting object placed in a tri—layered
cover/film/conductor environment is investigated. The device is printed on the dielectric
film layer, residing in the cover region, as shown in Figure 3.1. This type of object is
often characterized as a microstrip antenna element or a microstrip resonator, depending
on the desired application.

Investigation of the properties of this type of device started in the 1950’s, and has
continued through the present. A thorough survey of microstrip antenna element
technology from its inception through 1981 is given by Carver and Mink [3], while a
similar survey of microstrip array technology is found in Mailloux et a1. [4]. References
[5-11] refer to later papers on" these subjects, although this list is by no means
comprehensive. References [34-36] refer to papers treating the conducting objects as
microstrip resonators.

A wide variety of mathematical techniques have been employed in the above

mentioned works to analyze these devices. Rigorous methods involve utilizing the exact

33

 

 

///,m

Figure 3

 

 

 

 

CZViducitng

\m /\\\\\/\ \\\\\\ //\//\\\\\/

 

 

 

 

 

 

 

 

 

 

 

OOOOOOOOO

 

 

 

 

 

Green’s
EM phe
of Somi
is descr
manner

In t
EM int:
point fc
establisl
in the n

In E
the funr
a sum 0
for train
are defr
found t
frequent
techniql

In E
excited
“311mm

incident

 

 

 

Green’s functions for the layered media, and result in conceptually exact descriptions of

EM phenomena. These methods may be computationally inefficient, due to the presence
of Sommerfeld integrals in the Green’s functions. In this chapter, a mathematical model
is described which involves the rigorous Green’s functions in a numerically efficient
manner.

In the next section, the fundamental electric field integral equation (EFIE) describing
EM interactions in the layered environment is formulated. This equation is the starting
point for the analysis to be presented in this chapter, as well as the basis for the well-
established methods which will be used for comparison. The later methods are described
in the next chapter.

In Section 3.3, the singularity expansion method (SEM) is presented, and applied to
the fundamental EFIE. The motivation for representing the unknown device current as
a sum of pole singularities is developed by considering the singularity expansion method
for transient scattering from free-space objects. Natural resonant modes (or eigenmodes)
are defined, which are characteristic of the device structure. Coupling coefficients are
found which relate the device response to an impressed field for near—resonant
frequencies. Later, it will be shown that results of the SEM agree with those of other
techniques.

In Section 4, the SEM is applied to the specific example of a microstrip dipole
excited by a nearby transmission line. The electric field of an isolated microstrip
transmission line is found, and used as an approximation to the impressed electric field

incident upon the dipole. The validity of approximating the impressed field by the

35

 

 

 

 

unpertui
detailed

wave 111'

3.2 I

In tl
current .

study of
Con

dipole,

impress:
Perfectly
conditio
{ham

form of

 

 

 

unperturbed field of an isolated transmission line is assessed. The results of the SEM

detailed in this chapter are presented in Chapter 4, along with results from other full—

wave methods and measurement.
3.2 FORMATION OF THE ELECTRIC FIELD INTEGRAL EQUATION

In this section, the electric field integral equation (EFIE) for unknown device surface
current I? is formed. This equation is the fundamental integral-operator equation for the

study of microstrip-based circuits.
Consider Figure 3.2, which shows a microstrip device, specifically a microstrip

dipole, embedded within the tri-layered integrated electronics environment. An

impressed source .7 maintains electric field E1, which excites surface currents on
perfectly conducting device surface S, producing scattered field Es. The boundary
condition for tangential E at conducting surface S requires that £1555”) =0, where
f is a unit tangent vector at any point on surface S. Expressing scattered field 13” in the

form of equation (2.20) leads to the EFIE for unknown current I?

. - - 'k . -,
t'(kc+VV')fG(F[?’)°K(F’)dS’ = -]—‘t°E (F) v 755. (1)
n

S

C

36

 

 

Figure :

 

 

 

 

 

 

 

 

 

ZL/ZIZK'ZZIZ
IllZ J
as
E [39'
57 E'
(:1
2f. 5
E5 Dipole

Coever

 

e/////////

W72; Fig.1. t\\\\\\\\\\\\

 

 

 

Figure 3.2:

 

Microstrip device excited by an impressed source J

37

 

 

 

 

 

 

 

Altern:

the EF

Equatil

numeri

3.3

EF

microsr
These <
limited
devices

Kn

represe

Where 1

natural.

amplitu

Alternatively, the scattered field may be expressed in the form of equation (2.21), and
the EFIE becomes
A ”e.._./ *—./ / jkc“"i... _.
rfG ( Ir )°K(r )dS = ——t'E (r) v reS. (2)
S "c
Equation (2) is useful for theoretical manipulations, whereas equation (1) is used for

numerical computation.

3.3 SINGULARITY EXPANSION OF DEVICE CURRENTS

EFIE (2) is to be solved for unknown surface current K. It is well—known that

microstrip antenna and resonator elements exhibit very frequency selective responses.
These devices interact strongly when excited at or near their resonant frequency, but have
limited responses at other frequencies. This feature has initiated representing these
devices as lossy cavities, and other physically insightful models.

Knowledge of the resonant characteristics of microstrip elements leads to the
representation for the device surface current near resonance as
APEPU’)

(w -op)’

120; (a) :3 (3)

where (up is a complex natural frequency, 1 is the order of the pole at w = up, 12; is the

natural-mode current distribution associated with the p‘“ natural-mode, and AI) is the

amplitude of the p“I natural mode. It will be shown later that the order of the pole

38

 

 

 

 

singula
Eq.
order t1

8801111

The re;

natural-

motival

which i

3.3.1

The
as a me
motivat
dominal
shape (
corresp,
Singular

The

Scatterei

 

singularity is unity [37].

Equation (3) describes the surface current near a single resonant frequency. In
order to describe the current over a wider frequency range, the current is represented as
a sum of terms of the form of equation (3),

iota» = 2:

q (0 -<->,,)'

The representation for surface current (4) will be used in EFIE (2) to quantify complex

natural-mode frequencies wq and their associated amplitudes Aq. Before proceeding, the

motivation for surface current representation (4) should be placed in the proper context,

which is the subject of the following section.

3.3.1 MOTIVATION OF CURRENT EXPANSION : THE TRANSIENT
SINGULARITY EXPANSION METHOD

The transient singularity expansion method [18—20] was developed in the early 1970’s
as a method to characterize the response of a scatterer to a transient excitation. It was
motivated by the observation that the transient response of an object appears to be
dominated by a few temporally-damped sinusoids, which are characteristic of the size and
shape of the responding structure. The Laplace transform of a damped sinusoid
corresponds to pole pairs in the complex frequency plane, leading to the frequency-plane
singularity representation for scatterer current.

The experimental observation of the time-domain, transient current response of a

scatterer leads to

39

 

 

 

 

 

 

 

 

 

or equr

where .

modal .

where .

This pri
for the

For
in the nr
comes“
the steal

of pole.

 

N
KIT—it) = 2 Anal?) e o"'cos(<.>nt + (1)") (5)
n=1

or equivalently
2N

Em) z ZanEnG’ks": (6)

n=1
where Sn = on +jcon is the complex natural-mode frequency of the n‘h mode, and En is the

modal distribution of current. Defining the bilateral Laplace transform pair [38]

(re-jun
1 St
E f F(S)e d?

a —j~

fit)

F(s) = [Me "“dt

where s = a +jm, equation (6) may be written in the complex frequency plane as

This provides the desired motivation for the frequency-plane, pole—singularity expansion
for the surface current in the case of a transient excitation.

For excitations at a single frequency, the region of interest in the s-plane would be
in the neighborhood of a single point. For sinusoidal steady—state excitations, only modes
corresponding to poles near points s =jw will be excited. Therein lies the motivation for
the steady-state singularity expansion of current. Surface current (4) consists of a sum

of pole—terms, which may be truncated after one or two terms to represent current for

40

 

 

 

 

 

 

excitati
efficier
agree \

It:
130,8)
as a s
singula
free-Sp
branch
work c
in App
and rel
frequer

will be

establis

In I
tYpicall

method

 

excitations near a single frequency. Evaluation of one term of (4) is found to be an
efficient representation for the device current near resonance, leading to results which
agree with other methods.

It should be noted that, in general, the complex s-plane may contain singularities of
K(?,s) other then simple poles. The time-domain current I?(f‘,t) will then be expressed

as a sum of contributions from poles, branch points, and possibly entire function
singularities (singularities at infinity). It has been shown that for finite-sized objects in
free-space, the object response has only poles as singularities. Other objects may require
branch point and entire function singularities, as well as pole singularities. The present
work concerns conducting objects placed in a non-homogeneous medium. It is shown
in Appendix D that branch point singularities are present in the complex frequency plane,

and relate to surface-wave propagation. Hence for a complete singularity expansion of
frequency domain current 1?, singularities other than just poles would be required. It

will be shown, though, that (4) yields results that agree quite well with other more

established methods in the resonance range, justifying its use.

3.3.2 DETERMINATION OF NATURAL MODES

In this section, the defining relation for natural modes is obtained. These modes are
typically defined by the source—free solution to EFIE (1) or (2) (E i =0). An alternative

method is followed here [37], which provides more physical insight into the problem.

41

 

 

 

 

 

For frl

Since

indete:

homog

with n

oomph

3.3.3

 

Singularity expansion (4) is substituted into EFIE (1), leading to

Fe»

A A u —o .k A —0‘
——° t' G‘(?li”)~k (F’)dS’ z —J—‘rE‘(F) v 763. (7)
_ q
to) s T]

c
For frequencies to zap , it is obvious that the p‘h term in the sum (7) becomes unbounded.
Since 5 i is regular at these frequencies, the p‘h integral term must vanish to produce an
indeterminate form. Therefore, modal current distribution 12;, must satisfy the
homogeneous EFIE

tA-fG‘(F|F’;m).IE°P(F’)dS’ = o v res (8)
S

with non-trivial solutions only for o) = cop. Equation (8) defines the p‘h natural mode with

complex natural frequency (up.

3.3.3 DETERMINATION OF EXCITATION AMPLITUDES

The excitation amplitude for natural—mode current, A q, found in current expansion

(4), relates the amplitude of the q‘h natural mode to the impressed excitation. These

amplitudes are determined from fundamental EFIE (2) and current expansion (4) upon
invoking reciprocity of the Green’s dyad kernel, G:p(f’l?’)=G;a(F/|F) [37].

The integral operator

42

 

 

 

 

 

 

 

which p

where r:

u=op l

The leac

Singular

leading 1

 

fds fem-tn}
S
which performs the tA-{u-} operation, is applied to EFIE (2) yielding
.. .. _. 'k .. ..-
fds’ K(?’)-fG‘(7’|F;m).kp(r)ds = -J—‘fkp(r)-E‘(7)ds
s s no 3

where reciprocity of G‘ has been invoked. Expanding (3‘ in a Taylor’s series about

0) =<op leads to

 

fds’ EM)- [0 (co-mp) + ...J-Ep(r)ds =
s s my

 

‘(F’Inmp)+ 66 (a, (LIN!) 0))
-’—k° [12,,(7) q? ‘(r)ds.
11,,

The leading term vanishes due to (8), consequently

 

fdsl [go-.4) f 6G ‘(7 llr; (1))

 

(m—mp) + 4.1,};ng =

(1)-(1),

 

 

’k .. _,
-]—‘ka(F)~E ‘(r)ds.
11c 3
Singularity expansion (4), with poles of arbitrary order lq , is exploited in the above,

leading to

____fdslkq.(’-:l)f 66 e0- Ilr; (1))

«(w— Ziql')

((0 -mp) +

0’0),

'kp(i") dS =

 

 

 

'k - ,.
-J—‘fkp(f’)°E ‘(fi0)ds.
n. s

43

 

 

 

 

Inthe

inthe

One n

11:1“

where

The al
in the
consid
((0 -o
that 112

to rela

large c
leads t

SEMI

 

In the limit (1)—«op, the term ((1) ‘01,) can annul at most a simple pole for the q=p term

in the sum over q. This establishes the simple nature of the q=p pole.

At this point, two viewpoints may be adopted, leading to slightly different results.
One method would be to take the limit of the above expression, and note that only the
q=p term is non-vanishing, resulting in coupling coefficient

lim jk,

A _ d _
P 0) (up ncCp

 

[156)]? ‘(rjmms (9)
S

where natural-mode currents are normalized such that (reciprocity invoked again)

 

slé'p(?’)dS’. (10)

0:0,

_ - _. aG‘GIF’w)
C1D - de kp(r)f—am———
s s
The above coupling coefficient is seen to be similar to a "class- 1" coupling coefficient
in the transient SEM literature [18]. Alternatively, the frequency range (1)sz is

considered. All terms q;ép in the sum over q are small due to the presence of the term
(0 '14)), which does not cancel with the corresponding pole term. Moreover, it is found
that natural frequencies (04 are widely separated from each other (0n=20n_1), leading
to relatively large denominators ((1)-o) q). The combination of small numerators and
large denominators for the q¢p terms indicate that their sum may be neglected. This
leads to a quantity similar to a "class 2" coupling coefficient as defined in the transient

SEM literature [18]
jk

c I? (r)-E"(?,w)ds (11)
118i "

 

A,=-

44

 

 

 

 

 

 

 

constar

becoml
summz
on " til
differe
dissert

Se
coeffic

impres

simple

due to

task.

impres:
theory
devices

No

 

with normalization (10), valid for frequencies a) zwp. The differences between equations

(9) and (11) are minor, since the source excitation EiG‘ZQ) and the normalization

constant Cp(<o) are assumed regular in the frequency range of interest. The two results

become identical when evaluating residues at the poles in the transient response current
summation, but the class—2 coefficient leads to a better physical interpretation of "turn-
on" time, which is important in the transient case. For steady-state excitations the
differences between (9) and (11) are not important, although for the remainder of this
dissertation, the term "coupling coefficient" refers to expression (1 1).

Several observations can be made regarding equations (10) and (11). Coupling

coefficient (1 1) is seen to be an overlap integral of the p‘h-mode natural current with the
impressed field. It is found that natural-mode currents EPG’) can be modeled by fairly

simple expressions, hence any difficulties associated with evaluation of integral (11) are

due to the form of the impressed field. For plane-wave excitation, the simple form of
ET?) yields an easily evaluated integral. Excitations due to infinite transmission lines

will involve one-dimensional Sommerfeld integrals, leading to a more difficult numerical
task. When the source of excitation is another finite size microstrip device, the
impressed field will involve two-dimensional Sommerfeld integrals. A perturbation
theory is developed in Chapter 5 which simplifies the analysis for nearly-identical
devices.

Normalization constant C? involves the frequency derivative of the Green’s dyad

45

 

 

 

 

rhea
tedious ma

to be evalr

The cr
suitable f0
and (11)

transmissir

an express
impressed
dimension
analytical]

The r
transmissi
0f total 1e
an angle l
amIllitude

The follo‘

Cllrrents i

 

 

G‘(F|i";w) . This derivative may be evaluated analytically, in a straightforward though

tedious manner. Although Cp involves two-dimensional spectral integrals, it only needs

to be evaluated once for a given mode and device.

3.4 SEM ANALYSIS OF THE MICROSTRIP DIPOLE

The coupling coefficient (11) and normalization constant (10) are in a compact form
suitable for analytical manipulations. It is the aim of this section to explicitly obtain (10)

and (11) for the example of a microstrip dipole excited by a nearby microstrip

transmission line. It is found that the natural—mode current 12;,(F) can be represented by

an expression that leads to closed-form evaluation of the spatial integrals in (10). The
impressed field in (11), due to the microstrip transmission line, is found to involve one-
dimensional spectral integrals, where again spatial integrals can be preformed
analytically.

The configuration to be considered is shown in Figure 3.3. A microstrip
transmission line of infinite length and width 2w, is located along the x-axis. A dipole
of total length L=2l and width 2wd is located a distance d from the transmission line, at
an angle 6. Coupling coefficient (11) along with normalization constant (10) relate the
amplitude of the natural-mode current to the excitation provided by the transmission line.

The following analysis neglects the effect of the transmission line field maintained by

currents induced upon the microstrip line by dipole current K and its associated electric

46

 

 

 

Z
Y Dipole A

 

2Wd Z
M
O
2:11):t
E Microstrip Line V— —————— fi X §
1‘

 

Coever

 

WgtiiériW

 

 

 

\\\\\\\\\\ \\\\\\\Vjfi round pl onek

 

Figure 3.3: Microstrip dipole excited by a nearby transmission line.

 

47

 

 

 

 

field.
from
Figur
show.
coupl
dipoll

comp

dipolr
chang
for Si
indica
be ac<
the u
excita

F1
the 01]
is not
0f the

neglec

 

field. This approximation is found to be valid for dipole elements spaced sufficiently far

from the transmission line. This can be seen from the experimentally measured data of
Figure 3.4 (Details of the experimental method can be found in Chapter 7). Figure 3.4
shows the real resonant frequency and quality-factor (Q—factor) of a 5.0 cm parallel-
coupled (0 =0) dipole excited by the transmission line as a function of
dipole/transmission line separation. The Q-factor, which is proportional to the complex

component of frequency by (afar/20, is a measure of the degree of coupling of the

dipole to the transmission line. It can be seen that the real resonant frequency does not
change substantially as the dipole position is varied. The Q-factor changes considerably
for small dipole/line spacing, but becomes relativity constant for d>.40 cm. This
indicates that for d < .40 cm mutual coupling in this dipole/ transmission line system must
be accounted for. When the dipole is located beyond d=.40 cm, the principal field of
the unperturbed transmission line should be sufficient to represent the impressed
excitation.

Furthermore, the radiation pattern of the dipole is principally normal to the plane of
the dipole, and is zero in the plane of the dipole in the far field [39]. Although this work
is not concerned with separations which would place the transmission line in the far field
of the dipole, knowledge of the radiation pattern of the dipole qualitatively motivates

neglecting the effect of the dipole field upon the transmission line for sufficient spacing.

48

 

 

 

 

 

 

3.4.1

 

 

3.4. 1 N ORMALIZATION CONSTANT

 

 

 

 

 

 

 

3.4.1

Th1
integral

found l
to (8),
definin]
and on]

Fig
associar
Theser
pulses i
with thl

very sir

fOI eve]

 

 

3.4.1 NORMALIZATION CONSTANT

The normalization constant (10) is a four—dimensional integral, with two spatial

integrals and two spectral integrals (associated with G 3). Natural—mode currents I}; are

found from the solution of (8). A pulse function/Galerkin’s MoM procedure is applied
to (8), to allow the unknown current freedom to assume any form required by the
defining homogeneous EFIE. The details of this procedure are covered in Chapter 4,
and only the results will be presented here.

