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This is to certify that the

dissertation entitled

ANALYTICAL AND EXPERIMENTAL TECHNIQUES FOR THE
ELECTROMAGNETIC CHARACTERIZATION OF MATERIALS

presented by

Michael John Havrilla

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Electrical and
Computer Engineering

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Date 19 April 2001

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ANALYTICAL AND EXPERIMENTAL TECHNIQUES FOR THE
ELECTROMAGNETIC CHARACTERIZATION OF MATERIALS
By

Michael John Havrilla

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical and Computer Engineering

2001

ABSTRACT
ANALYTICAL AND EXPERIMENTAL TECHNIQUES FOR THE
ELECTROMAGNETIC CHARACTERIZATION OF MATERIALS
By

Michael John Havrilla

Electromagnetic material characterization is the process of determining the
permittivity and permeability of matter. This process is predominantly employed in
stealth and integrated-circuit technologies with the aid of the analytical Nicolson-Ross-
Weir (NRW) formulation. The increasing demands of industry have rendered the NRW
technique invalid under certain conditions due to theoretical violations, leading to
erroneous results. In rectangular waveguide measurements, for example, it is assumed
that sample material is comprised of a single layer only, that walls are perfectly
conducting and that no gaps exist between the sample and conducting boundaries.
However, in the industry environment, samples are often multi-layered due to material
integrity and high-temperature measurements lead to sample-to-wall gaps and involve
waveguide metals that are typically poorly conducting. In addition, high-temperature
strip and microstrip field applicators also involve imperfectly-conducting boundaries,
leading to gross errors in the material characterization process. This dissertation provides
several techniques to accommodate these errors.

Chapter 2 provides two methods, the direct and deembed techniques, for
characterizing materials that are embedded in multi-layered samples. Although both

formulations utilize wave-transmission matrices, it is shown that the direct method must

be used if sample homogeneity is to be accurately monitored. Errors due to sample-to-
wall gaps are accommodated in Chapter 3 by regarding the waveguide as
inhomogeneously filled in the cross-sectional plane with LSM and LSE propagation
modes supported in the sample/gap regions. This analysis leads to corrections in the
scattering parameters and ideal TE”) propagation constant of a uniformly-filled guide.
Chapter 4 investigates the effects of waveguide wall loss by using a coupled-mode
perturbation theory which is based upon an impedance boundary condition at the
imperfectly-conducting walls. The result is a complex correction to the ideal TE“)
propagation constant.

Strip and microstrip field applicators having imperfectly-conducting boundaries
are investigated in Chapters 5-7 using a spectral-domain integral-operator formulation
with the aid of electric-field dyadic Green’s functions. The resulting electric field
integral equations, which follow from enforcement of impedance boundary conditions on
the imperfect strip conductors, are solved using a non-Galerkin’s Method of Moments
technique employing Chebyshev basis functions of the first and second kind. The
analysis in Chapters 5-6 and 7 leads to a correction in the ideal principal-mode

propagation constant for the strip and microstrip transmission lines, respectively.

In loving memory of my father

Albert L. Havrilla

ACKNOWLEDGEMENTS

I would like to express sincerest thanks to my committee members. My advisor
and mentor, Dr. Dennis Nyquist, has given generously of his time throughout the years.
His expertise, physical insight, enthusiasm and guidance made this dissertation possible.
It’s difficult to express in words just how grateful I am to have studied under such a
gifted and talented scholar. Dr. Edward Rothwell’s academic abilities, patience and
kindness will always be an example to me. Like many others, I am indebted to Dr. Kun-
Mu Chen for establishing a tradition of excellence in the electromagnetics research group
at Michigan State University. Dr. Byron Drachman also gave freely of his time and
provided valuable mathematical insight. I would also like to express my appreciation to
Dr. Lydell Frasch for his insightful comments. A majority of this dissertation was
supported by GE Aircraft Engines and I would like to thank Matt Kress and John Nelson
for their guidance and expertise in material measurements. Jeff Meese and Dennis Kelly
provided valuable computer consultation. The secretaries were very kind and made me
feel right at home.

The support of my parents, family and friends was equally important. Special
thanks go to my parents, Margaret and Albert, and to my colleagues and friends Chris
Coleman, Dave Infante, Mark Perrin and Dave Story. My friends from the marketing
group, especially Rose, were endlessly entertaining. Ken Noren, my friend and winter
mountaineering partner, and his wife, Deb, have been exceedingly gracious and
hospitable over the years. My loving girlfriend, Som, is a constant encouragement and

support to me. Finally, and most importantly, I am forever grateful to Christ Jesus.

TABLE OF CONTENTS

Table of Contents ..................................................... vi
List of Figures ......................................................... xi
Chapter 1
Introduction and Overview .......................................... 1
Chapter 2
Direct and Deembed Methods for Material Characterization ............ 6
2.1 Introduction ........................................ 6
2.2 Material Characterization for a Single-Layered Environment ........ 7
2.2.1 Overview ..................................... 7
2.2.2 General NRW Formulation ......................... 8
2.2.3 Computation of Constitutive Parameters for a TEM System . . . . 10
2.2.4 Computation of Constitutive Parameters for a Waveguide
System ....................................... 11
2.3 Geometry of Multi-Layered Environment ..................... 11
2.4 A-Parameter Description of a Multi-Layered System .............. 12
2.5 Direct Method of Constitutive Parameter Extraction .............. 16
2.6 Deembed Method of Constitutive Parameter Extraction ............ 17
2.6.1 Modification of A-Parameters for Deembedding ............ 17
2.6.2 Deembed Procedure for Constitutive Parameter Extraction ..... 19
2.7 General Comments and Experimental Results .................. 20
2.7.1 General Comments ............................... 20
2.7.2 Experimental Results .............................. 21
Chapter 3
Analysis of Sample-to-Wall Gaps in Rectangular Waveguide Material
Characterization Measurements ............................... 30
3.1 Introduction ......................................... 30
3.2 Review of a Uniformly-Filled Rectangular Waveguide ............ 31
3.3 LSM Mode Analysis for Bottom/Top Gaps .................... 32
3.3.1 Geometry ..................................... 32
3.3.2 Hertzian Potential Generating Function and EM Field
Components ................................... 33
3.3.3 Identification of the LSM Mode Characteristic Equation
for 71s»: ...................................... 34

vi

3.3.4 Perturbation Theory for Lowest-Order LSM Mode

Propagation Constant ............................. 39
3.3.5 Approximate Expressions for Scattering Parameters Using
Mode Matching ................................. 42
3.4 LSE Mode Analysis for Left/Right Gaps ...................... 48
3.4.1 Geometry ..................................... 48
3.4.2 Hertzian Potential Generating Function and EM Field
Components ................................... 49
3.4.3 Identification of the LSE Mode Characteristic Equation
for 7,58 ...................................... 50
3.4.4 Perturbation Theory for Lowest-Order LSE Mode
Propagation Constant ............................. 53
3.4.5 Approximate Expressions for Scattering Parameters Using
Mode Matching ................................. 54
3.5 Experimental Results .................................. 56
Chapter 4
Accommodation of Wall Loss in Rectangular Waveguide Material
Characterization Measurements ............................... 63
4.1 Introduction ........................................ 63
4.2 Review of Attenuation Correction for the Ideal TElo
Propagation Constant .................................. 63
4.3 Accommodation of Wall Loss Using a Coupled-Mode
Perturbation Theory ................................... 66
4.3.1 EM Field in Terms of the Transverse Electric Field .......... 66
4.3.2 Impedance Boundary Conditions on E in Terms of
Transverse Field é’ ............................... 68
4.3.3 Coupled-Mode Perturbation Theory .................... 69
4.3.4 Complex Correction to the Ideal TElo Mode Propagation
Propagation Constant ............................. 73
4.4 Experimental Results .................................. 75
Chapter 5
Green’s Function for EM Field Within an Imperfectly-Conducting
Parallel-Plate Environment .................................. 82
5.1 Introduction ........................................ 82
5.2 Geometrical Configuration ............................... 82
5.3 EM Fields and Helmholtz Equation for Hertzian Potential .......... 83
5.4 Spectral Representation of Principal and Scattered Waves .......... 84
5.4.1 General Formulation .............................. 84
5.4.2 Principal Wave Representation ....................... 87
5.4.3 Reflected Wave Representation ....................... 89

vii

5.4.4 Total Wave Representation ......................... 89

5.5 Computation of Spectral Coefficients from Limit of
Highly-Conducting Case ................................ 90
5 .5. I Introduction ................................... 90
5.5.2 Radiation Conditions .............................. 92
5.5.3 Tangential Boundary Conditions ...................... 92
5.5.4 Normal Boundary Conditions ........................ 94
5.5.5 Mixed/Coupled Boundary Conditions ................... 95
5.5.6 Limiting Case ac ——) —j0'c / a) ........................ 98
5.6 Computation of Spectral Coefficients Using Impedance
Boundary Conditions ................................. 100
5 .6. 1 Introduction .................................. 100
5.6.2 Geometry .................................... 101
5.6.3 Tangential Impedance Boundary Conditions ............. 102
5.6.4 Coupled Impedance Boundary Conditions ............... 103
5.7 Hertzian-Potential Dyadic Green’s Function .................. 106
5.8 Physical Observations and Limiting Cases ................... 108
5.8. 1 Introduction .................................. 108
5.8.2 Symmetric-Slab Waveguide Modes ................... 109
5.8.3 Wave Interaction Between Source and Field Points ......... 110
5.8.4 Limiting Case for a Perfect Conductor ................. 11 1
5.8.5 Limiting Case for a Source Over an Imperfectly-Conducting
Half Space ................................... 111
5.9 Electric Field Dyadic Green’s Function ..................... 112
5 .9. 1 Introduction .................................. 1 12
5.9.2 Principal Electric Field Dyadic Green’s Function .......... 114
5.9.3 Reflected Electric Field Dyadic Green’s Function .......... 117
5.10 Magnetic Field Dyadic Green’s Function .................... 119
Chapter 6
Analysis of a Lossy Stripline Field Applicator .................... 121
6.1 Introduction ....................................... 121
6.2 Geometry ......................................... 121
6.3 Development of the Lossy Stripline EF IE .................... 122
6.4 MOM Solution for the Lossy Stripline EFIE ................... 126
6.4. 1 General Formulation ............................. 127
6.4.2 Even-Mode MOM Formulation for Lossy Stripline ......... 129
6.4.3 Odd-Mode MOM Formulation for Lossy Stripline .......... 134
6.4.4 Quasi-TEM Characteristic Impedance ................. 135
6.5 Numerical Results ................................... 139
Chapter 7

Analysis of a Lossy Microstrip Transmission Line Having an

viii

Imperfectly-Conducting Ground Plane ......................... 160

7.1 Introduction ....................................... 160
7.2 Geometry ......................................... 160
7.3 Electric-Field Dyadic Green’s Function ..................... 161
7.3.1 General Formulation ............................. 161
7.3.2 Computation of Spectral Coefficients .................. 163
7.3.3 Identification of the Hertz Potential Dyadic
Green’s Function ............................... 169
7.3.4 Identification of the Electric Field Dyadic
Green’s Function ............................... 170
7.4 Development of the Lossy Microstrip EFIE ................... 172
7.5 MOM Solution for the Lossy Microstrip EF IE ................. 174
7.6 Numerical Results ................................... 176
Chapter 8
Conclusion ........................................... 1 87
Appendix A
EM Fields and Hertzian Potentials ............................ 192
A.1 Introduction ....................................... 192
A2 Maxwell’s Equations and the Wave Equations for E and Fl ........ 192
A3 Electric Hertzian Potential .............................. 193
Appendix B
Evaluation of the Fourier Transform Inversion Integral for the
Spectral-Domain Principal-Wave Green’s Function ................. 195
3.1 Introduction ....................................... 195
B2 Evaluation of G2” for y - y’ > 0 .......................... 197
3.3 Evaluation of (if for y — y' < 0 .......................... 199
B4 General Representation of G2” ........................... 200
Appendix C
Electric-Type Hertzian-Potential Boundary Conditions at a
Material Interface ....................................... 201
CI Introduction ....................................... 201
C2 Geometry ......................................... 201
C3 Hertzian Potential Boundary Conditions ..................... 202

ix

C.3.1 Horizontal Source .7 = JEJx ......................... 203

C.3.2 Vertical Source .7 = 52.1}, .......................... 205
C.3.3 Horizontal Source .7 = 2.], ......................... 206
C.3.4 General Source .7 = .72.], + szy + 2.12 ................... 207
C4 Summary ......................................... 208
Appendix D
Hertzian-Potential Impedance Boundary Conditions ................ 210
D] Introduction ....................................... 210
D2 Geometry ......................................... 210
D3 Hertzian-Potential Impedance Boundary Conditions ............. 211
D.3.l Horizontal Source .7 =11], ......................... 214
D.3.2 Vertical Source .7 = szy .......................... 215
D.3.3 Horizontal Source .7 = 2J2 ......................... 215
D.3.4 General Source J = 32.1, + w, + 2.1, ................... 216
D4 Hertzian-Potential Boundary Conditions at a Perfect Conductor ..... 217
D.5 Summary ......................................... 218
Appendix E
Overview Of Chebyshev Polynomials .......................... 220
El Introduction ....................................... 220
E2 Chebyshev Polynomial Properties ......................... 220
E.2.1 Fundamental Expressions .......................... 220
E.2.2 Parity and Recurrence Relations ..................... 221
E3 Integrals Involving Chebyshev Polynomials . , ................. 222
E31 Preliminary Formulae and Well-Known Relations ......... 222
E.3.2 Orthogonality Relations ........................... 224
E.3.3 Integrals Involving Tn(x) .......................... 224
E.3.4 Integrals Involving U n (x) ......................... 227

E35 Asymptotic Form of Integrals Involving Tn(x) and Un(x) . . . . 231

Bibliography ............................................... 232

Figure 2.1
Figure 2.2
Figure 2.3

Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8

Figure 2.9

Figure 2.10

Figure 3.1

Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6

Figure 3.7

Figure 4.1
Figure 4.2

Figure 4.3

LIST OF FIGURES

Multi-layered environment .............................. 12
Reflected and transmitted waves at a planar interface ............. 12
Wave description for shift of distance é ..................... 14
Introduction of zero-length air layers for deembedding ............ 18

Verification of layered analysis for Re{£,} using an Alumina sample . . 24

Verification of layered analysis for Im{£,} using an Alumina sample . . 25

Homogeneity interrogation of Re{£,} via direct method .......... 26
Homogeneity interrogation of Im {6,} via direct method .......... 27
Homogeneity interrogation of Re{£,} via deembed method ........ 28
Homogeneity interrogation of Im{£,} via deembed method ........ 29
Cross-sectional geometry for bottom/top gap analysis ............ 33
Cross-sectional geometry for left/right gap analysis .............. 49
Experimental results of a 0.8 mil top/bottom gap for Re{£,} ........ 58
Experimental results of a 0.8 mil top/bottom gap for Im{6,} ........ 59
Experimental results of a 1.9 mil top/bottom gap for Re{£,} ........ 60
Experimental results of a 2. 9 mil top/bottom gap for Re{£,} ........ 61
Experimental results of a 35 mil left/right gap for Re{£,} .......... 62
Defined unit vectors for a uniform waveguide .................. 77
Wall loss theory comparison for Re{8,} using Alumina ........... 78
Wall loss theory comparison for Im{£,} using Alumina ........... 79

xi

Figure 4.4
Figure 4.5

Figure 5.1

Figure 5.2

Figure 5.3

Figure 6.1

Figure 6.2
Figure 6.3
Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7
Figure 6.8
Figure 6.9

Figure 6.10
Figure 6.11
Figure 6.12
Figure 6.13
Figure 6.14

Figure 6.15

Figure 6.16

Conductivity profile of Re{8,} for a resistive-card material ........ 80
Conductivity profile of Im {23,} for a resistive-card material ........ 81
Parallel-plate environment with general 3D current .707) .......... 83

Parallel-plate environment for impedance boundary

condition analysis ................................... 102
Wave interaction between source and field points .............. 110
Cross-sectional geometry of the lossy stripline field applicator ...... 122
Convergence rate for propagation constant 4’ = ,6 — ja ........... 143
Convergence rate for surface current kJr .................... 144
Convergence rate for surface current k2 .................... 145
Comparison of MOM theories for surface current kz ............ 146

Full-wave and perturbation theory comparison for

attenuation constant a ................................ 147
Effect of ground plane conductivity on phase constant ,6 ......... 148
Effect of ground plane conductivity on attenuation constant a ...... 149
Effect of strip conductivity on phase constant ,B ............... 150
Effect of strip conductivity on attenuation constant a ........... 151
Effect of strip conductivity on surface current kx .............. 152
Effect of strip conductivity on surface current k2 .............. 153
Effect of frequency on surface current kx ................... 154
Effect of frequency on surface current k2 ................... 155
Effect of width/height ratio on phase constant ,6 ............... 156
Effect of width/height ratio on attenuation constant a ........... 157

xii

Figure 6.17
Figure 6.18
Figure 7.1
Figure 7.2

Figure 7.3

Figure 7.4
Figure 7.5
Figure 7.6
Figure 7.7
Figure 7.8
Figure 7.9

Figure 7.10

Figure 7.11
Figure 3.1

Figure B.2

Figure C.1

Figure D.1

Effect of refractive index on the attenuation constant a .......... 158
Effect of conductivity on characteristic impedance .............. 159
Background environment for the microstrip field applicator ....... 161
Cross-sectional geometry of the lossy microstrip field applicator . . . . 174

Full-wave and perturbation theory comparison for

attenuation constant a ................................ 178
Effect of ground plane conductivity on phase constant fl ......... 179
Effect of ground plane conductivity on attenuation constant a ...... 180
Effect of strip conductivity on phase constant ,8 ............... 181
Effect of strip conductivity on attenuation constant a ........... 182
Effect of strip conductivity on surface current kJr .............. 183
Effect of strip conductivity on surface current k, .............. 184
Effect of frequency on surface current k, ................... 185
Effect of frequency on surface current kz ................... 186
Evaluation contour for G57 when y — y’ > 0 .................. 198
Evaluation contour for Gzp when y -— y' < 0 .................. 199
Material interface for Hertzian potential boundary conditions ...... 201 .
Interface between material and good conductor ............... 211

xiii

Chapter 1

INTRODUCTION AND OVERVIEW

This dissertation deals with aspects of theoretical and experimental techniques in
electromagnetic material characterization, which is the process of determining the
permittivity and/or permeability of a material. Permittivity and permeability are specific
values that describe the effect that an externally applied electric and magnetic field has on
matter. Several disciplines rely heavily on the material characterization process;
including stealth, integrated circuit and agricultural technologies. For example, in stealth
technology, the permittivity and permeability describe how effectively a particular
material can absorb incoming radar signals. Due to the increasing demands on industry,
accurate characterization is vital if stringent design specifications are to be realized.

Typically, materials are first measured in the small-scale laboratory environment
prior to large-scale application. In stealth technology, it’s common for a radar absorbing
coating to encounter months or years of testing and design changes in the laboratory
before it can be applied to the structure of a vehicle. Several laboratory devices
(operating in their fundamental mode) can be exploited in the material characterization
process, including rectangular and circular waveguides, coaxial transmission lines, free-
field measurement systems, strip transmission lines, microstrip field applicators and
cavity resonators. In each case, the material must be machined to fit the particular
geometry of the given apparatus and a corresponding theory must be developed that

ultimately relates the permittivity and permeability of the sample to the experimental

measurement. Rectangular waveguides, strip transmission lines and, to a lesser extent,
microstrip field applicators are most often used in the industry environment due to the
ease of machining rectangular samples and the relative broadband nature of such devices.

As an overview, the material characterization process generally involves the
following steps. First, as mentioned above, a material sample is appropriately machined
to fit into the testing device. Next, the device is connected to a network analyzer, which
launches an incident wave towards the sample and subsequently measures the amount of
signal that is reflected from and transmitted through the material. These experimental
measurements are then correlated with suitable theoretical expressions and the
permittivity and permeability are computed using numerical algorithms. The
predominant algorithm used in industry is the Nicolson, Ross and Weir (N RW) technique
[11,[21-

The fundamental underlying assumptions in the NRW technique are as follows.
First, the material sample is assumed to be simple (i.e., linear, homogeneous and
isotropic) and have front and back interfaces that are coplanar. In addition, the sample is
assumed to be comprised of a single layer only. Furthermore, if a rectangular waveguide
is used, it is assumed that the waveguide walls are perfectly conducting and no gaps exist
between the sample and conducting boundaries. In the stripline or microstrip
environments, it is also assumed that the boundaries are perfectly conducting. However,
in the industry environment, these assumptions are often violated. Samples that are thin
and susceptible to warping must be attached to known substrates to facilitate
measurement, thus violating the single-layer assumption. Materials are frequently

inhomogeneous due to sample preparation methods. Samples that are measured at high

temperatures must involve field applicators made of special alloys that typically have
poor conductivities. Gaps can also occur in high-temperature applications due to
differing rates of thermal expansion for the sample material and metallic walls. These
NRW violations can lead to significant errors in the material characterization process and
render standard methods useless. The main objective of this thesis is to develop various
techniques to accommodate for and subsequently reduce these errors.

Chapter 2 presents two material characterization techniques for analyzing multi-
layered samples, the direct and deembed methods. Both methods utilize wave matrices
[3] for extracting the unknown layer parameters from the known substrate layer(s). It
will be shown that the direct method has a distinct advantage over the deembed method
when monitoring for sample inhomogeneity. Chapter 3 addresses the issue of sample-to—
wall gaps in rectangular waveguide-based measurements [4]-[6]. These gaps are
analyzed by regarding the waveguide as inhomogeneously filled in the cross-sectional
plane with longitudinal section magnetic (LSM) and longitudinal section electric (LSE)
propagating modes supported in the sample/gap region [3]. It will be shown that this
leads to a correction to the ideal TElo mode propagation constant and interfacial
reflection and transmission coefficients, thus allowing for accurate determination of the
constitutive parameters for the sample material. Chapter 4 accommodates rectangular
waveguide wall loss [7]-[13] by invoking an impedance boundary condition at the
imperfect conducting boundaries, which leads to a coupled mode perturbation theory that
is subsequently specialized to single mode operation [3]. The result is a complex
correction to the ideal TEm propagation constant. It will be demonstrated that this

complex correction is critical in the material characterization process via comparison

with the standard power-loss method [3], [14]-[19]. Therefore, Chapters 2-4 present
techniques for accommodation of NRW violations when using rectangular waveguides in
electromagnetic material characterization measurements.

The focus and intent of Chapters 5-7 are to account for imperfect conductors in
stripline and microstrip field applicators using a full-wave analysis in the Fourier
transform domain [20]-[24]. First, in Chapter 5, the Green’s function for the fields
excited by a general 3D current source immersed in a stripline background environment
is developed using two different methods. The primary method involves the use of
Hertzian potential boundary conditions [25]-[27] for a symmetric slab waveguide in
which the outer cover regions are allowed to become highly conducting. The secondary
method utilizes Hertzian potential impedance boundary conditions, which are developed
in Appendix D. Although both methods produce identical results, it will be shown that
considerably less effort is required when utilizing Hertzian potential impedance boundary
conditions.

The lossy stripline field applicator is analyzed in Chapter 6 by specializing the
general 3D current source of chapter 5 to an infinitely-long strip surface current
symmetrically located between imperfectly-conducting plates. An EFIE (electric field
integral equation) is subsequently developed [28] by satisfying an impedance boundary
condition on the strip conductor. The EFIE is solved numerically using a MoM (method
of moments) technique [29] and the principal-mode propagation constant and
corresponding surface current distribution are identified and examined. Chapter 7 is
exclusively devoted to the lossy microstrip field applicator. Similar to Chapters 5 and 6,

the electric-field dyadic Green’s function is developed for a general 3D current source

immersed within a microstrip background environment. An EFIE is formulated and
solved by confining the 3D current to a strip conductor, implementing impedance
boundary conditions on the surface of the imperfect strip and invoking the MOM
technique. A complex propagation constant, vital to the material characterization
process, is identified and investigated, along with the strip surface current density.

Chapter 8 provides conclusions and future recommendations of study.

Chapter 2

DIRECT AND DEEMBED METHODS FOR MATERIAL CHARACTERIZATION

2.1 Introduction
The well-known Nicolson-Ross-Weir (NRW) [1], [2] algorithm is predominantly

used in industry for computing the permittivity and permeability (a, ,u) of an unknown
single-layered material from forward measured scattering parameters 511,821.

Constitutive parameters can also be computed using the reverse S-parameters S22, S12. It
is common practice to use both sets of S-parameters since a comparison between them
can provide a measure of sample isotropy and homogeneity, which are important criteria
in the NRW technique. Unknown materials which are subject to warping or bending are
frequently attached (through deposition, spraying, etc.) on a known substrate layer to
facilitate measurement. In this case, the standard NRW method cannot be employed
since it can only handle single-layered environments. Although ignoring the substrate
layer would allow application of the NRW technique, experience shows that this leads to
gross errors.

Wave transmission matrices (i.e., A-parameters) [3] can be utilized to account for
the multi-layered environment so that the constitutive parameters of the unknown layer
can be properly determined. Two schemes for extracting the material parameters of the
unknown layer will be discussed, the direct and deembed methods. It will be shown
through comparison that only the direct method is reliable for monitoring the important

property of sample homogeneity.

2.2 Material Characterization for a Single-Layered Environment
2.2.1 Overview
The overall scheme of the material measurement process is to experimentally

obtain the sample S-parameters (Slefp ,Sfip ) and compare them with their theoretical

expressions (Sf)? ,S’hy ). Mathematically, the above condition leads to the following set

of coupled equations

thy (mu—5y) Sff ”((0): 0

(2.1)
S'hymw) 55" .P=(w) o

The constitutive parameters (a, ,u) in (2.1) can be determined analytically using the NRW

technique or numerically using a two-dimensional Newton’s root search. If the sample is
non-magnetic, then the permittivity can be computed using S” or S21 , that is

’;(w, e)— Sffpm): 0 0r S";"(w, e)— ngp (w): 0 (2.2)
A one-dimensional root-search must be used in (2.2) since no closed-form solution exists.
A Similar scheme applies if S§2p ,5?” are measured. That is, if the material has both
dielectric and magnetic properties, then the following coupled equations must be solved

(either analytically or numerically) for the reverse direction

SW» 5 m- 55%) = 0

(2.3)
Sf’Ww a 111—Swan) = 0

If the material is non-magnetic, then the permittivity must be computed numerically

using either of the following relations

S’hy(w g)- -59xp(w)= 0 or S’gv(w,£)-Sf§p(a))=0 (2-4)

A review of the analytical NRW technique is given next for the benefit of the reader.

2.2.2 General NRW Formulation
It can be shown (see section 2.4, for example) that the theoretical (forward) S-

parameters of a single-layered planar material are

R(l—P2)
5"” = —— (2.5)
“ 1-R2P2
2
thy _ P(l—R )
521W ‘26)

where R = R(a),£,,u) is the interfacial reflection coefficient and P = P(w,6,,u) is the

one-way phase delay and attenuation through the material. Equation (2.1) suggests that

(2.5) and (2.6) can be written
Sexp_R(1_P2) 27
S... _ Pa-R2) 2 8
21 _ 1_R2P2 ( ' )

A closed-form inverse solution exists for the above nonlinear equations. That is, R and

P can be solved in terms of 51"?” and 8;?” . Once these values have been determined,

the constitutive parameters (a, ,u) can be computed. The inverse solution can be obtained

by first solving (2.7) for P2 and (2.8) for P , leading to

 

2 R _ exp
p = ”ex (2.9)
R(1— RS] ,P)

553;" (1 - R2102)

P:
1--R2

 

(2.10)

Substituting equation (2.9) into the P2 term of (2.10) results in

2
exp exp
P=—§—‘— :> 102... _Sz__ (2.11)
1— RSf’fP 1— RSffP

Equating (2.9) and (2.11) gives

 

 

2
R _ Sexp Sexp R _ Sexp_ Sexp2
“ — ——'— :> '1 ( (2.12)
R(1— RSlexp ) 1— R53” R 1— RS “‘0
Cross-multiplying and grouping terms leads to the following quadratic equation
exp exp 1
R2—2QR+1=0 , Q: (5“ )2 (52‘ )2 + (2.13)

 

253‘”

whose solution is

R=Qi([Q2—l (2.14)

The proper root choice is based on the requirement that the magnitude of the

interfacial reflection coefficient must be less than unity for a passive material, that is,

[R] < 1. Equation (2.14) produces two roots. One will have a magnitude greater than

unity and the other root must have a magnitude less than unity. To see this, let

R1 = Q+‘/Q2 —1 and R2 = Q—(lQZ —l . Note the following relationship between the

 

 

tWOI'OOtS
L_1 1 Q VQ "lzg ,/Q2— —=R2 (2.15)
”I 9+ng —1 Q+JQ2 —1Q— JQ: —1
therefore
l_l=_‘_=|1r I (216)
R, |Rl| 2 '

Now that R and P have been related to 53‘” and S5“? in equations (2.14) and (2.11),
the permittivity and permeability can be determined. The details of this calculation are

provided in the following two sections. Note, a similar analysis holds for the reverse S-

exp ‘ exp
parameters S22 and S12 .

2.2.3 Computation of Constitutive Parameters for a TEM System

If the sample (having length 8 ) is measured using a transverse electromagnetic
(TEM) field applicator (i.e., coaxial, stripline, free-field, etc.) operating in its
fundamental mode, then the interfacial reflection coefficient R and one-way propagation
term P are related to the relative permittivity and permeability ( 8,, p,) in the following

manner

R__Z—Z0_z—l Z EL

 

 

 

 

_ _ , z = __ = (2.17)
Z+ZO z+1 Z0 8,
_7[ II'IP . . .
P=e , 7=-—,—=J)6=1(cz2/c)‘/8r#r =Jwy/c (2.18)
Solving the above equations for y and 2 produces
z = EL = 133 (2.19)
8, l—R
y: W), =(_C=JC'“P (2.20)
160 (06
Therefore, 8, and p, are computed using the relations
y ,clnP l—R] .clnP(l+R]
g 1: — = , I“ = Z = 2.21
’ z Jw€(1+R ‘1 y jwe l—R ( )

10

2.2.4 Computation of Constitutive Parameters for a Waveguide System
If the sample is measured using a rectangular waveguide operating in the principal
TElo mode, then R and P are related to 8, and ,u, in the following manner (assuming

no gaps exist between the sample and perfectly-conducting waveguide walls)

 

R-z_-1 , 2-1-me (2.22)
2+1 20 Jwflo/70 7

2 2 2 2 2 2 2 2 2 2 2 ”2
7 =kc —k , 70 =kc “k0 , k0 :0) Eoflo , k =kogr/Jr , kc 3—2‘ (2.23)

a

_ lnP
P=e 75 , yz—T=\/kc2-kg8,p, (2.24)
Solving (2.22) for ,u, and (2.24) for 8, gives
#r = 21. =_ln_P(fli_] (2.25)
70 2’05 l-R

k3 —y2 __ k3-(1nIJ/r)2

8,=——

flrkd — kgln_P(1+__R__)
yof l—R

 

(2.26)

2.3 Geometry of Multi-Layered Environment
Figure 2.1 shows the multi-layered environment under consideration. The
system, which is assumed to be immersed in free space, is comprised of N layers and

N +1 interfaces. The thickness, effective complex permittivity and permeability of the

ith layer are 61-, p,- and 6,. The terms ci,b,- are the complex wave amplitudes of the
incident and reflected waves immediately to the left of the 1'” interface. The interfacial

reflection and transmission coefficients at the I'm interface are R, and T,- .

11

 

 

 

 

 

 

 

 

 

01 92 01 0N
« >§ y -n————>

“.1. 22. ii. EN. 9:; £192

<— 4— <— <— <— <—

b1 b2 bi bN bn+1 bN+2
R12, R22, RI'Z RN21 RN+Q

T1 T2 TI 7 TN TN+1 F

(30,110) (31,111) (gal/12) (Salli) (3NallN) (80,110)

Figure 2.1 Multi-layered environment.

2.4 A-Parameter Description of a Multi-Layered System
Wave transmission matrices or A parameters describe the relationship between
incident and reflected wave amplitudes at a prescribed input terminal plane to those at a

prescribed output terminal plane. The general A-parameter formulation is developed as

follows [3]. Consider a wave 8] incident on an interface from the left and a second wave

b5 incident on the interface from the right, as shown in Figure 2.2.

flNT12C1
R101 1 szI2 )0;
<——-; ——>
bl { T21b2 ( I
‘ ve/ ‘ b2

 

(31,111) (32,142)

Figure 2.2 Reflected and transmitted waves at a planar interface.

12

If R1,T12 and R2121 are the respective interfacial reflection and transmission

coefficients experienced by waves cl and b5 , then the following relations prevail

 

1 . 2 ,
. : C1=—02——b2
02:71261+R2b2 T T
, => ‘2 ‘2 (2.27)
b1=R1C1+T21b2 R1 . T12721—191“ ,
1:“02+———b2
12 T12
where (using continuity of the tangential electric and magnetic fields)
Z —Z
R =—R = 2 1,T =1+R ,T =1+R =1—R 2.28
1 2 Zz+ZI 12 1 21 2 1 ( )

Substitution of (2.28) into (2.27) leads to the matrix expression

6'} _ 1 1 RI C'z
[ml-5dr. 1M ‘2”)

Thus, (2.29) describes the relationship between the forward and reverse traveling waves

(c],b1 ) immediately to the left of the interface to the forward and reverse traveling waves
(c'2, b5) immediately to the right of the interface.

Before considering the cascade connection of N sections, it must be shown how

the waves (c'z,b§) are related to waves (cz,b2) that are located a distance 6 from the
interface as depicted in Figure 2.3. Since the region is assumed to be linear,

homogeneous and isotropic, a simple relationship exists between the waves, namely

62 = cge’hf and b; = b25725 . Therefore, the relationship between (c'2,b§) and (c2,b2)

can be written
6'2 :62” 2 [C’] = em 0 , [CZ] (2.30)
b5 =b2eT7zf b2 0 [725 b2

13

 

 

 

 

( Q )
02’ 02
—> —)
*1— {—
b2 b2
(919%) (32,142)

Figure 2.3 Wave description for shift of distance 6 .

The A-parameter relationship between waves (cl,b1) and (c2,b2) is obtained by

substituting (2.30) into (2.29), leading to

cl __1_ e72! R1e_}’z€ [CZ]_ A11 A12 [CZ] (23])
b1 T12 Rlehf e425 b2 A21 A22 bz .

The overall A-parameter description of the multi-layered system of Figure 2.1 can now

be obtained by generalizing (2.31), leading to the desired result

151' _ ifi sys sys
b, H T, Rieh’r e—rr‘r . bN+2 A3}? A3}; bN+2

Afyls A33 N+1 em R.e-7i[i N+1 " ’
=Hl ‘ = (2.33)

Ai’i’ A52}; i=17I Rien" emf“ i=1 A21 {22
where
i=Zi—Zr_1 , 7}:1+R, (2.34)
Zi+Zi—l

14

Note that the upper limit on the product is N +1 since there are N +1 interfaces and

because the relationship between (c1,b1 ) and (CA/+2 , bN+2) is desired.

As an example, consider a single-layered sample of length 6 immersed in free

space. Since there will be two interfaces, equation (2.32) specializes to

[i — tfi
[cl]=l£[i e7: Rte 7 [C3]: A” A12 [63] (2 35)
bl i=1 7} Riey’g’ e-m" b3 A21 A22 b3

Where Rlz—R2=R, Ti=l+R, T2=1—R, 71=}’, y2=y0, €1=€ and 62:0. The

wave matrix for the system is therefore
A11 A12 _ 1 e”6 Re'fl 1 1 -R (236)
A2, A22 —1+R Rerf [75 l—R —R 1 '
Carrying out the matrix multiplication and letting P =5” leads to the following A-
parameters for a single-layered sample

[A11 A12] 1 [1_R2P2 —R(1—P2)]

— (2.37)
A21 A22 R(1—P2) P2 —R2

_ P(l — R2 )
The S-parameters of the above single-layered sample can be obtained by using the well-

known relations [3]

[S11 511:; A21 A11/‘122—I‘121A12 (2.38)
S21 S22 A11 1 "A12
A A 1 —S
“ '2 =_1_[ 22 ] (2.39)
A21 A22 S21 Sn S21512—511522

Although (2.39) is not required here, it has been given for the sake of completeness.

Substituting (2.37) into (2.38) produces the familiar result

15

R(1—P2) S _P(1—R2)

—- (2.40)
1- RZPZ 1— R2P2

SII=Szz= , 21-12-

The direct and deembed methods will now be discussed.

2.5 Direct Method of Constitutive Parameter Extraction

The direct method is one scheme for determining the constitutive parameters of an
unknown layer immersed in a multi-layered system. The overall procedure for the direct
method is as follows. First, a guess for the permittivity and permeability of the unknown
layer is provided. Next, the theoretical A-parameters of the known/unknown layers are

calculated and the overall system wave matrix is determined using (2.33), that is

_ N+1 _
4.4.4- —> A‘ , .133; = H A’ (2.41)
1 1 1 1 ,1 1
Note that the permittivity and permeability of layers 0 and N +1 are (80,;10) and the

length of the N +1 layer is 8 N H = 0. The theoretical A-parameters of the system are

then converted into the theoretical S-parameters of the system using (2.3 8), namely
[Ag’g] —> [5’33] (2.42)
The theoretical S-parameters of the system are then compared to the experimentally

measured system S-parameters using the relations

SW (w,a.y)—S“P (w)=0
ll,sys ll,sys : 8f,,Uf (243)

533,. (60,8, 1:) — 551;, (w) = 0

for the forward direction and, independently,

16

82,2),st (‘09 £9 .u) _ 82.2%,“ (0)) = O

h
Sltgsys (ma 5, .u) _ Slegfsys ((0) = 0

=> 5,.)1, (2.44)

for the reverse direction. The permittivity and permeability values for the forward
direction and reverse direction are numerically iterated until conditions (2.43) and (2.44)
are satisfied. If the unknown layer is non-magnetic, then the following forward and

reverse relations are used

sf’l'fm(w,e)-Sf{§ys(w)=o or Sgfifsys(w,e)—S§fgys(w)=0 } :3 8, (2.45)

sthy

th
22.5ys (‘0’ 8) — S22€sys (0)) = 0 or S y

12.5ys

(w,e)—Sf§§ys(w)=0} => e, (2.46)

Comments regarding the advantage/disadvantage of the direct method will be delayed

until the deembed method has been discussed.

2.6 Deembed Method of Constitutive Parameter Extraction
2.6.1 Modification of A-Parameters for Deembedding

In the deembed method, the constitutive parameters of the unknown layer are
extracted by first mathematically removing the known layers until only the unknown
single layer is remaining. The NRW procedure is then invoked and (8,p) are
analytically computed. Since the NRW technique assumes that free space exists on either
side of the single-layered sample, the multi-layered system must be mathematically
viewed as having infinitesimal artificial air layers on either side of the unknown layer.
These zero-length air layers will be introduced between all layers, however, since this is
computationally easier to implement. Note that because the A and S-parameters are

based on tangential fields, the validity of the above procedure is provided by continuity

l7

of the tangential electric and magnetic field components. The artificial air layers are

shown in Figure 2.4.

 

 

01 02 II 0i II II 0N 0N+l:O
“r" 11"**"11 11"“+
Cl 02 OH C-+1ll CNII CN+1
_. -4. 11.1 ‘44 Au 3
H 2. 4+: eH ‘H .13
b1 b2 bill bi+111 lel bN+l
(€0,110) 11 11 11
4/ :1 :: ::
(80,110) (€1,111) (€2,112) 5 ”(8,11,)” ”(34,154) (80,110)
CT“ 11 11 ll

 

 

 

 

 

 

 

5

1th layer

Figure 2.4 Introduction of zero-length air layers for deembedding.

Previously, the A-parameters of the 1'” layer described the relationship between
the waves (ci,b,-) immediately to the left of the i'h interface to the waves (c, +l,b,- +1)
immediately to the left of the 1+1 interface. In the deembed technique, the waves
(ebb 1') immediately to the left of the i’h interface must be related to the waves (ci+1,b,.+l )

immediately to the right of the 1' +1 interface. This A-parameter modification is easily

accomplished using the result of equation (2.33), leading to

87'!" Rte—7151 1 e705 _Rie")’o§

 

(2.47)

21' = lim _

where the interfacial reflection coefficient between layer i and the infinitesimally thin air

18

layer is

z—ZO

l

Rf:
Z,-+Z0

 

(2.48)

Taking the limit in (2.47) produces the following appropriate A-parameter modification

for the deembed scheme

1 1-18122 —R.<1—P.-2)

I l

2" =———— 2.49
[ ] 4(1-R?>R,-<1—42) Pf-Rf ( )

where
Pr = em" (2.50)

is the one-way propagation delay and attenuation within the i’h layer. This result should

have been anticipated from (2.37).

2.6.2 Deembed Procedure for Constitutive Parameter Extraction
The deembed method for material characterization is accomplished through the
following procedure. First, the A-parameters of each known layer are computed using

(2.49), that is
e,,a,.,7, —+ [A’] ...i¢u (2.51)

where the tilde notation has been dropped for convenience. Note that the A-parameters
of the unknown material, which will be designated as layer u , cannot be computed. The
experimentally measured S-parameters of the system are then converted into A-

pararneters using (2.39), namely

[5:55] —> [.4313] (2.52)

19

Next, the unknown layer of the system is isolated by mathematically removing the known

layers using the following inverse matrix multiplication

14:1#1141=1A1114'114711411141

1471411141"')1A:—;:1[fi141‘)

i=u-l i=N

(2.53)

where N (not N +1) must be used due to the modified A-parameter formulation. The
experimental S-parameters of the unknown layer are obtained from (2.53) with the aid of

(2.38), that is

[Afj‘P] —> [55W] (2.54)
As a final step, the permittivity and permeability of the unknown layer can be computed

analytically (if both 8 and p are desired) or numerically (if the unknown is non-

magnetic) using the knowledge of section 2.2.

2.7 General Comments and Experimental Results
2.7.1 General Comments

Now that both methods of parameter extraction have been discussed, several
comments are in order. First, the method of using A-parameters for layered media is
exceedingly powerful since it is valid for both TEM and TE”) field applicator systems.
Thus, the techniques discussed in sections 2.5 and 2.6 are applicable for the commonly
used coaxial, free-field, stripline and rectangular waveguide systems. The second
comment is in regards to how the A-parameters were defined and referenced for the
entire multi-layered system. In both extraction schemes, we were ultimately interested in

obtaining an overall relationship between the forward and reverse waves immediately to

20

the left of the first layer and the forward and reverse waves immediately to the right of
the last layer. The reason we are interested in the wave amplitudes outside the multi-
layered structure is that this is where the S-parameters are accessible via measurement.
Another comment is that the direct method requires an initial guess for the permittivity
and permeability, whereas the deembed method does not. This can be a drawback since
numerical root-search algorithm’s are typically very sensitive to initial guesses.
However, it will be demonstrated that the direct method must be used if sample
homogeneity is to be accurately monitored. One suggestion would be to invoke the
deembed method to analytically obtain an initial guess and then utilize it for
implementation of the direct method. The final comment is that if the known and
unknown layers are perfectly homogeneous, then the two techniques should yield
identical results. In the industry environment though, samples are rarely perfectly

homogeneous and thus the direct method should be exploited.

2.7.2 Experimental Results

One experimental result to discuss is the verification of the multi-layered analysis
using a rectangular waveguide field applicator. Consider a non-magnetic Alumina
sample having a thickness 1? =25 inches. Figures 2.5 and 2.6 show the permittivity
computations (real and imaginary based on 52, only) for the sample when it is measured
independently, that is, the stand-alone measurement. Next, a multi-layered system is
formed by inserting the Alumina sample between a non-magnetic piece of Teflon and

Zirconium Oxide (2,02). Thus, the three-layered {system has Teflon as its first layer,

Alumina as the second layer (treated as the unknown) and Z, 02 as the third layer. It will

21

be assumed, from previous measurements, that the known properties for Teflon and

2,02 are sit/=2.09—j0.001,(2,,,=0.1261n. and 8Z'=3.81—j0.015,£’zr=.114in.,

respectively. Figures 2.5 and 2.6 show the extracted permittivity values for the Alumina
layer if the direct method is used (based on 52] only). In addition, the uncorrected
values for the Alumina layer are also shown when the known layers are ignored. It is
evident that the direct method does indeed accurately account for the layered
environment. The results are erroneous if the extraction method is not utilized. For
example, Figure 2.6 falsely shows that the Alumina sample is highly lossy when the
Teflon and Z, 02 layers are blatantly ignored.

The other significant result to discuss is the advantage that the direct method has
over the deembed method for monitoring sample homogeneity. Suppose we construct a

two-layer system using the above mentioned Teflon, 2,02 and Alumina materials.

Layer 1 will consist of the Teflon and Z ,02 samples and will be treated as the unknown

layer. Layer 2 is comprised of the Alumina sample and will be assumed to have the

known properties 8, :9.65, E =.25in. Layer 1 has been made inhomogeneous on
purpose to see whether both techniques can detect this inhomogeneity. Figures 2.7 and

2.8 show the results of the direct method of parameter extraction for the real and
imaginary parts of the computed permittivity based on S“ (for the forward direction) and
$22 (for the reverse direction). Similarly, Figures 2.9 and 2.10 show the results of the
deembed method of parameter extraction. The direct method clearly reveals that the
unknown layer must be inhomogeneous since the S“ (forward) and $22 (reverse)

measurements are drastically different. In comparison, the deembed method shows

22

virtually no difference between forward and reverse measurements, even though it is
known that layer 1 was intentionally made inhomogeneous. Thus, although the direct
method requires an initial guess, it has a clear advantage over the deembed method for
detecting sample inhomogeneities. It should be noted that this issue of monitoring for
sample inhomogeneity is important in the industry environment since it frequently occurs

due to the limitations of the various methods of sample preparation.

23

ALUMINA MULTI-LAYERED MEASUREMENT
12 ”

 

 

 

 

 

 

 

Teflon/AI203l2,02
— Stand-alone
11 ~ ------- Extracted
——- Uncorrected
10 L g c a 0’
M 9 t
:2"
g
8 )—
——§_‘
7— T“~-
‘\‘
“\‘
\“~
6 T ~‘~~~‘
5 . . “'1
2.5 3.0 3.5 4.0
Frequency (GHz)

Figure 2.5 Verification of layered analysis for Re{8,} using an Alumina sample.

24

ALUMINA MULTl-LAYERED MEASUREMENT
Teflon/N203/2,02

 

4 L — Stand-alone
r ------- Extracted
3 r -—— Uncorrected

 

 

 

lm{8,}
l
l
l

_.L

-2-

 

 

-5 l I r l
2.5 3.0 3.5 4.0

Frequency (GHz)

Figure 2.6 Verification of layered analysis for Im{8,} using an Alumina sample.

25

DIRECT METHOD (Forward vs. Reverse)

 

 

 

 

 

 

 

15.00 '
— Direct-forward (S1,)
------- Direct-reverse (822)
11.75
1
flw: >-
3; 8.50
a:
5.25 r
L ...........................................................................................
2.00 ‘ ‘ '
2.5 3.0 3.5 4.0
Frequency (GHz)

Figure 2.7 Homogeneity interrogation of Re{8,} via direct method.

26

DIRECT METHOD (Forward vs. Reverse)
Teflon-2,02/Al203

 

10 '
— Direct-forward (S11)
Direct-reverse (S )

 

 

 

Im{8,}

 

 

-5 ‘
2.5 3.0
Frequency (GHz)

Figure 2.8 Homogeneity interrogation of Im{8,} via direct method.

27

DEEMBED METHOD (Forward vs. Reverse)

 

 

 

 

 

Teflon-Z,OzlA|203
6 -
Deembed-forward (S11)
5 _ ....... Deembed-reverse ($22)
4 -
23“-
?7 3 I
Dd

 

 

 

2.5 3.0 3.5 4.0
Frequency (GHz)

Figure 2.9 Homogeneity interrogation of Re{8,} via deembed method.

28

DEEMBED METHOD (Forward vs. Reverse)
Teflon-2,02lAl203

 

 

Deembed-forward (S1 1)
....... Deembed-reverse ($22)

 

 

 

Im{8,}
‘3
1

 

 

2.5 3.0 3.5 4.0
Frequency (GHz)

Figure 2.10 Homogeneity interrogation of Im{8,} via deembed method.

29

Chapter 3

ANALYSIS OF SAMPLE-TO-WALL GAPS IN RECTANGULAR WAVEGUIDE

MATERIAL CHARACTERIZATION MEASUREMENTS

3.1 Introduction

In rectangular waveguide material characterization measurements, gaps
commonly occur between the sample and waveguide walls as a result of imprecise
machining of the sample. These gaps are exacerbated in high temperature measurements
due to the differing thermal rates of expansion for the waveguide and sample materials.
These gaps can influence the accuracy of measured constitutive parameters because
higher order modes are excited, resulting in a shift of the ideal TEIO mode propagation
constant and changes in the ideal wave impedance and interfacial reflection and
transmission coefficients.

Sample-to-wall gaps will be analyzed in this chapter by regarding the waveguide
as inhomogeneously filled in the cross-sectional plane with LSM and LSE propagation
modes supported in the sample/gap region [3]. The longitudinal section magnetic
(LSM ) modes accommodate bottom/top gap geometries, whereas longitudinal section
electric (LSE ) modes accommodate left/right sample-to-wall gaps. Characteristic
equations for the corresponding propagation constants will be derived and solved

numerically to determine shifts from the ideal TE“) propagation constant of a uniformly-

filled guide. A modal analysis is utilized to obtain (under small gap conditions)

approximate expressions for the wave impedance and interfacial reflection and

30

transmission coefficients. This is done by considering a single TE“, mode incident upon,

and reflected from, the sample and only a single LSM or LSE mode inside the sample
region. Approximate expressions for the scattering parameters are also obtained using

wave matrices. Theoretical and experimental results will be given and discussed.

3.2 Review of a Uniformly-Filled Rectangular Waveguide

Consider a rectangular waveguide, having width a and height b , operating in its
principal TE“, mode and unifonnly filled in the cross-sectional dimensions by a planar
sample having thickness 6 (it is assumed that the sample is linear, homogeneous and
isotropic). If the scattering parameters Sff” and S33” are experimentally measured, then
the following scheme is utilized for computing the relative permittivity and permeability
(8,,p,) of the sample. First, an initial guess for 8, and ,u, is provided. Next, the

propagation constants in the free-space ( 70) and sample regions (y) are computed using

70=\/ke2-k(i a 7=\/kc2-k2 (3.1)

where

(0 71'
[(0:0) 80/10:: , k=k0"£,fl, , kc=z (3.2)

The wave impedance in the free-space (20) and sample regions (Z ) are determined next,

followed by the interfacial reflection coefficient (R) and one-way phase delay and

attenuation term (P )

Zo ___ Jwflo , Z = Ja’flollr (3‘3)
7’0 7

31

R_Z-ZO

_ , P=e-7’ (3.4)
Z+ZO

The final step is to calculate the theoretical S-parameters and compare them with the

experimentally measured values, that is

R(1-P2) ,, P(l-R2)
SW = —— , s’ y = —— (3.5)
” 1—122192 2' 1—192192

51”,” (a), 8,,,u, ) — sffpm) = 0

thy exp (36)

The parameters 8, and p, are iterated using a two-dimensional root search algorithm
(such as Newton’s method) until (3.6) is satisfied within the desired/specified accuracy.

If the sample is non-magnetic, then ,u, =1 and 8, can be computed using the following
S{?y(w,e,) — Sff”(a)) = 0 or Sgfima, e,) — S§{P(w) = 0 (3.7)
The procedure for computing 8, and ,u, when gaps are accommodated is similar except

that the expressions for 7 , Z , R and P must be modified (as will be demonstrated).