Figure 3.5 shows complex resonant modes in the wavevector-plane, and their
associated current distributions, obtained by the pulse function MoM solution of (8).
These results were found by using 40 pulses over the dipole half—length, although fewer
pulses lead to similar results. Even and odd modes are found to alternate, beginning
with the principal first even mode. It is seen that the various current distributions are

very similar to sinusoidal functions. This motivates modeling the modal current as

 

h.

 

mtz
an CO —2'l—
Kp(i") = t 2 (12)
\i “’d
for even modes; n= 1,3,5 or
an sfl' TUE—:l
KP(?) = t 2 (13)
1- i -
w

 

 

 

Acev
\A
0C
0
3
we
0
L
.m
w:
0
COM
0
mm
_0
0
N;

F.
lgun

 

 

 

 

2.60 — —100
2.54 i C, Q ~94
2.48 _
2.42 I
2.36 -

2.30

 

2.24

Quality Foctor (Q)

2.18

   

2.12

Reol Resonant Frequency (f0)

 

2.06 ~45

 

 

14L 1;! 1 gr! 1 l I

oo - . . - - . 40
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Seperotion d (cm)

 

Figure 3.4: Measured real resonant frequency and Q—factor of a dipole excited by a
microstrip transmission line as a function of dipole/line separation.

49

 

 

Figure

 

 

1.0 1
0.8 r
0.6 r
0.4 -
0.2 -
0.0 -
~02 -

Current

.04 .
_o_5 .
_08 ..

 

 

0.08
0.07

l'r‘j

0.06
0.06 -
0.05 -
0.04 -
0.03 -
0.02 -
0.02 -
0.01 l

 

0.00

Noturol Modes

>< Even Modes
0 Odd Modes

 

     

 

 

 

 

_1 .0 ._L 1 1 1 ._; J _1‘0 . 1 L 1 1 1 1
—1 .0 —O.8 —O.6 -0.4» —O.2 0.0 0.2 0.4- 0.6 0.3 1.0 —-1.0 -O.8 -0.6 —0.4 —O.2 0.0 0.2 0.4 0.6 0.3 ‘l .O
Antenna length

Figure 3.5:

Microstrip dipole eigenmodes and their associated current distributions
obtained by pulse-function MoM solution, 40 pulses. '

 

51

 

 

 

 

 

 

for c

equa
com]

be v1

is pr

Indiv:

where

for odd modes; n=1,2,3. The unit vector f is oriented tangential to the dipole. Figure

3.6 shows the comparison between the pulse function MOM current distribution and
equations (12,13) for the n=1 first even/odd modes, and Figure 3.7 shows the same
comparison for the second even/odd modes. In each example, agreement is found to
be very good. The square-root edge singularity in (12,13) is included to model well-

known behavior at the dipole edges.

With 13;, known, the normalization constant (10) may be found. Evaluation of

G'" = aé‘(r|?’;o)
60) «1:0
P

 

is preformed by term by term differentiation of the electric Green’s dyad 0‘.

Individual terms of G" are found to be

 

 

 

 

 

r _ 2 V
Rtbcwl'O’W/h-l— +R¢/ e-p‘wyl)
p. 1 p,
G" i
t ‘3 . _ , _ ' 2
G)! = fdedC em”) “1‘”. Cbc‘“1(y+yx)+i]+cle-mw’n
:. .. zone. _ p. 1 .
b2 _ J
c IY‘y/i+l]e pcly yl
. . P.

 

 

 

where be =nc p.060 and derivatives of reflection and coupling coefficients (2.18) are

52

LOZ
N.:0Cc
#CvaLDO Um

QE<

0U3t_

 

 

 

 

Mm

*\

Normalized Current Amplitude

 

 

 

_10 1 1 1 11.. 1 1 1 1 1 1 1 1 ,
—1.0 —0.8 —0.6 —0.4-—0.2 0.0 0.2 0.4 0.6 0.8 1.0

Normalized Distance (z/l)

 

Figure 3.6: Comparison of nullspace current distribution (pulse function MoM) and
approximate current distribution (eq’s. 12,13) for the first even/odd

modes

53

®UJt_QE< 0:01.130 D®N:OELOZ

Figure

 

 

 

1O :' /-i.\
0.8 - '
0.6 -

 
 

0.4 ~
0.2 -

0.0 .4 ——————— —— —————

  
  

m

—0.2 \1

—0.4 —

Normalized Current Amplitude

—0.6 - 1p

 

-—0.8 —

 

3',“ 1 1 1 1 1 1 1 1 1 1

.0 1 ‘ ' ' . . .
—1.0—O.8 —O.6—O.4—O.2 0.0 0.2 0.4 0.5 0.8 1.0
Normalized Distance (z/l)

 

Figure 3.7: Comparison of nullspace current distribution (pulse function MOM) and
approximate current distribution (eq’s 12,13) for the second even/odd

modes.

54

 

 

 

 

with N

respect

Sp;

current

and

lead to

 

 

 

R.’ = _1_ /_er"/
2" 1 z“
2 2 e’ /
C/= -2(Nfi.-l) bcw+ch +chh
zhze p. 2e 2"

 

 

with N1, Z ‘, and Z h defined by (2.19). In the above, primes denote differentiation with

a_4

respect to (1); e.g., A’=
8(1)

 

U=Up

Spatial integrals over dipole surface S may now be evaluated, using natural-mode

current distribution (12) or (13). The integrals

wd etjEx
f —dx = (wdrr)Jo(Ewd)
2

 

w, 1_[i
wd
and
I (nrrl)sin(1215)cos(Cl)
[COS(£ZEl£)e #2de : ___—__— : 11(nsl’C)
_ £2 + 21.
1 (2 Cl)(2 Cl) (14)
1
, 335 :‘(z = j(—1)"(2nlrr)sin(Cl) = I l
lismi 1 ie 1 dz 1 (nfi+Cl)(mt-Cl) 201,0
lead to

55

 

 

for ev

been r

3.4.2

Tl

curren

provid
discus:
It

transm

Where

Similar

 

a: “[2
C = 4ff 2. b3®(R1+1)
p o o 2(2fl)2 Pc

 

 

b.._w 2+2_ lsin(0) 2
p. P.
/

. pinata? - 141nm] + A’sin’wl}

(15)

1,1 2
(wdnngacosrewd) {23,113} d6 111

for even/odd modes where J0 is the 0“1 order Bessel function [40]. Equation (15) has

been put in polar form by the transformation (2.28).
3.4.2 COUPLING COEFFICIENT

The coupling coefficient (11) relates the amplitude of induced natural resonant—mode
current 12; to the impressed excitation E". The field of a microstrip transmission line

provides the desired excitation, under the assumptions excluding mutual coupling

discussed earlier.
It has been found that the even-mode current distribution on a x—directed microstrip

transmission line in the propagation regime is efficiently represented by [41]

Euro-l.) = if: M +22 “10.44)T2,,,.1(Z/W,)v1‘(Z/W‘)2 e -jflx (16)
71:0

"’0 \/1 - (Z/W)2

where T. are the Chebyshev polynomials [40]. The odd—mode current is given by a

similar expression. Propagation eigenvalues B are found by a numerical root—search of

56

 

 

 

a cou;

nulls;

propa

nume:

electr.

Substi

of [42

result:

The or

Where

 

a coupled set of spectral EFIE’s, and amplitude coefficients a {i are obtained as the

f, (M)

nullspace of the solution matrix. For the example considered here, only even

propagation modes are considered. For narrow transmission lines (w,<).o), the

transverse component of current may be ignored for field computations, although the
numerical root-search for propagation eigenvalues includes both components. The
electric field of the transmission line is found from equation (2.21) as

-1- - ‘mc (3‘ ~ - .1: ")dS’ (17)

E (r) — TI (rlr’) “(r .

c 5

Substituting the x-component of transmission line current (16) into (17), and making use
of [42]

fe-Mii+i)dx = 21:6(13+E)

results in field components
E{x}(r‘) = e71” f {£EE;}N(C)eKZdC. (18)

The coefficients in (18) are given by

10:.2 — 13002, + 1) + 13219.61

41rjr1) nczeopc

 

Y(C) =

[R,+1-P,C]15C

. 2
41:10) nc eopc

Fro =
where R‘, C, and pc are evaluated at E=-D. The term N(C) arises from the spatial

57

 

 

 

 

L__ __. ___- ___

integra

The cl

leads

when

Cheb

wher

The

of st

integration transverse to the transmission line as

M!“ n

I
N(C) = f Z———a2‘"’”T2"(Z[We-K222. (19)

-w‘ "=0 ‘1 1 _ (Z l/w,)2

The change of variable

= 27w!
dz~ = dz’lw,

 

leads to

°° ’ T (z‘)
N(C) = 2w‘a m 2"—cos(Cw,z')dZ ]

where the integration and summation have been interchanged and the even nature of the
Chebyshev polynomials has been exploited. The integral identity [43]

1
dz 11:
T (z)cos(az) = ('1)"—J (a) [a>0]
t "' r— 2 ”'

 

1-z2

where .Im(a) is the m‘h order Bessel function leads to

 

N(C) = nw,2a2(n,l)(-l)"J2u(Cw,).
n=0

The field distribution arising from (18) is shown in Figure 3.8, for a representative set
of structural parameters at an operating frequency of 8.95 GHz.
The expression for electric field components (18) is valid for a x-directed

transmission line. In order to allow for the impressed field to have an arbitrary

58

o03:aE< 2.11;qu

_

Figure

 

 

  

  

 

 

 

2000 l 1 1
1600 i : i A EX
‘ 1
- IMicrostrip : 0 E2
1
1200 - 1 (211.) l 1
- | | 1
111 800 — :
"O - 1
.3 400 — 1
a ' 1
E 0 — 1 1
< _ | |
1
E —400 - 1 l
.‘2 _ 1 1
L1_ 1 1
l —800 - 1 1
L1_l . I 1
—1200 — I X l
- l l
—1600 - : b:
- 1 1
_2000 1 1 1 1 1 1 l 1 l 1 1 1 1 1

 

-2.0-1.6—1.2—0.8-0.4 0.0 0.4 0.8 1.2 1.6 2.0

Transverse Coordinate (cm)

Figure 3.8: Electric field distribution of a microstrip transmission line, principal
even propagation modes, x-component of current.

59

 

 

orient:

system

Figure

lead to

orientation, the field components are assumed to be referenced to a local coordinate

system (xl’,zl’), as shown in Figure 3.9.

(x,z)=<lo,-a)

\
[X

 

 

 

 

 

Figure 3.9: Local and global coordinate system used for field component evaluation.

The translation and rotation of coordinates
x1 = x—b
21 = z+a
x1, = xlcos(0) +z,sin(0)
z,’ = z,cos(e)—x,sin(e)
lead to the field components with respect to the global coordinates (x,z)

}(,) = , -111....19)+..111e)1j{cos(e)Y(0 ~sin<e)F(0}Nm

E {11 3111(6) Y(C) + cos(B)F(C) (20)

. e -jxcsin(e)ejzceos(e) e 1211 d C.

60

 

 

 

 

 

 

 

 

 

 

where

3.5

descn“

utilizi1
difficr
solutit
teChoir
metho

device

 

The coupling coefficient (11) is found from the impressed excitation (20) and natural-
mode current (12,13) as

‘k . I ,l, .
A, = - I c f {‘]o("‘1“"d)ejc‘i {1:E:,1,::;}[Sm(e)Y(C)+cos(6)F(C)]

 

C
”2° on
+ J (or w )e ‘1'“ 11("’l’°“) [sin(0)Y(C) —cos(0)F(C)] dC
° 2 d 12(n,l,a4)
where
or, = —B cos(0) — C sin(0) a3 - —B sin(0) + C cos(0) (22)
a2 = —[3 cos(0) + C sin(0) a4 = —[3 sin(0) - C cos(6).

3-5 W

The electromagnetic interactions of microstrip integrated electronic devices are

described by an electric field integral equation (EFIE)
11 ”c.....] ".../ /_ jkce"r'_. _. 23
z-fG (rlr )'K(r )dS — -—t°E (r) v res ( )
1'l
S c

utilizing the rigorous dyadic Green’s functions presented in Chapter 2. The numerical
difficulty associated with evaluation of these functions hinders the straightforward
solution of eq. (23) by the method of moments (MOM), and prompts an alternative
technique. Following the methodology of the well-established singularity expansion

method (SEM) which quantifies the EM responses of free-space scatterers, the microstrip

device surface current is written as a sum of pole singularities in the complex (1) -plane,

61

 

 

 

 

 

 

 

when

assoc

For 5
surfa1

defini

with 1

With 1

Greer

integr

leadin

Where

169:1») z 2 M (24)

q (d) - (1) q)
where cop is a complex natural frequency, 12; is the natural-mode current distribution

associated with the p‘11 natural-mode, and AP is the amplitude of the p11 natural mode.

For steady—state excitations, only one or two terms Of (24) are needed to represent the
surface current near resonance. Exploiting current (24) in EFIE (23) leads to the

defining relation for natural resonance modes
f°f0‘(F|F’;w)-Ep(f")dsl = o v F135 (25)
s

with non-trivial solutions only for (1) =wp. Equation (25) defines the p‘h natural mode
with complex natural frequency (up.

Amplitude coefficients A q are found by applying reciprocity of the electric dyadic
Green’s function to EFIE (23), subsequent to testing with an appropriate natural-mode
integral Operator. The Green’s dyad 0‘ is expanded in a Taylor’s series about co=cop,

leading tO

.k ‘ -.
J C fkp(F)‘E‘(?,w)dS (26)
cCpS

Ap=-

 

where the natural-mode currents are normalized according to

62

 

 

 

 

used

mode

..A'

C, = {d3 Eptr). {3535501 .

w=<op

EP(F’)dS’. (27)

The electromagnetic response of a microstrip dipole excited by a microstrip
transmission line is studied. The unperturbed field Of an isolated transmission line is

used to approximate the impressed excitation, leading to explicit expressions for natural-

mode current amplitude AP.

63

”a W‘-

 

 

 

4.1

11
nncro
press
induo
defini
solud.
EFIE
appro

arejn

genes
solver

a gem

 

 

 

 

CHAPTER FOUR

FULL-WAVE SOLUTIONS OF THE FUNDAMENTAL EFIE AND
EXPERIMENTAL AND THEORETICAL VALIDATION OF THE SEM
THEORY

4 . l INTRQD [JCTIQN

In Chapter 3, the fundamental EFIE which quantifies electromagnetic interactions in
microstrip circuits was developed. A dominant-mode singularity expansion analysis was
presented which leads to an approximate representation for device surface currents
induced by an impressed excitation. Pole-singularity terms were also seen to lead to the
defining relation for natural modes, which is conventionally found by the source-free
solution to the defining EFIE. In this chapter, full-wave solutions to the fundamental
EFIE are presented, which provide a comparison to the singularity expansion
approximation. Results Obtained by the methods detailed in this chapter and Chapter 3
are presented, and compared with measurements.

EFIE (3.1) can be solved by the method of moments (MOM) [44]. The MOM is a
general technique to transform an operator equation into a matrix equation which can be

solved on a computer. Salient features of the MOM are summarized here, beginning with

a general Operator equation,

AX (1)

ll
h<

 

 

 

 

 

where

unknt

with

R .

N’1

and i:

where

andL

weigh

methc

weigh

Thisl

 

 

 

where A denotes an Operator and X is the unknown to be determined for a given Y. The

unknown X is expanded in a set Of known expansion (or basis) functions x,,

N
X40152 “1'":
1-1

with unknown coefficients 01,. This approximation is substituted into (1), resulting in

N
2 111,111:i = Y”. (2)
is]

In (2), Y N is the solution Obtained from the approximate expansion X zXN. The residual,

RN, is formed as

RN=AXN-Y=YN-Y

and is weighted to zero with respect to a weighting function

i,

<RN,W,.>=O, i=1, ---,N (3)

where the bracket notation indicates a suitable inner product [44,45] such as
<u,v> = fu(z)v(z)dz
L
and L denotes the domain Of the inner product. A common procedure is to choose the
weighting functions equal to the expansion functions, which is known as Galerkin’s
method. Implementation of the MOM then requires choosing appropriate expansion and
weighting functions which will result in an accurate and efficient numerical solution.

This has been discussed by many authors, more recently by Sarkar et al. [46].

65

 

'—- __ ~ - __.: .-_~.—-.su-t£.;."....- 43...». dwgfium:r .~._~. _

 

 

 

pach
funcn

1111

me u
rdad

ofcu

bash
accur

exdh

V& d

Pfiha

 

 

 

Lawr. —‘_ y- r . M .

The solution Of integral equations for microstrip antenna problems (dipoles and
patches) have been investigated in many papers [6—1 1,47 -5 2]. Various expansion
functions have been used, and Galerkin’s method is usually implemented. Two different
MOM solutions are developed in this chapter, using different basis functions.

In Section 4.2, EFIE (3.1) is transformed into a Hallen’s form integral equation
(HFIE) [53], and subsequently solved with sub-domain basis functions. It is believed that
the use of the Hallen form IE for microstrip circuits is new, and a discussion Of its
relative merits is included in Section 4. The HFIE can be solved in general for any type
of current, or even/odd modes can be specified analytically.

The next section presents a solution of the EFIE by the MOM, with entire-domain
basis functions. It is found that an appropriate choice of basis functions results in good
accuracy with a small number Of terms, and just one term is Often satisfactory for
excitations near a natural resonant frequency.

In Section 4, a comparison between the solution of the EFIE with entire-domain basis
functions and that of the HFIE with sub-domain basis functions is presented. It is found
that each method has advantages for certain applications and disadvantages for others.
Convergence studies for both methods are presented, and the method of numerical
integration used for each solution, introduced in Section 2.4.2, is discussed.

Numerical and experimental results are presented in Section 5. Characteristics of
transmission line fed dipoles such as frequency response and induced current amplitude
vs. dipole/transmission-line separation are studied. The differing theoretical methods

presented in this chapter and in Chapter 3 are found to agree with each other, as well as

66

 

- ...- .fl:gemm.mfi_-l;:€1 .'_'. - _.~~.--y

 

 

 

 

 

with

loss 1

4.2

thek

isin

diffei

For a

Whicl

 

 

 

with measured results. Power dissipation due to space and surface-wave radiation, Ohmic

loss and dielectric loss is discussed.