3.3 LSM Mode Analysis for Bottom/'1’ op Gaps
3.3.1 Geometry

The geometry for the bottom/top gap analysis is depicted in Figure 3.1. The
spacing of the bottom gap in region 1 is d, and the spacing of the top gap in region 3 is
(12. Both gap regions are assumed to have constitutive parameters (80,;10 ). The height

of the sample in region 2 is t1 and has constitutive parameters (8,, p, ). The width and

height of the rectangular waveguide are a and b , respectively. The thickness of the

sample along the z -axis is 6. The waveguide walls are assumed to be perfectly

32

conducting.

_b —
T Region3 (capo) 1d.

    

Region 1 (80,110) i dI
y=0 -| I :
x=0 x=a

 

Figure 3.1 Cross-sectional geometry for bottom/top gap analysis.

3.3.2 Hertzian Potential Generating Function and EM Field Components
An electric-type Hertz potential having only a y-component is sufficient to
generate all EM field components (i.e., LSM modes) necessary for satisfying boundary
conditions, that is
if = fifl3(x,y)e_7”“z = 5’11(X)gt(y)e"7”’"z (3-8)
for a forward traveling wave in the i'h region, where it is assumed that propagation is
along the z -axis. Note that this particular choice is justified by the uniqueness theorem.
Also, it is common knowledge [3] that the propagation constant y)”, in each region must
be identical, thus having a subscript i on 7,5," is not required). Since it is assumed that no
sources exist in the sample or gap regions, the electric-type Hertz potential satisfies the

source-free Helmholtz wave equation

V272? +13”? = 0 (3.9)
l l I

33

where k] =k3 =k0 and k2 =k. The EM field components satisfy the well-known

relations (see Appendix A for details)

1?, = jweyxir'f (3.10)

VXPI, (3.11)

 

Substitution of (3.8) into (3.10) and (3.11) leads to the following field components

 

H= —jws- [mm-(n+2 af" ( ax)]g.(y)e‘"mz (3.12)
{4,973 316 ,ggx>+.,..aa].a
(3.13)

_ 2713",]; (x) M} 8—71er

ay

Note, equation (3.12) reveals that PI only has components in the longitudinal x — 2
plane, hence the LSM terminology. The above wave equation and field components are
utilized in the development of the characteristic equation for propagation constant 7,5,, in

the next section.

3.3.3 Identification of the LSM Mode Characteristic Equation for 7,3,,

The LSM mode characteristic equation for 7,”, can be identified by solving the

wave equation in the gap and sample regions and enforcing boundary conditions at the

waveguide walls and air/dielectric interfaces. Substitution of (3.8) into (3.9) and dividing

by f,-(x)g,-(y)e_7’“"z leads to the familiar separation of variables result

34

1 62f(x) 1 azg-(y) 2 2
' ‘ = — k- 3.14
fxx) 6x2 + gi (y) ayz (ylsm + 1 ) ( )

 

Equation (3. 14) can only be satisfied if the first and second terms are constants, that is

 

 

2
1 a fig") =—k§; (3.15)
fix) ax
2
1 a gig” =—k,-i (3.16)
810’) 6y
where the constraint equation is
k; + k; = yin, + k} (3.17)

Equations (3.15) and (3.16) are ordinary differential equations which have the following

well-known solutions
f,-(x) = A,- sin kux+ B,- cos kux (3.18)
gi(y)=Cisinkiyy+Dicosklyy (3.19)
It will be shown that f,-(x) is identical in each region but g,- (y) is not. Therefore, it is

mathematically convenient to introduce appropriate shift factors in the definition of

g,- (y) as follows

gl(y)=C1 sinhy+D1coshy ...O<y<d1 (3.20)
g2(y)=C2Sinl(y—d1)+chosl(y—d1)...d1<y<d1+tl (3.21)
g3(y) 2C3 sinh(b—y)+D3 cosh(b—y) ...dl +tl < y <b (3.22)

where the wavenumbers along the y direction are
kly =k3y =h , ka =1 (3.23)

Boundary conditions on E and [I can now be implemented with the aid of(3. 12)

35

and (3.13). The tangential electric field at x = 0,a (for all y, z ) must be zero, thus
Eia(0,y,z)=0 , Eia(a,y,z)=0 ...a=y,z (3.24)
Substitution of (3.13) into (3.24) produces the following condition

f,(x)l - O (3.25)

x=0.a _
The tangential electric field at y = O,b (for all x, 2) must be zero, that is
E1a(x,0,z)=0 , E3a(x,b,z)=0 ...a=x,z (3.26)

Substitution of (3. 1 3) into (3.26) results in the general relations

6810’) :0 6330’) :0 (3.27)

9

6y y=0 6y y=b
The tangential electric field must be continuous at y = d1 ,d1 +t1 (for all x, z ), thus
E1a(x,dl,2)=E2a(x,dl,Z) , E2a(x,d1+tl,z)=E3a(x,dl+tl,z) ...a=x,z (3.28)

Inserting (3.13) into (3.28) produces the following

 

 

 

 

6g] (y) = 6g2 (y) ag2(y) = 683 (y) (329)
6y y=d1 0y y=d1 6y y=d1+h 6y y=d1+t1
fl(x) = f2(x) = f3(x) = f(x) = Asin kxx+Bcos kxx (3.30)

Equation (3.30) physically requires that the phase velocity along the x direction must be

the same in the sample and gap regions for boundary conditions to be satisfied (same

applies for propagation along the z direction since 7,3," is the same in each region). The

final boundary condition is that the magnetic field must be continuous at y = dl,d1 +1]

(for all x,z), thus
Hla(x,dl,2)=H2a(x,dl,Z) , H2a(x,dl+tl,Z)=H3a(x,dl+tl,Z) ...a=X,Z (3.31)

Substitution of (3. 12) into (3.31) leads to

36

g1(d1) = 5r82(d1) . 5r82(d1 +11) = 83(d1 “1) (3-32)
The LSM mode characteristic equation is obtained by inserting (3.20)-(3.22) and

(3.30) into the above various relations. First, substitution of (3.30) into (3.25) produces

the result
f(x) = Asin kxx = Asin(m7rx/a) ...m =1,2,3,... (3.33)
where
k. = E. (3.34)
a

is the same in the sample and gap regions. If the coefficient A is absorbed into the
coefficients of g,—( y) , then f (x) can be written as

f(x) =sin(m7rx/a) ...m =1,2,3,... (3.35)
Note that f (x) = 0 when m = O and is therefore of no interest since it leads to a trivial

solution. Substitution of (3.20) and (3.22) into (3.27) leads to

ngho 2» C.=o g1<y1=choshy (3'36)
Eaglzo 2. C3=0 g3(y)=D3cosh(b—y) (3.37)

Inserting (3.36) and (3.21) into the first relation of (3.29) gives

QM
5y

= 5820’)

:> — Dlh sin hd, = C21 (3.38)
y=a', 6y

 

 

y=d,

Solving (3.38) for C2 allows g2 (y) to be written as
820’) = -Dl §sin hdl sinl(y—d,)+ D2 cosl(y—d,) (3.39)

Substitution of (3.39) and (3.37) into the second relation of (3.29) results in

37

032 (y) = 5830’)
6y y=dl +tl 6y y=dl+ll

 

 

:> — Dlh sin hdl cos It, — D21 sin II, = D3h sin hdz (3.40)

 

 

Solving (3.40) for D3 and inserting this result into (3.37) leads to the following
expression for g3 (y)

Dlh sin hdl cos It] + D21 sin [I]
h sin hdz

 

g3(y) = - cos h(b - y) (341)

Substitution of (3.36) and (3.39) into the first condition of (3.32) gives

 

81(d1) = 5r82(d1) 3 D1 905 hdl = grDZ D2 = D1 COShdl (3-42)
Insertion of (3.42) into (3.39) and (3.41) leads to
1 h . .
g2 (y) = D] [E— cos hd, cosI(y-d1)—751n hdl sm l(y — do] (3.43)
h sin hd, cos II, + —1— cos hdl sin It]
8r
83 (y) = "'Dl COS h(b - y) (344}

 

h sin hdz

As a final step, (3.43) and (3.44) are substituted into the second relation of (3.32), giving

h sin hdl cos It, + —1— cos hd] sin ltl
6

cos hdl cos It] — fl sin hdl sin [II = — _ ’ cos hdz (3.45)
l h srn hdz

 

Multiplying (3.45) by the factor ark sin hdz and using the trigonometric identity

sin hd, cos hdz + cos hdl sin hdz = sin h(d, + d2) produces the characteristic equation

82 2
1 cos hdl cos hdz — ’

 

sin hdl sin hdz )sin II, + grh sin h(dl + d2 ) cos II, = 0 (3.46)

where, upon using (3.17), (3.23) and (3.34), the wavenumbers h and l are

 

 

h=\[y,§m+kg—(m7r/a)2 , I=\/y1im+k2—(m7r/a)2 (3.47)

38

Since h=h(7,sm) and I=I(y,sm), equation (3.46) constitutes a transcendental
equation for 713m which must be solved numerically. An infinite number of solutions
exist for the LSM mode characteristic equation (i.e., n = 1,2, 3,... ). The gap geometry,
dimensions of the waveguide and the frequency of operation all determine whether a
mode is propagating (71m = j film) or evanescent (71m = alsm ). If the sample is lossy,

no true cutoff exists and all the modes will be propagation modes (however, the modes

that exist above the frequency of operation will decay rapidly). Note that, for the mn’h

LSM mode, the indices m and n give a measure of the field variation along the x and
y directions, respectively. Also note that when d1,d2 —> 0 (2) t1 —-> b), equation (3.46)

simplifies to sinlb =0 :> l=fi7r/ b for fi=0,1,2,.... This, of course, is a well-known

result of guided wave theory.

3.3.4 Perturbation Theory for Lowest-Order LSM Mode Propagation Constant

It was mentioned in the previous section that an infinite number of roots exist for
the characteristic equation for 7,3,". If the gaps between the sample and waveguide walls
are small compared to the dimensions of the waveguide (and frequency of operation),
then the lowest-order (i.e., LSM 11) mode propagation constant should be dominant (the

higher-order mode propagation constants will lead to waves that are highly evanescent).

It should also be anticipated that this propagation constant will only be slightly perturbed
from the ideal T E10 propagation constant ( y) for a guide uniformly filled with the

sample material, that is

r2... z 72 +673... (3.48)

39

where

y2 = (7r/a)2 —k2 (3.49)
Since the algorithm for finding the various roots of (3 .46) is sensitive to the initial guess,

the perturbation expression in (3.48) will be utilized to obtain an accurate initial guess for

mm. Note, the perturbation yZ‘m z 72 +67%," is used instead of mm z y +6715”, since
the wavenumbers in (3.47) depend on rim and not 7,8,".
A perturbation formula for 6%,," can be obtained by first substituting (3.48) and

(3.49) into (3.47), resulting in the following expressions for h2 and 12 (for m =1)

h2 = 1.5—1.2 +5713... = hf + hf, (3.50)
12 = ayfsm (3.51)

where
hf = 1.3—1:2 , h; = 6%," (3.52)

Next, the Taylor series expansions for sin x and cosx , that is

2
sinxzx , cosle—x? ...x<<1 (3.53)

are utilized in (3.46), leading to the following simplifications (assuming that hdl, hdz

and It] are small compared to l)

2 2 2 2
lcoshdl coshdz sinltl zlztl [l—h—j—III-h—zflz—J (3.54)

2 2 2 2
lcoshdlcoshdz sinlt] szyfm , C=t][l-fl“—2fl][l-%‘b—] (3.55)

40

2 2
— r h sin hdl sin hdz sin 1:, s —d,dzz,g,2h4 s 416121.334? — 2.112121133123123, (3.56)

 

2 2
— r h sin hdl sin hdz sin 1!, z -A + D5713":

 

(3.57)
A : dldztlgzhg , D = —2d1d21183h3

2 2
grhsm Md! + d2)COSl’1 z 8.172011 +d2)[ -1—21]

(3.58)

 

2 113112 2
z£r(dl+d2)hu +8,(d1+d2) 1- 2 5715m

arh sin h(d1 + d2 ) cos [I] z —B + E5739,"

2 2 (3.59)
B : —gr(dl +d2)h3 , E :£r(dl +d2)[ "fllzt—l]

Substitution of (3.55), (3.57) and (3.59) into (3.46) leads to the desired result

A+B
—A+B+C+D+E52 =0 :5 62 =——— 3.60
( ) ( )713m 715m C+D+E ( )

7E2.” erz +673... (3.61)

The result of (3.61) is used as the initial guess in the algorithm for finding the

lowest-order root of (3.46). The reader is reminded that this perturbation result is only
valid if the gaps are small. Note that It] is always much smaller than unity since
I =5715m, thus the Taylor series expansions in (3.53) are always valid. This is not
necessarily the case for hdl and hdz. However, h can be broken into an unperturbed

and perturbed portion and the sin and cos angle formulas invoked. It can be shown

(although algebraically tedious) that this leads to a better approximation to 673m , and

hence a more accurate initial guess.

41

Now that we have developed a scheme for computing the propagation constant

mm , we can compute the one-way propagation term P by using the second term of (3.4)

with 7 replaced by mm. This expression is not exact, but offers a good approximation if
the gaps are small. However, experience shows that a more adequate model for the
interfacial reflection and transmission coefficient R,T and wave impedance Z can be
developed by using mode matching, as will be discussed in the following section. Once
these expressions have been found, the general technique of section 3.2 can be invoked

for the material characterization process.

3.3.5 Approximate Expressions for Scattering Parameters Using Mode Matching
The objective of this section is to obtain expressions for the scattering parameters
S11?" and S5“ . Approximate formulas can be obtained using wave matrices if the

interfacial reflection and transmission coefficients are computed at the front ( Rlsm,T,sm)

and back (Rhmjbm) sample interfaces. The front sample surface is comprised of an
air/dielectric interface, whereas the back sample surface is a dielectric/air interface.
Expressions for R,sm,7}sm and R,sm,i}sm will be found using mode matching. An

approximate expression for the equivalent wave impedance in the region 0 < z < E will
also be obtained using the above analysis.

A general formulation for computing the reflection and transmission coefficients
at an air/dielectric interface is as follows. First, assume that only a single mode is
incident upon the interface and N modes are reflected and transmitted. Thus, the total

tangential fields for z < 0 and z > 0 can be written as

42

N
E], = urge—7'2 + Za;é'ne7"z

n=l

H ,=alh1e 7'z—Za; h" e7"

n=l ,

 

N Itm ‘
= Z flame—7" 2
n=1

N ..l [rm
_ 4* SM —)’n 2
— 2b,, h" e

n=l

 

~Ism ”ah 13m

where en , hn , en

air and dielectric regions. Enforcing continuity at the z = 0 interface leads to
N N 1
+ - — - + _. sm
0, el + Zane" = 2b,, en
n=l n=l

lb+hhm

t:
~+
iMz
11 :M2

If the above equations are divided by af and the operators

jéj-uds, [131.4% ...j=1,...,N
CS CS

are applied to (3.64) and (3.65), respectively, then the following relations prevail

N N
IEj-éldS+ZRn IEj-énds=ZTn jéj-éfmds ...j=1,...,N
CS "=1 CS "=1 CS

-¢ —0 N -0 —o N, —o
Ihj-hlds—ZRn Ihj-hnds=ZTn jh h’smds =1,...,N
CS "=1 CS "=1 CS
where
a‘ b)r
Rn=—’;L 9 Tn_—n+ ’ Rlsm_Rl ’ Tlsm_Tl
01 01

43

(3.62)

(3.63)

are the normal modes for the electric and magnetic fields in the

(3.64)

(3.65)

(3.66)

(3.67)

(3.68)

(3.69)

It will be assumed that the normal modes are normalized in the following manner

, - l...n=j - - C-— ---n=j
CS ] CS ...n¢j
Thus, equations (3.67) and (3.68) may be written
6.11-FR}. ___:Mj 1’ "9N (3.71)
N
Cilafl—CJJRJ.=ZN,,,T,, ...j=1,...,N (3.72)
where
M. = ".-”’S”’d N. = fi-ii’smd 373
1,, el en 5 , 1,, j n s (. )
cs cs

The factors 61-, and C115}, are the matrix forcing terms, thus (3.71) and (3.72) can be
more suitably written as

N

R} -ZM,,,T,, =’5fl
"=11, > ...for j =l,...,N (3-74)
C1] R +ZN MT =5jC11

 

or in partitioned matrix form

12 ‘l’ll’iHil m»

where I is the N x N identity matrix, C is a N x N diagonal matrix with elements C1] ,

M and N are NxN matrices with elements Mjn,Nj-,, and R,T,Bl,Bz are NxI

column vectors given by

44

R1 Tl ‘1 C11
[R]: 5 , [T]: 2 , [8,]: 0 , [132]: 0 (3.76)
Note that the expected result of R WT n = 0 for n $1 when dl,d2 —> 0. That is, coupling
to higher-order modes disappears when the gaps are removed.

In a similar manner, a general formulation for computing Ennis", can be found

by considering a dominant propagating mode incident upon a dielectric/air interface. The

total tangential fields in the dielectric (z < 0) and air (2 > 0) regions are

Elt_ _ _afellsme- 71"" 2+ "2.1 ~--o1sme7f,""z l

 

> ...z < 0 (3.77)
17,- mm. 2h
N .
E2, = 25:5,,e-7"z
";' ) ...z > 0 (3.78)
H2, = 25;hne_7"z
n=l )

 

Enforcing continuity at the z = 0 dielectric/air interface leads to

~1+e ~llsm + Zan ~"' “‘13"! =Zb+én (3.79)
n=l
—. ‘N -. N ~ —0
(if/45'" — 531,?“ = Zbghn (3.80)
n=l n=l

Dividing the above equations by ("If , and using a similar set of steps as in the previous

1:: 21:11.21

45

paragraph, produces the result

where Rn =&;/Eil+, Tn =b;/&l+, RIM, =R1, Tlsm =7] and

R1 T1 M11 N11

[R]: s ,[f]: 5 ,[1‘3’1]=— s ,[132]= 5 (3.82)

RN TN M N] N N]
As a specific example, consider a single incident, reflected and transmitted wave
(i.e., N = 1). If the incident wave is the T E10 dominant mode, then the well known

tangential fields in the region 2 < 0 are

E , ___ +—° _7lz+ _E 712
1,(x 2) 0191009 a1 1(x)e } ...z<0 (3.83)

Hm) = 48(on —a1_’;1(x)e7lz

where

 

é]=y,/-2—sinfl , lip—ii —2—sinfl , yl=\/(7r/a)2—kg' (3.84)
ab b a

The tangential fields in the region 2 >0 for the lowest-order LSM mode (i.e., the

LSM 11 mode), upon examination of section 3.3.2-3.3.3, are

... -°1 _ lsm
E2!(x’yaz)=bl+elsm(x’y)e 71 z

_. _. [m .z > 0 (3.85)
H2t(xaysz) = bl+hllsm (x’y)e-71 2
where
+ _ D1 ab 71.2 lsm 2
b1 - Z —2— a—z—(h )
(3.86)

 

~llsm=j> 2 sin— axglsm(y) , Ellsm=_xl~_ [1 _2_Sin7[_gllsm(y )
V— ab Ziab‘mml ‘1

Ist 2 .
m — II/ a 8 ...gap region
2“ 0047' ) (1...) , siy)={ ° . (3.87)
jw£(y)7 a ...sample region

 

46

3, costh ...0 <y <d]

glls'" =4 coshldl cosll(y—d,)— 8;,“

1
£,h, sin hid] cos lltl +1] cos hid] sin [It]

L hl Sin hld2

 

Sinh1d1 Sinll(y-d]) "'dl <y<dl +1] (3.88)

 

cosh1(b—y) ...dl +11 <y <b

 

 

 

hi=J(ri’”")2+k§—<n/a>2 , I.=./0{~‘”’)2+k2—(rr/a)2 (3.89)

The matrix equation (3.75), for this special case, reduces to
= (3.90)
C11 N 11 T1 C11

1

where

 

 

C“: [amp—7 (3.91)
cs 20
1b,
M 11 = Iél 'Eismds = 3 181m (y)dy = A (392)
CS 0
J' " "Ism 1 b gllsm(y) 1
N“: hl-hl ds= dy=—— (3.93)
CS [’20 0 ZismO’) 2021:};
The solution to (3.90) is
Zeq --Z0 1
R1=R,S,,, =—Z’ei;"-—Z , Tl =T,,,,, =;(1+R,,m) , 2;?" = Azgffn (3.94)
lsm + 0
Note that the following expected result occurs when d1, d2 -—> 0 , namely
1-Z_Z° , Tl=1+R1 , zgin=A-Z;/{,=1.Z=Z (3.95)

_Z+%
It should be noted that the relations in (3.94) are approximate for several reasons.

First, the fields in the region 2 < 0 are independent of the variable y , whereas the fields

47

in the region 2 > 0 are functions of y . Secondly, the x -directed tangential electric field

for the LSM 11 mode has been ignored (i.e., assumed to be negligibly small). Finally, and
most importantly, the expressions are approximate since the modal expansion of the
reflected and transmitted fields has been truncated to a single mode.

In a similar manner, the interfacial reflection and transmission coefficients
Izsmflsm at the dielectric/air interface are found by considering a single LSM mode

incident upon, and reflected from, the interface and a single TEIO mode transmitted into
the air region. The result, with the aid of (3.81), is

Rlsm = _Rlsm ’ 738m = AG + Rlsm) : AG — Rlsm) (3 '96)
If the sample region has length 6 , then an application of the wave-matrix approach of

chapter 2 leads to the desired scattering parameter expressions

 

 

2 2
5111," 2 R15," (1 -’ Plsm) Slsm = Plsm (1 — Rlsm) (3 97)
2 2 ’ 2 2 °
1 — Rlsm Plsm 1 - R13"! PIS’"
where
_ lsmf
Plsm = e 71 (3.98)

Of course, these scattering parameters are only approximate since RM , T13", and PM,

are not exact.

3.4 , LSE Mode Analysis for Left/Right Gaps
3.4.1 Geometry
Figure 3.2 shows the cross-sectional dimensions of a sample inserted into a

rectangular waveguide having left/right gaps. The spacing of the lefi gap in region 1 is

48

d3 and the right gap spacing in region 3 is d4. Both gap regions are assumed to have
constitutive parameters (£0,210). The width of the sample in region 2 is t2 and has the

electromagnetic properties (8,, ,u, ). The width and height of the rectangular waveguide
are a and b . The sample has a thickness 8 and the walls of the waveguide are assumed

to be perfectly conducting.

 

 

Any
y=b —
Region 1
(809140)
<—>
d3
y=0 I
x=0

 

Figure 3.2 Cross-sectional geometry for lefl/right gap analysis.

3.4.2 Hertzian Potential Generating Function and EM Field Components
The following magnetic-type Hertz potential is sufficient to generate all EM field

components necessary for satisfying boundary conditions for the left/right gap geometry
7?!‘ =ifl.’;(x,y)e"mz =£f.(x)g.(y)e"m’ (3.99)
where the Hertz potential satisfies the source-free Helmholtz wave equation
V277,.” + 1.35? = 0 (3.100)

with kI =k3 =k0 and k2 =k. For magnetic-type Hertz potentials, the EM field

49

components satisfy the relations

 

E", =—jw1.,.vX;z°,." (3.101)
H,=— .1 VXEI (3.102)
Jaw,-

Substitution of (3.99) into (3.101) and (3.102) leads to the following field components

‘° . A A6 i — .z
E,- =my.f.-(x)[m..g.<y)+z—ga%]e 7'” (3.103)

 

 

2
1‘51.- = {-ff.(x)[a 8"?) +r§.g.<y)]+9 egg") 5“”
6y 6y (3.104)
61100
6x

 

_Zylse

81-00} e_7lsez

Note, the electric field only has components in the longitudinal y—z plane (i.e., the

plane parallel to the gap-sample interface) and are therefore called LSE modes.