4.2 HALLEN—FORM SOLUTION WITH SUB-DOMAIN BASIS FUNCTIONS

The EFIE (3.1) relates the unknown surface current on a microstrip device, KO") , to

the known impressed electric field, as

. .. .. ’k . -.
Mk: +VV°>fG(Fl?’)°K(F’)dS’ = -J—‘toE‘(r) v FeS. (4)
S n1:
The integral term in (4),
R1?) = [definite/my (5)
S

is in the form of a magnetic vector potential. Equation (4) can then be written as a

differential equation for the vector potential

. - 'k . -.
t~(kc2+VV-)R(f‘) = -{—9t°E'(i"). (6)

C

For a narrow, z-directed dipole,
I‘d?) and?)
1‘ =2

which leads to the sealer differential equation

67

 

. nglzm.;zv_inamm-a' . -

 

 

 

with

andt

point

is ad

when

and

 

62 _. _. ikc 1- _.
[13+51—,]R,<r>+r.<r>= 76-511) <7)

with

R,(?) = [(GP+G,')K,(F’)ds’

 

3
ya’
0') = ‘ Kz("’)dS’
R’ iayaz’ '

and the Green’s dyad components are understood to be functions Of both source and field

points, e.g.: G;=G;(?|?’). The term

60'
k3 f——‘K,(r’ms’
3y
S

is added and subtracted from the LHS of (7), resulting in the forced differential equation

[kf+—§2—)L(?) = F(f‘) (8)
622
where
L(f’) = f GSKZ(?’)dS’ (9)
S
and

60' jk .
1x?) = k2 .——‘K(?’)ds’-—£E‘(?)
“C 6y ‘ n. z
(10)
aa' 7 i'"
am i”) = 6’0] 7’) + G,'(?| F’) + ——§—l——) .

68

 

 

 

 

Thel

when

justif

when

when

repre

for 31

term

integr

Thel

to tha

The homogeneous solution of (8) is given by

Lh(i’) = C1 cos(kcz) + Czsin(kcz) (11)

where Cl and C2 are treated as constants although they are actually functions of x,y. The
justification for treating Cl and C2 as constants is as follows:

Consider the homogeneous differential equation

[’63 +£Juxw) = 0 (12)
Z

where L(x,y,z) is defined by (9). Equation (12) can be solved easily to yield

f GSKZ(F’)dS’ = C1(x,y)cos(kcz) + 2(x,y)sin(kcz) (13)
S

where Cl and C2 are unknown functions of x,y. Making use of the spectral

representation for Green’s components (2.17), equation (13) may be written as
[few em e_P°yH(}.)dZA = C1(x,y) cos(kcz) + 2(x,y) sin(kcz) (14)

for source points on the film layer surface y’ =0 , and assuming field points yzO. The

term HO.) comes from the coefficients of the Green’s dyad components and the spatial

integration as

 

R - C . .
11(1) = (1+ ‘ pc )fe_llee'J‘z’Kz(x/,z/)dS/.
2(21t)2 c

The functional dependence of the LHS of ( 14) on x,y can now be studied, and compared

to that Of the RHS. Since the original EFIE is valid only for field points FES, equation

69

 

 

 

 

(14)
atx=

expa

wher
integ
term

(15),

the v

Simi

Equa

Equa

 

(14) is limited to the same region. For narrow dipoles oriented along the z axis, centered
at x=z=0, the field point variation in x will be minimal (x=01:5x). The LHS may be

expanded in a Taylor’s series about x=0,
Q 2 . -
LHS(x,y,z) = ff [1 +jEx "' £242 +«-] 8101 e pcyH(A-)d2A (15)

where derivatives with respect to x can be taken inside the spectral integral since the
integrand is continuously differentiable in x. Equation (15) may be written as a sum of
terms, where it is seen that small variations in x about x=0 result in small variations in
(15), assuming, of course, that the spectral integral converges to a finite value. Since
the variation in the RHS of ( 14) as a function of x must be the same as that of the LHS,

the terms Cl and C2 must change very little with x and may be treated as constants.
Similar arguments apply to the y variation, and in fact y=0 is usually implemented.

The particular solution of (8) is given by
1 Z
LP(F) = FfF(x,y,z=z’)sin[kc(z-z’)]dz’. (16)
c 0
Equations (9)-(11) and (16) combine to yield the desired IE

f G, [qr-0115’ = C,cos(kcz) + 28in(kcz) +
S (17)

 

sin[kc(z -z /)]dz ’.

1:2,

c
0 S

 

k} [Bananas/“Lam
6y ‘ kn ‘

 

C c

Equation (17) is the general form Of the HFIE for microstrip dipoles.

70

 

 

4.2.1

andi

funcl

be cl
func

form

of or

into

func

Whe1

4.2.1 MOM SOLUTION OF THE GENERAL HFIE

Equation (17) can be solved by the MOM, after choosing an appropriate set of basis
and weighting functions. Two general classes Of basis functions exist. Sub-domain basis
functions (SBF) exist only over subsections Of dipole surface S, and are zero everywhere
else. Entire-domain basis functions (EBF) exist over the entire range of S, and should
be chosen tO model the vanishing Of current at the device ends. In this section, pulse-
function (PF) sub-domain basis functions are used as both basis and weighting functions,
forming a pulse-function Galerkin’s solution.

Consider a dipole of width 2wd and total length L=21 which is centered at the origin

of coordinates (x,z) along the z axis, as shown in Figure 4.1a. The dipole is subdivided
into 2N sections, each of width 26. The current in (17) is expanded in a set of pulse

functions (PF),

N a P (2)
K , = _L_"___
‘(x Z) 211 x 2 (18)
wd

with unknown amplitude an. The square root edge singularity condition is incorporated

in ( 18) to model well-known behavior of the current. The PF ’3 are defined by

. 1 lz-Z. l<5
P"(z) - {0 otherwise

where 25 is the size of the partition, as shown in figure 4.1b. The weighting function

71

 

 

13wc

Figu

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A
-t 8
3%] >Z
—-> <—
26
81(2)
(0.) A
1.111-
41111'1114111111117\Z
25-0 Zn Zn+0
(l0)

 

Figure 4.1: Microstrip dipole subdivided into segments for pulse function expansion
(a). Pulse function distribution (b).

72

 

 

is at

The

T6011

 

is applied to HFIE (17) resulting in

(”'41 N

“’41 ‘
ffdzd,_f%_(Z)_ H G. 2 Edi-1,12%)-

'Wd'l l—[ifi _w‘fl n=-N 1-[£)2
\i “’4 \ \J “’4

 

 

 

 

z ”’41 r
BGC N anPn(y) - i (19)
kJ f f E ———dyax’-—J—E,(7) sin[kc(z-z’)]dz’
0 -w -I 6y ""N l 2 kcnc
d 1_ x—.
(Wd] iz=z’

 

C,cos(kcz) -Czsin(kcz) * ‘ 0'
l

The order of integration and summation may be interchanged, since the sum is finite and

 

all integrations are assumed to be convergent. Exploiting the sub-sectional nature of the

pulse functions reduces the integrations over 2 such that

“’4 1 W4 z,,+b
f ff(x.z)P,(z)dzdx = f f f(x,z) dzdx.
-w,-l “W4 ‘11"

The spectral integrals associated with the Green’s function components are evaluated in

rectangular form, as detailed in Section 2.4.2. The integrals

73

 

 

 

 

 

 

 

and

lead

The

isi1

etjEx
f ———dx = (wdrt)Jo(Ewd)

-W1 1 _ [if
\ W111

 

 

and

f e 0411311111905 -z’)]dz’ = _2i_2_{kc[cos(cz) -cos(kcz)]

0 c

ijrkcsintrz) - rsrntk,z)1}

lead to the matrix equation

sin(k 6) (20)

N
E aan-2(wdrr) kc {Clcos(kczm)+Czsin(kczm)} = 3,".

n=-N c

 

The term

as . 2 '-
Mm, = 16de cos{C(z,,-z,,,)lsmc(fa)51(C)+
0

 

 

32(4)
(k3 - c2)

 

_ $111095) sm§C5)[COS(CZn)COS(kam) (21)

k

C

 

52(0 }

4 .3 in(Cz )sin(k )]
k S n sz (k3 - (2)

C

is in the form of a 2-dimensional spectral integral, where

74

 

 

and .
integ
poly:
integ
dime
simp
all n
puls
samr

entri

Sect

For

 

 

 

°‘ (1 +R -p C)
S(C)= (w W’s ' c d
1 ii a“ 0( W11) 2(21t)2pc
(22)
'° K20
._ 2 1:
52m — f (wmraow) 212102“

and J0 is the 0‘11 order Bessel function [40]. The above two terms arise from the spatial

integrations over the transverse coordinate, and are approximated by Chebyshev
polynomials [54] over ranges of C that might be encountered in performing the spectral
integral in (21). Evaluation Of matrix entries (21) is then reduced to performing a 1-

dimensional spectral integral involving the approximated functions S1(C), 32(C), and

simple trigonometric functions. Since S1(C) and S2(C) are approximated only once for

all matrix elements, this method is increasingly efficient as the number of sub-sectional
pulse functions increases. Later, a entire-domain basis function MOM solutiOn to the
same EFIE will be Obtained. This solution requires a relatively small number of matrix
entries, and it was found for this method that the polar integration scheme discussed in
Section 2.4.2 is preferable.

The RHS of (20) is given by

W41 F

B... = f f 11,11) - j fE.‘<x.y.z=z’)sin1k.(z-z’)idz’ 424x
x

-w‘-1 2‘ 'chlc 0 (23)
l “H '
W4

. . . . . I _. . .
For the arbitrarily oriented incident field Of a nucrostnp transmrssron line, E; (r) 18 given

 

 

75

 

 

by (3

whe1

last j

Expi

 

by (3.20). With (3.20), equation (23) becomes

B. = - ’ deN(C)(w.n){ 1

(763-110

 

 

kn

CCO

[kc [icos(zma,)sin(a,a)
“ 1

- -§-cos(zmkc)sin(kcb)] +j [leaf—Silly“ 081110115)

c 1

- at1k3sin(Z.,.k..)Sin(k.<‘3)J] 'Jo(a3wd)ej‘dlsin(6)Y(C) + c08(6)F(C)]

(24)
1
(k3 - as

+

 

C

k [icos(zma,)sin(a26) - k£cos(zmkc)sin(kc6)]
d2 c

 

- j (keg-sin(zmazkinmzb) - a27‘2—sin(zmkc)sin(kc5)]]
a

2 c

'Jo(a.wd)e " “(8111000) - 008(6)F (01}

where “1-4’ Y(C), F(C), and MC) are defined in Chapter 3.

The system of equations (20) has 2N equations and 2N+2 unknowns. The first and

last pulse amplitude can be set to zero, since the current must vanish at the dipole ends.

Exploiting a_N=aN=0 leads to a (2N x 2N) system of equations,

c, l
Kt-mr-N) K(-N)N “emu ‘ B-N
. . = , (25)
KN(-N) KNN “(N—1) BN

 

 

76

 

 

 

 

whic

and

are:

shor

Clll’l

which may be solved be standard matrix methods. Matrix entries are given by

 

 

r m = -N, ...’ N
Minn for n: -N+1, ...’ N_1
sin(kce) = -
Kmu = 1 ‘2(Wdfl)008(kczm) k for ’2: -x: 9 N
. sin(kcfi) m -N, ..., N
1 -2(wd1r)sm(kczm) ’6, for n —N

 

and Bm is given by (24). The solution of the above system of equations yields pulse

amplitudes an , from which the current distribution may be constructed. Natural modes

are found by a numerical root search of (25) when the forcing term vanishes, e. g.:

det[Km] = 0.

4.2.2 SEPARATION OF THE HALLEN EQUATION FOR EVEN/ ODD MODE
SYMMETRY

HFIE (17) may be separated into even and Odd modes with respect to z. It can be

shown that both terms of (17) which don’t involve C1 and C2 are even/Odd in z for

current Kz even/odd in z. This leads to the conclusion

C1 = 0 for Kz(x,z) = Kz(x,-z) (even modes)
C2 = O for Kz(x,z) = -Kz(x,-z) (odd modes).

Exploiting symmetry about 2:0 leads to the HFIE for even/odd modes,

77

 

 

whe1

and:

resul

whe:

Mat

 

 

 

 

W11 1
{Z} _, C cos(kc z)
.1 f Gs K102“ I = Cisin(kcz) +

'Wd 0

z “’4 l 6Gr{;} .
kc]; f f a; Kz(i")dS’- 1:111 Ez(i’) sintkc(z—z{)1 dz’

-w0 cc
4 z=zl

 

 

where

/

(x,y,z Ix .y’.z’)) = G,(x,y,z lx’,y’,z’) i G,(x,y,z lx’,y’,-z’)

.121

r
C

 

and similarly for the term. Following a procedure similar to that of the last section

results in the matrix system

 

 

(1,
K11 K1111 31
= 3 (27)
“(N-1)
Km ' KNN C1 BN
. {2}.
where
{2} "1:1, ..., N
M11111 «for ":1, ..., [v-1
K = 1
11111 cos(kcz,,,) sin(kcb) for m=1, ..., N
4094“) sin(kczm) kc n =N.

 

M . . 1:}
atnx entnes M," are

78

 

 

and

whe

dim1

whe

4.3

[ha

don

 

Mm = 321d<{costtz )cos(Cz ,)S——mc(f5)[s 5.10.149 ]

 

 

(1.2—:2)

(28)
'Cos(Cz )°°S(kczmz \ sin(Cfi) sm(kc 0) 52“)
C k. (k3- -c2)
and
M... = 32de 3111th )3sz )sm ((6) 5110+ Szto

C (kf-cz)

(29)

 

-sin(Cz,l )sin(k 2111/
‘5 kc k.<k. — c2)

,sin(ca)sin(k 5) (32(6) }
where S,(C) and S2(C) are defined by (22). The matrix entries are evaluated as 1-
dimensional integrals, where S,(C) and S2(C) are approximated by Chebyshev

polynomials as was done in the last section. The forcing term 3111 is given by (24),

where 1111 runs over only half of the dipole.

4.3 ENTIRE-DOMAIN BASIS FUNCTION SOLUTION OF THE EFIE FOR
MICROSTRIP DIPOLES

In this section, EFIE (4) is solved directly by the MOM, without first converting it

to a HFIE. Entire-domain basis and weighting functions are used, to form a entire-

domain Galerkin’s solution.

For a narrow, z—directed dipole, the sealer EFIE is found from (7) as

79

 

 

Curr

whe1

and

resui

natu
for 1
only

and

 

[. +_]?}(GP+G[)K(F’)&’&’+ 7j63G;K K(F’)dx’dz’=- jk‘E(f’). (30)
ayaz n .

"Wd‘l “W4 “Ia)’ c

Current Kz(x,z) is expanded in a series of entire-domain functions

Kz(x,z) = ii +
n=1,3 ,5 |1_wid]

where the first sum involves only odd values of n. Applying the weighting functions

ams 711151

 

«M:

(31)

2

3

wd

 

 

 

irrz
a, COS '31—
W.(x.z) =
x 2
1- _
W4

 

 

and

 

aj SE]. in?

W,-(x.z)) =
ll—[E—
wd

result in a linear matrix system which may be solved for the current amplitudes.

N

 

Current expansion (31) was chosen because each term in the sum closely models a
natural-mode current distribution. Pozar [48] used similar entire-domain basis functions
for the study of rectangular antennas, and found that good results were obtained with
only one or two modes. This was also found to be the case for the work described here,

and so as an example a two-term solution is described below.

80

whe

The

cut

 

 

For a current modeled as the first two even-mode (or odd-mode) terms of (31), the
2x2 matrix system is obtained as
A11 A12

al Br

 

 

 

 

 

 

= (32)
A21 A22 “2 32
where matrix entries are
u g e
A” = f f d6 d). 4(w,,u)210 [Acos(6)wd] pi}
o o (33)

. kfa +R,) + Azsin2(6)(ch -R, - 1)
2(21:)2 . °

 

The above integrals have been converted to polar form by the transformation (2.28).

This was found to be the most efficient integration technique when only a few matrix

entries need to be evaluated. The term Pi] is defined as

Pf, = I,(1,A)1,(1,x) P‘; = 20.20120,»

n

Pf, = I,(1,A)I,(3,A) Pf, = 2(1,).)I,(2,A)
19;, = I,(3,A)I,(1,A) P; = 12(2,)L)I2(1,A)

P' = I,(3,A)Il(3,).) P2; = 2(2,).)12(2,}.)
with

(mrl)s T)c:s[lsin(6)l]

 

11(n,)t)=

[£112 +Asin(6)l 2 illt—-Asin(6)l]

 

 

81

 

 

The

sim

wit

j(-1)"(2nlrr) sin[}tsin(6)l]
[m + Asin(6)l] [mr - rsin(e)z]’

 

[202,20 =

The above results were presented without detail or intermediate steps, but are derived in

a fashion similar to the HFIE equations. The evaluation of the forcing function B, is

similar to that of the coupling coefficient presented in Chapter 3, and is given by

.k 9 ' 8
B.- = "JTCI {10(“1wge1‘d T.{°}(a.)[sm<e)Y<o+cos<e>F(o1
C 0
(34)

+ Jo(a2wd)e ““4 2}{°}(a4)[sin(6)Y(C) 'COS(9)F(C)I}dC-

where
T1‘(x) = Il(l,x) T1°(x) = 12(1,x)

T;(x) = 11(3,x) T,°(x) = 12(2,x)
with “1-4: Y(C), F(C), and MC) defined in Chapter 3. Natural even/odd resonant modes

are obtained by a numerical root-search of the homogeneous equation

det[AU] = O.

The above example of a two-term current expansion has been generalized to
accommodate N terms, with a corresponding increase in matrix size. It has been found
that higher order terms contribute very little to the current expansion, and at most a two
term current is needed. This is especially true near a natural resonant frequency.

A two-term current which consists of an even-mode term and an odd—mode term has

also been investigated. This combination is not very practical for the near resonance

82

 

 

 

pra

ah

it i:

the

per

der

cm

 

 

 

regime, as it was found that the amplitude of the term not associated with that resonance

practically vanished, as might be expected.

Other single-term currents were also investigated. These included

cos(k¢z) -cos(k¢l)

Kz(x.z) =
1M

 

x

where kc = kc, kefi' with kc being the cover wavenumber and kefl' the wavenumber for

a homogeneous region of effective permittivity [48].

Since a single term EBF solution requires the least amount of computational effort,
it is desired to determine which current, equation (35) or single terms of (31), provides
the most accurate results. To compare different single term current functions, a
resonant-mode root-search was preformed for the first even resonance, where the film

permittivity varies. Table 4.1 contains a comparison between a single even term of (31),

denoted as I, and (35) with a =c denoted as II and a =efl denoted by III. Table entries
are the normalized complex resonant wavenumbers, kol =(kr +jk,)l , for a 1.0 cm x .01

cm dipole over a dielectric film of thickness t=.05 cm and permittivity ef=3.78.

83

 

 

 

 

 

 

 

Tat

nur

of
cm

1'81

4.‘

int

diJ

 

 

 

 

 

 

 

 

Table 4.1: Comparison of resonant wavenumbers obtained by different single-term
current distributions
5, I II III
1.5 (1.335,.00284) (1.332,.00286) (1.334,.00288)
3.5 (.9662,.00121) (.9640,.00123) (.9660,.00123)
5.5 (.7961,.000640) (.7945,.000646) (.7957,.000648)
7.5 (.69277,.OOO390) (.6915 ,.OOO395) (.6925 ,.OOO395)
9.5 (.62149,.00026l) (.6205 ,.000265) (.6212,.000265)

 

 

 

 

 

 

It was found that all three current functions converge to the approximately the same

 

resonant wavenumber. When the cover wavenumber was used in (35), the numerical

root-search required two more iterations than either I or III, which required the same

number of iterations. Forced response current amplitudes, apaz, found by the solution

of (32) were also found to be approximately the same. It can be concluded that either
current I, II, or 111 could be used to model the desired current distribution. For the

remainder of this dissertation, the single-term, entire-domain basis function solution of
the EFIE is accomplished with current I.

4.4 COMPARISON OF MOM SOLUTIONS

Two full-wave solutions of equation (4) have been presented, based on different

 

integral equation forms. The Hallen-form integral equation (HFIE), (17), is derived

directly from the fundamental electric field integral equation (EFIE), (4). The HFIE is

84

 

 

 

 

 

 

potential is re

asmoothing
sensitive to

adjacent puls
functions in
solution of a

Either
domain basis
which may r
computation
times.

A benefi
current, if er
relatively un‘
almost certaii

Entire-dc
somewhat, fr
For example1
functions. Pl

solved by on

 

 

 

generally considered to be a more stable integral equation than the EFIE. This is

because the HFIE is based on the vector potential, not on the electric field. The vector
potential is related to the integral of the electric field, where the integration tends to have
a smoothing effect leading to greater numerical stability. Accordingly, the HFIE is less
sensitive to discontinuities in the current expansion, such as is found at the juncture of
adjacent pulse—functions. For this reason, pulse—functions are viable for use as expansion
functions in a HFIE solution, but may lead to numerical instability when used for the
solution of a EFIE.

Either type of integral equation may be solved with either entire-domain or sub-
domain basis functions. Sub—domain basis functions often lead to large solution matrices,
which may require a large amount of computer time to fill. For problems involving the
computation of Sommerfeld integrals, this may lead to prohibitively long computation
times.

A benefit of sub-domain basis functions is their ability to correctly model any
current, if enough subdivisions are used. Thus, for a problem where the current is
relatively unknown, a pulse-function solution along with a convergence study would
almost certainly yield the correct current distribution.

Entire-domain basis functions have the advantage that if the current is known
somewhat, functions may be chosen to correctly model the current in a very few terms.
For example, the resonant modes of the microstrip dipole are very similar to sinusoidal

functions. Problems involving frequencies in the resonant frequency regime can be easily

solved by one or two terms of sinusoidal functions, such as equation (31).

85

 

 

 

 

 

 

This process

a considerab
domain basi
the polar for
entry. As a
(EBF) soluti
basis functio:
In order
for the first r
the film are t

was used, i.€
the real reso
dashed linei

wavenumber

considerable

 

In section 2.4.2, two different integration schemes were presented to evaluate the
spatial integrations associated with the dyadic Green’s functions. One method
accomplished the integration in polar coordinates, while the other method utilized
rectangular coordinates. It was found that when only a few integrations were needed,
the polar-coordinate integration is preferable. The rectangular coordinate integration
requires evaluating a one—dimensional integral as a function of some parameter, and then
approximating (or interpolating) that function as part of the integrand of another integral.
This process is not efficient when only a few integrations need to be evaluated. When
a considerable number of integrations are to be performed, such as the case with sub-
domain basis functions, the rectangular coordinate method is much more efficient than
the polar form, since the function approximation is done once and used for each matrix
entry. As a result, the polar form integration is used for the entire-domain basis function
(EBF) solution of (4), and the rectangular form integration is used for the sub-domain
basis function (SBF) solution of (17).

In order to compare the two solutions, a convergence study based on the root-search
for the first even resonant mode was undertaken. Physical parameters of the dipole and
the film are the same as described in Section 4.3. The even-mode solution of HFIE(17)
was used, i.e., the homogeneous solution of (27). Figure 4.2 shows the convergence of

the real resonant wavenumber, k,l, as a function of the number of pulses used. The

dashed line is the solution of a single term of (31). It can be seen that the real resonant
wavenumber converges to a value close to the l EBF solution, but requires a

considerable number of pulses. Figure 4.3 is a similar plot of the imaginary resonant

86

 

 

 

 

 

 

wavenumber
single term 0
of the EBF s
Additionally

to be

 

one or at m
solution me
not be dism
numerical er
problems ex
dipole orien‘
resonant fre
of the HFIE
EBF solutio

A furthr
4.6 and 4.7

film permitt

 

wavenumber, where again the dashed line is the solution of the EFIE obtained with a
single term of (31). Figures 4.4 and 4.5 are the corresponding plots for the convergence
of the EBF solution. It can be seen that excellent results are obtained with a single term.
Additionally, the nullspace current amplitudes for the three-term expansion were found

to be

at 1.0000
(12 = 0.0160 .
a3 0.0084

This demonstrates that the second and third terms may be neglected.

The convergence study would indicate that, at least for resonant-mode root searches,
one or at most a few terms of (31) used in a EBF solution of EFIE (4) is the better
solution method. The importance of the sub-domain basis function of HFIE ( 17) should
not be dismissed, though. This solution, along with various root-searches and other
numerical experimentation, led to the motivation for current expansion (31). Also, many
problems exist for which the current is expected to have an unusual shape, such as a
dipole oriented at an arbitrary angle to a source of excitation, and excited away from a
resonant frequency. For these types of problems, the sub-domain basis function solution
of the HFIE should be performed, and the unknown current quantified so as to see if an
EBF solution is appropriate.

A further comparison between the two solution methods is demonstrated in Figures
4.6 and 4.7. These figures show the real and imaginary resonant wavenumber versus

film permittivity. The pulse-function solution was the even-mode solution with 20 pulses

87

 

 

 

mrdodgg

Figure 4.2:

9&3 ubQEjC®>O>> SEOCOmmVK

 

 

 

*1

 

1.06 -
1.04

1.02

0.98
0.96

0.94 -

A
—L
x

v

&_
<1)
Q
E
3
C
(D
>
O
3
4.2
C
O
C
O
(f)
G)
01

0.92 r

 

5 9 12 16

LLLJ_I

19

23

. 27 .
Pulses (per A/2) .

Figure 4.2: Convergence of pulse-function MoM solution for real resonant

wavenumber.

88

 

 

 

C
VA

1.86

4
7.

2 O 7 5 3 4| CC PC
6 R v. VNV. A/._ 4|. 0. OO. 7
1r 1 1| 4| 1 ..| O O

A_ EV L®QE3C®>O>> ycocommm

0.64

Figure 4.3;

 

 

Resonant Wovenumber (k; I)

0.88

0.76

0.64 '
5 9 12 16 19 23 27 30

Figure 4.3:

 

 

 

i_l r l Lli_| r l l L [11 Ill

34' 37 41
Pulses (perk/2)

Convergence of pulse-function MoM solution for imaginary resonant
wavenumber.

89

 

 

0.95(

.43
9.9.
00

C LXV LWD

0.94l

3
9
O
E

2 2 4| 1|
9. 9. 9. 9.
O O O 0

0.9.73‘

3C®>O\S yCOCOw0K

m.
0

Figure 4.4:

0.950 -
0.946 -
0.942 -
0.938 -
0.934 —
0.930 -
0.926 -
0.922 _

0.918 —

Resonant Wovenumber (kr I)

0.914 -

 

r L

0.910 '
1. 2.0 3.0

Entire—Domain Bosis Functions

Figure 4.4: Convergence of entire-domain basis function MoM solution for real
resonant wavenumber.

 

9O

   

 

 

1|
VA

1.2C

E
all
4|

1! z i r\ << 1 - l-
.l l .l .I 0 0 O 0

C _xv L®QE3C®>O>> yCOC0m®K

Figure 4.5;

 

X10
1.20

1.10
1.08
1.06

1.04

A
_\<

V
L.
(D
_Q
E
3
C
(1)
>
U
B
4..)
C
O
C
O
U)
(1)
0:

Figure 4.5:

 

—3

a)

 

2 3
Entire—Domain Basis Functions

Convergence of entire-domain basis function MoM solution for
imaginary resonant wavenumber.

91

 

 

 

 

per 1/2, W
permittivity
is seen to go
is still well

environment

Since th

_ general chec

was accomp
method.
The re:
comparison
over a grout
Cylindrical:
an effective
manner, the
obtained by
plane) mod<
solved by a
SPace Greer
wavenumbe
sePartition n

Signifies the

 

 

per M2, where it refers to the wavelength in a homogeneous region of effective
permittivity eefl. The EBF solution consisted of the first even term of (31). Agreement

is seen to good throughout the range of permittivities. This demonstrates that the current
is still well-modeled by a single term of (31), as a parameter of the surround
environment is varied.

Since the differing solution methods are seen to agree for the homogenous case, a
. general check on the overall accuracy of the resonant—mode solution is in order. This
was accomplished by comparing the solution of the EFIE with an independent solution
method.

The resonant modes of coupled, free-space thin cylinders was available for
comparison [55]. By considering the film permittivity to be unity, a microstrip dipole
over a ground plane can be considered to be two coupled, planer dipoles in free space.
Cylindrical and planer antennas can be compared by considering the planer dipole to have
an effective radius aeff=w/2, where w is the half-width of the planer strip. In this
manner, the resonant modes obtained from EFIE (4) can be compared with those
obtained by the free-space solution for the anti-symmetric (constrained by the ground
plane) mode of a coupled—dipole system. The free-space coupled dipole problem was
solved by a pulse Galerkin’s solution of a coupled set of EFIE’s, which involve the free—
space Green’s function. Figure 4.8 shows such a comparison, where the real resonant
wavenumber is plotted against the imaginary resonant wavenumber, for values of dipole
separation ranging from d=0.2 em up to d=8.0 cm. The plot labeled "microstrip pole"

signifies the single—term EBF solution to (4). Agreement is seen to be excellent over the

92

 

 

 

0000

C evC L®QCCDC®>O>> yCOCOmmm

r0.
0

AI

Figure 4.6:

 

 

,2: O 1~—EBF (cosine)
- D Pulse Function

0.9 -
0.8 -

0.7 -

 

Resonant Wavenumber (kr l)

0.6

 

05 L l pLLL+I 1 ;;l 4L 1 1 r l

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Film Permittlvity (8f)

Figure 4.6: Real resonant wavenumber versus film permittivity.

93

 

 

 

4|
VA

3.0C

r4
_/.
2

’— E E C ’1. I. e E
4. .I. oo. 6. 5. O. 7. 4.
2 2 1 | | 4| 0 0
9 iv L®DE3C®>O>> wCOCOmwmm

fl.
0

FigUre 4.7:

x 10 ‘3
3.00 -
2.72 -
2.4,,“ O 1—EBF (cosine) .

_ D Pulse Functions l
2.16 —
1.88 -

1.60-

1.32
1.04

0.76

Resonant Wavenumber (k1 I)

0.48 ~

 

 

0.20 1 l r l r I r 1
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Inga—l 14 J

Film Permittivity (er)

Figure 4.7: Imaginary resonant wavenumber versus film permittivity.

 

94

   

 

 

0.4E

/( r. tr. 4 i It ,1 5L 1 ,l
0 0 0 0 0 0 0 0 0 0 4
m
u
.We
F

C _ XV L®DE3C®>O>> yCOCOm®K

 

O Microstrip Pole
Free-Space Pole

 
  

Resonant Wavenumber (k1 l)

 

000 L1 1 144—: L r l #14 A

0.75 0.86 0.961.061.17 1.271.381.48 1.59 1,691.80
Resonant Wavenumber (kr I)

Figure 4.8: Comparison of free-space, coupled dipole natural modes and microstrip
modes.

95

 

 

entire rang
signifies th
when ef=1
Agreen
numerical
integration:
Figure 2.7
solution de
are accurat1
integration
the HFIE-t
space solu1

verified.

The six
in this Chap
numerical 1

With exPeI-i

entire range of (1. Similar agreement was found with the HFIE-based solution. This
signifies that the electric dyadic Green’s function (2.24) reduces to the proper result
when ef=l.0.

Agreement between the above solution methods provides a validation check for
numerical integrations performed in the solution of IE’s (4) and (17). Numerical
integrations in the EBF solution follow contours in the complex lambda-plane shown in
Figure 2.7. Agreement between the EBF solution and the independent free—space
solution demonstrates that the numerical integration methods used in the EBF solution
are accurate. The HFIE-based solution follows similar integration contours, although the
integration is performed in rectangular coordinates as detailed in Section 2.4.2. Since
the HFIE-based solution agrees with the EBF solution, and with the independent free-

space solution, accuracy of the rectangular coordinate integration technique is also

verified.

4.5 RESULTS

The singularity expansion theory (Chapter 3) and the full-wave methods presented
in this chapter should provide results that agree in the resonance regime. In this section,
numerical results obtained using the above methods are compared with each other, and

with experimental data where applicable.

96

 

 

 

 

4.5.1 CU

The SI
of nearly-r1
device cun
modeled i1
approximal

It was
frequency
modes was
pulse-funct
functions (2

The c1
Figure 4.9
transmissio
Cm thick w
0%, ~5%,2
at resonant
resonance 1
ampiituder

Figure

the current

4.5 . 1 CURRENT DISTRIBUTION

The SEM theory was proposed in Chapter 3 as an efficient method for the analysis
of nearly-resonant microstrip device interactions. For this method to be successful, the
device current must be modeled accurately. The current on a microstrip dipole was
modeled in Chapter 3 with simple sinusoidal functions, and the validity of this
approximation is studied here.

It was stated in Chapter 3 that the theoretical current distribution at a natural-mode
frequency was very similar to a sinusoid. The current distribution for the first four
modes was shown in Figures 3.5, 3.6, and 3.7. These distributions were based on a
pulse-function MoM solution of the HFIE, equation (25). It was seen that the sinusoidal
functions (3.12, 3.13) closely model the full-wave solution.

The current distribution at several frequencies near resonance is investigated in
Figure 4.9, for the case of a parallel-coupled dipole a distance of .75 cm from a
transmission line. The dipole is 5.0 cm long and 0.1588 cm wide. The film is 0.0787
cm thick with permittivity (2.2-j.00198). The current at a frequency of +10%, +5%,
0% , -5 %, and -10% of resonance is shown, along with the measured current distribution
at resonance. It is seen that the current distribution doesn’t seem to change from its
resonance value, for frequencies at least 10% away from resonance. Beyond 10%, the
amplitude response of the dipole is negligible, hence those frequencies aren’t of concern.

Figure 4.10 shows the same data for a perpendicular-coupled dipole. It is seen that

the current does change slightly with frequency, although at resonance the current

97

 

 

 

 

Amplitude

Figure 4.9;

 

All Frequencies
0 Measured (Resonance)

1.0 —
0.9 1
0.8 '—
07 l
0.6 1
0.5 C
0.4 1
0.3 L
0.2 i
0.1 1

.

Amplitude

 

0 rlrJL114 IL | l J_i_l

i—1.0 —O.8 —0.6 —0.4 —0.2 0.0 0.2 0.4 0.6 0.8 1.0

z/L

Figure 4.9: Current distribution near first even resonance for a parallel—coupled
dipole.

98

 

 

 

0.9 i
0.8 3
0.7 i
0.5 i
0.5 1
0.4 3
0.3 3
0.2 3
0.1 3
0.0 i

Amplitude

 

Figure 4.1(

 

1.0 r
0.9 -
0.8 —
(I) _
U 0.7 -
§ 0.6 —
3 0'5C 0 Resonance
< 0.4 - A 407’
- U —5Z
03: 0 +57.
0.2 — + +1 OZ
0.1 —

 

0.0
—1.0 —0.8 —0.6 —0.4 —0.2 0.0 0.2 0.4 0.6 0.8 1.0

z/L

Figure 4.10: Current distribution near first even mode for a perpendicular-coupled
dipole.

99

 

 

 

 

distribution
frequencies

sinusoidal 1

4.5.2 FR]

The fit
singularity
presented i

induced CU]

cm parallel
a SBF solu
comparisor
function is
Agreement
resonant-m
normalized
resonance,

In orde
the Strip or
addition of

term iS deg

 

 

distribution is sinusoidal regardless of orientation. Since the dipole’s response at
frequencies just 5% away from resonance is practically negligible (see next section), the

sinusoidal distribution (3.12) should be sufficient.

4.5 .2 FREQUENCY RESPONSE

The frequency dependence of a dipole’s surface current is obtained approximately as
singularity expansion (3.4). This current should be in agreement with full-wave solutions
presented in this chapter, as well as experimental measurements. Figure 4.11 shows the

induced current amplitude as a function of normalized cover wavenumber (kcl), for a 1.0

cm parallel-coupled dipole located 1.8 cm from a transmission line. The MoM solution,
a SBF solution for even modes of (27), is compared to results from the SEM theory. A
comparison between the SEM theory and the EBF MoM solution with one expansion
function is shown in Figure 4.12, for the same physical configuration as in Figure 4.11.
Agreement between the differing methods of solution is excellent over the entire
resonant-mode frequency regime. It should be noted that in both figures, curves were
normalized by the same value, which was obtained from the SEM method at the peak of
resonance.

In order to compare theoretical and measured results, the imperfect conductivity of
the strip conductors must be accounted for. This effect modifies EFIE (4) with the
addition of a term involving the skin-effect surface impedance [56]. The addition of this

term is described in Appendix E. Figure 4.13 shows the induced current amplitude as

100

 

 

 

0.

AI

9.
0

no
0

7.
0

000

003:0E<

3.
0

2
0

1|.
O

00
0

Figure 4.1

 

1.0 ~
0.9 —
0'8: o SEM
0.7 — E1 MoM (SBF)
0.6 - i
0.5 -

0.4 -

Amplitude

0.3 -
0.2 -
0.1 _

 

 

0.0 7' 1 1 1 a 1 1
0.96 0.97 0.98 0.98 0.99 1.00 1.01 1.02 1.02 1.03 1.04-

Wavenumber (kc/kc r)

Figure 4.11: Comparison between SEM theory and PF MoM solution for current
amplitude vs. wavenumber.

101

 

 

 

 

0.

/||

00.
0

DO.
0

./.
0

0 5 A.
O. O. O.

O

Figiue 4.12:

 

 

 

1.0
0.9 —

 

0.8 — 0 SEM
0-7 - D MoM (EBF)

0.6
0.5
0.4

Amplitude

0.3
0.2
0.1

 

 

DO 1111] 11 1111 IJIALlLI l

0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05
Wavenumber (kc/kC r)

Figure 4.12: Comparison between SEM theory and EBF MoM solution for current
amplitude vs. wavenumber.