3.4.3 Identification of the LSE Mode Characteristic Equation for 7,3,,

The form of fI-(x) and g,( y) for the magnetic-type Hertz potential, determined

by solving (3.100), is

fI(x)=AIsinpx+BIcospx ...0<x<d3 (3.105)
f2(x)=Azsinq(x—d3)+Bz cosq(x—d3) ...d3 <x<d3+t2 (3.106)
f3(x)=A3sinp(a—x)+B3 cosp(a—x) ...d3 +12 <x<a (3.107)
gI-(y)=CIsinkIyy+DIcoskIyy (3.108)

where the constraint equation and wavenumbers along the x direction are given by

k; + k; = 7,1, + k,-2 (3.109)

50

k1x=k3x=p , k2x=q (3.110)

The following boundary conditions for fI(x) and g,( y) prevail

E,ang(x,0,z)=E,a,,g(x,b,z)=0 :> 5359—) :0 (3.111)
6y y=0,b

E,a,,g(0,y,z)=E,ang(a,y,z)=0 :5 fI(O)=0 , f3(a)=0 (3.112)

E...g(d;,y,z>=E...g(d;,y,z) 2 fi(d3)=#rf2(d3) (3.113)

Emngw,+t;,y,z)=E,a,,g(d,+t;,y,z) :> ,urf2(d3+tz)=f3(d3+t2) (3.114)

 

 

 

 

~ _ ~ 6 x 6 x
H...g(d3 ,y,z)=H...g(d;.y,z) 2 44—) =14) (3.115)
' ax x=d3 ax x=d3
- _ ~ 6 x 6 x
ng(d3+t2,y,z)=Hmng(d3+t§,y,z) 2 12(_) =fl (3.116)
ax x=d3+t2 ax x=d3 +12
In addition, the relations in (3.113)-(3.1 16) also require that

81 (y) = 820/) = 83 (y) = 80) = C sin kyy + Dcoskyy (3-117)

The LSE characteristic equation is identified by substituting (3.105)-(3.107) and

(3.117) into the above boundary conditions. The result is

 

535M =0 :> g(y)=coskyy=cosfl ...n=0,1,2,... (3.118)
6y y=0,b b
fI(0)=0 :> BI=0 fI(x)=AIsinpx (3.119)
f3(a)=0 :> B3=0 f3(x)=A3sinp(a—x) (3.120)
. sin d
fI(d3)=#rf2(d3) => A151npd3=flr32 =9 32=A1 ”p 3 (3-121)
. Sin d3

f2(x)=Azsrnq(x-d3)+AI cosq(x——d3) (3.122)

 

r

51

#rf2(d3 “2) = f3(d3 “2) =>

 

 

 

' 3.123
p,(Azsinqt2 +Al smpd3 cosqtzj=A3 sinpa'4 ( )
,u, . sinpd3 .
f3(x)= . Azsmqt2+A1 cosqtz smp(a—x) (3.124)
s1npd4 ,
M 4712—3)— : Alpcospd3=A2q :> A2=A1M (3.125)
ax x=d3 ax x=d3

 

 

f2 (x) = A1 [£- cos pd3 sin q(x — d3 ) + -1— sin pd3 cos q(x — d3 )] (3.126)
q l1

r

 

f3(x) = A1 . ’ (Ems pd3 sin qtz +isin pd3 cos qtzjsin p(a—x) (3.127)
sm M4 4 #r

af2 (x)
0x

= aux)

x=d3+12

=> (3.128)

 

 

x=d3+12

p cos pd3 cos qtz — _q_ sin pd3 sin qtz =

r

(3.129)

 

_ pflf cos pd4 [Boos pd3 sin 6112 + ‘l—Sin Pd3 C05 ‘15]
sm pd4 q #r

Multiplying (3.129) by the factor qsin pd4 / pp, and using the trigonometric identity
sin p(d3 + d4) = sin pd3 cos pd4 + cos pd3 sin pd4 leads to the following LSE mode

characteristic equation

 

 

 

2
[ p cos pd3 cos pd4 — q 2 sin pd3 sin desin qtz
’ (3.130)
+-g—sin p(d3 + d4)cosqt2 = 0
where
p=\/y,ie+k§-(m/b)2 , q=\fy,7;e+k2—(mr/b)2 (3.131)

52

The above eigenvalue equation has an infinite number (m =1, 2,3,...) of roots for 7,58 and

must be solved numerically. Note that (3.130) implies that q = m7r/ a when d3,d4 —> 0 ,

as expected.

3.4.4 Perturbation Theory for Lowest-Order LSE Mode Propagation Constant
An initial guess for the lowest-order LSE (i.e., the LSEIO) mode propagation
constant can be obtained using the perturbation approximation
72.. ~72 +5712... , 72 =(zr/a)2-k2 (3.132)

Substitution of (3.132) into (3.131) leads to the following binomial approximations for p

andq
p=pu+pp , q=qu+qp (3.133)
where
p =(y2+k2)“2 p :57]; :5 q =57ge (3134)
u 0 ’ p 2p. ’ " a ’ p 2‘1.

Steps analogous to those in section 3.3.4 lead to the result

ess A+B+C
7E: 2J72+57§e , 67§e=—— (3.135)
where
2 2
d
— [1_pu 4]=_puqut2ClC2

_ puqu (d3 + d4) 1_ q3t22
#r 2

(3.136)

53

2 2
D = %[&C1C2 ‘ puqudgcl + C2 {FEE—1362]]
qu

2
Ez—fifI—‘l’gi 3+q—“ (3.137)
2,11,. p“

2 2
szuqu(d3+d4) _12_+L2 1_M_ "t22
2% p. q. 2

3.4.5 Approximate Expressions for Scattering Parameters Using Mode Matching

 

It can be shown by a set of steps similar to those in section 3.3.5, that the

approximate scattering parameters for the left/right gap geometry are (for N =1)

Sl/ie = Rise (1 - Fife) Slse = Plse (1 - Rlie)
2 2 ’ 2 2
1 — Rise P138 1_ R1531)!“

 

 

(3.138)

These scattering parameters are found using the wave matrix formulation which requires

the computation of the interfacial reflection and transmission coefficients Rlse,7}se at the

front sample interface and Elsejke at the back sample interface. As in section 3.3.5, the

above coefficients can be computed using (3.75) and (3.81), except that M

j", N j" are

given by

J" J
CS

M- = E- ~é’feds , NJ." = [ii]. Jifeds (3.139)

CS
As a specific example, when N =1, the matrix equation for the air/dielectric
interface reduces to (3.90). The tangential fields in the region 2 < O are given by (3.83)-
(3.84) and the tangential fields in the region z > O, with the aid of sections 3.4.2-3.4.3,

are

54

 

 

 

 

 

 

_. 7130
E x, ,2 +8 lse x e 71 z
”24 y P b1 1 ( y) ...z>0 (3.140)
H21(x,y,2)=b1 +h11se(x,y)e—71 z
where
+ . [SQ p ab
b, =A11wfloflr71 -—(/-- (3'14”
q 2
-1... (x) y ,2 bfllseoc) , h,’“(x )__x £21133“ ) /a_2b fl’“(x ) (3.142)
__ 1 ...gap region
fn’”(x)=#,-(x)fi’“(x) , yr<x)={ - (“43)
y, ...sample region
r .
M ...0<x<d3
pl/lr
illse =4 COSP1d3Sin‘110‘ d3) ‘11 s1np1d3 cosql(x d3) ...d3 <x<d3+tz (3.144)
#r pllur
Sln.p](a—x) (cos p1d3 Sin (1112 + q] sin p1d3 COS (11,2) ~--d3 +t2 < x < a
5m pld4 P1,“;-
60 x
Z’“(x )=’ if, ) , #(X)=#o#r(x) (1145)

71
The factor C“ is given by (3.91) and the matrix elements M“ and N” are

a

 

M” = [8] -Ellseds = 3 sin(7rx/a)fllse(x)dx = B (3.146)
CS a 0
0 Ise
~ ~ ,8 sin 7rx/ a x 1
N“: [hyhf ds= 2] ( lesem ( )dx =——e7 (3.147)
CS “20 o (x) ZOlee

Therefore, the solution to (3.90) is

29" Z 1
R,=R,,e= _-Z’:—e—° , T1=T,se=—§(1+R,se), fizz-132,315 (3.148)

lse

55

Similarly, the solution for R1“, and 73,“, at the dielectric/air interface is
Rise = _Rlse 9 flse = B“ + Rise) = 30- Rise) (3°149)
An application of the wave-matrix procedure leads to the desired result (3.138) where

1),, = 671““ (3.150)

Note, since PM = —R,se and E3373,” =1— RE“, , the wave-matrix procedure leads to

the result in (3.138). However, for the case N >1, these relations may not be valid and

the general wave-matrix result must be used, that is

 

11+sz Pfi‘
= ~ _ S =___ 3.151
” 1+RRP2 2' 1+RRP2 ( )
11+sz P1—R2 1—171'2
22=-——.. 2 Sn: (. X- 2 ) (3.152)
1+RRP TT(1+RRP)

where R,T,Rj' are representative of the LSM or LSE interfacial reflection and
transmission coefficients at the front and back sample planes and P is equal to em” or

e-y’“. One can see that, when R = —R and 77‘ =1—R2, the anticipated scattering

parameters are obtained.

3.5 Experimental Results

The gap analysis measurements for an Alumina (A1203) sample are discussed in
this section in order to verify the theoretical development in the preceding sections. In
the top/bottom gap measurements, an Alumina sample having length 13 = 0.05 inches was
initially machined to accurately fit into an ideal X-band waveguide having a width of

a=0.9 inches and height of b=0.4 inches. The permittivity of this sample was

56

computed using the ideal procedure in section 3.2. Next, the sample was machined so
that the height was reduced by 0.0008 inches (i.e., 0.8 mils). Thus, the height of the
sample was reduced from 0.4 inches to 0.3992 inches. Note, the width of the sample (0.9
inches) was not altered. The sample was then reinserted into the waveguide and it was
noted that there was a 0.5 mil gap at the bottom and a 0.3 mil top gap. The permittivity

was then determined with and without implementing the gap correction (for N=I) of

section 3.3. Figures 3.3 (Re{£,} vs. freq.) and 3.4 (Im{gr} vs. freq.) clearly show the
validity of the gap analysis. The sample was then removed, machined to a height of
0. 398] inches and inserted back into the waveguide. Again, the permittivity was
computed with and without gap correction and subsequently compared to the ideal case.
Figure 3.5 shows that the analysis of section 3.3 mitigates the gap error. In addition,
Figure 3.6 demonstrates the effects of a 2.9 mil gap. In this case, the gap correction
accommodates an error of approximately 5%.

Figure 3.7 reveals that left/right gaps do not significantly alter permittivity
measurements of non-magnetic materials. Although the left and right gaps were 17.5
mils, the difference between the gap and no-gap correction was negligible. This is an
anticipated result since the electric field is exceedingly small near the walls at x=0,a. It is
anticipated that the permeability profile of a magnetic material would be significantly

altered by the presence of left/right gaps.

57

ALUMINA COMPARISON
Ideal vs. 0.8 Mil Gap

 

 

 

 

 

 

 

10.0 '
9.8 ~ _ Ideal
--------- Gap (wo/ corr.)
9-5 ” ------- - Gap (wl corr.)
9.4 ”
, _ _ _ _ _ _________________ -..... ........... _ _ ______
9.2 r
A """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""
\‘2’ 90 1-
o n
ad 1
8.8 r
8.6 * Bottom Gap=.5 mil, Top Gap=.3 mil
1
8.4 ‘
8.2 ’
L
8,0 x l L l l
8 9 10 11 12

Frequency (GHz)

Figure 3.3 Experimental results of a 0.8 mil top/bottom gap for Re{8,} .

58

1.0

0.8

0.6

0.4

0.2

Im{e}

-0.2

-0.4

-0.6

-0.8

-1.0

 

ALUMINA COMPARISON
Ideal vs. 0.8 Mil Gap

 

— Ideal
--------- Gap (wol corr.)
....... . Gap (wl corr.)

 

 

 

”...OF'
_0-0-'~0--0-¢-0“-0-¢-0.0-...-.-.-§.O-~.O-|~O-'~l-..O-l-.-U-I-O-I.l-Ow-Ou-I~-.'-'

..___—__——————————-————
-0.0

Bottom Gap=.5 mil, Top Gap=.3 mil

 

9 1o 1 1 12
Frequency (GHz)

Figure 3.4 Experimental results of a 0.8 mil top/bottom gap for Im{8,}}.

59

Re{s}

10.0

9.8

9.6

9.4

9.2

9.0

8.8

8.6

8.4

8.2

8.0

ALUMINA COMPARISON
Ideal vs. 1.9 Mil Gap

 

 

 

 

 

 

 

 

_ — Ideal
--------- Gap (wol corr.)
f ------- - Gap (wl corr.)
— Bottom Gap=.5 mil, Top Gap=l .4 mil
9 1o 11 12
Frequency (GHz)

Figure 3.5 Experimental results of a 1.9 mil top/bottom gap for Re{8,} .

60

10.0

9.8

9.6

9.4

9.2

9.0

Re{e}

8.8

8.6

8.4

8.2

8.0

ALUMINA COMPARISON
Ideal vs. 2.9 Mil Gap

 

 

 

 

 

 

 

 

L — Ideal
--------- Gap (wol corr.)
— ------- - Gap (w/ corr.)
_ ................................ W— ... . .............................
1. .....................................................................................................
r Bottom Gap=.5 mil, Top Gap=2.4 mil
8 9 1o 11 12

Frequency (GHz)

Figure 3.6 Experimental results of a 2. 9 mil top/bottom gap for Re{8,} .

61

Re{a}

10.0

9.8

9.6

9.4

9.2

9.0

8.8

8.6

8.4

8.2

8.0

ALUMINA COMPARISON
Ideal vs. 35 Mil Gap

 

 

 

 

 

 

 

 

. — Ideal

I --------- Gap (wol corr.)

” -------- Gap (w/ corr.)

l

r Left Gap=17.5 mil, Right Gap=17.5 mil

1

l

8 9 1o 11 12

Frequency (GHz)

Figure 3.7 Experimental results of a 35 mil lefi/right gap for Re{3,} .

62

Chapter 4

ACCOMMODATION OF WALL LOSS IN RECTANGULAR WAVEGUIDE

MATERIAL CHARACTERIZATION MEASUREMENTS

4.1 Introduction

Imperfectly conducting walls can influence the accuracy of measured constitutive
parameters in rectangular waveguide material characterization measurements. Wall loss
effects become significant in the presence of materials having large permittivities or
permeabilities and in high-temperature applications in which the waveguide material has
relatively poor conductivity. The finite conductivity induces mode conversion and results
in a shift of the ideal TE“) mode propagation constant.

Wall loss effects can be accounted for using a coupled-mode perturbation theory
(specialized to single-mode operation) which is based upon an impedance boundary
condition at the imperfectly-conducting waveguide walls. The result is a complex
correction to the ideal TElo propagation constant. Experimental results will be given and
a comparison will be made with the standard power-loss method of attenuation

correction.

4.2 Review of Attenuation Correction for the Ideal TE") Propagation Constant
An expression for the conductor loss attenuation correction ac to the ideal TE")
mode propagation constant can be obtained by considering the rate of decrease of power

propagated along a finitely conducting line [17]. If the power at z = 0 is P0 , then at z it

63

is P, (z) = Poe-2a“; . Consequently, the conservation of energy requires

(112(2) _

——2.Pz(z) => —P’(——z—) ”‘2 0) :1)”

C 2P (2) 2P,(z= 0) 2PO

 

I1<z>=- (4.1)

where P10 is the power loss/unit length at z = 0 due to finite conductivity and P0 is the

total axial power transport at z = 0. That is, the power loss along the line is equal to the
rate of decrease of the power pr0pagated along the line.

If the conductor loss is small, the fields will not be substantially perturbed from
their loss-free values. Thus, an approximate expression for ac can be obtained by using

the known fields of the loss-free case. Finite (i.e., good) conductors exhibit a surface

Zc = RC + ch = (1+ j) [”0 (4.2)
20".

and the tangential fields satisfy the boundary condition

impedance (see appendix D)

 

—.

E =23; x-szg 212 . (4.3)

tang
Therefore, the power loss/unit length (i.e., the power transported normally into the

conducting surface/unit length) can be written

I

PIO=—_ C£Re{E tangx Htang ng J‘J‘ZR C{E tang fithang}dZdl
C0

2 (4.4)

~ R
1::=—29J|K|2dl=—2-61M dl

 

 

 

R ..
dl=-§£- 6” H tang

tang2

where C is the periphery around the waveguide, 11 is the unit outward normal pointing

from the conductor into the guiding region and E, F! are the unperturbed TE”)

ang 3 tang

tangential fields relative to the conducting surface at location 2 = 0. The total axial

64

111

C0

 

 

power transport at z = 0 is

ab ab
p,=%C£22{2,27.2 212 3:7001312 H|2Hy_RI,20MH|HxHy (4.5)

where 1:3,, H, are the unperturbed TE“) transverse electromagnetic fields at z = 0,

R10=Re{Z10}=Re{jwp/y,0} is the real part of the TB”) wave impedance with

 

710 = \[(It/ a)2 — k2 and E, = —Z102 x H, is the relationship between the transverse field
components.

The well-known unperturbed fields for the TElo mode at z = 0 are

 

 

 

          

 

E = 54A sin ?
fi=—xZ—Tgsin%-M%cos? (4.6)
The power loss/unit length at z = 0 for the TEM) mode can therefore be written
P,O = % j|ng d]: R? 1 [HM 2 dl (4.7)
C 232.0»
where
2 0 y.z=0 2 (02.14“ 2 2a

7:
a2

 

 

 

dy=c

— 4.9
x=a.z=0 2 (Uzflfl. ( )

b
2 d1: £ jH,H;
2
0

8..
/—\
D'-

N
\__/

R 2
—Zc- C]; IHtang

 

 

         

RC ~ 2 _Rc" _Rc AA" 0 2:2
76“ng all—7 63—sz 271071o+‘2— (410)
3

65

 

 

 

x,z=0

b

2 4

dl=£€szHz 2 ,
2 0 a) pp

R - R AA" 222
70 Hng dy=-?" [b—] (4.11)
C.

Therefore, the power loss/unit length is

 

AA" .. 7:2 2:2
P10 =Rc 2 .[“710710+‘2_+b—2'] (4-12)
w 1“.“ a a

The total axial power transport at z = 0 is

 

AAI ab ..
{(771071) (4-13)

Thus, the attenuation constant due to conductor loss will be
2 2
a at 71' 71'
R — + — + b ——
P10 c[2710710 2a a2]
a —

c _ 2P = ab .
0 1114371071 J

 

 

(4.14)

which is in agreement with Collin [17] when the material medium inside the guide is

lossless.
The overall perturbed propagation constant 7 can therefore be written as
7:710+ac (4.15)
Note that the power-loss method only results in a real correction to the ideal TElo mode
propagation constant. It will be shown in the following sections that a coupled-mode
perturbation theory can be utilized to obtain a complex correction to the ideal propagation

constant.

4.3 Accommodation of Wall Loss Using a Coupled-Mode Perturbation Theory

4.3.1 EM Field in Terms of the Transverse Electric Field

66

A guide with lossy walls will support hybrid modes, that is, the modes of
propagation are combinations of TE and TM modes. Therefore, both h, (the generating
function for the TE modes) and e, (the generating function for the TM modes) are

involved. This combination is required to generate all field components necessary for

satisfying boundary conditions. However, it can be shown that the transverse field E (or

Ii) alone is sufficient to generate all field components necessary for implementing
boundary conditions.

Consider the electric and magnetic fields of a hybrid mode with propagation

constant 7 , namely
1?: = awe—72 + 22,031?” (4.16)
H = i1°( [2)2‘72 + 2h,(;2)e‘" (4.17)
where 5,1: are transverse fields, ez,hz are longitudinal fields and ,5 =£x+ 52y. Since
V-E =(V, +26/62)-E =0 in a source-free simple medium, we obtain the following
longitudinal electric field component

8 =lV,-é (4-13)

where V, =x6/6x+ jzé/éy. Substitution of (4.16) and (4.17) into Faraday’s law,
equating transverse and longitudinal field components and using the vector identity
V, x2e, =—2xV,ez in conjunction with (4.18) leads to the following result for the

magnetic field components

 

zx(V,V, ~§+yzé°) (4.19)

67

h, =——1—2.v,xé (4.20)

jaw
Equations (4.18)-(4.20) show that all field components can indeed be generated by E . In

addition, the field E satisfies the Helmholtz equation V217? +k2E = 0, which separates

into the transverse and longitudinal wave equations given by
V}é+(yz+k2)é=0 (4.21)

V32, +(y2 +k’-)e, = 0 (4.22)

4.3.2 Impedance Boundary Conditions on E in Terms of Transverse Field 5

The coupled-mode perturbation theory requires boundary conditions on the

tangential components of E at the surface of the imperfectly-conducting waveguide
walls based on ('3’ alone. If ii is the unit normal pointing from the conductor into the
guiding region, f is the unit tangent in the transverse plane and 2 is the unit tangent

along the axial direction (where fxfi = 2 as shown in Figure 4.1), then the tangential

d

boundary conditions can be found by investigating the relations f-E,,,,,g and é-Emg.

Note, of course, that 61-20%, = f- “8’72 will have a transverse component only and
2 - Emg = eze'7z will have only an axial (i.e., longitudinal) component.

The first boundary condition on E can be obtained by taking the dot product of f

and equation (4.3), noting that 1‘ fixflmng = fxfi-ng = 2-H,a,,g =hze’73 and using
the result from (4.20), leading to
. ~ . . ~ . - Z, ,. a
r-E,a,,g=Zcr-an,a,,g :> r e=-—.——z-V,xe (4.23)
160/1

68

The second boundary condition on E is obtained by dotting 2 into (4.3). The left-hand

side of this relation, with the aid of(4. 18), is

 

 

 

21%,, = 2.2-73 = 1V, 22‘” (4.24)
7
The right-hand side of 2 dotted into (4.3) is
223 - fix ”mg = 202x 9-fi,,,,,g .—. ace-Hm, = def-52‘” (4.25)
Substitution of (4.19) into (4.25) gives
—Zcz°-l-z.e"’z = — _Zc 2-2x(V,V, -é+y2§)e_7z
a)
Z J ”7 Z (4.26)
= — 6 2x 2-(V,V, @4722)?” = , c fi-(V,V, -é°+yzé)e_7z
1501‘”

since 2x 2 = —fi. Using the identity V,V, -E = V, x V, xé’ +V,2§ and (4.21) allows (4.26)

to be written as

.26 fi-(V, xV, xE—k2§)e'7z (4.27)
10.117

A.A '° __ A... _72
Zcz an,,,,,g— Zcr he —

 

Thus, the second boundary condition on E is

2-13‘,,,,,,, -_— Z,‘-fix Hm :5 V, 2' =,—ZC—f1-(V,xV,x'é—kzé’) (4.28)

Note, when the walls are perfectly conducting, Z, = 0 (since 0', —+ co) and the boundary

conditions of (4.23) and (4.28) simplify to the expected well-known result

= 0 ...for a, —-> oo (4.29)

ml

2-E=O ..
g or nx
e

4.3.3 Coupled-Mode Perturbation Theory

The next step in the analysis is to expand 8' in terms of the normal mode

69

functions for the same guide with perfectly-conducting walls, that is
E = Zanén (4.30)
n=1

The sum is extended over all the TE and TM modes, an is an unknown amplitude
coefficient and 5,, is the transverse electric field for a given mode in the lossless (i.e.,
ideal) guide which satisfies the wave equation

V323,, + (rfi “(2)5, = 0 (4.31)
where 1“,, is the ideal-mode propagation constant. It will be assumed that the normal
mode functions 3,, have been normalized in the following manner

1 = '
ja,.a,,as:5,,,= " 1, (4.32)
CS 0 ...n¢j

Hence, the mode functions form an orthonormal set. Note, 2, satisfies a wave equation

similar to (4.31), that is
via, + (F3 + k2)é°j = 0 (4.33)
The unknown amplitude coefficients are given by

a, = j 252,218 (4.34)
CS

where E is the actual transverse electric field in the lossy guide and 5,, is the transverse
electric field of the n”I mode in the ideal lossless guide. This result is obtained by dotting
E, into equation (4.30), integrating over the guide cross-section and using (4.32).

The coefficients an and hybrid-mode propagation constant 7 can be determined

in the following manner. First, scalar multiply (4.21) by E,- and (4.33) by E , subtract the

70

two resulting equations and integrate over the guide cross section. This leads to

(y2 47;) jaéjds =(72 —r3.)a, = [(8935, —é‘, -V,2é')dS (4.35)
CS CS

The surface integral on the right-hand side of (4.35) can be converted into a contour

integral using the vector relation [3]
j (ii-V32 — 21° - V3§)ds
CS

- _. g - - _. - , (4.36)
=(l)[(fixA)-(V,xB)+fix(V,xA)-B+(fi-A)V,-B—(fi-B)V,-A]dl
C

where 13 is the unit normal pointing from the conductor boundary into the guiding region.

Letting Z = E,- and B = E in (4.36) allows (4.35) to be written as

(72 —rj.)a, =<fi[(9xa,)-(V, xé’)+fix(V, xé’j)-E+(fi-é’j)V, -é’-(fi-E)V, -é,]d1 (4.37)
C

The tangential electric field components for the lossless guide are zero on the boundary,

that is
‘ 7 - — 1 V 7 - 4
nxej—O , ezj_T‘_- ,-ej—0 (.38)
J
and equation (4.37) therefore simplifies to
(72 -rj.)a, =gS[ax(V, xéj).é+(fi-é,)v, -é‘]dl (4.39)

C

An expression for the first term of the contour integral in (4.39) can be found by
using the relation V, XE, : —2ja),uhzj (from Faraday’s law) and equation (4.23). This

produces, with the aid of ii x 2 = 2 , the result

fix(V, 2(5):; =—ijh,,£.é = zch,,.2.(v, x5) = -,Z—C(V, xé,)-(V, x25) (4.40)
14271

The final step involves the substitution of (4.40), (4.28) and (4.30) into (4.39), leading to

71

Z . - . - . -
(72-F§)aj =-j7:1<j[(V,xé‘j)-(V, xé’)—(n-e,-)(n-V,XV, xe—an-e)]dl (4.41)
C

 

 

 

Z °° _. _.
(72—F3)aj =— . C 2a,,(f)[(V,Xej)-(V,Xen)
10).” ":1 C (4.42)
—(fi-§j)(fi-V,><V,xé’n—kzfi-é’nfldl ...j=1,2,3,...
Equation (4.42) can be cast into the following matrix form
lMllal=101 (4.43)
where
M,"=ficflcfiKV,xéj)-(V,xén)—(fi-Ej)(fi-V,><V,xé’n—kzfi-é’nfldl ...j¢n (4.44)
C
2 2 Zc 4 4 2 ~ ~ 4 2. - -
Mji =7 —l‘j+jw‘u¢[(V,er-)-(V,Xej)—(n-ej)(n-V,XV,Xej—k n-ej)]dl (4.45)
C
a,

[a] = 02 (4.46)

Note, when the walls become perfectly conducting (i.e., Z C = 0), equation (4.42) implies

that 72 = F3 as expected.
The above result represents an infinite set of equations for the unknown
coefficients an. An examination of (4.43) shows that a solution for the an exists only if

the determinant vanishes. This leads to a characteristic equation that has an infinite
number of roots for 2,2 , which corresponds with the infinite number of hybrid modes that

may exist in the lossy guide. For each root of 72 , a set of coefficients an is obtained,

and it is these coefficients that determine the field configuration of the particular hybrid

72

mode.