102

 

 

 

 

 

 

a function 1
cm from a
amplitude t1
zero surfac
normalized
Agreement
comparison
normalized
then for 1111

the microst

4.5.3 VA
DII
The va

is also pred

This can

exPerimenl
line is use
exPerimenl
from the t]
for the cas

Separation!

 

 

 

 

a function of normalized wavenumber for a 5 .0 cm parallel~coupled dipole located 1.0
cm from a transmission line. Plots are shown comparing the SEM predicted current
amplitude to the measured current amplitude. The theoretical results account for the non—
zero surface impedance by the methods of Appendix E. The experimental curve was
normalized to unity, and so only the correct bandwidth can be compared with theory.
Agreement is seen to be good over the entire frequency range considered. For
comparison, theoretical results ignoring the finite surface impedance are included,
normalized to unity. It can be seen that the bandwidth of this curve is much narrower
then for the others, suggesting that ohmic losses must be considered to properly model

the microstrip dipole. This is discussed further in Section 4.5.4.

4.5 .3 VARIATION OF DIPOLE CURRENT AS A FUNCTION OF
DIPOLE/TRANSMISSION LINE SEPARATION

The variation of dipole current as a function of dipole/transmission—line separation

is also predicted by singularity expansion (3.4), through the coupling coefficient term AP.

This can also be compared with results obtained through full—wave methods and
experiment. As discussed in Chapter 3, the unperturbed field of an isolated transmission
line is used as the approximate excitation in the theoretical methods. It was shown
experimentally that this should be a good approximation when the dipole is separated
from the transmission line by a sufficient distance, which was found to be fairly small
for the case examined. Therefore, theoretical and experimental results should agree for

separations beyond that critical value. Figure 4.14 shows the amplitude of a 5 .0 cm,

103

 

 

0.9

0.8

6 5 4.
O. 0 0
0030L_QE<

DC
0

 

 

Figure 4.1

 

1 .0 t 71.
,1.

"" a SEM

0.8 — .
_ i . + SEM (ZI =0)
0.7 e l 0 Measured
_ ll
0) — 0
_O 0.6
:5 ' 1|
1: 0.5 ~
Q— |- 0
E 0.4 — n
< _ u
0.3 - o “
_ 0
0.2 - ”90 o
_ g’m § E
0.1" acce’a'fl ac“. =

 

 

 

l 1 l 1 l 1 1 1 l 1 I I I 1 4L 1 1

0.0 '
0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05
Frequency (f/fo)

Figure 4.13: Comparison between SEM theory with and without finite conductor
impedance accounted for, and measured Q-curve.

104

 

 

 

 

 

parallel—cor
normalized
quality fact
the curves i
then the cri
It shoul
of the tran
constant, t1
Only the R
This also p1
system Q-f
approximal
when norrr
Figure
curve was
This was d
for Perpenl

leave only

4.5.4 LO

Figure

 

 

parallel-coupled dipole vs. dipole/transmission-line separation. Each curve was
normalized to unity at a separation distance where the dipole/transmission—line system
quality factor became within 10% of its isolated value. It is seen that agreement between
the curves is good for separations beyond the critical value, and poor for separations less
then the critical value, as expected.

It should be realized that the main significance of this figure is to verify the accuracy
of the transmission line field, found by equation (3.20). Since the frequency is held
constant, the LHS of matrix system (25) or (27) doesn’t change with separation distance.
Only the RHS, which involves the transmission line’s electric field, varies with distance.
This also provides complementary verification for monitoring the dipole/transmission-line
system Q-factor to indicate when the unperturbed field of the transmission line is a good
approximation to the actual impressed field, since the predicted current amplitudes agree
when normalized at this critical separation distance.

Figure 4.15 is a similar plot for a perpendicular—coupled dipole. For this case, each
curve was set individually to unity at a small value of transmission-line/dipole separation.
This was done because the induced current amplitude falls off very sharply with distance

for perpendicular-coupled dipoles, and to normalize at a sufficient separation value would

leave only a few data points to compare.

4.5.4 LOSS CONSIDERATIONS

Figure 4.13 showed the need for correctly accounting for ohmic losses due to

105

 

 

 

 

Amplitude
3

Figure 4_1

 

 

22— .
20: I O MeasuredAnwfldude
- : a SEM Amplitude :94
t8 : A Measured<3 -89
| _
1.6 _ i ,85
14 ' '
_g —78
12 1
{g -72
§_10. - 7
—6
.< 08 _
06 —62
0.4 -56
02 —m
0.0 111 L 1 l 1 1 J_l r 1 L g, 4 4 1 45

 

 

 

 

Separafion ds(cnfi

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00

Figure 4.14: Experimental and theoretical current amplitude vs. separation for a

parallel-coupled dipole, with measured Q—factor.

106

 

Quality Factor

 

0.

0.

0.0.0.

0030.:QCC<

0.

0.

0.

Figure 4.

 

 

 

Measured
SEM

 
   

0.6 -
0.5 -

Amplitude

0.4 -

 

 

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Distance 08 (cm)

Figure 4.15: Experimental and theoretical current amplitude vs. separation for a
perpendicular-coupled dipole.

107

 

   

 

 

imperfect
properties.
loss, and
effect of v

cm dipole

Table 4.2

Each C881
conductor
finite con
the finite
Appendix

dissipated

Power (11

imperfect conductors. By including this effect, and by varying the dielectric film
properties, loss mechanisms associated with space and surface-wave radiation, dielectric
loss, and conductor loss can be studied. The following table entries demonstrate the
effect of varying the film and impedance parameters on resonant wavenumber, for a 5.0

cm dipole over a t=.0787 cm film.

 

 

Table 4.2: Effect of dielectricand ohmic loss on the complex resonant
wavenumber
e, Zi ko 1 Description
(2.2,0.0) =0 (l.l4l96,.000378) PD, PC

 

(22,00) #0 (l.l3916,.003183) PD, 1c
(2.2,-.00198) =0 (l.l4l96,.000802) ID, PC
(2.2,-.00198) ¢0 (1.13916,.003607) ID, IC

 

 

 

 

 

 

 

 

Each case is given a descriptive set of letters: PD=perfect dielectric, PC=perfect
conductor, ID=imperfect dielectric, and IC=imperfect conductor. When Z‘=O, the
finite conductivity of the copper dipoles was ignored. For the two cases listed as Z #0,
the finite conductivity of the dipoles was accounted for by the methods described in
Appendix E. The imaginary wavenumber is related to the Q and hence the power

dissipated (Pd) and the energy stored (E,) by

 

Q: r = rs. (36)

Power dissipated is then

108

 

 

 

 

EL

Assuming 1

table, the r

where C =1
d=dielectr
the total d

found as

Thus it is
small, whj
resonators
thickness,

Which the

where it i

SUbStrates,

Pd = 20),.ES. (37)
Assuming that the stored energy remains constant as parameters are varied in the above

table, the power dissipated due to all loss mechanisms is obtained ast“ = .003607C

where C =2ES and r=space radiation, s=surface—wave radiation, c=conductor loss, and

d=dielectric loss. This represents the total power dissipated by the dipole. Normalizing
the total dissipated power to unity, dissipated power due to other mechanisms can be

found as

1. PD, PC: P§‘=.1048

2. PD, IC : PJSC=PJS+PJ=PJ =.7777

3. ID, PC: Pg“ =1); +P;‘=»Pf = .1176
Thus it is seen that conductor loss is the dominant factor. Radiation losses are very
small, which agrees qualitatively with Belohoubek et a1. [57], who studied microstrip
resonators. It was stated there that radiation increases with increasing substrate
thickness. This was verified by increasing the substrate thickness to t=.315 cm, for
which the dissipated powers were found to be

1. P;‘=.6445

2. P; = .2810
3. Pj=0749

1.0

where it is clear that radiation has a dominant effect on the dipole’s losses for thicker

substrates.

109

 

 

 

 

Full-v
to the SE]
equation 1

The ti
circuits is

equation (

which is 1
for the dig
EFIE

resulting .

is solved

expanded

 

4.5 W

 

Full-wave solutions to the EFIE presented in Chapter 3 are obtained, and compared
to the SEM theory. The method of moments (MoM) is used to transform the integral
equation into a matrix equation, which can be solved numerically on a computer.

The fundamental EFIE which quantifies electromagnetic interactions in microstrip
circuits is solved by two methods. EFIE (4) is transformed into a Hallen-form integral

equation (HFIE)

f GSKZ(?’)dS’ = c1 cos(kcz) +C2sin(kcz) +
S

2 r .
kcf f aG‘Kz(f")dS’— J Ez(f’)
0 S 0y kt‘nc

sin[kc(z-z ’)]dz’.

2:7,,

 

 

 

 

which is solved with a pulse-function, Galerkin’s MoM solution. A complete solution
for the dipole current is obtained, as well as individual solutions for even/odd modes.
EFIE (4) is also solved directly, without converting to the Hallen form. The

resulting equation,

 

w w I r .
Isa—‘92— fi<GP+G')K(F’)dx’dz’+ [if—G“ K(r"")dx’dz’=-J—k‘E(i-')
c 622 _wd._1 t Z _wd_layaZ2 2 nc z

is solved by an entire—domain basis function (EBF) MoM solution. The current is

expanded as

110

 

 

 

which is 1
current re
Conv
obtained '
generally
is more r:
The <
frequenci
distributil
resonance
dipoles.
Resui
is found 1
exPerime
dipole/tr:
is again 1
Pow:
108s, and

but radia

 

N a "COS!:%Z; M ans ms;
Kz(x’z) = E + 2

m=1 x
1- __

 

 

 

 

which is used for testing also. It is found that one or at most a few terms of the above
current result in good accuracy.

Convergence studies are performed on the various methods of solution, and results
obtained with each method are compared. It is found that the EBF solution method is
generally preferable for near—resonant frequencies, although the pulse-function solution
is more robust for the purpose of correctly modeling the unknown current.

The current induced upon a microstrip dipole by a transmission line is studied, for
frequencies corresponding to resonance and near resonance. It is found that the current
distribution remains sinusoidal for parallel-coupled dipoles excited up to 10% away from
resonance, and that nearly sinusoidal currents were found for perpendicular—coupled
dipoles.

Results are presented for the frequency response of the dipole current. Agreement
is found to be good between the various solution methods of this chapter, Chapter 3, and
experimental methods. The amplitude response of the dipole current as a function of
dipole/transmission—line separation is studied by the various methods, where agreement
is again found to be good.

Power dissipation is discussed in terms of space and surface—wave radiation, ohmic
loss, and dielectric loss. It is found that ohmic losses are dominant for thin substrates,

but radiative losses become significant as the substrate thickness increases.

111

 

 

 

 

 

5.1 Iii

The 1
dipole cm
are studie
is a full-1
will be u:

Sectit
devices.
singularit
of EFIE’
modes. 1
interact s
isolated ]

 

CHAPTER FIVE

COUPLED MICROSTRIP DEVICES

5. 1 INTRODUCTION

The preceding chapters described various methods for the analysis of a microstrip
dipole excited by a nearby transmission line. In this chapter, coupled microstrip dipoles
are studied using an approximate, dominant-mode singularity expansion. Also presented
is a full-wave MoM solution for coupled dipoles, which, along with measured results,
will be used to validate the coupled—mode perturbation approximation.

Section 5.2 presents the approximate singularity expansion method for coupled
devices. The current on any element of the coupled system is approximated by a
singularity expansion similar to equation (3.4). Exploiting this current in the coupled set
of EFIE’s describing the system then leads to the defining relation for natural system—
modes. The assumption that only elements with nearly identical resonant frequencies will
interact strongly prompts expanding the Green’s dyad in a Taylor series about the
isolated resonant frequency, leading to an algebraic set of system-mode equations

describing the coupled system.

112

 

 

 

 

Sectio
with entir
presented
isolated e
accuracyi

Sectic
coupled-d
experimer
"critical s
which res
as the dip
throughor

for separz

5.2 M

The l
environm
The n“I di
in Figure
longitudir

Syste

 

Section 5 .3 presents a MoM solution to the coupled-system problem, implemented
with entire—domain basis functions (EBF’s). Similar full—wave solutions have been
presented for rectangular patches by Pozar [48], where it was found that, as in the
isolated element case, only one expansion function is required to provide adequate
accuracy in the resonance regime.

Section 5.4 presents numerical and experimental results for the investigation of
coupled-dipole system-modes. Agreement between the two theoretical methods and
experiment is found to be good for dipoles separated by an adequate distance. This
"critical separation" is found to be quite small, and can be inferred to be the point at
which results from the approximate theory diverge from those of the full-wave theory,
as the dipoles are brought together. Experimental results agree with the full-wave theory
throughout the entire range of dipole-dipole separations, and with the approximate theory

for separations beyond the critical distance.

5.2 APPROXIMATE PERTURBATION THEORY FOR COUPLED DEVICES

The N coupled dipoles are located in the cover layer of a conductor/film/cover
environment, at the film/cover interface. The film is of thickness t and permittivity 5,.
The n‘11 dipole is of total length L, and half length In. A system of two dipoles is shown
in Figure 5.1. The dipoles are separated from each other by transverse distance d,“ and
longitudinal distance d,2.

System eigenmodes of an N-dipole system are based upon the coupled set of electric

113

 

 

 

 

 

field integ

for all F6

are the or
EFIE’s (l

multi-dev

5.2.1 N

The:

natural-n

similar f1

mode cur

Hm dipole

Where to

ammitud.

field integral equations (EFIE’s)

 

N
552 [d‘(7|r’)-Kn(?’)ds/ = °£ it?) (1)
S

for all FESM, m =1, ...,N, where in is a unit vector tangent to the m‘h dipole and kvnc

are the cover layer wavenumber and refractive index, respectively. The coupled set of
EFIE’s (1) provide the fundamental resource for the investigation of EM phenomena in

multi-device systems.
5.2.1 NATURAL SYSTEM-MODES

The singularity expansion for isolated element currents (3.4) was based upon isolated

natural—mode currents 1340') and resonant frequencies (eq. It is reasonable to assume a

similar form for the current on each element of a coupled system, where the natural

mode current is replaced by EMU) , the eigenmode current of the q“1 system mode on the

n‘” dipole. The current on the rim dipole can be approximately represented as

Eamzzw (2)
" q (w—wq)

where wq is the q‘11 complex natural system-mode frequency and am, is the natural-mode

amplitude. Exploiting expansion (2) in the coupled EFIE’s (1) lead to

114

 

 

 

 

Figure 5

 

dgr

l

 

 

Dipole 1

Figure 5.1: A system of two coupled dipoles.

115

 

 

 

" 1'
Since E

vanish at

system

with non-
frequency

The 1
impractic
mode pe:

coupled—5

5.2.2 c

Natu

Where kn

kIlOWiedg

 

 

Since E." is regular at 0) amp and the LHS has a pole there, the integral sum must

vanish at up to produce an indeterminate form. This results in the homogeneous EFIE

system

N
5.5, fé‘tr‘lr’mrfmer’ms’ = o m=1,...,~ (3)
n= Sn

with non-trivial solutions for w = to which defines the q"‘ system mode with natural

q,

frequency 0),, and element current distributions knq’

The direct solution of (3) by the MoM is examined in Section 3, but becomes
impractical for systems composed of a large number of interacting devices. A coupled—
mode perturbation approximation, based upon the isolated natural-mode currents of

coupled-system elements is consequently prompted.
5.2.2 COUPLED-MODE PERTURBATION EQUATIONS

Natural system-mode frequencies wq can be obtained from the direct solution of (3),

where knowledge of the isolated device characteristics is not required . Alternatively,

knowledge of each system element’s isolated resonant frequency and current distribution

116

 

 

 

can be us

The t

is applied

Exploitin,

Mzamle

with non-

w=wq do

For near]
identifier

leading [1

The lead:

can be used to obtain an approximate perturbation solution for system-modes.

The testing operator

[as 1352(8)-

3-

is applied to EFIE’s (3), where 13:: is the resonant current on the m‘h isolated dipole.

Exploiting reciprocity of G‘G‘IF’) and making use of the coupled-mode approximation

-v

knq saw/2:2) leads to the algebraic set of system—mode equations

N
E C1400", =0 m =1,2,...,N (4)
n=1

with non-trivial solutions for current amplitudes am only for complex natural frequencies

w=wq determined from det[C,‘f,,(co)] =0. Coupling coefficients Cg” are identified as
0,3,,(0) = fds’ E30") -f 677/180) exams. (5)
SE S-

For nearly-identical dipoles, the operating frequency regime of significant interaction is

. . *‘ 0 .
identified as w z (of: z 053,). A Taylor’s series expansron of G ‘ about of"; is prompted,

leading to

nq

_. .. ”e _. _. o 80‘ (0) .-(0) _.
0.1.(0)= 49' 1.90500 (r’lr;o£.3)+‘a‘.7| ...‘“"""'1"“' "Mmds' (6)
sn Sn “M

The leading term vanishes for n=m, by (3.8) for the resonant current on the m"' isolated

117

 

 

 

device. '1

where, b)

When n1

0)
tarot.)

Exploitin

mode per

which de

element 1

Natl
to HOn-tr
devices,

the avera

 

 

 

 

 

device. The coupling coefficient for the n=m term becomes

Cn‘fmm) z C:m[<o -(.)SZ] (7)

where, by reciprocity of 0‘,

"q ._ «0) 4 , 86171700) :10) — l
cm(o)~sfds kmo) [T a) kmo’) dS. (8)

(
u S. 0M

 

When mm, the leading term in (6) is non-vanishing. The term proportional to

(o) - 052) is consequently rendered second-order small, leading to

C5. = [d5 ’ 5217’) - [d ‘(F’Ir‘mffh 73,5330”) d5. (9)
s, s

Exploiting equations (7) and (9) in the system—mode equations (4) leads to the coupled-

mode perturbation equations

[w—wg]aMC:m + Z: C,,‘f,,anq = 0 ...for m=1,2,...,N (10)

which depend only upon the constant coupling coefficients (8) and (9), and the isolated-

element resonant frequency.

Natural system-modes are obtained from the solution of det [C,:,,(m)] =0, which lead
to non-trivial solutions for amplitudes a”. For the case of two nearly—degenerate
devices, the system-mode frequency is obtained as to =516, where (3 =(w1+w2)/2 is

the average of the two isolated resonant frequencies and

6 = ,lA2+(C,,C,,)2

118

 

 

 

 

with A =|

For two
(12 = 10,.

the two 5

Thee

isolated d

in the ne
equation
repeated
entries in
developrr

(5.22).

 

with A =(m1- (15)/2. The ratio of natural-mode currents is found as

———: — +—.

“2 _ A ( A )2 C21
a, C12 C12 C

12

For two identical elements, ml=wz=wo and C12=C21 such that w=woiC12 and

a2 = 2:0,. These system modes correspond to antisymmetric/ symmetric coupling between

the two system elements.

The example to be considered is a system of two coupled microstrip dipoles. The

isolated dipole current Eng) is modeled by (3.12, 3.13), which was found to be accurate

in the near-resonant frequency regime. The coupling coefficient (8) is identical to
equation (3.10) for isolated dipoles, and its implementation, equation (3.15), is not
repeated here. For n¢m, coupling coefficient (9) is identical to off-diagonal matrix
entries in the single-term EBF solution of the coupled-dipole system of EFIE’s. This
development is presented in the next section, and the resulting equation is given by

(5.22).

5.3 MoM SOLUTION FOR COUPLED DIPOLES WITH EBF’S

 

The coupled set of EFIE’s (l) is solved by an EBF MoM solution, to provide a
comparison to the approximate coupled-mode theory presented in the last section.

The current on the nlh dipole is expanded in a set of EBF’s

 

 

 

 

 

resulting

for all Fe

elements,

operator

Which is
The 1
coupled (

is 0f leng

has lengt

coupled 1

J
1H?) = Zaan .(F) n=1,...,N (12)

resulting in

-jk
11c

”M? ~‘E’(r)

 

N
2‘".sz (f1?)- Ea“! K ”(r/)ds’ =
n=1 3"

for all 765,”, m =1, ..., N. It is assumed that each dipole has the same number of current
elements, (I). The above system has N equations and NJ unknowns. Testing with the
operator

[as Km,(r“)-
S.

which replaces the fm- operation leads to

N J _
EE‘17-}deEmz(F)'f6e(7|F/)'/;,,j(f")dS’ = J c
C S.

n=1j=1 S

 

m = 1,...

“.6810st --{ ,

   

which is a (JN)x (JN) system of equations.
The above system of equations is solved to determine the system resonances of two—
coupled dipoles. Consider the coupled dipole system shown in Figure 5 .2. Dipole one

is of length 21,, width 2w,“ and is located at the origin of coordinates x,z. Dipole two

has length 212, width 2wd2, and is located at the origin of local coordinates (x1’,zll). The

coupled set of equations ( 13) become

120

 

 

 

Figure

 

 

 

 

Z1
X
1 )9
Dipole 8 A/ Z
(X’Z):(dsl’d52) 1
‘F. A, z

 

Dipole 1

Figure 5 .2: Global and local coordinate systems of a two-dipole system.

121

 

 

deE11(F)'fée'E1j(F/)dsl : deE11(7)°fée'E2j(7/)dsl a“
SI SI 51 52

o
_____________________ l--------------—---—- __ _ l=1,...,J
' o ' = 1,...,J.
fdskfl(f’)-fG‘°klj(F’)ds’ : fdsIEzj(7)-fé‘-E2j(?’)dsl a2,
52 51 S2 52
(14)

It should be noted that terms in (14) are represented in block-matrix form. Entries in
each block are designated by (l, j), and each block is of size (JxJ). When each dipole’s
current is modeled with a single term (1:1), (14) will have the form of a 2x2 matrix
system.

The diagonal blocks in (14) are found to be the reaction of each dipole with itself.
These terms are independent of the position of the other dipoles, and are found to be
identical to the (l, j) matrix entries found in the J-term EBF solution for single dipoles
presented in Chapter 4. These terms are given by (4.33) for the case of two EBF’s. If
the two dipoles are identical, the diagonal blocks are the same.

The off-diagonal blocks represent an overlap integral between the current of one
dipole and the field due to another dipole, evaluated at the first dipole’s location. This

relationship can be written as
[as Ea(f')-f§"EB(F’)ds’ .. fds Ea(F)'E°B(F)
s‘ s, s,
where ED is the field due to the i3" dipole and I}; is the current on the a‘h dipole. To

evaluate the above, the field of dipole B is found in terms of a local coordinate system,

122

 

 

 

 

 

and then translated to the coordinate system of the a“ dipole. This results in elementary
evaluation of the spatial integrals associated with EBF’s such as (4.31).

As an example of evaluating an off-diagonal term, consider dipole [3 to be positioned
at the origin of local coordinates xll,zl/, 7"” affix; +2121] +59), and have current
fimo=1gmq
The electric field (2.21) then becomes

62
S” ax; 621/

 

 

6G '
GP +6: + aye )Kfl01o1/l) ds //

 

fimo=fi
(15)

7
C

y GP+GJ+6G
5y

P

 

+zif kf<GP+Gl>+
SI

K961”) dS ”

 

 

where the double-prime notation designates the source-point coordinates, and the y-

component of field is not of consequence. Expanding the current as either even EBF’s

 

mtzl’
N an COST
e I /
Kp(x1:zl) = E (16)
n=1,3,5 / 2
x1
1- _

 

or odd EBF’s

i d]
M amS "TIT
qwm=2 (m

m=l x/ 2
wd

123

 

 

 

 

and exploiting the integral form of Green’s dyad terms G; (2.17) leads to

 

E{‘:1}(F’)= fdedCZ

n):Jo(wd€)eJEx|e 1‘21 2 I 1}(n, UPC) {x1 [E ( (FCC -R‘ — 1)]

+ 2“, [1:30 +R,) + (2(ch ~12, ~ 1)]}
(18)

where for even modes the sum is over odd terms and for odd modes the sum is over all
terms. The quantities I,(n,l,<‘.’) and 12(n,l,C) are defined in Chapter 4 (equation 4.14).

The above field of dipole D can be translated to the coordinate system of dipolea
(ie., x,z; F = £x+iz+9y) by the rotation and translation of coordinates

x1 = x—d
21: z-ds

NA.

= (x-d1)—cos(6) (z- d2)sin(6)
z} — (x- d1)sin(6)+(z- d2)cos(6)
xcos(6) - zsin(6)
z“: = 2sin(e)+z*cos(e)

A,
x1

resulting in the z-component of field

210(W42 E) ejxx ejzweIE(d,13in(e)‘d"°°3(9»

 

29¢) =ffd€dc:
-I€<d.isin<e)d.1cos(e» 19)
e i] i}(n, ,{le —sin(6)[EC(ch— —R- —1)]

MI

+cos(e) [kfa + 12,) + (2(ch —R,~1)]}.

124

 

 

 

 

 

 

 

 

in terms of the coordinate system associated with dipole a , where

x = Ecos(6)+Csin(6)
q: = C 005(6) - Esin(6).

cos|n—21;£!
ids ‘
2
_i
w

Testing with

(20)
s ‘
d
for even modes, or
s rm %]
d3 ' (21)

(a
H
.—'—.
Eh
‘—I
N

for odd modes results in the matrix block

A{:} _ f] d (wdzwdlnz) j:(d,,sin(e)-d,.cos(e» -j((d,.sin(e)+d.,cos(e»
a6 " Ede: WJO(W¢2 E) Jo(deX) e e
-.. 7‘

(22)
N N

—' -R —1
2 1{;}(m,z,,¢> "211902.59 { sm(6)[EC(ch , )]

m=1

+ 003(6) [kfa +1?) + (2(ch —R, — 1)] }

Other matrix entries are evaluated in a similar manner. It can be seen that each matrix
block is the reaction between two dipoles, or one dipole with itself, and doesn’t involve
any other dipoles that may be present. Thus, (14) may be easily generalized for N dipole
systems by adding the appropriate blocks, which will be of the same form as those in

(14).

125

 

 

All matrix entries are converted to polar coordinates in the spectral plane, which was
found to be the most numerically efficient form if only a few integrations need to be

performed. The polar coordinate transform was originally discussed in Section 2.4.2.

5.4 NUMERICAL AND EXPERIMENTAL RESULTS FOR COUPLED DIPOLES

The approximate, coupled-dipole perturbation theory should agree with the full-wave
MoM solution for various systems of coupled dipoles. Experimental results are also
obtained by methods described in Chapter 6, and a comparison of results is presented in
this section. All results obtained with the MoM used 1 EBF. This results in resonant
system—modes which agree with experimental results.

The coupled system of two identical, parallel coupled dipoles has been investigated.

The physical configuration is shown in Figure 5.2, where d,2=0, 6 =0 degrees, and (1,1

varies. The dielectric film has permittivity ef=2.20 —j.00198 and thickness t=.0787 cm.

Two L=5.0 cm dipoles of width wd=.0784 cm are located on the film layer, separated
by a distance d“. Figure 5 .3 shows the real resonant system wavenumber, normalized
by the isolated resonant wavenumber, as a function of d,,. All three methods
(approximate perturbation, full-wave MoM, and experimental) agree very well for
separations beyond a "critical separation" distance of about .25 cm. For separations less
then this critical value, all three methods agree for the symmetric modes (bottom set of
curves), but do not agree for the antisymmetric modes (top set of curves). For the

antisymmetric modes, the perturbation approximation diverges from the MoM and

126

 

 

 

1.090

0.992
0.978

0.964

Resonant Wavenumber (kr/ko)
3
M
O

: 1::1

— x ::1

'_ 0 Perturbation
_ >1: X Measured

_ D MoM

 

 

LlllLlL4lLi4lilllLLl

50 ‘
0.10 0.25 0.40 0.56 0.71 0.86 1.01 1.16 1.32 1.47 1.62

Figure 5.3:

 

Separation dsi (cm)

System—modes for two identical, parallel coupled dipoles.

127

 

 

 

 

 

 

 

 

 

experimental results, whereas the latter two agree qualitatively though not quantitatively.
It is sensible that the perturbation approximation breaks down for very close spacings,
since the currents on the coupled-dipoles are expected to be significantly perturbed from
their isolated states. It is also reasonable that the symmetric mode would be easier to
model for small dipole-dipole separations, since this configuration is analogous to one
thicker dipole. Two closely spaced dipoles at the anti—symmetric mode frequency have
equal but opposite currents, and a complicated interaction is expected. It should be noted
that the theoretical curves were normalized to the same isolated resonant wavenumber,
and the experimental points were normalized to the measured isolated resonant
wavenumber. These isolated wavenumbers differed by 1.42%.

Figure 5 .4 is a 3-dimensional plot of the current amplitude on one dipole of a two-
coupled-dipole system versus separation and frequency. This data was obtained
experimentally for the system of identical, parallel-coupled dipoles considered above.
It is seen that the symmetric/antisymmetric modes are clearly discemable for small
separations, and that there is little response at other frequencies. As the dipole—dipole
separation (d,,) increases, the frequencies of the two modes coalesce into a single
frequency, that of the isolated dipole.

As a further study of parallel-coupled dipoles, resonant system-modes are studied at
a fixed transverse separation, d,1=.l6 cm, as longitudinal separation d,2 is varied. All
other physical parameters are the same as in the above. Figure 5.5 shows
symmetric/antisymmetric modes versus longitudinal separation. It can be seen that the

mode—splitting increases initially, and as the separation is further increased, the modes

128

 

 

 

.0

Rude
0.7 1

NM”
0.3

Figure 5 .4:

 

 

 

Measured parallel-coupled dipole response vs. frequency and

129

 

 

 

 

 

 

 

::

   
   

“220 1::
1.178 O MOM
A Perturbation

1‘1 36 Measured

1.094

1\ T—r fl] 71* [‘77—]

 

_x

.010 ________________________________
0.968 ‘
0.926

0.884

Resonant Wavenumber (kr/ko)

0.842

 

gilngLJILJILLIIIILJ'

00
0.00 0.54 1.08 1.62 2.16 2.70 3.24 3.78 4.32 4.86 5.40
Separation dsz (cm)

Figure 5.5: Resonant wavenumber vs. longitudinal separation dsz.

130

 

 

 

 

 

 

 

 

approach the isolated resonant mode, which is represented by the dashed line. This is
in agreement with physical intuition, since for sufficient separations the two dipoles do
not overlap each other at all, and little coupling would be expected.

Figure 5.6 shows the real resonant system wavenumber versus dipole separation d,,,
for two parallel-coupled, unequal dipoles. The plot is normalized by the average of the
isolated dipole’s system wavenumbers. The physical parameters are the same as in
Figure 5 .3, except that the two dipoles have length L1=5 .0 cm and L2=4.5 cm. It is
seen that the system-mode wavenumbers are split symmetrically about the average
wavenumber, corresponding to symmetric/antisymmetric coupling. Again, results from
all three methods agree for the symmetric mode, but the perturbation approximation
disagrees with the experimental and measured results for the antisymmetric mode at very
small separations.

The system-mode resonances of two coupled dipoles is shown in Figure 5.7, as the
angle between them varies. The longitudinal displacement is d,2=2.6 cm, and the
transverse separation is d,l=-.l6 cm. The relative angle between the dipoles, 6, is
varied from 0 to 70 degrees. All other physical parameters of the board and dipoles are
the same as in Figure 5.3. It can be seen that the maximum coupling exists between
dipoles when 6 =0 degrees, and that the coupling decreases as 6 increases until the
dipoles are virtually uncoupled. The resonant system wavenumber is normalized by the

isolated dipole’s resonant wavenumber, kg.

131

 

 

 

Resonant Wavenumber (lo/ROW.)

1.08

1.06

1 .04

Figure 5.6:

 

1::1

    
 
 

_ E::l

_ 0 Perturbation
D MoM

_ Measured

 

 

LL L P4 L L41 L4 L4 l 1 L; l 141 1 l

2
0.10 0.25 0.40 0.56 0.71 0.86 1.01 1.16 1.32 1.47 1.62

Separation dsi (cm)

Resonant system-modes for non-identical, parallel-coupled dipoles.

132

 

 

 

 

 

 

 

 

1.199 '
1.150 1::
1.120

Perturbation
Measured

1.080
1.040
1.000
0.960
0.920 —

0.880 ’

Resonant Wavenumber (kr/ko)

 

0.840

 

0800 l I 4 A 4‘ 1 1* 1— ' l I l l 1
O 10 2O 30 4O 5O 50 7O

Angle 0°

 

Figure 5.7: Resonant system—modes as the relative angle between two dipoles is
varied.

 

 

 

0

 

5 .4 SUMMARY

Resonant system-modes of coupled microstrip dipoles are studied. A perturbation
theory is developed based on the coupled set of EFIE’s which rigorously describe the .
system.

The current on the n‘h dipole can be approximately represented as

K(r, (.0)= ~02 —""—— k’WG) (23)
q (w- wq)

where wq is the q‘“ complex, natural system-mode frequency and am is the natural-mode

amplitude. The above current is utilized in the coupled set of EFIE’s (1), leading to the

defining relation for natural system-modes

1 ,..., N (24)
m
n:

N
t“ :1 [G‘(?|F/;w)-knq(r’)ds’ = o m
S

with non—trivial solutions for w = wq, which defines the q‘h system mode with natural

frequency wq and current distribution EM

Coupled-mode perturbation equations are developed by testing the coupled set of

homogeneous EFIE’s (24) with'

fds 553%)-

S.

where 13(0) is the resonant current on the m” isolated dipole. Exploiting the coupled—
"'4

134

 

 

 

 

 

 

mode approximation 1?,” zanqlgsg) , and expanding the Green’s kernel in a Taylor’s series
about the isolated element’s resonant frequency, leads to the perturbation equations

mqmm

[w-0f210 5" + 2 Gina” = 0 ...for m=1,...,N (25)
where 5:" and C3,, are coupling coefficients which depend only on the isolated

element’s resonant frequency and current distribution.

A MoM solution of EFIE’s (1) with entire—domain basis functions is presented, to
provide a comparison to the perturbation approximation. A numerical root-search
provides system resonant frequencies. It is found that the approximate perturbation
theory leads to results which generally agree well with the MoM solution. The
perturbation theory requires significantly less computational time then the full MoM
solution, and thus was found to be an efficient technique.

Measurements are made to validate both methods. Experimental results are found

to agree with the two theoretical solutions.

135

 

 

 

 

 

 

 

CHAPTER SIX

EXPERIMENTAL METHODS

6-1 W

Experimental methods used in the investigation of the electromagnetic properties of
integrated electronic devices are presented in this chapter. Experimental measurements
have been made in order to: i) investigate an isolated dipole’s EM characteristics, ii)
quantify the dipole/transmission-line separation needed to neglect secondary coupling
effects, iii) validate the approximate dominant-singularity-based analysis of transmission-
line/dipole coupling and iv) confirm the perturbation approximation theory for coupled
dipoles. Additionally, the relative merits of different experimental methods is studied
and discussed.

Theoretical investigations of microstrip devices (transmission lines, dipoles, etc.) are
described in a great many papers, although relatively few describe experimental
procedures in great detail. References [58-62] consider this topic, although the main
focus of these is microstrip transmission lines.

The experiments were performed on microstrip circuits applied to a printed circuit
(PC) board, which consists of a thin dielectric film layer backed by a copper ground

plane. The dielectric film was RT/duroid 5880, which is a glass microfiber reinforced

 

 

 

 

 

 

 

PTFE composite, available from Rogers Corporation. The board was 16”x 10”, with
1/2 oz. electrodeposited copper on one side and unclad on the other. Electrical and
physical properties of the board were as follows:

Dielectric constant @ 10 GHz: 2.20:0.02

Loss tangent @ 10 GHz: 0.0009

Dielectric thickness: 0.07874 cm.

Circuit devices were formed on the dielectric film layer with commercially available
gum-backed copper tape (manufactured by GC Electronics), in widths of 0.3175 cm and
0.15675 cm. The 0.3175 cm tape was used to form microstrip transmission lines of Zo
z 42 ohms. The use of copper tape allows unlimited flexibility in the positioning of
circuit elements, while conserving resources. This is especially important for the
investigation of the effect of physical separation on coupled dipole performance, which
would require many circuit boards to be etched with various dipole-to-dipole separations.
It is assumed that EM properties of the copper tape are similar to those of an etched
copper conductor.

Two different instruments were used to measure the EM properties of microstrip
circuits. A Network analyzer, Hewlett Packard (HP) model 8720B, was used to perform
swept frequency measurements of both reflection and transmission parameters. A vector
voltmeter, HP model 8508A, was used to perform single—frequency measurements of
transmission parameters. The network analyzer was used for all measurements except
in the determination of the induced current distribution on the dipole, where the increased

sensitivity of the vector voltmeter proved useful. Both of the above instruments have

 

 

 

 

 

terminal ports designed for coaxial connections. Hence, some additional circuitry was
needed to excite the device-under-test (DUT) and receive its response. This circuitry
consisted of small E—field probes, or transmission line segments. The E-field probe
was constructed using rigid (solid-jacketed) 50-ohm microcoaxial cable, with .030 inch
outside diameter. At one end of the microcoax, approximately 1 mm of the outer jacket
was removed, leaving the center conductor and insulation intact, to form an insulated
monopole probe. The other endyof the microcoax was terminated in a SMA coax
connector, to which the measurement instrument’s cables were attached. The probe was
inserted through holes in the PC—board so that the truncated outer jacket abuts the ground
plane, as shown in Figure 6.1. Solder was applied to this joint to insure good electrical
contact. The insulated center conductor continues past the ground plane, into the
dielectric film layer, to sample the vertical component of electric field. The center
conductor was often allowed to protrude into the cover region slightly, which resulted
in a stronger received signal then obtained with probes confined to the film region.
Transmission line segments were also used to excite and receive energy from the
microstrip dipoles, forming transmission line (T—line) probes [62]. The wider copper
tape of width 0.3175 cm was applied to the dielectric film layer to form microstrip
transmission line segments of Z0 z 42 ohms, as shown in Figure 6.2. Copper tape was
not available in widths which would correspond to 20 z 50 ohms. One end of the
transmission line was left open, with the open end located a distance d, from the DUT.
The other end terminated in a SMA coaxial connector. The center pin of the connector

protruded through a hole in the PC-board into the cover region, piercing the copper tape.