4.3.4 Complex Correction to the Ideal TEm Mode Propagation Constant

In practice, the expansion in (4.30) must be truncated in order to obtain a finite
number of equations. Equation (4.42) reveals that all the coefficients are small except
those for which 72 z F2. , thus an approximate expression for 72 can be found by

including only a single mode in the expansion. Since we are interested in how the
propagation constant of the TB“) mode is perturbed, this suggests making the following

choice

é°j=é°n=él=jz,/a—2bsin% , 1"1=}’10=\l(7r/a)2—k2 (4.47)

where E, and I] are the ideal TE") transverse electric field and propagation constant,

respectively. For this single-mode case, equation (4.43) reduces to

Mna, =0 :> M“ =0 (4.48)
Substitution of (4.45) into (4.48) leads to the following approximate expression for the
hybrid-mode propagation constant

72 z )3, + 672 (4.49)

where the complex correction 6y2 to the ideal TEm propagation constant ylzo is given by

 

5y2=—:cfl¢[(V,x'e’l)-(V,xé’,)—(ii-é’l)(f1-V,XV,xé’,—k2f1-é',)]dl (4.50)
C

1'
If a rectangular guide having width a and height b is considered, then integration

along the bottom (contour C, where ii = j) , y = 0 and d1 = dx), right (contour C2 where

ri=—£, x=a and d1=afy), top (contour C3 where fi=—j‘2, y=b and dl=dx)and left

73

(contour C4 where 73 = 2 , x = 0 and d1 = dy) boundaries leads to the following result

1W:x51>-<V1><é‘1>—<fi-é1><fi-V.WM4242201209

 

C1
a k2
= IKV‘XEU'WIXél)_(5"§1)(5"vzXVzX§1—k25"51)lly=0dx=72_
0
[[(v,xal).(v,x2,)_(2.2,)(2.v,xv,x2,-k2,~,.2,)] dl
C2 x=a
b 2
-. - 2n
= I(Vt xel).(V, xel)lx=ady=—T
0 a
k2
1W1xéD-(V.xé1>—<fi-é.><fi-V,xv,x4428-601! J"?
C, y-
_. .. . _. A _. 2,. _. 271'2
[[(V,Xel)-(V,xel)—(n-el)(n-V,XV,xel—k n-el)]lx=0dl=—a—3—
C4

Substitution of (4.51)-(4.54) into (4.50) produces the final desired result

2 2
672: z, [47! +212)

 

 

 

_jwy a3 b
2 2 2
2 Z 47! 2k 2 71’ 2
7=J710‘ .c[ 3 + J , 710:7-1‘
160;: a b a

(4.51)

(4.52)

(4.53)

(4.54)

(4.55)

(4.56)

Note, the ideal propagation constant is recovered if the walls are lossless (i.e., Z, = 0 ).

The coupled-mode perturbation theory has several advantages over the power-loss

method. First, the coupled-mode perturbation theory resulted in both an attenuation and

phase constant correction to the ideal TE”) mode propagation constant whereas the

power-loss method produced only an attenuation correction. The additional phase

correction is important in material characterization measurements. Secondly, it is

74

observed that the attenuation correction term from the power-loss method approaches
infinity at the cut-off frequency since axial power transport ceases. The more rigorous
coupled-mode perturbation theory shows that infinite values of attenuation do not occur.
From a physical viewpoint, the presence of finite conductivity results in a coupling
between the TE and TM modes, and propagation along the guide does not stop at a
particular frequency. Hence, there is finite axial power transport, resulting in finite
attenuation. Lastly, the coupled-mode perturbation theory can easily accommodate
attenuation due to degenerate modes while the power-loss method cannot. If two or more
modes are present simultaneously, the power—loss method can be applied to each mode
individually, provided the axial power transport and power loss are the sum of those
contributed by each mode. That is, the power-loss method can be applied if the modes
are not strongly coupled. However, strong coupling does occur when the modes are
degenerate, thus the power-loss method breaks down. Mode degeneracy is inherently

accounted for in the coupled-mode perturbation theory.

4.4 Experimental Results
Figures 4.2 and 4.3 show a comparison between the power-loss and coupled-

mode perturbation theories for an Alumina material characterization measurement. The

Alumina sample (of thickness 8 = 0.1314 cm) was initially placed inside a silver-plated

rectangular waveguide holder having conductivity 0,. = 6.1x107 [S /m] and the

permittivity was subsequently computed. This measurement constitutes the ideal case in
Figures 4.2 and 4.3. Next, the sample was removed from the ideal silver-plated sample

holder and placed into a high-temperature rectangular waveguide having a conductivity

75

0', = 3x 105 [S / m]. The scattering parameters were then measured and the permittivity
computed using no wall loss correction, the power-loss method of attenuation correction
and the coupled-mode theory of attenuation and phase correction. Figures 4.2 and 4.3
reveal how critical the additional phase correction can be in material characterization
measurements.

The second set of measurements in Figures 4.4 and 4.5 demonstrates the
sensitivity of the material characterization process on the conductivity of the waveguide
holder. A high dielectric resistive-card sample (having a thickness of .004 in.) was

placed into the silver-plated rectangular waveguide holder and the real and imaginary

permittivity was computed using the conductivities of 0",. = 6.1e7 , 0', = 184 and

0', =1e3. Figures 4.4 and 4.5 show that errors on the order of 10% can occur if precise

knowledge of the waveguide conductivity is not known. Thus, wall loss correction can

be a critical factor in the material characterization process.

76

N)

C!)

Figure 4.1 Defined unit vectors for a uniform waveguide.

77

Wall Loss Theory Comparison

 

 

 

 

 

 

 

11.0 _ Alumina Measurement (Cc = 3e5 S/m)

10.8 _' ............................................................................................

10.6 l

10.4 ~ _______________________________________________________
9.. 10.2 ~
9:" — Ideal
: 10‘0 ,- """""" No Correction
M 93 - ------- - PL Thy. (att. corr.)

' » ----- CM Thy. (Att./Phs. Corr.)

9.6 ~

9.2 -

9.0 1 1 . .

8 9 10 11 12

Frequency (GHz)

Figure 4.2 Wall loss theory comparison for Re{8,} using Alumina.

78

Wall Loss Theory Comparison
Alumina Measurement (CC = 3e5 S/m)

 

 

 

 

 

 

 

3 ,.
l
2 -
1 _. .............................................................................
I} _ ................................................................................
r" 0 ‘~“— ..............................................
E T """""""
-1 ~ —— Ideal
--------- No Correction
_ ------- - PL Thy. (att. corr.)
'2 ----- CM Thy. (Att./Phs. Corr.)
-3 ‘ ' 1
8 9 10 11
Frequency (GHz)

Figure 4.3 Wall loss theory comparison for Im{£,} using Alumina.

79

360

220

200

Conductivity Profile

_ Resistive-card Material

 

 

 

 

 

 

l 1 1 l l

8 9 10 11 12

Frequency (GHz)

Figure 4.4 Conductivity profile of Re {3,} for a resistive-card material.

80

Im{8,}

Conductivity Profile

-350 Resistive-card Material

 

 

 

 

 

 

 

-370
-380
-390
_400 J l _l_ l
8 9 1o 11 12
Frequency (GHz)

Figure 4.5 Conductivity profile of Im{8,} for a resistive-card material.

81

Chapter 5

GREEN’S FUNCTION FOR EM FIELD WITHIN AN IMPERFECTLY-

CONDUCTING PARALLEL-PLATE ENVIRONMENT

5.1 Introduction

In this chapter, the Hertzian-potential and electric and magnetic-field dyadic
Green’s functions [30] are found for a general 3D current source immersed within an
imperfectly-conducting parallel-plate environment. This analysis will be utilized in
Chapter 6 to understand the effect of wall loss for a stripline field applicator. The above
Green’s firnctions will be determined using two methods. The first method consists of

analyzing a symmetric slab waveguide and investigating the limiting case where the outer
regions (having conductivity 0', and permittivity 5,.) become good conductors
[ac MOE, >>1 :> a, = 5,.(1 — joc lwéc) —9 —j0'c / a) , where 6,. is the effective complex
permittivity]. The second method utilizes Hertzian-potential impedance boundary

conditions.

5.2 Geometrical Configuration
Figure 5.1 shows the geometry of the parallel-plate environment. Regions 1 and 3

comprise the outer portion of the structure and region 2 the inner portion. A general 3D

electric current source .7 is immersed only within region 2. The structure (but not the

current) is invariant along the x and z axis. The total height of the inner region is 2h

and is centered about y =0. The effective complex permittivities of the outer (i.e.,

82

cover) regions are assumed to be identical, that is, 8] = 83 = a, = 5,.(1 — ja, No.2,) , and

the effective complex permittivity of the inner region is 62 = 6‘ = 2(1- jo / 602') . All three

regions are assumed to be non-magnetic (,u, = ,uz = ,u3 = ,uo ).

 

y
Reflected -
g wave Regnonl (e814)
y=
§Principal
wave
Reflected
g; wave Region2 (82110) ——>X
V
§Principal
wave

 

=—h

é Reflected
wave

Region 3 (8c9l‘l0)

Figure 5.1 Parallel-plate environment with general 3D current 7(7).

5.3 EM Fields and Helmholtz Equation for Hertzian Potential

The electric and magnetic-field dyadic Green’s functions Ge (FlF') and (7"(FIF')

can be identified by using a Hertzian potential 7? as an intermediate step (see Appendix

A for further details). One reason for using a Hertzian potential is that Ge and (7” are

more readily determined by first solving the Hertzian potential wave equation

—0

V22+k2” =——j]— (5.1)
1608

and then computing E and if using

83

E = 1222? + V(V-fi) (5.2)
I? = jweriE (5.3)

The alternative is to identify (7" and Ch by directly solving the wave equations for E

and 17, that is
V217: + kZE -_- jam] — —,—1—V(V-.7) (5.4)
1608
v21? +1221? = —v x .7 (5.5)

The Hertzian potential approach is easier to implement since a simpler relationship exists

between if and J in (5.1) than 2,1? and J in (5.4) and (5.5).
Another motivation for using a Hertzian potential is that 7? is less singular than

E or 17. It will be shown that this leads to, in a mathematically straight-forward

manner, expressions for G" and Ch that are valid both inside and outside the source

region. Thus, the electric and magnetic-field dyadic Green’s functions will be found by

first determining if using (5.1) and then computing E and H from equations (5.2) and

(5.3).

5.4 Spectral Representation of Principal and Scattered Waves

5.4.1 General Formulation

As mentioned in section 5.3, the solution to equation (5.1) needs to be

determined. This solution can be found using the superposition of a principal wave if”

emanating from the source .7 in unbounded space (having effective complex permittivity

84

a) and waves 7'1" that are reflected from the introduction of boundaries at y = ih with
the source removed (see Figure 5.1). This method is often used in the solution of
differential equations, where the principal wave is called the particular solution and the
reflected wave is referred to as the homogeneous (or complementary) solution. The total

solution for if in each respective region will therefore be

7;] = 51’ (5.6)
if, = 72‘; +55 (5.7)
7‘53 = 7?; (5.8)

where ff and 7736 (,B = 1, 2,3) satisfy the respective Hertzian potential wave equations

 

Wafer/(2272', 312:8 (5.9)
2
V2 —-r 2-or _
flfl'l’kflfl'fl —O (510)

where k,2 = k32 = k3 = wzacyo , k22 = k2 = wzapo and £2 = a. The principal wave

contributes only in the inner region since it has been assumed that .7 is strictly confined

within region 2. Although the structure is symmetric about y = O , the current is generally

not, thus no mathematical simplification can be used in the analysis that follows.
Equations (5.9) and (5.10) can be decomposed into 3 scalar equations each, thus

solutions must be found for the following equations

 

.. - J (F)
V27!” r +k27rp r =— a 5.11
2a( ) Za( ) 1.08 ( )
szgam + kgngam = o (5.12)

85

where a = x, y,z and ,6 = 1,2,3. The Fourier transform domain method will be
employed here to solve these equations. The background structure is invariant along the

x and 2 directions, which prompts transformation on those variables using the generic

2D Fourier transform pair

 

i(i,y)= j j f(F)e‘fI°7dxdz (5.13)
.. l 00 00 ~ -° 3L;
f(r)=(2fl)2 iiflmw c122 (5.14)

where Z =£§+2§ (2 x12 :10}: =52 +52) , F =JEx+yy+éz and d2). =dé‘d5. Trans-
formation with respect to y is avoided at this point so that boundary conditions can be

enforced at y = ih. Applying the Fourier transform differentiation theorem, equations

(5.11) and (5.12) become

 

 

6222541,» 2 p ~ J <1 y)

_ 7; ,1, =_ a ’ 5.15
6y, pz 2.,( y) jam ( )
azfipad,» 2- -

6y, —pprr,3a(/I,y)=0 (5-16>
where p1 =p3 =pc =(llz—kcz , p2 =p=\//12—k2 with Re{pfl}>0 chosen and
afadm): ] Infa(F)e‘ji'Fdxdz (5.17)
7%(21', y): j Inza(F)e'jZ°Fdxdz (5.18)
Judy): j [Ja(r)e‘f1"dxdz (5.19)

86

The general form of the solution to equation (5.15) in the central region (i.e., region 2)

and equation (5.16) in regions 1, 2 and 3 will be investigated in sections 5.4.2 and 5.4.3.

5.4.2 Principal Wave Representation

The principle wave satisfies equation (5.15) for all unbounded space (i.e., in the
absence of boundaries), as dictated by the superposition method of solution. This
prompts transformation on the y -variable in equation (5.15), leading to (with the aid of

the differentiation theorem)

2 ~ 2 - i 21‘,
-n2n§.(2,n)—p2n§a(2,n)=——“.(—”) (5.20)
1606‘
where
§§a<Ln>= ji§a(1,y)e‘f’"dy (5.21)
Ja(i,n)= Ija(1,y)e‘j”ydy (5.22)

Solving for 755a in equation (5.20) produces the result

ja(1,77)/jw£ __ ja(1,77)/jw£

”PM, )= — . .
”2 ’7 (272+p2) (n+1p)(77-1p)

 

(5.23)

Since p = p(§,§) is not a function of 77 and Re{ p} > 0, this implies that 77 = ijp are
simple poles in the upper and lower-half of the complex 1] -plane, respectively.

The inverse transform must now be taken to get back to the complex 2. -plane so
that the y -variable is present for implementing boundary conditions. The inverse

transform is calculated using the formula

87

- ~ 1 °° : - - 1 °° j(1,n)/ja)e -
up (z,y)=— 2:” (2,n)e”’ydn=— 0’ . . emdn (5.24)
2" 2” l, 2" 2” i(77+JP)(77—JP)

 

The Fourier representation of jg is
fad, n) = «l J. (1’, y')e'j’""dy' = Hawk—”yer (5.25)
—00 y'
The limits of integration in (5.25) have been truncated since jab-11y) = 0 outside the
source region. Also, y' has been used in (5.25) as the dummy integration variable
instead of y so as not to confuse it with the functional y-dependence of the em“ term in

equation (5.24). Substitution of (5.25) into (5.24) and interchanging the integral signs

(valid if the analysis accurately describes a real physical problem [31]) leads to

.. ~ ~ .. ,J 21' ,
”§a(/1,y)=le”(l;yly)“—(.wgi)dy (5.26)
y. 1
where
eery-y')

 

(5.27)

. . dn
('7 + Jp)(n - 11))

~ ” I ~ _° I 1 a)
G§(l;yly)=Gf(l;y-y)=ZZ-I

is the spectral-domain principal wave Hertzian-potential Green’s function. Equation

(5.27) may be evaluated using Cauchy’s Integral Theorem. The result is (see Appendix B

 

 

 

for details)
- _. - - e-Ply-y'l
0501» y') = qu-y') = 2p (5.28)
The representation of the principle wave in the complex ll. -plane is therefore
-p|y-y'| " ” .
- e e J A, ,
may) = I ..g y ) dy (5.29)

. (08
y 1

88

where y is the field point, y' is the source point and p is the spectral-domain

propagation factor.

5.4.3 Reflected Wave Representation
The spectral-domain representation of the reflected waves for each region is
obtained from (5 .16). The well-known solution of this second-order partial differential

equation is
afiad, y) = Wga()1°)e"’fiy +Wfl‘a(}1’)e”/’y (5.30)

where Wflia are the complex amplitude coefficients of the up and down-going reflected

waves, respectively.

5.4.4 Total Wave Representation
The spectral-domain representation of the total waves in each region is obtained

by superposing the results from the previous sections. This leads to the following

 

{E10, zfifa = WIZe-pcy +W1;ep‘y ...h <y <oo (5.31)

" —"P ‘r — e—plfll j“ ' W” ‘Py W‘ W h h 532

”2a —7[2a+7[2a —J _ dy + 208 + zae ...—' <y< ( . )
y, 2p 1608

723a :figa = Wire-pd +W3_aep‘y ...—CD <y ('47 (5.33)

where the functional dependence in the above equations has been dropped for notational

convenience. Note that

— '... > '
ly-y'|={y,y y y, (5.34)
y—y...y<y

89

thus, equation (5.32) may be written (for the purpose of implementing boundary

conditions) as
72:20, = VJe-py +W223—py +W2—aepy ...y'<y<h (5.35)

£20, = Vgepy +W2+ae-py +W2}epy ...—h < y < y' (5.36)

where

 

dy' (5.37)

are associated with up and down-going waves launched from the source.

It is observed in equations (5.31)-(5.33) that there are 18 unknown spectral-
domain coefficients, thus 18 boundary conditions must be enforced to guarantee a unique
solution. These spectral coefficients, Wflia , will be determined in the following sections

using the two methods discussed in section 5.1.

5.5 Computation of Spectral Coefficients from Limit of Highly-Conducting Case
5.5.1 Introduction
One approach that will be used to determine the spectral coefficients Wig for the

imperfectly-conducting parallel-plate environment of Figure 5.1 is to first consider the

solution to a symmetric dielectric slab waveguide and then look at the limit where the
outer dielectrics become good conductors. If Q = EC (1 — jac / cage) is the effective
complex permittivity of the outer (i.e., cover) regions, then the appropriate specialization
is that 6“. -—> —jac / a) if ac / (05¢ >>1 (i.e., if the conduction current dominates over the

diSplacement current in regions 1 and 3).

9O

The solution to the above symmetric slab waveguide can be found by enforcing

the following spectral-domain radiation and boundary conditions (see Appendix C)

 

 

madly —. oo) = 723a(}1',y —> —oo) = o ...a = x,y,z (5.38)
6.518(111) = £fi2a(i,h) ...a = x, z (5.39)
57820, (21' ,—h) = 3,723,, (21' ,—h) ...a = x, z (5.40)
a, ———6”'a(’1’h)=e————a”20(’l"h) ...a =x,z (5.41)
5y 5y
a 07r2a(/1,—h) = ac 5713a(4.-h) ...a = x, z (5.42)
6y 6y
sci.y(2°,h) = 3fi2y(1,h) (5.43)
5782,0134) = gcfi3y(Z,-h) (5,44)

aa,y(2',h) _ 6fi2y(i,h)
6y 6y

 

=[1-(8/5c)][j§772x(15h)+j§fizz(iahl] (5.45)

8,82,01,42) afi3y(Z,—h)
6y 6y

Equation (5.38) represents the radiation conditions. Equations (5.39)-(5.42) are

 

=[1—(a./e)][7443.<1,—h)+maxi-11)] (5.46)

taIlgential boundary conditions, (5.43)-(5.44) are normal boundary conditions and (5.45)-
(5.46) are mixed/coupled boundary conditions. Since (5.45) and (5.46) are coupled
b(Dundary conditions (i.e., contain both tangential and normal components), this suggests
enforcing boundary conditions in a specified order. The easiest and first boundary
Condition that will be implemented is (5.38). Equations (5.39)-(5.42), the tangential
boundary conditions, are enforced next, followed by the normal boundary conditions

(5.43) and (5.44). The coupled boundary conditions (5.45) and (5.46), which are the

91

most difficult to implement, are enforced in the final step. Note, there are 18 boundary

conditions as required.

5.5.2 Radiation Conditions
The radiation conditions in (5.38) are the easiest to implement and lead to the

following result when applied to (5.31) and (5.33)
771a(;1',y—>oo)=o :> m; =0 ...a=x,y,z (5.47)
723a()1°,y—>—oo)=o 2 W3; =0 ...a=x,y,z (5.48)

since Re{pc} > 0 (a result of the medium being assumed to be lossy, i.e., Im{kc} < 0).

Therefore, the spectral representation of the Hertz potential consists of only an up-going
reflected wave in region 1 and only a down-going reflected wave in region 3 (as shown in

Figure 5.1), that is
filo, = fifa = W126” ...h < y < 00 (a = x, y, z) (5.49)

£30 = 735a = Wilepcy ...—00 < y < —h (a = x, y,z) (5.50)

5-5.3 Tangential Boundary Conditions
The first tangential boundary condition (at the y = )1 interface) is enforced by

SIlbstituting (5.49) and (5.35) into (5.39). This leads to the following expression

ecu/IZe-p‘h = £(VJe-ph + Wile—W + Wfaeph) ..a = x,z (5.51)
Solving (5.51) for W1; leads to

W1; = flepc” (Vge‘l’h + Wzge'P” + Wz‘aep”) ...a = x, z (5.52)
8C

92

Therefore, upon substitution of (5.52) into (5.49), the spectral-domain representation for

ifla becomes

.. 8 .. .. _ _ _
7:10, =—e PAY “(Vote P”+W2‘;,e Ph+WZaePh) ...a=x,z (5.53)

8c

The second tangential boundary condition (at the y = —h interface) is implemented

through the substitution of (5.36) and (5.50) into equation (5.40). The result is

W3; = iep‘h (Va-e'ph + WZZeph + Wide—pk) ...a = x, z (5.54)

5c

.. _ a _ _ _ _
7:30, =W3aep‘y =—ep“(y+h)(Vae ph+W2+aeph+W2ae pk) ...a=x,z (5.55)

5c

Substitution of (5.53) and (5.35) into the third tangential boundary condition

(5.41) leads to

pc (Vote—pk + WZ‘Lae—ph + Wfaeph ) = p(VJe'ph + Wire—”h -W2—aeph) ...a = x,z(5.56)

Solving (5.56) in terms of W2; gives

W2}, = -R e‘ZP” (V; + W23) ...a = x,z (5.57)
where
R = ——-pc "1’ (5.58)
I). + p

is the interfacial reflection coefficient. The last tangential boundary condition is

implemented by substituting (5.36) and (5.55) into (5.42), producing the result

p(V;e’P" — Wzgep” + W221i”): p, (Va‘e'Ph + Wgaep” + Wz'ae‘l’h) ...a = x,z (5.59)

Solving (5.59) in terms of W2; gives

93

eth
W2; = —'Va_ —-—R——’W2:r ...a = x,z (5.60)

The spectral coefficient W2; can now be explicitly determined (for a = x,z) by

equating (5.57) and (5.60). The result is

W+ = 19212—4”th —Re_2phV(;
2‘1 2 -4ph
I—R e

 

x,z (5.61)

The spectral coefficient W2; is found by substituting (5.61) into either (5.57) or (5.60),

leading to

2 -4ph - -2ph +
R e Va —Re Va

Wza = 1 _ RZe—4ph

 

...a =x,z (5.62)

Those familiar with guided wave theory will note that the denominator of equations
(5.61) and (5.62) can be factored into the form l-R2e_4ph =(1+R e-th)(1-Re’2”h).

This is significant because the eigenvalue equation 1: R e_2ph = 0 identifies the expected
poles of the even/odd TE surface-wave modes of a symmetric slab waveguide. A more

detailed discussion will be provided later.

5.5.4 Normal Boundary Conditions
The first normal-component boundary condition is implemented by substituting

(5.49) and (5.35) into relation (5.43), leading to

WI; zaiepc" (Vy+e_ph +W23e-P" +W2',ep”) (5.63)
C
~ 8 — _ —h — h — h — h
my zg—e 1"” ’(V;e P +W2+ye P +W2yep ) (5.64)

C

94

Similarly, substitution of (5.36) and (5.50) into the second normal boundary condition

(5.44) produces the result

— 8 h — — h h — — h
W3yzg—epc (Vye P +W2+yeP +W2ye P ) (5.65)
C
.. _ c a c h — - h h - - h
n3y =W3yepy =:ep 9+ )(Vye P +W2+yep +W2ye P ) (5.66)
C

5.5.5 Mixed/Coupled Boundary Conditions

The final two remaining spectral coefficients, W24"), and W2"), , can be determined

from the coupled boundary conditions. Inserting (5.64) and (5.3 5) into (5.45) gives

5 + —ph + -ph — ph
—pC£—(Ve +W2ye +W2ye )+

 

y
c - (5.67)
— h — h - h a
p(Vy+e p +W2+ye ‘0 —W2yep )=A(1—8—]
C
where
A=j§Ax+j§Az= 568
15(Vx+e—ph + Wire-pk + foeph )+ jC(Vz+e’ph + sze’p" + Wz—eeph) ( ' )
Solving (5.67) for W;y and letting N 2 = 5/ 8C leads to
_ __ —ph 2 _
W2; = —R e-ZPPVy+ — R e"2phW2+y + e (1:, 1) A (5.69)
ch + p
where
- p N 2 -p
R = -—c—-i—— (5.70)
ch + p

95

is the interfacial reflection coefficient. The expression in (5.70) occurs in the analysis of

a current source in the presence of a half-plane where pCNZ + p = 0 identifies Zenneck
surface-wave poles. It will be shown later that Zenneck surface waves do not exist for
the parallel-plate environment.