138

 

 

 

 

 

Probe Device

W/W///

\\ \\ \\ \\\ \ \\\\\\\§

\\\ Ground
Solder Joint Plane

 

 

 

 

 

 

__Center
Outer Conductor
Jacket 4— Dielectric

Figure 6.1: E—field probe structure used in measuring microstrip device
characteristics.

139

 

 

 

 

PC Board

T-LMe
Probe

 

i 0 j

l-Connector
Center Pm DUT

(Solder Connection)

 

 

 

 

Figure 6.2: T-line probe structure.

 

140

g 2 .

 

 

 

 

 

Solder was applied to this connection, and also to the connection between the outer
conductor of the connector and the ground plane, to insure good electrical contact.

Section 2 describes the study of "isolated" dipole characteristics, such as the natural
resonant frequency and quality factor. Different measurement schemes are presented and
compared, and some typical results are shown.

Section 3 describes the investigation of transmission-line—fed dipoles. The mutual
interactions in a dipole/transmission-line system are assessed by measuring the change
in dipole Q as a function of dipole/transmission-line separation. The forced current
distribution on the dipole is measured, as well as the relative induced current amplitude.
The amplitude and Q-factor are investigated for differing dipole positions and orientations
with respect to the transmission line.

Section 4 describes the measurements made to confirm the approximate perturbation
theory for coupled dipoles, which was presented in Chapter 5. Swept frequency
measurements are made to ascertain the frequency response of a coupled dipole system,

allowing for the determination of system-mode frequencies.

6.2 ISOLATED DIPOLE RESONANT CHARACTERISTICS

The experimental study of an "isolated" dipole is intrinsically more difficult then that
of a dipole coupled to another device. When making measurements, care must be
exercised in order to separate the device’s characteristics from those of the measuring

system. This is especially true for "isolated" device measurements, since the device can

 

 

 

 

 

 

 

 

 

never be truly isolated from the measurement system. Coupled device systems are
generally less sensitive to interactions with the measurement system, since mutual
interactions among the individual devices may often dominate over the interactions
between a small probe and the circuit devices.

Two characteristics of the isolated dipole were investigated: i) real resonant
frequencies and ii) Q—factor, which is related to the imaginary resonant frequency. It
was found that the real resonant frequency is an easily measured parameter, and is
insensitive to interactions with the measurement system. The Q-factor exhibits
considerable sensitivity to dipole/measurement-system interactions, which is expected
since this coupling allows power to be transferred from the resonant dipole to the
measurement system.

The experimental investigation of the real resonant frequency may be accomplished
in a number of ways. E-field probes may be used to excite the dipole, and to receive the
dipole’s response, or sections of transmission line may be used in place of the E-field
probes. Both measurement schemes are depicted in Figure 6.3. Swept frequency
measurements of the port-to-port transmission coefficient (S21) are made with the network

analyzer. Typical data resulting from this measurement is shown in Figure 6.4, for a 5.0

cm dipole. Peaks of transmission indicate the position of natural modes, at f, z Re{fn} ,
where f,l is the complex natural-mode frequency associated with the isolated dipole.
Measured resonant frequencies fr were found to agree to within 2% of values obtained

by the full-wave methods described in Chapter 4. It was found that the real resonant

 

 

 

 

 

 

 

 

 

T—Lme

 

 

 

 

 

Probes PC Board
1
Q1
Port 1 ”_Ols Port 8
[o l
[Connector DUT

Center Pm
T—Line Probe Method

Port E-Field Probe Method

Port ’ E-Field
1 ' Probes

 

 

 

Figure 6.3: Investigation of isolated—dipole resonant frequency and quality factor
using T-line and E—field probes.

 

143

 

 

 

 

 

0.9
0.8
0.7
0.6
0.5 -
0.4 l
0.3 1
0.2 L

0.1 -
O O 1 . 1 L 1 g; "~ W1

.50 1.55 1.80 1.95 2.10 2.25 2.40 2.55 2.70 2.85 5.00
Frequency (GHZ)

1 [*1 1‘1 [*1 1 1

S21 (mag)

   

 

 

—.I

Figure 6.4: Typical results for transmission (Sn) measurements made on a isolated
microstrip dipole.

144

 

 

frequency (peak of $21) was relatively insensitive to the coupling between measurement
probes and the dipole, since increasing the dipole-probe separation did not change the
measured resonant frequency. The position of the probe along the dipole (at the dipole’s
end, center, etc.) did influence which modes were observed. Certain modes, even or
odd, would not be "found" for some probe positions, although most probe positions
resulted in the observation of most modes.

The approximate placement of the probes to observe a particular mode can be
determined by considering the current distribution of the mode of interest. The E-field
probe actually provides a voltage proportional to the local charge distribution along the
dipole. If the approximate current distribution is known, then the expected charge

distribution can be found by

an(z)

= —.(l) . (1)
82 1 p5

This indicates that the probes should be placed where the greatest rate of change of the
current occurs, since the induced charge will be maximum there. As an example, the
current distribution of the first even mode associated with an isolated microstrip dipole

is shown in Figure 6.5, obtained by the MoM solution described in Chapter 4 (20 pulses
per lei/2). Also shown is the expected charge distribution, obtained by (1). It can be
seen that the logical place to position the probes in order to observe the first even mode
is near the dipole ends. Similarly, probes should be positioned near the dipole’s center

to observe the first odd mode, and so on.

The experimental investigation of the Q-factor may accomplished by two

145

 

 

 

 

 

 

 

Current
Charge

1.0—
as
as
a7
as
a5—
a4:
a3:
a2:
011

OO .1 1 14 C4 .11LL1 1441111411

'—ro -aa —a6 —a4 —a2 00 02 04 as as LO

    

Amplitude

 

Position Along Dipole

Figure 6.5: Theoretical current and charge distribution (magnitudes) for an isolated
microstrip dipole.

146

 

 

 

fundamentally different methods. One method consists of attempting to critically couple
the device to the measurement system [58]. The unloaded Q, Q, is then related to the
measured Q, Q,, by Qo=2 Q1. This technique suffers from the difficulty of finding the
probe position which achieves critical coupling. This is also a fairly narrow-band
procedure. Alternatively, measurements may be made on devices that are very loosely
coupled to the measurement system, such that Q: Q, [58,62]. This procedure is simple
and wide-band, although the loose coupling results in low power levels of the
measurement signals. A brief comparison of these methods appears in [58] for the
investigation of microstrip transmission lines, where it was found that the two methods
agreed to within three percent. The latter technique was implemented using transmission
line segments as shown in Figure 6.3.

The experimental procedure for determining the isolated dipole’s Q is as follows.
A dipole of dimensions 5.0 cm x .159 cm was placed on the dielectric film layer.
Microstrip T-line probes were located perpendicular to the dipole as shown in Figure 6.3,
with their open ends very near the dipole. The probe—to—probe transmission, 82,, was
measured for frequencies near the real resonant frequency of the device, resulting in a
figure similar to Figure 6.4. The Q-factor of the dipole’s resonance was recorded. The
open ends of both T—line probes were then trimmed back with a razor blade, to increase
the probe-to-dipole separation. The new Q-factor was found, and the process repeated
until the dipole’s Q stopped changing. This Q was then considered the unloaded Q of
the dipole, since it was unaffected by further increases in probe~to-dipole separation.

The quality factor of two different dipoles was measured, and compared to theoretical

147

 

 

 

results. Table 7.1 contains the theoretical resonant wavenumber for two dipoles of
different widths, along with the theoretical and measured Q-factor, Ql and Qm,

respectively. The dipoles were of length L=5 .0 cm.

Table 7.1: Theoretical and measured Quality factors

 

Dipole kol Q k, Q... % A
‘7 2k,
wd=.0794 cm (1.139,.00360) 157.91 156.8 0.7

wd=.1588 cm (1.115,.00309) 180.67 173.5 4.0

 

 

 

 

 

 

 

 

 

 

The theoretical values were obtained by the EBF MoM solution discussed in Chapter 4,
and the finite conductivity of the copper dipoles was accounted for. It can be seen that
good agreement was obtained between theory and measurement. Accounting for the
finite conductivity of the copper was found to be critical in order to obtain agreement
between measurement and theory. For example, the theoretical Q-factor of a perfectly

conducting dipole of width wd=.0794 cm was found to be 711.9, which yields a 78% A

compared to the actual measured value.

6.3 TRANSMISSION LINE FED DIPOLES

In Chapter 3, the theory of a dipole excited by a microstrip transmission line was
developed. The impressed field was assumed to be the unperturbed field of an isolated
transmission line, which neglects the secondary coupling effect of nearby objects on the

transmission line currents. In this section, the validity of that assumption is examined

148

 

 

experimentally. Techniques to measure the forced current distribution and relative
current amplitudes are also presented. Comparisons between measurements and theory

have been presented in Chapter 4.

6.3.1 NEGLIGENCE OF SECONDARY COUPLING EFFECTS

It is desired to experimentally quantify the transmission-line/ dipole separation needed
in order to neglect the secondary coupling of the dipole field with the transmission line.
In order to investigate the above, a transmission-line/dipole system was constructed, as
shown in Figure 6.6. The transmission line was excited at one end by the center
conductor of a coax probe, and the other end was connected to a 50-ohm matched
termination through another connector. This resulted in a traveling wave on the
transmission line.

The dipole was located near the transmission line, a distance d, away. An E—field
probe was located near the dipole, and the transmission line to dipole transmission (S2,)
was monitored, beginning with the dipole positioned close to the transmission line. The
dipole’s Q—factor was recorded. The dipole and E—field probe were then moved as a unit,
further away from the transmission line. Care was taken to insure that the dipole-to-
probe separation did not change, and that the probe was located at the same point relative

to the dipole as previously positioned. The dipole’s Q-factor was then found, and the

process repeated until the Q~factor stopped changing. Data for this experiment can be

found in Figure 3.4.

 

 

 

 

 

PC Board

 

LO _ 1 (D l

t—Input 0151 Matched
Termination

 

 

 

Figure 6.6: Measurement system for the investigation of transmission-line/dipole
interactions.

 

   

 

 

 

 

 

 

 

It should be noted that in the above experiment, the actual Q-factor of the
transmission-line/dipole system was not being measured, since coupling to the E-field
probe was still relatively strong. The relative Q-factor was being measured, as the
transmission—line/dipole separation was varied. This is the important quantity in the
above experiment, though, and indicates how the transmission line and dipole mutually

interact.

6.3.2 FORCED CURRENT DISTRIBUTION

In this section, experimental methods to measure the current distribution induced
upon a microstrip dipole by a nearby transmission line are described. Measurements
were made using E—field probes, which sample the local charge distribution along the
dipole.

The experimental setup is depicted in Figure 6.7. A signal generator provides a
sinusoidal steady-state signal to port A of a directional coupler. The input wave is split
by the directional coupler, and appears at ports B and C. The output of port C is sent
to a vector voltmeter, to provide a voltage reference. The output of port B provides the
excitation for a microstrip transmission line, which is terminated in a (nearly) matched
load impedance. The resulting EM field of the transmission line excites currents on a

nearby microstrip dipole, which is the quantity to be determined. Holes were drilled
through the PC board along the length of the dipole, into which E-field probes were

inserted. The vector voltmeter monitors the voltage induced upon the probes, where it

 

 

 

 

Signal
Generator
Directional

LL——:>
Coupler

50 Dhm
Termmafion
ReFerence $gnal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 Output _
.l r———r*
A B
Vector [i 'E:
Voltmeter ,",,N
:‘:::::::::::::_l
E-Field
Probes
PC Board

 

 

 

Figure 6.7: Experimental set-up for measuring microstrip dipole current
distribution.

152

 

 

 

 

 

 

was compared with the reference voltage. In this manner, the induced charge on the
dipole is measured relative to a reference value, for various positions along the dipole.
A fortran program was written to integrate the charge distribution to provide the

desired current, using

erz) = ~jw f p.<z) dz. (2)

The measured charge was interpolated by a cubic spline [54], and then integrated to yield
the current.

The lower limit of integration in (2) was left as a for generality. Let one end of the
dipole be at z=-l, and the other end at 2:1, with its center at 2:0. The charge could

be integrated as

Jztz) = ~11» f ps(z) dz for all z (3)
-1

although this method results in errors accumulating as the integration proceeds, and leads
to erroneous results for regions 2 approaching l due to incomplete cancellation (of
contributions from the two dipole halves) occurring in the integration. A better method

was found to be

an - —jw f p.(z) dz for zso
-l

(4)

12(2) -jw f p5(z) dz for z>0
1

which does not rely on complete cancellation, at least for the first even mode.

153

 

 

Integrating from two different positions can be thought as providing different phase
references for the current, but results in correct magnitudes.

The verification of this method (integration of an interpolated charge distribution)
was accomplished by considering some theoretical results obtained from the MoM
solution described in Chapter 4. A complex-valued amplitude distribution was obtained
for a dipole fed by a transmission line at its resonant frequency. This current distribution
was interpolated by a cubic spline, and differentiated to provide the charge using equation
(1). This charge profile was compared to the measured charge distribution, where
agreement was found to be good. The theoretical charge was then integrated using
equations (3) and (4), to obtain the current back again. It was found that equation (3)
resulted in nearly correct magnitudes, and correct phases. Equation (4) resulted in
correct magnitudes and nearly correct phases, which is expected. In this way, the
numerical procedure associated with (3)-(4) was tested, as well as the measurement
procedure involving the vector voltmeter to obtain the charge profile.

The experimentally measured current distribution is shown in Figure 4.9, for a
parallel-coupled dipole at resonance. The width of the copper tape used to construct the
transmission line actually resulted in a 42 ohm transmission line, so some standing waves
were expected since the measurement system was 50 ohms. Additionally, reflections will
undoubtedly occur at the transition to the microstrip. It was found that these standing
waves do not interfere with measurements made at the resonant frequency, although they
disturb the induced current distribution at other frequencies. For this reason,

measurements were only made at the resonant frequency of the dipole.

154

 

 

 

 

 

6.4 COUPLED DIPOLES

 

An approximate theory for coupled microstrip devices has been presented in Chapter
5, along with the full MoM solution for coupled dipoles. Natural resonant system modes
are found to split about the isolated devices’ resonant modes. For the case of two nearly-
degenerate dipoles, the system modes can be classified as symmetric and antisymmetric,
which refers to the direction of current on the two dipoles. It is the aim of this section
to describe the experimental method used to measure these system modes.

The experimental setup for the determination of coupled dipole system modes is
shown in Figure 6.8. An E-field probe was used to excite the structure, slightly off
center from dipole number 1. A second E-field probe was located near the end of dipole
one, and the probe-to-probe transmission was measured. As was the case for isolated
dipoles, peaks of transmission indicate the presence of system modes. Typical results
Lof such a measurement are shown in Figure 6.9, for the case of two identical parallel
dipoles, L=5.0 cm, separated by d,=.281 cm. This measurement system allows for
freedom in changing the second dipole’s position, relative to the first dipole. Since
system modes are shared by both dipoles, only one dipole need be monitored, which
enables the probe position to remain stationary when the position of the second dipole
is changed. Results of these measurements are presented in Chapter 5. It was found that
the probe—to—probe transmission vanished when the dipoles were removed, so the

measurement system didn’t contribute significant errors to the response of the dipoles.

155

 

 

 

 

 

 

 

PC Board

E-Field
Probe

Dipole /

D09 :2 .
:3 Ids

Dipole

Two

 

 

 

Figure 6.8: Experimental configuration for the investigation of coupled-dipole
characteristics.

156

 

 

 

 

 

1‘0 _ Symmetric mode

0.9 -
0.8 -

Anti—Symmetric Mode

Amplitude

 

 

1 1 l 1 1 AL |

0.0 1 l L L a I 1 l 1 l 1
.50 1.65 1.80 1.95 2.10 2.25 2.40 2.55 2.70 2.85 3.00

Frequency (GHz)

_.\

Figure 6.9: System—modes of two coupled, 5 cm dipoles separated by .281 cm.

157

 

 

 

6.5 W

Experimental methods used in the investigation of microstrip dipole properties have
been described. Measurements to characterize isolated, coupled, and transmission-line-
fed dipoles have been made, using both E-field and transmission line probes.

An isolated dipole’s real resonant frequency has been measured, where it was found
that the probe type and degree of coupling to the measurement system are relatively un-
important. The quality factor has been measured using transmission line probes, while
trying to very loosely couple the dipole to the measuring system. A lack of coupling to
the measurement system would, of course, lead to vanishing of the measured signal, so
the isolated dipole is characterized by finding the loosest coupling that yields a
measurable signal.

Transmission-line—fed dipoles have been studied to determine the separation needed
to neglect secondary coupling of the dipole field upon the transmission line. This
condition was assessed by measuring the relative Q-factor of the dipole/transmission-line
system, as their separation was gradually increased. It was found that the dipole’s real
resonant frequency changed little as the separation varied, but the Q—factor changed
considerably for very close spacings, results for which were presented in Chapter 3.
Both parallel and perpendicular coupled dipoles were investigated.

The induced current distribution upon the dipole was examined by measuring the
charge distribution along the dipole. The charge was interpolated, and the current was

obtained as

158

 

 

 

 

 

 

Z
Jztz) = -jw 19,0) dz
a.
where the correct value of a was discussed.

Coupled-dipole system modes were found with swept transmission measurements
between E-field probes located near the coupled dipole system. Since the transmission
between probes vanished when the dipoles were removed, the response of the

measurement system by itself didn’t appreciably affect the measurements.

159

 

 

 

 

 

 

CHAPTER SEVEN

CONCLUSIONS AND RECOMlVIENDATIONS

An integral—operator formulation for the analysis of the electromagnetic properties
of microstrip devices in the near resonant frequency regime has been presented. This
approximate theory was proposed as an efficient method of analysis to quantify the
dominant interactions in integrated electronic systems. This formulation was based on
the rigorous dyadic Green’s function which characterizes the layered microstrip
environment, and was found to be computationally efficient compared to other full-wave
methods. Systems composed of microstrip dipoles were studied as an example of
applying the general method.