Substitution of (5.36) and (5.66) into the second coupled boundary condition

(5.46) gives
_ _ h h _ _ h
p(Vye P “Wit-yep +W2ye P )—
(5.71)
_8_ “"Ph +ph —-ph_ f._
pc EC (Vye +W2ye +W2ye )—B[£c I]
where
The solution of (5.71) in terms of W2} is
219" pk _ 2
W,—, = -V; _£.__W,+y .e__<12_N_>_B (5,3)
R ch —p

The spectral coefficient W5} is explicitly determined by equating (5.69) and

(5.73). The result is

 

 

_ ‘3Ph 2 -ph 2
R2e_4phV;_ R e_2phV-+ Re (l— N )A+Re (l- N )B

2 2
+_ pN+p pN -p
W2, _ 1_ch—4ph (5.74)

 

Substitution of (5.74) into either (5.69) or (5.73) leads to the following expression for the

spectral coefficient W2}

96

R2 -4phV- _R- -zpth._ e P"(1- N2)A_ R e‘3P”(1-N2)B

2 2
— p N H? p N -p
W2y=1_R§e_4pR (5.75)

 

The spectral coefficient W2“; can be cast into a more physically meaningfiil form (as will

be shown later) via substitution of (5.61), (5 .62), (5.68) and (5.72) into (5.74). The result

is, after some algebraic effort,

 

 

 

 

 

 

 

 

 

 

W2+y =W2+yx+W2+W+W2+yz (5.76)
where
R e‘3P"(1—N2) ng Re‘Pha— N2) 153
—4 h — -2 h -
W. = ch2+p pczN -p :93??? ”VI-(3+ e pVx (577)
Zyx 1_Rze-4ph (1_R2e—4ph )(1_R2e-4ph) .
Rze‘4PhV+-Re‘2P"V‘
+ _ y y
W2”, _ 1—R2e-4P" (5.78)
Re 3P’—”(1 N2)ng +‘Re P"(1- N2)j§B~
—4 h — —2 h -
W.» = pN2+p pNz-p =C;z+e sz+-C+ 9 ”V2 (579)
2yz 1_§28—4ph (1_R2e~4ph)(l_RZe-4ph) .
CH: jRR(1—N2)(1-R)(1—R/R)

.0 N2 +p

(5.80)
C._ _ jé‘m— N2)(1— R)(Re‘4P”— —1/R)
ch2+p
+._ . R(l-N2)(l-R)(l-R/R)
Cy: "14 2

p N HR

(5.81)

R(1- N2)(1— R)(Re"4P”—1/R)

c;;= j;
ch2+19

 

Similarly, substitution of (5.61), (5.62), (5.68) and (5.72) into (5.75) leads to the

following expression for W2}

97

Wz—y = Wz-yx + W230, + WZ—yz (5.82)

 

 

 

 

 

 

 

 

 

where
—ph 2 - _2 —3ph 2 -
e (1-N Met/1.1.1? e (l-N )JéBx h M
—- -4 — -+ — +
W2. =_ ch2+p ch2_p =nye pVx —nye pVx (5 83)
yX 1__ R23-4ph (1 _ RZe-4ph)(1_ ‘R‘Ze—4ph)
RZe-4pth— _ R e-thVy+
W2” = 1_ R264” - . (5.84)
e‘P"(1-N2)jCAz + Rze‘3P”(1—N2)7482 4 h 2 h
-— — - —+ — +
W2- =__ ch2+P chZ—p =C)’ze P VZ —C)’Ze P V2 (585)
y. 1_ RPe-PPP (1 —Rze“’P")(l —R2e"‘P">
c; = —C;; , c; = —C;; (5.86)
(3;; = -C;;+ , C; = —C;; (5.87)

5.5.6 Limiting Case 60 —) —jac /w
The limiting case of when the outer dielectrics become good conductors is

discussed in this section. Only the spectral coefficients W21; will be examined since we

are specifically interested in the Green’s function for the central region. If the outer

regions are good conductors, then the following relation prevails

8. = "c -113. 0% (5.88)

since ac >> wéc (i.e., the conduction current dominates over the displacement current).

The tangential coefficients W21; ((1 = x,z) for the symmetric slab waveguide, repeated

here for convenience, are

98

 

W23, _ 1 R2;_4PR P ...a=x,z (5.89)
where
R=__§c:£ (5.90)
C

If (5.88) is inserted into (5.90), then the interfacial reflection coefficient R reduces to the

following expression

O'HZ —p =__—c1p/0'Zc

 

R: (5.91)
a". Zc +p -cl+p/0' ZR
since (see Appendix D)
. 2g 2
llmac pc- — limor —kc limor (DJ-k2: ma mJ—a) ecyo
aRa-j; £—->— jd; £-+—j — £—>—j—
(5.92)
limo pcz ijacflo = ”:1?"
a a-jifi

where it has been assumed that k3 >> 12 (when ,1 becomes large in the inversion

integral the contribution to if approaches zero due to the Riemann-Lebesgue lemma
[26]). Therefore, the tangential spectral coefficients in region 2 are given by (5.89)
where the interfacial reflection coefficient R is now given by (5.91).

The normal spectral coefficients in region 2 for the symmetric slab waveguide are

given by equations (5.76)-(5.87). If so —+ —j0'c la) then, using (5.92) and N 2 =5/ac ,
the following relations are established

£_jcoa

 

—> 0 (5.93)

99

lim ch2 = 0,2,, 1‘0" = ngZc (5.94)

.0'.
R ,_j . ac
(l)

 

Rm R: lim chz-pzwaZc-p=_l-jwsZc/p
gc_,_j-Zc_ 6.6-jif- pRN2 +p jwsZc +p l+jw£Zc /p
(l) (U

 

 

(5.95)

since ac >> (08 is assumed in (5.93). Note that Zc cc 1/ (lac whereas N 2 cc 1/ ac , that is,

terms on the order of 1/ ac are kept, while terms on the order of 1/ ac are neglected.
Substitution of (5.92)-(5.95) into (5.80), (5.81), (5.86) and (5.87) leads to

__ . R1—R 1—R/R
C;:=—ny =J§ ( )( )

 

 

 

'weZ +
_ J :4 hp _ (5.96)
..- .5. . R(1—R)(Re P —1/R)
ny =_ny =16 .
ngZc+p
C; =-C;z_=jCR(ITRX1—R/R)
1w£Zc+p
(5.97)

R(1— R)(R e‘4P" -1/ R)
jwaZc + p

 

c; = —C;;‘ = j;

where R and R are given by (5.91) and (5.95), respectively. The second method for
determining the spectral coefficients is presented next and will be compared to the above

limiting case.

5.6 Computation of Spectral Coefficients Using Impedance Boundary Conditions
5.6.1 Introduction

The second method for computing the spectral coefficients is based on the
spectral-domain electric-type Hertzian-potential impedance boundary conditions (see

Appendix D for details)

100

1 67220, (1,12)

 

 

 

7? 1,12 =— a=x,z 5.98
20( ) 0'ch a); ( )
£2,101.41): 1 a”2“(’1”h)...a=x,z (5.99)
UCZC 6y

air'zydm) , ~ _. , ~ - . ~ -
___—6y = —]waZczr2y(l.h) — Jéflzxafi) —JC7?22(/1’h) (5100)

6,22,07,42) . ~ _. .. .. .. -
ay = Jwé‘ZcfrzyMr/v) - 157:2.(2. -h) - 147mm. 41) (5-101)

Equations (5.98) and (5.99) are tangential boundary conditions, equations (5.100) and

(5.101) are coupled boundary conditions and 26 =(1+ j)‘/wp0/20'c is the intrinsic

impedance of the imperfect conductors. The radiation conditions are built into the
impedance boundary conditions and therefore do not appear here as they did in section
5.5.1. In addition, computation of the spectral coefficients in the conducting regions is

not required since the tangential coefficients are essentially accounted for through the

intrinsic impedance Zc and the normal coefficients are zero (refer to Appendix D). Thus,

there are only 6 spectral coefficients, W21; for a = x, y,z , that must be computed in

comparison with the 18 required for the more complicated method in section 5.5.

5.6.2 Geometry
The geometry that depicts the Hertzian-potential impedance boundary condition

approach is shown in Figure 5.2. The top and bottom conductors have a conductivity of

0". and permeability yo , or equivalently, an intrinsic impedance of Zc. Region 2 is the

central/material region and has an effective complex permittivity 8 and permeability #0.

101

“h

The reflected waves in the conductors are not shown since they do not need to be

computed, as previously mentioned. The unit vector ti points out of the conductor and

into the material region.

 

Zc
y:
Principal l
wave ii: _9
Reflected
wave 8,110 ___)x
v _ . A A
Pnncrpal n = Y
wave
=.. h ,

 

Figure 5.2 Parallel-plate environment for impedance boundary condition analysis.

5.6.3 Tangential Impedance Boundary Conditions

The first tangential impedance boundary condition (at the y = h interface) is

enforced by substituting (5.35) into (5.98). This leads to the following expression

Vge'P” + Wgae‘P” + Wz'aeP” = L(V;e‘l’" + Wzge‘l’" —W2';,eP") ...a = x,z (5.102)
C C

Solving (5.102) in terms of W2; leads to

W2; = —Re'2p" (V; + W23) ...a = x,z (5.103)

where the interfacial reflection coefficient R is given by

102

R=ach—'p _l—p/O-CZC

 

— (5.104)
0'ch +p l+p/0'ch

Substitution of (5.3 6) into the second impedance boundary condition (5.99) gives

Va'e'ph + Wgaep" + Mae—P” = L(Va‘e‘Ph — W2+aeph + Wide—pk) ...a = x, 2 (5.105)
C C

Solving (5.105) in terms of WZ-a leads to

eth
W2_Clz_VC.Z-_TW2+0 ...a=x,z (5.106)

The spectral coefficient W2”; can now be explicitly determined (for a = x,z) by

equating (5.103) and (5.106). The result is

W, = R264”th —Re'2P"V;
20 2 -4 h
I—R e p

 

...a=x,z (5.107)

The spectral coefficient W2; is found by substituting (5.107) into either (5.103) or

(5.106), leading to

2 —4ph -_ —2ph +
W27, = R e 1 V2, 54:}? Va ...a=x,z (5.108)
— e

 

Note that (5.107), (5.108) and (5.104) are in agreement with (5.89) and (5.91).

5.6.4 Coupled Impedance Boundary Conditions
The first coupled impedance boundary condition is implemented by inserting

(5.35) into (5.100). This results in the following relation
p(V+e‘P" + Wgye'P” — W2_yeph ) = ngzc (V;e_ph + Wye—P” + W2;eP" ) + A (5.109)

y

where A is given by (5.68). Solving (5.109) for Wz’y leads to

103

W;, = —R‘ e‘ZPhV; — R (2"th — —.———-— (5.110)

where the interfacial reflection coefficient 17 is given by

jcanc —p __l—jwa‘Zc/p
jwsZc +p 1+jm£Zc /p

R:

 

(5.111)

Substitution of (5.36) into the second coupled impedance boundary condition (5.101)

produces the result

p(V;e‘P" — szvep” + Wye—P" ) = ngzc (we—P" + nyeph + Wz'ye_ph )— B (5.1 12)

where B is given by (5.72). Solving (5.112) for W27), leads to

th ph
W2;=—V;—3—_— 2+y+—fl— (5.113)
R jwaZc - p

The spectral coefficient W243, is explicitly determined by equating (5.110) and
(5.113). The result is

Re'3PhA + Re‘PhB
jwaZc + p jcanc — p
1" RZe-4ph

 

Eze’4phV; _ Ee'zpth' +

+
W2}, =

 

(5.114)

Substitution of (5.114) into either (5.110) or (5.113) leads to the following expression for

the spectral coefficient W2”),

e—phA RZe—3phB
jcanc + p _ jcerc —p
l- [Fe-4”

 

§2e_4pth— —§e'2phV; —
Wz—y =

 

(5.115)

The spectral coefficient W2“; can be cast into a more physically meaningful form via

substitution of (5.107), (5.108), (5.68) and (5.72) into (5.114). The result is, after some

algebraic effort,

104

W2+y = W2+yx '1' W230, + W2+yz (5.1 16)

where

X

jwaZc + p jwaZc — p = C;x+e-4pth+ ‘C;;e—2pth_

 

R e‘3phng, + R e’Phng

 

 

 

 

 

 

 

 

W+x = _ _ 5.117
2y 1_R2e-4ph (1_R2e-4ph)(1_R2e-4ph) ( )
R2e_4phV+ — §e_2phV-
+ _ y y
Wzyy — 1—R28—4ph (5.118)
Re'3phjCAz + R-e’phngz 4 h 2 h
W+ _ fweZc+p jwaZc-p _C;,+e‘P Vz+‘C;z—e_p V; (5119)
Zyz 1_R2e-4ph (1_R2e-4ph)(1_R2e-4ph) .
. R’ l—R 1—R/R' _ , 111—R Re‘4Ph—1/R
q; =1: ( . X ) , C; =14 ( .X ) (5.120)
160526 + p 160.94 + p
_ _ _ _ __ _ -4ph_ _
C; =1,; R(l R)(l R/R) , C; =1,; R(l R)(Re UK) (5121)

 

 

ngZC + p jcanc + p
Note that equations (5.120)-(5.121) are in agreement with the limiting case (5.96)-(5.97).

Similarly, substitution of (5.107), (5.108), (5.68) and (5.72) into (5.115) leads to the

following expression for W23,

Wz-y=W2_yx+W2_yy+W2—yz (5.122)
where

e-P” 15A): +RZe-3ph1'53x __ _4 h _ -+ —2 h +
_jwaZc-t-p jwe'Zc—p _nye P Vx -nye P V):

1_ RZe-4ph — (1_ Rze-4ph)(1_ RZe-4ph)

 

 

 

W2_yx = (5.123)

RZe’4phV; - If e'ZPhV;
Wm = 1_ R284” (5.124)

 

105

e'phjé’Az + Eze'3phjCB

Z

_ jCUEZc + p ngZc _ p _ Ck—e-‘ipth- -C;z+e_2phVZ+

 

Wz—yz =

 

1_ EZe-‘(ph _ (1_ Rze-4ph )(1 _ RZe—4ph)

~— _ ++ —+ _ +—
ny “_ny 9 ny __ny

c; = —C;; , c; = —C

+ _
yZ

(5.125)

(5.126)

(5.127)

Equations (5.126)-(5.127) are also in agreement with the limiting case (5.96)-(5.97).

Now that the spectral coefficients have been found (using the method in section 5 .5 or

5.6), the Hertzian-potential dyadic Green’s function can be identified.

5.7 Hertzian-Potential Dyadic Green’s Function

The Hertzian-potential dyadic Green’s function G(F|i" ) can be identified by

initially examining the tangential and normal components £2,472, and iizy.

Substitution of (5.28), (5.37), (5.61) and (5.62) into (5.32) leads to the following result

for the tangential components

~ " ~ _' r "r _° 1 j i, I r
”2a(l,y)= I[Gp(/1;y-y )+Gaa(/1;y,y )]—“§.—w-g—)dy ma=x,z
y!

where the spectral-domain principal and reflected Green’s functions are

G;a(1;y,y') =

-pIy-y' |

Gp(1;y-y') = 55(1;y-y') =
2p

 

Kile—WI —Re"’¢2 +R2e'p‘i3 —Re'p¢‘
2 p(1 - 8254”")

 

...a =x,z

¢1=¢1(y,y')=4h+y-y'

¢2=¢2(y.y')=2h+y+y'

106

(5.128)

(5.129)

(5.130)

(5.131)

(5.132)

¢3=¢3(y,y')=4h—y+y' (5.133)
¢4=¢4(y.y')=2h-y-y' ‘ (5134)
where ¢,- (y, y') are associated with phase-delay distances. Substitution of (5.28), (5.37),

(5.76)-(5.79) and (5.82)-(5.85) into (5.32) leads to the following result for the normal

 

component
~ _. ~P ... r ”r " , jy(i>y’) ,
”2y(’19y)=jl:G (lay—y )+ny(l,y,y )]—aF—dy +
y, (5.135)
JG; (1;); y,y 3&4}; + [0" (x1;y,y ):_]____2(’13y )dy
1608 jam
y’
where
~ 4 C++e _m— 1'95 +C“e‘w39+ ‘ p,
or (.1; ,y.)_ we 32 y_ of mag“, (5.136)
342 y 2p(1—R2e4fi§(l—R2e' 413‘

[fie—m - Re'm + Ifze—P¢3 — Re—m"
2 p(1 — 82.24”)

 

ny(l;y,y')= (5137}

The spectral-domain Hertzian-potential dyadic Green’s function 50:; y, y') is
revealed by using the above results and writing the spectral-domain Hertzian potential for

region 2 in vector form, leading to

ad y): [Galyw 5’” ’y ——)dy' (5.138)
jwe
y
where
G<i;y,y'>=GP(1;y—y')+é'<1;y,y') (5.139)
5P(21'; y— y') = if?!) = iGP£+ 96P9+2GP2 (5.140)

107

G’(Z;y,y')=2G;,2+9G; x+9G;,y 9+9G;, 2+2ng 2 (5.141)
The Hertzian-potential dyadic Green’s function is identified by using the following

Fourier transform relations

 

722(7)=(2 )2 j [22(21',y)ef“d22 (5.142)
5(1, y'): I [Jane-J'Z'F'dx'dz': j IJ(F')e_j1'F' dx'dz' (5.143)
—oo—oo x'z'

Substitution of (5.138) into (5.142) and using (5.143) leads to the following expression

-.

n2(r)= VIGU If") J(r -——)dV'= I[Gp(r— —r ')+G’(r|F')] Jr —r—)dV' (5.144)
(08 V, ng
where the Hertzian-potential dyadic Green’s function is
G(r r|F')=(—2-l)— j 1 GM, y y )eJ’W" )dZ/I (5.145)

5.8 Physical Observations and Limiting Cases
5.8.1 Introduction

In section 5 .8, some observations will be made to ensure the physical
reasonableness of the above analysis. In addition, a couple limiting cases will be

investigated to see if the above theory reduces to expected (i.e., well-known) results.

First, the terms 1—Rze"4p" and 1—Eze“4p” will be examined to show that they lead to
the anticipated TE and TM modes of a symmetric slab waveguide. Next, the terms like

those in equation (5.130) will be identified as waves interacting with the conducting

108

boundaries. The limiting cases when the conductors become perfect (ac —> co) and when

the top imperfectly-conducting plate is removed will also be investigated.

5.8.2 Symmetric-Slab Waveguide Modes
The characteristic equation l—Rze“”’" = 0 can be shown to identify the TE
surface-wave modes of a symmetric slab waveguide in the following manner.

Multiplying the above equation by eZPh and factoring gives

e2ph — RZe—ZP” = 0 :> (eph + R e—P”)(eP" - R W”) = 0 (5. 146)
Setting each term in (5.146) equal to zero and using (5.58) leads to the following set of

characteristic equations
.2” + lee-P” = 0 :5 (p, + p)eP" +(pc — p)e'P" = 0 (5.147)

eP" — Re‘P” = 0 :> (p. + p)eP" -(pc — p)e‘P" = 0 (5.148)
Equations (5.147) and (5.148), after being multiplying by l/ 2 (for convenience), can be
grouped as follows

eph-l-e-ph eph—e_ph

2 +1, 2 :0 :> pccoshph+psinhph=0 (5-149)

Pc

eph -e_ph eph +e—ph

2 +p 2 =0 :> pcsinhph+pcoshph=0 (5.150)

Fe

The eigenvalue equations in (5.149) and (5.150) are well-known [26], [32], [33]. They

identify the even and odd TE surface-wave modes since they are even and odd in p ,

respectively. A similar analysis shows that 1— Eze’Jph = 0 leads to the well-known [26],

[32], [33] even and odd TM surface-wave mode eigenvalue equations

109

chzcoshph+psinhph=0 (5.151)

pCNZ sinh ph+ pcosh ph = 0 (5.152)

5.8.3 Wave Interaction Between Source and Field Points

In this section, the terms of equations (5.129) and (5.130) will be examined for
physical insight. Consider the geometry in Figure 5.3. Wave 0 travels from the source
point y' directly to the field point y and traverses a distance y — y'. Hence, wave 0 is
associated with the principal wave of (5.129). Wave 1, which experiences two interfacial
reflections and travels a distance 4h+ y — y' , is associated with the first term in (5.130).
Wave 2 is associated with the second term of (5.130) since it experiences only one
interfacial reflection and travels a distance 2h+ y+ y'. A similar discussion shows that
waves 3 and 4 are associated with the third and fourth terms of (5.130). Note that
waves 1-4 experience reflection from the top (y = h) and/or bottom (y = —h)

conductors and thus are associated with the reflected Green’s function.

 

 

 

 

Figure 5.3 Wave interaction between source and field points.

110

5.8.4 Limiting Case for a Perfect Conductor
If the conductors become perfect (ac —) co ), equations (5.130), (5.136) and
(5.137) reduce to the correct/known result [34]

-p¢l _e—p¢2 +e-p¢3 _e—p¢4

~ ~ 8
Gr (1;.y9y') =
.... 2.00—6"”)

 

...a =x,z (5.153)

G;a(2:;y,y')=0 ...a=x,z (5.154)

e-pé + e—p¢2 + e_p¢3 + e_p¢4

 

6’ (2,9,9) = (5.155)
W 2p(1—e'4P”)
since
lim R=l , lim §=—1 (5.156)
(Tc—>00 ac—mo

5.8.5 Limiting Case for a Source Over an Imperfectly-Conducting Half Space
Another known case is the problem in which a current source is situated over an
imperfectly-conducting half-space [3], [25], [27]. In the parallel plate environment of
Figure 5 .1 (or 5.2), the half-space geometry can be achieved if the top plate is removed.
However, if h is allowed to approach infinity in Figure 5.1, both plates would reside out
to infinity since the top plate is at h and the bottom plate is at —h. This difficulty can be
avoided if the structure is shifted upwards by a distance h so that the bottom plate is
located at 0 and the top plate at 2h. The geometrical shift can be accomplished
mathematically using the following change of variables
i=y+h => y=i—h (5.157)

7=y'+h => y'=9'—h (5.158)

111

Substitution of (5.157) and (5.158) into equations (5.129)-(5.134), (5.136),

(5.137) and taking the limit as h —> 00 leads to the following known result

 

 

 

p g _ _' e-pIP-yl
G (rhy-y): 21) (5.159)
.. a Re-PWW)
G;a(2;9,9')=—T ...a=x,z (5.160)
~ _. c+-e—P(7+7)
G;a(4;17,7)=— y“ 2 ...a=x,z (5.161)
p
.. - - 4919+?) 2_ -pG+7)
Gg(z;i7,9')=—5e———=—(”CN ”8 (5.162)

2p 2p(10.N2 +p)
Of course, the working variables y, y’ can be recovered by a second change of variables
y = 9, y' = 7 . Equation (5.159) constitutes the principal-wave contribution directly from
the source to the point of observation. Equations (5.160)-(5.162) are the reflected waves
that emanate downward from the Source, experience reflection at the half-space
boundary, and travel upwards to the observation point. The reader familiar with guided
wave theory will recognize the familiar Zenneck surface-wave [26] contribution in

(5.162), for example. A Zenneck wave doesn’t exist in the parallel plate environment

since the factor ch2 + p doesn’t explicitly occur in the denominator of (5.130), (5.136)

or (5.137).

5.9 Electric Field Dyadic Green’s Function
5.9.1 Introduction
The electric field dyadic Green’s function can be identified by examining the

following relation for the electric field, that is

112

-o

E=k22+vv-2 (5.163)

It is convenient to break (5.163) into two parts, a principal and scattered part, where

E=EP+E’= [Ge(F|F')-.7(F')dV’= [[Gep(F—F')+G”(F|F')]-](F')dV'(5.164)
V.

I!

EP = kzir'p +VV-7?” = [Gep(F—f")-j(7')dV' (5.165)
V.

E” = kzir" +VV-7'r" = [G”(F|F’)-J(F')dV' (5.166)
V.

It must be mentioned that no mathematical difficulties arise if the VV- operator is

applied in the computation of E' in (5.166) since the integrand of it" is well behaved.

This is not the case in (5.165), as discussed by [35]-[38]. Recall that

fip=ir°f(7)= [Gp(F—F')-JL-)-dV' (5.167)
V, 1606
where

-ply-y'l _- , ..
e em“r )d2/1 (5.168)

 

 

.. a g, _ 1 °°°°..
GP(r—r)—(2fl)2_£i1 2p

A simple passing of the VV- (more precisely, the 62 /6y2) operator through the integral

term in (5.167) is not allowed in this case since the integral would become singular due to
the absolute value term I y — y' I . In order to carefully handle this source point

singularity, integration over y’ must be broken into two parts as follows

y-6 h
y,:ali§i+[ [+ [ ]=PV_[ (5.169)

—h y+5 y

113

and is known as evaluating the integral in a Cauchy Principle Value sense. However, this

can be done only if the excluded portion doesn’t contribute, that is

     

y+6 oo oo Ie—ply—y 'l g .. _,
lim 1 1"(’-’)d’2(’)Jy'=0 (5.170)
6—20" y—6 (2 jcoa

Since the integrand in (5.170) is continuous, the contribution from the excluded part is
indeed zero and the integral in (5.167) can be evaluated in the Cauchy Principle Value

sense, namely

2”: PVV[GP(r—;'-)J——(’)dV'= [[[PV]GP(7—)1¥2dy']mt'dz' (5.171)
1025 . H y 1608

Note, any differential operator involving x or 2 can be freely passed through the
volume integral in (5.167) since the integrand is well behaved in those variables. Also,

since the integrand is continuous, Leibnitz’s rule for differentiation guarantees that the
G/Gy operator can also be freely passed through the volume integral. It will be
demonstrated (via Leibnitz’s rule) that, because the limits of integration in (5.171)

depend on y and the integrand becomes discontinuous after one differentiation with

respect to y , the 62 /Gy2 operator introduces an additional term that would be absent if it

were simply passed through. Note, the PV notation will be assumed but dropped for

convenience.

5.9.2 Principal Electric Field Dyadic Green’s Function
The principal electric field dyadic Green’s function can be identified by inserting

(5.167) into (5.165), resulting in the following

114

GP = —1— “2180524. 9k’GP9+ EkZGP2)-]dV'+
1608 V,

 

 

 

 

 

 

 

 

2 p 2 p 2 p _.
—.1—[£an 2+2aG 9+2“; 2 -JdV'+
1608 V. 6x 5x6)» (’9sz
(5.172)
2 p 2 p 4 2 *
—1—[9"G 2+9“; 2 -JdV'+—l—a— 9GP9-JdV'+
jam V, Gny 6‘sz jaw ayz V,
2 p 2 p 2 p -
——’—j 250 2+2“; 9+2“;2 2 -JdV'
1608 V, 626x (9sz 62
Leibnitz’s rule for differentiation is [39]
hm b(y) .
a I I a 9 I ab Ba
5; [f(y.y)dy = [Melly+f[y.b(y)]—a(yi)-f[y.a(y)]—;;yl (5.173)
am am
and when applied to the 62 /Gy2 term in (5.172) produces the following results
52: [9GP9.]dV'-[ a [9—aGp9 de' dx'dz' (5174)
2 — _ ' °
6y V' x'z' ayy' 6y
666p» ay“’aGP~ hGGP~
—[9——-9-de'= lim — 9—9-de'+ [9—9-de' (5.175)
ay,. ay “may _, 6y 9+5

where

 

 

 

66” 66” F-F’ 1 °°°° ,~ —. , -*,;_;«
69 = :99 )=_(2 )2 I [psgn(y—9)GP(,1;y—y)ef’< >212). (5.176)
7t —oo—oo

The right-hand side of (5.175) leads to, with the aid of (5.173)

6’0” 4
lim 692

a -* -r A -' *r
— 5(P—P))"J(P,J’) (5-177)
6—)0’ay

 

y“".aGP. ~ "5.
___. .Jd':
iy 69 y _[y

 

6(2—2'12-Jtfi',» (5.178)

115

where the generalized function identity [23], [40]

6(5-15')=(2;)2 I :Ioe 1"" ”.1221

 

has been used. Therefore, it is concluded that

 

— GP JdV':
669212” 9 VI9

y[62Gp(r— r )

ay2

—6(F-—F’) 9-J’dV'

(5.179)

(5.180)

The principal electric field dyadic Green’s function can be deduced by combining the

results of (5.180) and (5.172), leading to

Gep— - xGep x + xGep y + xGepz +

ij x+yGepy+yGepz+zGepx+zG py+zGepz

where

 

 

2 2 2
G$=1 szp+aGP ’Gepz l GGP,G:§____1_GGP
6x2 ’0’ jwa 6x6)» jaw Gsz

jaw

Gep_;azGP ep__ 1 6sz
y" jaw ayax ’ ”- jws ayaz

 

 

 

 

 

 

 

2
G55 = fi-l—[kZGP + 62:}: -6(7 4)]
1608
Gep ___ L 626p ep = L 626p ,Gep— = 1 kZGP + 620p
2x jaw 625x , Zy jcoe 6sz 27' To»; 2
‘3sz 2 21' 2
=- GPeJ (' ’ )d 21
6x’ <2 1211’
626p Ha’GP 1

 

 

axay ayax (22 >2

116

 

I I jépsgnw- y ')G”e”l ("’ ’d24

(5.181)

(5.182)

(5.183)

(5.184)

(5.185)

(5.186)

a’GP _ a’GP_ 1

 

 

_ Gp jA-(r—r)d2/1
axaz 222x _Mz )ZII‘K e

 

(3sz _a’GP_ 1 ,1
s n GPeJ 1’ ”(1221
2922 UGsz 12” )2111419 8 (y- y')
6sz
IIP ZGpe jl-(r— r')d21
ay’ (2202..
asz l 2~ 1.-
_= 0% J V ”(1221
622 (22: )2 ii;
~ ~ _, -p|y-y'|
GP: 2 I IGpemr ”)dz/l , G”=G”(4;y—y')=——e
(2704,17

 

 

 

 

5.9.3 Reflected Electric Field Dyadic Green’s Function

(5.187)

(5.188)

(5.189)

(5.190)

(5.191)

In an analogous set of steps as in section 5.9.2, the reflected electric field dyadic

Green’s function can be identified by inserting the expression for it"

result is

where

G" — -xGex;x + xG” y + x0952 +

yGegx-l-yGe Wy+ijg2+2GZx+2G§yfl+2G§zrz

xx

6x2 6x69

 

2 r 620’
GE; =;[k26;x+a G + yx]
1606‘

 

 

 

2 2 r
Ger___1___aG;y Ger_ 1 [6 Gyz+azG2z]

xy'jwe GxGy ’ "Z—jam My 2va

117

into (5.166). The

(5.192)

(5.193)

(5.194)

er_
Gw—

626;,

jam

1

er
02x

2 r
[k ny+

_ 1
jam;~

Zr
00”

 

2

er 1 2 r
] , Gyz=—[k Gyz+

2 2 r
00;.wa

 

er _
Gzz -

520;. _
ax2 _

 

 

60"); ___

5x5y

1 .
= (2”)2 1:1 16

1608

1

azax azay

_;_[k20;2 +

 

_(220

0000

—a)—w

06;,

5y

_ (fie-N1 —C;;e'P¢2 -C;;e'p¢3 +C;;e‘P¢4

 

(3ng + 520;):
2 ayax

2r
ac;yz

62G’
+

ZZ

 

jaw ay2

1 ,

620;,
2

er_

”7'52 azay

1

Zr
00),:

azay

+
62

oo 00 ~ .* _. fl
2 I IngLejZ.(r—r )dZ/l

ayaz

2r
16G”,

 

ej’HF' )dZ/l ...a = x,y,z

 

 

2 r
0 GW

2(1 — RZe-‘W )(1 — '1? 22.2-4”")

lye—M —§e—p¢2 —§2e-p¢3 + fie”1254

 

620;. _
6x62

1

2(1 —

 

_(270

 

6})2

020,;

 

ayax=

(2

—oo—oo

10000.
21116

72') —oo—oo

118

'R—ze-4ph )

2 a} a} .546; e ji-(F-F’) d2 2.

6:53: eji°(7-7')d2/1

...a =x,z

1

(5.195)

(5.196)

(5.197)

(5.198)

(5.199)

(5.200)

(5.201)

(5.202)

(5.203)

(5.204)

(5.205)

 

 

 

 

 

 

 

 

626‘; 1 LG-me -F —F'
~I’ 2 -p¢l _ —p¢2 _ 2 -P¢3 -p¢4
66W =_R e Re zR—fh +Re ...a=x,z (5207)
6y 2(1—R e P )
2 r
2
aZGra 00 00 Gya j}. (F—F') 2
azay _(2”1—®)2 I ij§% e d A. ..a: x,y,z (5209)
2 r °° °° .~ 4 4.
6002:: = (2 )2 1 1525.528’11'” )dz/l (5'21”)
2 7F —oo-oo

5.10 Magnetic Field Dyadic Green’s Function

The principal and reflected magnetic field dyadic Green’s functions are found by

substituting if” and it" into the expression for H , that is

HP = janxifp = thP(F—F')-J(F')dV' (5.211)
V.
H’ = janxiE' = jéh'(7|7').](7')dV' (5.212)

pf,

The principal and reflected Green’s functions, th and GI” , for the magnetic field are

(the minor details are left for the enjoyment of the reader)

 

 

é’PP = 801;; y+xthz + ythx+ ythz + szPx + ithy (5.213)
p
G’P: —GjP_ — -66 =_1 2 j j 140%“ 0" )d’vl (5.214)
62 (272')_0°

119

GE: —G£=

hp-
Gyz _

CI.

613;:

6hr:

21

6G'a

_ hp__
Gly—

 

@Pz

6G p
6x

 

-6G;x , G);—
62
60’
Ghr: xx
yx 62
5G5); _acg;r
6x 6y ’

 

62

6G;a

22m )2 1 1146; aefi'V‘P'W/l ...a=x,y,z

 

 

 

—_=(2n)2...
60’

62 =(271r)—oo
66x =71m )_...

l

 

6x

 

)2] I1vsgn(y—y')€~?”ej’i‘(F “7 )d22 (5.215)

1
"...a”?

 

60;”,
62

 

hr__
,0”—

hr
zy"

120

j‘lngpej/l-(F—F)d21

 

hr _ 662:2 _ 60};
’ xz 6y 62
_a%:
6x
=a_G—;’y Gh' -_- 63;:
6x 6x

2 :0" ljgcrejl-(F— F')d2’1

2 I 11.5628 jA(F—F)d21

031% __aa ejl-(F— F')d2/1" .a_ xz

”2 I 03-1502! ej1(rr)d2/i.. ..a: x,y,z

(5.216)

Ghr— - xGh'x + xGh'y + xGh’z + yGh'x + yGh’z + th'x + thzy'y + thzrz A (5.217)

(5.218)

(5.219)

(5.220)

(5.221)

(5.222)

(5.223)

(5.224)

(5.225)

Chapter 6

ANALYSIS OF A LOSSY STRIPLINE FIELD APPLICATOR

6.1 Introduction

The analysis of an imperfectly-conducting strip transmission line [41]-[44]
symmetrically located between two imperfectly-conducting plates is considered in this
chapter. Specifically, the effect that the lossy conductors have on the principal (i.e.,
dominant) discrete-mode propagation constant and respective surface current will be
investigated. Understanding the nature of the principal mode is fundamental to the
material characterization process (as well as to many other applications), and thus
provides motivation for this study.

The first step in the analysis is to specialize the general 3D current source of
chapter 5 to an infinitely-long strip surface current symmetrically located between
imperfectly-conducting plates. Next, an EF IE (electric field integral equation) is
developed by satisfying an impedance boundary condition on the strip conductor. As a
final step, the EF IE is solved using a MoM (method of moments) technique and the
principal-mode propagation constant and corresponding surface current distribution are

subsequently identified.

6.2 Geometry

The geometry of the lossy stripline field applicator is shown in Figure 6.1. The

structure is comprised of two imperfectly-conducting plates located at y = ih and an

121

imperfect stn'p conductor located in the center at y = 0. The top and bottom plates are
assumed to be infinite in length along the x,z directions and have properties (ac,Zc ).

The infinitesimally—thin center strip of width 2a has properties (0's,Zs) and is assumed
to be infinite in length along the z direction. The center strip is embedded in a non-

magnetic material characterized by the material parameters (5, yo ).

Conductor (002C)

 

y=b
Conductor (OS,ZS)
m (3,110) ___)

k—Za—A x

Conductor (0C,ZC)

 

y=‘

Figure 6.1 Cross-sectional geometry of the lossy stripline field applicator.

6.3 Development of the Lossy Stripline EFIE

The EFIE for the discrete-mode propagation constants of the stripline field
applicator is obtained by first introducing an excitatory current it into the stripline
environment that maintains an electric field E". This impressed field will induce a

surface current if on the strip conductor that subsequently maintains a scattered field

given by

122

F')-IZ(F')dV'.—. j IGe(x,y|x',0;z—z')-K(x',z')dx'dz' (6.1)

 

53(2) = [690'
S,

where G" is the electric-field dyadic Green’s function developed in chapter 5. The

source I? for the scattered field is confined to y’ =0 and is invariant along the z

direction, thus Ge = Ge (x, y|x', y' = O; z — z' ). In addition, I? will only be a function of x

and z and will have no normal component (since the strip conductor is infinitesimally
thin), that is

12(7) = £K,(x, z) + 2K, (x, z) (6.2)

As a last step, the EFIE for the lossy stripline is developed by invoking the

following impedance boundary condition on the imperfect strip conductor

limf-[E = 2.12] ...v x,ze s (6.3)
y—>0

where S is the surface of the strip conductor, f = x, 2 is the unit tangent vector and E is

the total field
E = E" + E‘ (6.4)
The limit as y —) 0 must be used for convergence reasons and will be discussed shortly.

Substitution of (6.4) and (6.1) into (6.3) produces the EFIE result

a CD
limf- ( j 6P(x,y|x',0;z—z')-12(x',z')dx'dz'—Zsi<°(x,z) =-E"(7) (6.5)
y—>0 -a—oo

The above equation is actually a pair of coupled EFIE’s (coupled Fredholm equations of

the second kind) since it must hold for f = x and 2. Due to the form of Ge , the limit

cannot be interchanged with the integral in (6.5) since this would render it divergent.

123

However, the MOM technique will strengthen the convergence of the integral and the
limit can then be freely passed through, as will be shown in the next section.

Since the structure is invariant along the z-direction, it becomes computationally
advantageous (and necessary) to transform (6.5) on the z-variable invoking the following

transform pair

mm: I Ham-142212
‘°°w (6.6)
F(2.z)=-2-1— 1 “2,0221%;
”—00

with ,5 = xx + yy. The result, with the aid of the convolution theorem, is

a
11113;. I§P<x.y|x'.o;4)-E(x'.4)dx'4.20.4)=—é"(/3,4) ...VxeC (6.7)
y—> _a

where C is the periphery of the strip conductor (—a < x < a). Equation (6.7) represents

the EFIE in the axial Fourier transform domain (i.e., the complex 6 domain), the

solution of which, leads to the p’h discrete-mode propagation constant (p. It can be

shown [45] that near simple poles for the propagation constant (p , the spectral-domain

current has the following behavior (for a wave traveling in the :2 direction)

 

1im F(x,4’)z kp(x)

6.8
19:4,, 414p ( )

where 11:", is the eigenmode current associated with the p'h discrete propagation mode.

Substitution of (6.8) into (6.7) leads to

k,,(x x')dx _Z ic'poc)

 

1 ,0;——=—”"*, 6.9
git {88.29ny M); {pdx 4:4,, M104) ( )

124

or equivalently

 

Iim f- [gem x',o;4)-i€,.(x')dx'—Z.I?,.(x)=—<4:4,.>2‘(2,;) (6.10)

y—>0
4614,, “a

The impressed field Ei remains analytic near the pole singularity $4,, since, by its very
nature, is independent of and cannot be influenced by the guiding structure. Therefore,

near pole singularities 4' —-> $4,, , equation (6.10) reduces to

 

a
limf- jge(x,y x',O;$§p)-Ep(x')dx'—Z3Ep(x) = 0 (6.11)
y—)0 -a

Decomposition into 55 and 2 components leads to

 

 

[[82:0’)’ x', 0)kx(x') + 82 (x, yl x', 0)kz(x')]dX' - Zskx(x) = 0 (6.12)
I [g:.<x.y x'. 0119.0) + g; (x, y|x', 0)k. 0)] dx' - Z.k. (x) = o (6.13)

where the limit and 4' notation, as well as the subscript on if}, , has been dropped for

convenience. Equations (6.12) and (6.13) constitute the fundamental pair of coupled

EFIE’s for determination of the p’h

discrete-mode propagation constant ${p and
eigenmode current 12p.
The axial-transform expressions for ggfl (a,,B =x,z) can be identified using

chapter 5. The result is (using the relation 5C;i =§C§f and G; =GZ'Z in the

expression for C2,)

125

x',0)= I Cafiefé‘x'xwg (6.14)

 

gig (x, yl x', 0) = 82% (x, yl x', 0) + 8222 (x, y

 

 

 

 

 

 

 

 

 

Cafl(§,y,€)=C£fl(€,y,§)+C;/;(c§,y,4) (6.15)
2_ 2 -p|y|
Cf; =k 5 e (6.16)
j27rw£ 2p
, 1 ~, , . 6620» y'=0)
Cn= . (k2-52)Gxx(y|y =0)+1cf y I (6.17)
1275606 6y
-p|y|
C}; =Czfi =-— .64 e (6.18)
12mm 2p
, , 1 ~, , . 56' (y y'=0)
sz =sz:— . 6463(ny =O)—j§ yz I (6-19)
127mg 6y
2_ 2 -p|y|
2’: =k 4 e (6.20)
j27rw£ 2p
r l "r v . aér (y y'=0)
C2, = . (kz-czmoly =0)+J; y” I (6.21)
127mm 6y

Equations (6.12) and (6.13), upon using (6.14) and interchanging the spatial and spectral

integrals, may therefore be written in the following form

I angel?" I [anx (x') + szk, (x')] [jgx’dx' — 23k, (x) = 0 (6.22)
I dgefé" j [Cakx (x') + szkz (x')] (may — z,k, (x) = o (6.23)

6.4 MoM Solution for the Lossy Stripline EFIE

126

6.4.1 General Formulation

The dual EFIE’s in (6.22) and (6.23) can be solved using a MoM technique. The

first step is to expand the unknown currents using appropriate basis functions e 5,, (x)

with complex amplitude a fin , that is

”/3
kfi(x)= Z} aflneflnu) ...,6=x,z (6.24)
n:
Insertion of (6.24) into (6.22)-(6.23) and interchanging the summation with the spatial

and spectral integrals leads to

N

X

a,,,[ jCngneféchg—Z e:xn(x):|+ 2a,, [:0 [C ,zgznejéxdg]=o (6.25)

n=l

"=1 ..a) "=1 —00

Nx co . N1 00 .
Z a..[ I Cugxneféxdé} Z a..[ I szgzne’5‘d6—2.ez.(x)] =0 (6.26)

where (since x' is a dummy integration variable)
g)... =—gfl,.(6) — leach—16% -,l- efln(x)e 6de fl= x z (6.27)
-a -a
The second critical step in the MoM solution is to apply the following testing
operator to (6.25) and (6.26), namely
a
[tam(x){-}dx ...m=1,...,N,+N, ;a=x,z (6.28)
“'0
where tam (x) is an appropriate testing function and a = x,z when applied to the x,z
component equation, respectively. The result, afier interchanging the testing operator

integral with the sum and spectral integrals, is

127

 

NX

n=l

N2: 00 N: 00
Z axn[ szfzmgxndé] + Z azn[ J‘szfzmgzndé: + Dznzzn] = 0
"=1 —m "=1 —oo

or in matrix form

N

I

NZ
mn mn _
Z Axx axn+ Z sz am —0

n=l n=l
N N > ...m=1,...,N,+N,

X 2
mn mn _
Z Azx axn+ 2 A22 azn "

n=l n=1

 

where

afl = lim [Caflfamgflndg'l'Daaaafl "aa p: x z

fam :fam(§)= Itam(x)ej:xdx ~11:va

-a

1...a=,6

Dmg= “SZ J’am(x)ean(x)dx ’ 6afl={0 _aqefl

“‘0

Z ax ”l:: ICU fxmgxnd€+Dmn]‘FZ N: Zazn[°jicxzfxmgzndg]=0
"=1 —00

(6.29)

(6.30)

(6.31)

(6.32)

(6.33)

(6.34)

The choice of expansion and testing functions is usually strongly influenced by

physical and mathematical considerations. From a physical standpoint, the expansion

functions should closely model the behavior of the unknown currents so that only a few

expansion functions are required to efficiently obtain accurate results. In addition, a

prudent choice of expansion and testing fimctions will allow the integrals in (6.27), (6.33)

and (6.34) to be computed in closed form. Although obtaining closed-form integrals is

not absolutely necessary, it does significantly reduce computational effort and therefore

increases numerical efficiency.

128

 

The stripline considered in this chapter can support both even and odd modes
(about the variable x) due to its symmetry. This, of course, has implications on the

choice of expansion and testing functions. Even and odd modes are typically classified
relative to the longitudinal current k, (x). If k, (x) is even(odd), then the particular
mode supported is classified as even(odd). Consequently, k,r (x) must be odd(even) and

p3 must be even(odd) since the axial transform continuity equation is

6k

VI? = a; + jgk, = - 1'pr (6.35)

 

Thus, when choosing testing and expansion functions for even modes, one should select

even functions to represent eszm and odd functions to represent exmtm. Similarly, for

odd modes, empty, must be odd and exmtxn will consequently have to be even. In

addition, the axial and transverse surface currents have the following well—known

behavior [46], [47] near the edges of the infinitesimally-thin strip conductor

kx 0c pl/Z , kz 0C p—l/Z (6.36)
where p is the radial distance from the edge. Accordingly, one should select expansion
functions that closely model this edge behavior. It is known [45] that the principal mode
(which is the only mode being investigated in this chapter) is even. However, an MoM

formulation for both even and odd modes will be given below for the sake of

completeness.

6.4.2 Even-Mode MoM Formulation for Lossy Stripline
A common choice of expansion and testing functions employed when considering

lossless structures (known as Galerkin’s method [29]) for even modes is

129

e...(x>=t..(x)=Two/c011—(x/a)2 (6.37)

T2,,(X/d)

em (x) = I," (x) = ——
([1 —(x/a)2

where Tn (x) are Chebyshev polynomials [48], [49] of the first kind (see Appendix E for

(6.38)

an overview of Chebyshev polynomials). Note that Chebyshev polynomials of odd/even

order are odd/even functions, as seen from the relation [50]

T.(x)=(-1)"T.(—x) (639)
Therefore, em (x), em (x) are odd/even functions and have appropriate weighting
fimctions, as physically required. However, there are now mathematical difficulties
encountered in the lossy case that were absent in the lossless case. Namely, the integral

in expression (6.34) becomes divergent for certain m,n. To see this, consider the case

when m = n = O , namely

00 " " T2(x/a)
Dz, =—Z, Itzo(x)ezo(x)dx=—Zs [ 170(37)de (6.40)

-a -a

The c.o.v. (change-of-variable) SE = x/a leads to the following integral expression

00_ To(X) ~__
Dz, ——aZsI1 2dx-—aZs I1

1

 

14 d? (6.41)

—1
where the property To (x) =1 has been used. The above elementary integral is known to
be divergent. Therefore, Galerkin’s method is no longer applicable for lossy structures
and alternative expansion and testing functions must be sought.

The problem becomes tractable if Chebyshev polynomials of both the first and

second kind are used in appropriate combinations. For even modes, the following

130

choices should be made for the expansion and testing functions (where m, n = 0,1, 2,...)

emu)=(12....<x/a)\/1—(x/a)2

txm (x) = U2m+l(x/a)

72,,(x/a)

ezn (x) =
(’1 — (x/a)2

tzm (x) = T2m (x/a)

(6.42)

(6.43)

(6.44)

(6.45)

It is important that the weight functions are contained in the expansion functions since

they are representing the physical currents (and are based on the required edge condition).

Then, after the expansion functions have been determined, the testing functions are

conveniently chosen so that (6.34) is easily computed using known orthogonality

relations.

Insertion of (6.42)-(6.45) into the corresponding relations (6.27), (6.33) and (6.34)

leads to (after using the c.o.v. x = x/ a and the even/odd properties of the Chebyshev

polynomials)

l
f...<6> = jza [(12.21 (asincaaab‘z = j2a1;...<4a)
O
1
6.6) = 2a [szenoswaaaa = 201%..(661)
0

l
g..<6) = -12a [02..1(£>\/1— £2 smeared)? = -12aI§..(4a)
0

l ..
g..<6> = 2a [Ii/41%coseaxw = 2a1;..(6a>
0 — x

131

(6.46)

(6.47)

(6.48)

(6.49)

 

l
123;" = —az, [ Um. (ix/2”,, (5N1 — szzdsz = -aZ,I,,,,,, (6.50)
—1

 

l .. ..
0,3” = —aZ [ 72'" (”Tin“) dz = —aZ,1,_,,,,, (6.51)

3-1 41—522

where the superscript e,o means the resulting integral is even or odd in 5 (which can be
verified by trivial inspection of the above integrals). Remarkably, Appendix E shows
that the above integrals exist in closed form as desired. Thus, the above combination of
expansion and testing functions adhere to the physical and mathematical stipulations.
Substitution of (6.46)-(6.51) into (6.32) results in the following expressions for

the matrix elements

AS." = 113) 303C... (5,2601%. (561)1an («50)0'4’ - Zslxmn (652)
Ag" =1135180?sz(5,y,§)1fim(§a)1§zn(éa)d§ (6-53)
143" = — 113(1) 130:]sz (6 #61112," (660133." (450)61’6 (654)
Ag" =11:3)803C226,y,C)1§zm(§a)1§zn(§a)d€ - ZsIzmn (655)

since Cxx,sz are even in 5 and sz,sz are odd in 4‘. Note that one a can be dropped
because we are solving a homogeneous matrix equation. In addition, a careful

examination of Chapters 5 and 6 shows that Cxx ~ 5 , Q, =sz ~l and C22 ~1/é as

4‘ ——) 00. Thus, with the aid of Appendix B, it can be shown that the magnitude of the

integrands of A0, , that is '1 (Aw )I , have the following asymptotic behaviors

132

 

 

lim|I(Axx)|,l(sz)|~1/53/2 ; lim|1(A,,)|,|1(A,,)|~1/255”2 (6.56)
5"”) 5—->oo

The strength of this asymptotic behavior guarantees convergence and formally allows the
limit to be brought inside the spectral integral in expressions (6.52)-(6.55). Therefore,

the above matrix elements simplify to

AS." = 8a:[C.. (6.01;... (46124464442... (6.57)
AIS" = 1'86:an(6,4)12m(§a)1§zn(§a)d§ (6.58)
A3" =—18a:[C..(4.6)I;...(64)I§..(:a)dé (6.59)
Ag" = 843021401}... (5a)1§zn(«fa)dé‘ -Z.I.... (6.60)

where, after some algebraic manipulation, the functions C006 (5, 5) = C (1,6 (5, y = 0,5) are

 

 

 

 

 

1 A k2_ 2 2
€564.24): , e ( 5 ’+ 5 h (6.61)
14721060 1) achD
«54 A 1
sz( 9 :sz( s )=———_ ___— (6'62)
6 C) 5 C j47rw6‘De p achDh
1 A k2_ 2 2
sz(s‘,€)= , e ( 4 )+ 5 h (6.63)
147tw£D P 0'ch0
A=sinhph+ p cosh ph (6.64)
e p . h ,waZ
D =coshph+ s1nhph , D =coshph+1

c sinh ph (6.65)
p

C C

133

 

Note that the C 0,3 are all even with respect to p, hence the branch points at ,1: +k are
removable. In addition, the above matrix elements reduce to the known lossless case [34]
in the limit as ac —> oo , as anticipated.

As a final note, since the Chebyshev polynomials are indexed starting from zero,

the matrix equation subsequently should be written as

Nx-l ‘

Z Ag"a,,,+ NZ 21'3"..—

13:31 "=0 > ...m=0,...,Nx+Nz—l (6.66)
2 Agra + N2 A'Z'Qz =0

n=0 n=0

 

This also applies to the odd-mode analysis given below.