The dyadic Green’s function for tri-layered media was developed in Chapter 2, and
a thorough discussion of its singularities in the spectral plane was included.
Understanding the physical and numerical implication of these singularities was of utmost
importance in correctly evaluating the desired field quantities. Efficient evaluation of the
Green’s function was discussed, and numerical integration schemes were presented.

In Chapter 3, the singularity expansion method for integrated electronics was

developed. This method is based on the conceptually exact electric field integral

equation, which quantifies all electromagnetic interactions in integrated electronic

160

 

 

 

 

 

systems. The example of a microstrip dipole excited by a nearby transmission line was
considered as representative of a typical application of this method.

Other full—wave methods were developed in Chapter 4. These well-established
methods, along with experimental results, were used to validate the approximate
singularity expansion theory. Theoretical and experimental results were presented, and
found to be in good agreement.

Systems of coupled microstrip devices were considered in Chapter 5. An
approximate perturbation theory for coupled devices was presented, and applied to the
problem of coupled dipoles. This method was found to be very efficient compared to a
full—wave method of moments solution, which was also obtained to provide a comparison
to the approximate theory. Theoretical and experimental results were presented and
compared, where agreement was found to be good.

Experimental methods used to verify the theoretical results were described in Chapter
6. Various experimental techniques were described and discussed for measuring
characteristics of both isolated and coupled systems. Natural resonances were identified,
and results were found to agree well with theory. The forced response of a dipole
excited by a transmission line was investigated, and the approximations made in
developing the singularity expansion description of this problem were discussed.

An approximate, engineering theory for the efficient analysis of dominant interactions
in integrated electronic systems was considered. Viability of this method was
demonstrated for single devices and small systems. It is proposed that future work

examine the feasibility of applying this method to study increasingly complex systems.

161

 

The secondary effects of individual system elements on each other is accounted for by
the perturbation theory for coupled devices, but does not account for coupling back to
the original source of excitation. It is recommended that these interactions be

investigated theoretically, and their relative importance assessed.

162

 

APPENDIX A

 

APPENDIX A

ELECTRIC HERTZIAN POTENTIAL

In general, both electric and magnetic potentials may be defined. For the case of no
magnetic sources, a single potential is sufficient to uniquely define the fields, which is
the circumstance for this dissertation.

Equation (l.b) shows that the divergence of the magnetic field vanishes,
demonstrating the nonexistence of magnetic monopoles. This enables us to define

I? = free vxfi. (A-l)
Substitution of (A. 1) into (l.c) yields
vXaf-szr) = 0 (A3)

where k = to JR. Since the curl of the gradient of a sealer field vanishes, equation (A.2)

gives us
if = —V¢ +k2II (A3)
where ¢> is an arbitrary scaler field. Substitution of equations (A. 1) and (A.3) into (1.d)

and use of the vector identity

VxVxII = V(V-fI)—V2fl (A-4)

results in

 

(V2+k2)fi = '7 +v(V-fi+¢). (A5)
(1)6

163

 

 

 

 

Since a vector field is uniquelydetermined by its curl and divergence, the divergence
of II must be specified. Choosing a = -V°fI, which is the Lorentz gauge, equation (A.3)
becomes

E = V(v-fi) +k2fi (A.6)

where II is the solution to the non-homogeneous vector Helmholtz equation

(V2+k2)f1 = ’—j (A-7)
jcoe

obtained from (A5) by use of the gauge condition.

164

 

 

 

 

 

APPENDIX B

 

 

APPENDIX B

SPECTRAL REPRESENTATION OF PRINCIPAL GREEN’S DYAD

The Helmholtz equation for the primary component of potential is found in Appendix

 

A to be
vain/.211» = '_J (13.1)
joe
which can be written in sealer form as
—J
vzn’; +k2H‘; = , °‘ 03-2)
Joe

for a=x,y,z. The Green’s function G”(F|F’) is defined by
VZGP(F|r‘/)+k2GP(F|F’) = -5(F—F’) (13.3)

where 6(7-7’) is the Dirac Delta distribution [42]. Without loss of generality, a solution

/

for G”(F|i"’=0) is sought, and the final result shifted to an arbitrary F . Defining the

two—dimensional Fourier transform pair

_ 1 .. ~ :21“ 2 13.4)
GP _ __ P(1,y)e I d 1 (
(r‘) (2102 U s

165

 

g”(X.y)= ff GP(F)e*ji‘Fdxdz (13.5)
where X = 335 +£C is a 2-D spatial frequency, equation (B5) is substituted into (B.4),

resulting in

mm = ijdx’dz’ GP(F’)(2_1:)2 I] ext-(Fr) (121. (B.6)

From the above, it is clear that
—1—- ff ejx'fi'fl) d2). = 6(x—x’)5(z-z’) 03-7)
(2102 -...

by the sifting property of delta functions. Use of (B3), (B4), and the Fourier transform
property

9‘1{...}=o .. {...} :0
leads to
32 2 7 P " - _ (B 8)
-——P (4) g (40’) - 50’) -
ayZ
where p().) =]/ 12 -k2). The above one dimensional ordinary differential equation for

g” can be easily solved to obtain

 

gP(X,y) = . 03-9)

166

 

 

 

 

Equation (B.4) becomes, after shifting to an arbitrary F’ ,

ej). (F- -F’)e ”Pgly Y|

G”(rlr’)= ff 20:10)2 —————d21. 03-10)

167

 

APPENDIX C

 

APPENDIX C

HERTZIAN POTENTIAL BOUNDARY CONDITIONS AND THEIR

APPLICATION

I. Hertzian Potential boundary conditions:

The electric and magnetic fields are found in terms of the Hertzian potential in

Appendix A as
E = (k2+VV-)fi

H = jwerfI.

Separating (C.l) into rectangular components yields

 

 

 

an an

E' = kiZII. +£V'Hr H1; =j061‘ —£——yi
IX IX ax ay az
2 a H , 8HJr an,

E . k‘ ”Flam -,...., a. a
an an

2 a _ . y _ ,,

Eu F "1 HiFa—ZV‘I“ H. ‘ “its; “at

for the 1‘” layer.

(C.l)

(C.2)

Enforcing the continuity of tangential field components of (C2) as generated by the

oz‘h source component individually [17],
as

168

the general boundary conditions are constructed

 

 

 

1110: = 01221112,, or =x,y,z (C.3.a)

 

 

 

 

 

 

 

an. 2 an.
a = N ___“ azx’z (C.3.b)
ay 2‘ ay
an.._an.. z _(N221_,)anz.+anz. (c.3.c)
6y 3)’ ax 62
for the y=0 interface and
H20: = N322113a a =39)“ (C'3'd)
6 8H
112. = N322 a. 0,sz (C.3.e)
3)’ 3y
an,,_arr,y = _(N322_1) am, 3113,] (can
37 5)’ az

for the y=-t interface, where N,,-= —n ,j/n and n is the ith layer refractive index.
11. Enforcement of Boundary Conditions to Determine Weighting Coefficients:

The Hertzian potential in each region as given in Chapter 2 can be written in sealer
form as

—i‘-F’ -p(l)Iy-1"l _
f f e”" [j Q _‘L’ e ‘ dv’+W{,(i)e “my d2).
jwel 2mm

 

 

II "‘ .._.
l¢(r) (211 )2...”

(C.4.a)

169

 

 

 

112.,(7) = f f e“’ l W,‘ a,'()t)e”2“‘)’+W (we ”2“” 1 d2), (C.4.b)

(2n)2

 

H3.(F)= f f e" F [ W,‘(1)e”3‘“y ] (:21. (OM)

(2102-..
for a =x,y,z and Re{p,} >0 is chosen to satisfy the radiation condition.

Enforcing boundary conditions (C.3.a), (C.3.b), (C.3.d), (C.3.e) at y=O,-t for

tangential components of Hertzian potential leads to the linear system of equations

_era +N221(W2t¢ + W22) = V

 

N2
W1; + 2W2 (W2; — W22) = Va
”1 (c.5)
W,‘,e ”‘2 +W,'e ’2‘ -N,2,W,‘,e 'P3‘ = o
N322p3

t-t r t t-t
W,e”2- W, ”2- W,,e ”3 =0

11

 

p,

where

 

J Fl -jX-F’ 1’1)"
V = f .a( )8 e dV/
V [we] 2p1()t)

The system of equations (C5) is solved to yield

 

 

 

 

Tt
W2; = iVa
D!
T! R: 2p2t
W2, _ 12 32 V“
a 0‘
2 t
Wt __ 1 T2: T12R32 P2 V
1a 1 Dt a
r T,‘,e “’3 ”F"
W32: = 12 Va
Dt

 

 

where

D -1 R12R32e

 

 

 

 

 

 

P1 “P P ‘1’
R2: = P + 2’ R12 = 2 1
1 p2 p1+p2
2N2 2p
T23 = 211722 T112 = 2 l ((3.7)
pl +p2 N21 (p1 +P2)
" 2
R3; = P2 P3, sz3 = 2 P2
P2 +P3 N32 (P2 +P3)

Enforcing boundary conditions (C.3.a), (C.3.d), (C.3.e), and (C.3.f) for normal

components of potential leads to the linear system of equations

- W1; + N22,(W2‘y + W2; = V),

P r . .
ery+—Z(W2ty-W2y = VHF [151/KKK]
1’1 2 _ , (C.8)
Wztye 'p"+W2'yeP2‘-N32W3‘ye p3 = 0
Wztye'p21_ 2gepzt-23W3'ye -p3t = -G [ngx +jCVZ]
2

where

(N3. -1) 7;, [1 + 11.1,. '2’2‘]

 

 

P; D ‘
G = (N322 " 1) Tr‘szta e (p, 72):. 1’3:
P2 D ‘

 

J -jl°r’ 1912’
V =j' ,(rl)e e dV/

y ij1 2p1(4)

and D‘ is defined previously.

The system of equations (C8) is solved to yield

171

 

 

n ’3 "Pzt n -2
r _ T12R32e V +[R32N21C1J'CJ e-szt

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2y _ D]: Y D" [ngx+jCVz]
Tn N-ZC _RnC e-2p2t
: _ 12 21 1 21 2 . ,
W22 ‘ Dan ,, [JEVx +1: V2]
r 1! T13R32 Tznie -2” T2,; (ngN 212 C1 + C ) e 4”
= 2 . .
le 1* D" Vy+ 1+ D" [JfiVx+](Vz]
11’;ng - Tn (N-ZC _Ran e-szt) —
W3; = D" vy+ N322C2+ 23 2‘ 1D" 1 2 [jEqu'CVz] e‘P3 ”9‘
where
D n = 1-R2'1R3'3e '2”
2 2
n N21p1 “P2 n N32P2 "Ps
R21 = ‘3'": R32 = —2———'
"21131 +p2 N32P2 +P3
2 2
T2,; = ...—————-p2 , T1,; .1: pl
2 2
lepz +P1 N21 (P1 +P2)
2
T2; = ___—’2 p2
N32p2 +p3
C = N221(N221 ’ 1) T1t2 1 +1133‘2“3 -2p2t
1 2
N21p1+p2 Dt
C _ N322(N322‘1) Tit2T2ts
2 .. f
N322p2 +P3 D t
111. Determination of Hertzian Potential:
Rewriting coefficients in region (1) as
r —
Wla ‘ RtVa ((3.10)

W1; = RnVy+C[jEV;+jCVz]

172

 

 

 

where

t t t
T12R32T21 e -2p2t

 

Rt = R2tl + D t
I! II n
Rn = R2711 + T12R32T21 e-szt (C011)
D n
-2
C = C + T2’i[R3I2N21 C1 +C2] ~2p2t
1 D n
the total potential may be written as
111 = II‘;+11'1.

Equation (C. 10) is substituted into (C.4.a) to yield

_. H -. -*/
1(F): fawn”) oflr—ldv/
V Jwel

where

G'mr’) = 6%? F’) + d'mr’)

ép(r|'r*’) = fGP(?|?/)

 

 

 

co .4. __.-_‘l _ _y/
e11 (rr) 6 Pcl)’ I

mer’) = H 2(21t)2p d2).

 

173

 

 

 

 

r r -. _‘l ‘
G:r(rlr ) on Rik) ejx.(?_fl)e—pcov+yl)
Gn(f‘|F/)1 = ff (A) d21-

n 2
exam -~ cm 20“) c

L

 

 

 

For the case of a conducting region (3) (substrate), the reflection and coupling

coefficients become

 

 

 

13,01)
Z ”(71)
R01) _ 221m -p2tanh(p2 1‘)
n 26(1)
co _ 2( 21-1»:1
#00212)

where

2‘00 = N§1p1+pztanh<p2r>
Z “(1) = p1+p2coth(p2t).

174

APPENDIX D

 

 

 

APPENDIX D

EXISTENCE OF BRANCH POINTS IN THE COMPLEX FREQUENCY PLANE

Singularity expansion (3.3) for unknown surface current I? is given as a sum of pole

terms in the complex frequency plane. This sum of pole singularities constitutes the
dominant contribution to the current, although the sum is not a complete representation.
Other complex frequency-plane singularities are needed, and a complete representation

for the surface current would be

+ Milt») (D. 1)

 

where W021») is the contribution from other singularities. It is conjectured here that
branch—point singularities are present in the complex frequency-plane, although
numerical results indicate that they may not be important when compared with the pole

singularities.
The solution of EFIE (3.1) requires the evaluation of the dyadic Green’s function

presented in Chapter 2. Components of the reflected part of the Green’s dyad are given

by

175

 

 

r 1

G'FF’ ..

1 :E'.) 2" Rm ...... . (D 2)
GnO'Ir) = ff 11(1) 2 1d A .
GZG‘IF’). ° 0 CW 212m .

Y—

 

 

 

where X =££ +28; is a 2-D spatial frequency with V = £2 + C2 and dzl =d£dC . Equation
(D2) is found from (2.17) by the rectangular to polar transformation

A cost)
A sine

E
C
Wavenumber parameters are p, a/AZ —k,-2 with Re{p,} > O for i=s,f,c. Coefficients R"

R1,, and C are given by

pc —pf coth(pf t)

 

 

 

R,(l) =
2’11)
2
Rn(l) : Nfcpc-pftanhwft)
Z‘Q)
2 2-
C (A) = (Nfc l)pc
zh(x)ze(x)

where

2‘01) = N; p.+p,tanho,t>
2"(2) = pc+pfcoth(pft).

Consider preforming the inverse spectral integrals in (D2) by the method of contour
deformation. The integral may be found as a sum of integrations around the poles plus

a term resulting from integrating along the branch cut. A representative Green’s

component of (D.2) may be written as

176

 

 

F (A 6)
Z(A)

dAdG

 

II
085)
0% 8

where the order of integration has been interchanged. F(A,B) is the portion of the

integrand analytic within and on a circle Cp enclosing the pole at Z(A). Considering the

integration around a Cp leads to
=f f—Z—— F” (122166)

If contour Cp is made limitingly small, and F(A,B) is a well-behaved function in the

vicinity of the pole, the above integral may be written as

~f d6 F(Apfi) [2(1) (D.3)

The Taylor’s series expansion of Z(A) about A =Ap 1s

+... .
A=Ap

Retaining the first non-vanishing term (the leading term vanishes by definition) results

2(2) = zap) +(A - 2) £20)

 

in

 

21‘ A ,6
o 2 (1,) of

where

177

 

 

a
Z’A = —z
(p) ax (A)A

:Ap

The change of variables

A -Ap = 6e”
dA = je e” (1111
leads to the second integral in (D4) being evaluated as 21rj. Integral (D4) is found to

be

21:
_ 27v}
I — F A ,6 d6. (D5)
2’0) 1 (, )

 

By inspection of (D.2) and (D5), it is obvious that wavenumber parameterspc = A: —k,'.2

are involved in the frequency—domain expression for I =I(w) resulting from the surface-
wave contribution to the spectral integral. Hence it is shown that branch points are found
in the complex frequency-plane. Since (D.5) is only part of the spectral integral
evaluation, it is not clear what role these branch points may take. Also, the above
derivation was for the Green’s function by itself, before it is operated on by the spatial
integrals associated with obtaining the electric field. It is assumed that the singularities

associated with the Green’s dyad are shared by the solution of the integral equation

involving G.

178

 

 

APPENDIX E

 

 

APPENDIX E

MODIFICATION OF THE EFIE TO INCLUDE FINITE CONDUCTOR
SURFACE IlVIPEDAN CE

An electric field integral equation (EFIE) is derived by enforcing continuity of

tangential electric field components across an interface, such as a conducting surface.

Typically, an impressed electric field E excites currents on surface S, producing
scattered field is and internal field 175”" . When S bounds a perfect conductor, EW=O

and the boundary condition for tangential E requires that f-(E i+133" 5) =0, where f is a
unit tangent vector at any point on surface S. If the conductor has finite oonductivityfw =Z ‘13?)
where 13(7) is the total current at point F and Z i is the internal surface impedance. The

boundary condition for tangential field components then becomes f°(Ei+Es-Em) =0,

resulting in EFIE

C

.. _. 'k . _.,. g l... _, _,
{o f G ‘(FIF’)°K(F’)dS’ = --]—1i-5t'[E(r)+Z K(r)] ...v rES.
s (13.1)

For wires, 15"“:le where I is the total current and the impedance per unit length can

be found as [56]

179

 

 

z, = 3230 +1) (15.2)
41c a

where

2
capo

5:

 

is the skin depth, a =conductivity in (mhos/m), and a=radius of the wire. Equation
(E.2) has been found to be accurate when (file) is small, which is often the case for

good conductors.

EFIE (BI) is the same as the fundamental EFIE derived in Chapter 3, with the

addition of the surface impedance term. When considering resonance problems (5 i=0) ,

the term involving Zi should be subtracted from the LHS, resulting in modification of
the resonant wavenumber which reflects the finite conductivity of the object.

As an example of computing Z ’I?(F), the single—term, even EBF solution of EFIE
(BI) is considered. All terms are the same as derived in Chapter 4, with the addition

of the surface impedance term. For narrow strip dipoles, Z iK can be replaced byZ,I

on a circular dipole of equivalent radius and the current reduces to [(2), which can be

found as

180

 

 

 

 

 

 

(nnz)
wd COCOS _7
1(2): fJ(x,z)dx= f dx

”1 1'1

 

and becomes
um
I = w cos—.
(z) a, ,1: [21)

Testing with the integral Operator

n?!
”d I COS -—-

ff _lz]dzdx

N

 

h.

 

results in the term

T = 11' “[a0(wdn)lel]

C

 

which augments the equations that neglect the finite conductor impedance of the

conductor.

181

 

 

 

 

 
 

 

BIBLIOGRAPHY

 

 

 

 

 

 

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

 

 

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