6.4.3 Odd-Mode MoM Formulation for Lossy Stripline
In the analysis of odd modes, in which k, is even and k2 is odd, the following

choices should be made for the expansion and testing functions (where m, n = 0,1, 2,...)

e,,,(x) = U2,,(x / (1),/1 — (x / a)2 (6.67)

I... (x) = U2m(x / a) (6.68)
T (x/a)

e..(x>=—21fl—— (6.69)
([1 —(x/a)2

ll2m (X) = T2m+l (x / a) (670)

In this case, the matrix elements are

Ar." =84 [6..(4.4>I"2...(6a)i;. ,.-Z(§a)d6 .1... (6.71)
0

134

4.2:" = —78a I C... (4.01}... (46113.. (and:
0

42'2" = 78a [CMéCfifozm(éa)i§m(§a)d€
0

A3" = 8a ]C,, (.5, gig," (gafigm (56w: — 2.7....
0

where

l
izmca) = [112.00 cos<4asz>di
0
~ 1
12,65.) = [szmosineaodi
O
l
ig..(4a> = (0.46.1142 008(6a55)d5c
0

~ ‘ T (55)
10,,(5a) = 2Lsummed);
.. (1 m

I
I'm = [U2m(x)U2,,(£) l—izdfc
—l
l .. ..
izmn = JT2m+l(x)T2n+l(x) di‘

_, \ll-xz

6.4.4 Quasi-TEM Characteristic Impedance

(6.72)

(6.73)

(6.74)

(6.75)

(6.76)

(6.77)

(6.78)

(6.79)

(6.80)

A quasi-TEM (transverse electromagnetic) characteristic impedance for the even

principal mode can be identified for the imperfectly-conducting stripline by assuming

135

that the transverse current k, is negligible in comparison with the longitudinal current k2

(Section 6.5 will show that this is indeed the case), or equivalently

—o

kx z 0 :> K z 2kz ...quasi-TEM approximation (6.81)
Under these conditions, a unique characteristic impedance can be identified in the
transverse plane of the stripline field applicator (since k, = 0 implies that no longitudinal

magnetic field is produced to cause an additional EMF) given by

— __z_ —oo oo = @
ZO—ii(z) <z< :> Z0 -1“) (6.82)

where V(5) and 1(5) are the axial transform voltage and current of a single traveling
wave (only single-traveling waves will be supported on the stripline structure since it is
assumed to be infinite in extent along the guiding axis). The plus sign should be used if
the observation point is to the right of the source point (i.e., for a forward traveling wave)
and the minus sign if the observation point is to the left of the source point (i.e., for a
reverse traveling wave). Since this impedance holds true for all z , the expressions for V
and I can be formulated in the axial Fourier transform domain using the following

standard TEM definitions of voltage and current

”CF-[Email , I(6>=<]>E(x,4>-c7r' (6.83)
_ C

Since voltage is independent of path in the transverse plane, the integration

contour (from the center conductor to the ground plane) for the computation of V can be

conveniently chosen along x = O , thus if = jzdy. The displacement vector d—z: equals

261x since current flows along the longitudinal direction and is confined within the strip

and ground planes. When the center strip is positively charged and the ground planes

136

negatively charged, then V and I take the following form
It 61

V16) = Ie.(o,y.6)dy , 1(6) = I k.<x,6)dx (6.84)
O —a

Note that ey is the 51 component of the total electric field, that is

€y(0,y,C)=€;(0,y,C)+e;(0ay,C) (6.85)

where e3, is the impressed electric field and e; is the scattered electric field given by

(with the aid of Chapters 5 and 6)

x',o;4)-ic'(x')dx'= I g;10.y|x',o;6)-k.(x')dx' (6.86)

-a

 

a
e;(o,y,6>=2~ I may
—a

Near pole singularities, as already discussed, the surface current has the following

behavior (for forward/reverse traveling waves)

lir_n E(x,5)z kpm :5 lim kz(x,5)z kzpm

45.4,, 5:5,, 6 44,. 6 MP

 

 

(6.87)

Substitution of (6.84)-(6.87) into (6.82) and multiplying the resulting numerator and

denominator by 5 i 5 p produces the following expression

 

h h a '
I<;:4.)e;(o.y,6)dy+ I I g5.(o.y|x',o;6)~k...(x')dx'dy
20 =i lim 0 0‘“ (6.88)
(+16. “
I 19.0041:

—a
The incident field, in general, remains analytic at the pole singularities since it is
independent of, and cannot be influenced by, the guiding structure. Therefore, due to the

limit process above, the incident field is not implicated in the computation of

characteristic impedance for the discrete principal mode. Thus, in the limit as 5 —> $5p ,

137

the characteristic impedance takes on the following form

I

gyz (09 yl x'9 0; $4p) ' kz (x')‘t'dy

 

V
Z0: i7=+ a (6.89)
I kz(x)dx
—a
where the p subscript on k,p has been dropped for notational convenience and
N,—1

k, (x) = aznezn(x) , e," (x) — ——
"2:6 (I1 — (x/a)2

The computation of V involves spectral integration due to the presence of g;

and must therefore be handled numerically. However, an analytical expression for I
exists and can be identified as follows. First, substitution of (6.90) into the expression for
I in (6.89) leads to, with the c.o.v. 52 = x/ 61 followed by the c.o.v. 52 = x ,

1

N,—
1= a 2 am I TM” dx (6.91)

n=0 -1 l—x2

 

The c.o.v. x = c0819 and the use of the relation T2,, (cos 6) = cos(2n6) from Appendix E

leads to the following expression for the integral in (6.91), namely

1
T2n(x) ...n = O
J1_xdx=0I cos(2n6)d6= {0 m” i 0 (6.92)

Therefore, upon substitution of (6.92) into (6.91), the current reduces to the simple
expression

I = anazo (6.93)
where azo is the leading expansion coefficient for the longitudinal current k,. In

general, the computation of V will require knowledge of all the coefficients a,,,.

138

6.5 Numerical Results

In this section, numerical results of the MoM solution for the propagation

constant (5 = ,6— ja) and surface currents (kx,kz) are examined. Note, it will be

assumed that a = so , unless specified differently. First, it is important to determine how

rapidly the solution converges. Figures 6.2-6.4 show that the Chebyshev polynomials
insure excellent convergence properties for the propagation constant and surface currents
with only a small number of expansion and testing terms. Remarkably, only a single
expansion and testing function (i.e., the Maxwellian distribution) produces very accurate

results. In order to adhere to common notational convention, the following symbols will

be used. Ground-plane and strip conductivity are represented by the symbols 0 and

g

as , half-width and half-height are represented by w and t , and exponentiation is

represented by the letter e or d (for example, 1x106 =1e6 =1d6).

Next, it is important to check whether the non-Galerkin method developed here in
Chapter 6 produces the same result as the known Galerkin method [34] for the lossless
case (in which k, is identically zero and 5 = k). Figure 6.5 shows that both methods
produce identical results, as anticipated. In addition, the full-wave theory developed in
Chapters 5-6 compares closely with the well-known perturbation theory result [3], [18],
as demonstrated in Figure 6.6. Note, a perturbation and full-wave theory comparison for
the phase constant ,6 cannot be made since the perturbation theory only accommodates
for attenuation. Thus, an advantage of the full-wave theory is that it corrects for both
phase and attenuation (note, it was already shown in Chapter 4 that phase correction is

exceedingly important in material characterization measurements).

139

Figures 6.7-6.10 show the effects that ground-plane and strip conductivity have

on the principal-mode propagation constant. Since the surface impedance

Zg.s =(1+ j),Ia)yO/20g.s , the resistance and inductance of the stripline increases as

0g”, decreases. Elementary transmission line theory states that the phase constant ,6 is
proportional to inductance and the attenuation constant is proportional to resistance, thus
one should expect an increase in the phase and attenuation constant as the conductivity
decreases. An examination of Figures 6.7-6.10 demonstrates this anticipated result. In
addition, the strip conductivity has a more pronounced effect on the propagation constant,
as one would expect, since the current flow is restricted to a smaller region. This causes
resistance and stored magnetic energy (hence inductance) to increase, resulting in an
increase of a and ,6. The surface impedance model also predicts an increase in the
attenuation and phase constants as frequency increases. The increase in the attenuation

constant a at higher frequencies is clearly evident from Figures 6.8 and 6.10. The

increase in phase constant ,6 is not as clear due to the normalization with respect to k0.

Upon multiplication of k0 , the increase of ,6 at higher frequencies would be more
clearly evident, but is more cumbersome to present in graphical form due to the
exceedingly large values of ,6. Note, exhaustive results for a stripline having both
ground-plane and strip conductor losses are not given here for the sake of brevity
(perturbation theory predicts that the total attenuation is simply the sum of the two cases
presented here).

Figures 6.11-6.12 reveal the effects that strip conductivity has on the surface

currents kJr and k2. Figure 6.11 demonstrates that the transverse current k, increases as

140

strip conductivity 0', decreases. This is an expected result since lower conductivities
allow for more pronounced longitudinal fields in the conductor, which leads to larger
longitudinal field components (which are predominantly maintained by transverse

currents). Figure 6.12 shows that there is seemingly little effect on the surface current

k, , probably due to the method of surface-current normalization.

The effect that frequency has on the surface currents k, and k, is shown in

Figures 6.13 and 6.14, respectively. The surface impedance increases as frequency
increases, leading to a larger longitudinal field component, hence the increasing
transverse current density in Figure 6.13 is expected. Figure 6.14 reveals that the edge
singularity weakens for the longitudinal surface current k, as frequency decreases. This
is also an expected result since the edge condition is an induced EMF effect. As the
frequency approaches zero, the current density should become constant across the
transverse dimension of the strip.

Figures 6.15-6.16 demonstrate the effects of width/height variation. Decreasing
the width increases the magnetic field concentration (leading to higher inductance) and
also increases resistance. This manifests itself as increased phase and attenuation.
Increasing the width/height ratio has a predictable opposite effect. Figure 6.17 shows
that increasing the permittivity significantly increases the attenuation factor. This is
expected since higher dielectric materials reduce fringing (due to polarization and free-
charge annihilation at the dielectric/conductor interfaces) and consequently forces the
surface current to be more concentrated in the ground plane. The effect on the phase
constant (when normalized with respect to wavenumber k) was not so prominent and

was therefore not plotted. The last plot demonstrates the effect that conductor loss has on

141

the quasi-TEM characteristic impedance of a stripline having dimensions w: 25 mm ,
I: 17.38 mm at a frequency of f = lGHz. Elementary transmission line theory for a

lossless dielectric predicts that the Re{ZO} should be positive and increasing and the

Im{ZO} negative and decreasing as conductivity decreases since

20 = m = ELM—L. = i£_ji (6.94)
G +j60C ij C wC

and because the surface impedance model being utilized is

 

. . (0
2g, = kg, + 1L“ = (1 +1) 2:0 (6.95)
g,S

Figure 6.18 demonstrates this general trend.

142

fi-‘mmg
1 II .

Phase Constant B/k0

Convergence Rate for Propagation Constant C
og=os=1.e6 S/m, w=1.5 mm, t=.635 mm, f=1 GHz

 

 

 

 

 

 

 

 

1.1 — - 0.05
I
V t a i 3
1.0 - o 04
i + [We
0'9 F . . (We - 0.03
................. -I-I-I..
0.8T """"""" ' 1'00,
0-7 5 ~ 0.01
0.6 A l l ‘ 1 1 1 000
1 2 3 4 5 6

Number of Expansion/Testing Terms (NT)

Figure 6.2 Convergence rate for propagation constant 5 = fl — ja.

143

Attenuation Constant or/kO

 

Convergence Rate for Surface Current Kx
00020 f og=os=1.e6 S/m, w=l.5 mm, t=.635 mm, f=1 GHz
0.0018 '

 

£00016 ”
“50.0014 ” ....... . NT‘2

 

 

 

(D ......... =
“30.0012 ~ NT 3

30.0010 -

5

q, 0.0008 ~
> .

3 0.0006 -
a) .
cf.

 

0.0004 ‘

     

0.0002 I

Ulb'ounooIaII|:.;H-. .................
.fl -------------------
11111 an:1'1'-'I'15'-"I""'"‘.""'1" '. 1 . 1 . 1 ""."'41

 

 

0.0000 ‘ ' ' ' '
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Transverse Strip Distance x/w

Figure 6.3 Convergence rate for surface current k,.

144

Relative Amplitude of K2

1.0
0.9
0.8
0.7
0.6
0.5
0.4

0.3
0.2 ‘
0.1

0.0

0.0

l

fir f V

‘jfl

I

.—

Convergence Rate for Surface Current Kz

og=os=1.e6 S/m, w=1.5 mm, t=.635 mm, f=1 GHz

 

 

 

 

 

I
0000000
ca 0 OIDOIOOOIIIIIIIIDIIIOIIIIIOOUDIIII‘ -.-o‘o.

 

 

 

0.1 0.2

l 1 l 1 l 1 l 1 l 1 l _m I 1 l 1 l 1 I

0.3 0.4 0.5 0.6 0.7
Transverse Strip Distance x/w

Figure 6.4 Convergence rate for surface current k, .

145

Surface Current Kz - MoM Comparison
1-0 ’ og=os=1.e200 S/m, w=1.5 mm, t=.635 mm, f=10 GHz
0.9

 

1

0.8 L
0.7 ”

0.6 - — Galerkin Method
-------- Non-Galerkin Method

 

 

 

 

 

0.5 ’
0.4 “

I

0.3

Relative Amplitude of K2

1

0.2

I

0.1

 

ooi 1 l 1 l 1 1 l 1 l l 1 l l 1 l 1 J

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Transverse Strip Distance x/w

 

Figure 6.5 Comparison of MoM theories for surface current k, .

146

Attenuation Constant - Theory Comparison
0 =os=1.e6 S/m, w=1.5 mm, t=.635 mm

 

1.0 ' g

0.9 7

0.8 7 — Full-Wave Theory
0 7 _ ------- Perturbation Theory

 

 

 

Attenuation Constant or (Np/m)

 

 

 

0.0 l 1 A 1 l 1 J
1.00 8.25 15.50 22.75 30.00
Frequency (GHz)

Figure 6.6 F ull-wave and perturbation theory comparison for attenuation constant a.

147

Phase Constant Ground-Plane Conductivity Profile

 

 

 

 

 

 

 

1.02 F os=1.e200 S/m, w=1.5 mm, t=.635 mm
' — Gg=1£200 S/m
o -------- o =l.e7 S/m
34 8
E1. 1.01 * """"" og=l.e6 S/m
‘5’ ................... og=l.e5 S/m
{a .............
t: "1. ----------------
o -----------------------------------
o ..........................
a "‘----------.-.-.-.-.-.-II.'.'.'.'.'.'.'I'.'.'.'I'.'.'.‘I'.‘.'.;I'.'.'.':'.'.'.':'.‘.‘.':;'.'.'::.'.':;'.'.'::.:::.'.'.';'.'.':.'.'.'.'.°.'.:'.':.':'.':.
g 1.00
D...
0.99 ‘ J ‘ ‘
1.00 8.25 15.50 22.75 30.00

Frequency (GHz)

Figure 6.7 Effect of ground plane conductivity on phase constant fl.

148

Attenuation Constant or (Np/m)

Attenuation Constant Ground-Plane Conductivity Profile
os=1.e200 S/m, w=1.5 mm, t=.635 mm

 

 

 

 

 

 

 

2.0 — v
1.5 ~ ...............
"""""""" -— Gg=1£200 S/m
"""""""""""" ' Og=1.e7 S/m
1'0 ’ --------------- O'g=1.e6 S/m
------------- Gg=l.65 S/m
0.5 ................................................................
0.0 ‘ """""""
-0.5 . 1 1 . . ,
1-00 8.25 15.50 22.75 30.00

Frequency (GHz)

Figure 6.8 Effect of ground plane conductivity on attenuation constant a.

149

Phase Constant IS/ko

Phase Constant Strip Conductivity Profile

 

 

 

 

 

 

 

1.06 ' og=1.e200 S/m, w=1.5 mm, t=.635 mm
1.05
" -— os=l.e200 S/m
1.04 7 ------- - os=1.e7 S/m
------- 05:1.e6 S/m
1.03 _ """" 65:1.35 S/m
1.02 ~ .............
101 1— " - ........................................
1"“-------.-.-.-.-.-..11:1.17.11.'.'.'.'I'.'.'.'I'.'.'.'I;'.'.'I'.‘.'.‘.”.'.'.'.".'.'..'.';'.‘.'.':.:.':;'.:.';'.'.':;'.'.':;'.'.'.'.':'.'.'.'::
1.00
1
0.99 ‘ ‘ ‘ ‘ ' ‘ '
1.00 8.25 15.50 22.75 30.00

Frequency (GHz)

Figure 6.9 Effect of strip conductivity on phase constant ,6.

150

Attenuation Constant or (Np/m)

Attenuation Constant Strip Conductivity Profile
og=1.e200 S/m, w=1.5 mm, t=.635 mm

 

 

 

 

 

 

 

 

7 .—
5 ~ ..............
5 I ...............
4 :- ............ Gs=1.6200 S/m
. ,,,,,,,,,,,,,, 05:1.67 S/m
3 ” """"""""""""""" 65:1.66 S/m
’ """""""" 65:1.65 S/m
2 L ......................................
1 L" ......................................
.0 .. ..............
-1 1 l l J
1'00 8.25 15.50 22.75 3000

Frequency (GHz)

Figure 6.10 Effect of strip conductivity on attenuation constant a.

151

Surface Current Kx Strip Conductivity Profile

 

 

 

 

 

 

 

‘ og=1.e200 S/m, w=1.5 mm, t=.635 mm, f=10 GHz
- — os=l.d7 S/m
------- - os=l.d6 S/m
_ ------- os=l.d5 S/m '
f
r """"""""""""" s.‘
#1...;..:sl:: LLLL ;#-..I'-.‘:‘. l 1 l 1 l 1 l . l L J
0 0.1 0.2 I0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Transverse Strip Distance x/w

Figure 6.11 Effect of strip conductivity on surface current k,.

152

Surface Current Kz Strip Conductivity Profile
og=l.e200 S/m, w=1.5 mm, t=.635 mm, f=10 GHz

 

 

 

 

 

 

 

 

 

1.0 ‘ I
0.9 ‘
>12 0.8 ”
“-1 0.7 ‘
3 . — l.d5<os<1.d2008/m
B 0.6 r
E31 0.5 -
a, 0.4 ‘
.2. ’
E. 0.3 -
é .
0.2 '
0.1 '
0.0 11 l 1 L 1 l 1 l l 1 l 1 l 1 l 1 l 1 J

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Transverse Strip Distance x/w

Figure 6.12 Effect of strip conductivity on surface current kz .

153

0.0025

0.0020

0.0015

0.0010

Relative Amplitude of Kx

0.0005

0.0000
0.

l

l

l

I

 

Surface Current Kx - Frequency Profile

og=os=1.e5 S/m, w=l.5 mm, t=.635 mm

 

— f=1GHz
-------- f=10 GHz
--------- f=30 GHz

 

 

 

 

 

 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Transverse Strip Distance x/w

Figure 6.13 Effect of frequency on surface current k,.

154

 

Surface Current Kz - Frequency Profile

 

 

 

 

 

 

 

 

1.0 - og=os=l.e5 S/m, w=1.5 mm, F635 mm

0.9 P
Q 0'8 b — f=0.1GHz
‘15 0.7 7 ------- f=1 GHz
0 ‘ ------- - #10 GHz
3 0'6 " ------- f=30 GHz
'5'- 0.5 ~
5 ..
0) 0.4 ’— ‘71
.2. .4
E 0.3 ~ 0“,.
£2 ____,__-.—*‘

02 IUIam-n-nwwnunnusn519151515235;3:34:32:3131'. ----------

0.1 7

0.0 1 * I 1 1 1 1 1 l 1 1 1 1 1 1 . 1 4 1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Transverse Strip Distance x/w

Figure 6.14 Effect of frequency on surface current k, .

155

1.020
1.018
1.016
1.014
1.012
1.010
1.008

Phase Constant B/k0

1.006
1.004
1.002
1.000

Phase Constant - Width/Height Profile
_ og=os=l.e6 S/m, f=10 GHz

 

 

1 1 1 1 l 1 l l 1 L l

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Width/Height Ratio w/t

Figure 6.15 Effect of width/height ratio on phase constant 6 .

156

Attenuation Constant or (Np/m)

Attenuation Constant - Width/Height Profile

30 og=os=1.e6 S/m, fi—lO GHz

‘ j

l

2.8

l

2.6

I

2.4

I

2.2

l

2.0
1.8 “
1.6

T—l

T

1.4

 

1.2”

 

 

1.0 L 1 l 1 1 1 l 1 l 1 l . l 1 l
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Width/Height Ratio w/t

Figure 6.16 Effect of width/height ratio on attenuation constant a.

157

rmr—“a

Attenuation Constanta (Np/m)

Attenuation Constant - Refractive Index Profile

 

 

 

 

 

 

 

15 - w=1.5 mm, t=.635 mm, f=10 GHz
10 ~
' ..... ""w’ -— og=os=1.e7 S/m
”I”. -------- og=os=l.e6 S/m
I]... I """"" - og=os=1.e5 S/m
5 ;.. """""
0 l l l l l 1 I 1 l L l l I
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

Refractive Index

Figure 6.17 Effect of refiactive index on the attenuation constant a.

158

 

Characteristic Impedance - Conductivity Profile
w=25 mm, t=l7.38 mm, f=1 GHz

 

 

 

 

 

50-0 111;.“ 1.11.5 1.11..“ 1“...“ 11:; 96
104 105 106 10" 3“ 108 3 6109

Conductivity og=os (S/m)

Figure 6.18 Effect of conductivity on characteristic impedance.

159

 

Chapter 7

ANALYSIS OF A LOSSY MICROSTRIP TRANSMISSION LINE

HAVING AN IMPERFECTLY-CONDUCTING GROUND PLANE

7.1 Introduction

The analysis of an imperfectly-conducting microstrip transmission line [51]-[66]
situated over an imperfectly-conducting ground plane will be considered in this chapter.
Specifically, the effect that the lossy conductors have on the principal (i.e., dominant)
discrete-mode propagation constant and respective surface current will be investigated.
Understanding the nature of the principal mode is fundamental to the material
characterization process (as well as other applications), and thus provides motivation for
this study.

The first step in the above analysis is to (utilizing an electric-type Hertz potential)
find the electric-field dyadic Green’s function of the EM field for a general current source
embedded in a lossy microstrip background environment. Next, an EFIE (electric field
integral equation) is developed by specializing the general 3D electric current to a strip
surface current located at the cover/film interface and satisfying appropriate boundary
conditions on the strip conductor. As a final step, the EF IE is solved using a MoM
(method of moments) technique and the principal-mode propagation constant and

corresponding surface current distribution are subsequently identified.

7.2 Geometry

160

 

The microstrip background environment is depicted in Figure 7.1. The structure

is comprised of two main regions located above an imperfect ground plane
(—oo < y < —d ) having conductivity 0'1: and intrinsic impedance Zc. The cover
(0<y<oo) and film (—d<y<0) regions have properties (81,110) and (82,110),
respectively. The cover/film interface is located at y = 0 and the film/conductor

interface is located at y = —d. The general 3D electric current source density .7 is

assumed to be localized within region I .

 

Y
Principal
wave
av
Reflected Principal é
wave wave Regionl (8,,110)
y=0 :
Reflected Region2 (82,110)
wave
=—d

 

Conductor (oc,ZC)

Figure 7.1 Background environment for the microstrip field applicator.

7.3 Electric-Field Dyadic Green’s Function

7.3.1 General Formulation

Similar to the development in Chapter 5, an electric-type Hertz potential if and

161

the superposition method of solution can be utilized in determining the electric-field

dyadic Green’s function in region I . The total solution for if in each region will be

 

iil=iflp+iflr ...0<y<oo (7.1)
772:5; ...—d<y<0 (7.2)
where (for a = x, y, z)
V2751]; +k127r1’; = — :1“
16051 ...0 < y < oo (7.3)
szrl'a + klzzrfa = 0
(7.4)

V275,, +k227r§a =0 ...—d <y <0

The background structure is invariant along the x and z directions, thus Fourier

transformation on (7.3) and (7.4) leads to, with the aid of the transform pair of Chapter 5,

 

 

 

 

2~ ... 1
a 7Z1}; 2...], _ .10
2 —pl7tla — .
16051
2 >...0<y<oo (7.5)
a ir 2~r
éa-pl 111:0
2~r
6’2 2" =0. —d<y<0 (7.6)

 

where p13 = ‘IAZ -k€'2 with Re{ P12} > 0 chosen and 112 = 52 +5 2. The solutions of

(7.5) and (7.6) in the complex ,1 -plane are (based on the experience of Chapter 5)

'Prly‘y'l "' T 1
if", =55, +7212, = Ie JaQ’y)dy'+1V,;e-w +ngep‘y ...0 <y <00 (7.7)
J, 2P1 1051

01'

~ _ ~ ~r _ + — y + - y - py I
”la—251+7rla—Vae p‘ +Wlae p‘ +Wlael ...y <y<oo (7.8)

162

file: = 721’; +72]; = Va—ep‘y + lee-p'y + ngep'y ...0 < y < y' (7.9)
and
7‘22, = 76;, = Wzge‘l’zy + Wz‘aepzy ...— d < y < 0 (7.10)

where

 

1P1)” ~ _. '
Va: Ie Jed/15y )Cb/ (7.11)

,1 2P1 16081

The spectral coefficients Wli,W2ia are computed in the next section using appropriate

boundary conditions.

7.3.2 Computation of Spectral Coefficients
The spectral coefficients can be computed by implementing the following
boundary conditions (note, the ,1 dependence of the transform-domain Hertz potentials

has been dropped for notational convenience)

 

 

fila(y—>oo)=0 ...a=x,y,z (7.12)
721,,(0) = 1112152040) ...a = x, y,z (7.13)
270—17159-_-N2@2L(911.61:“ (7.14)
6y 6y
653(0) — 653(0) = 0 — 41211411110) + 1611110)] (7.15)
.. 1 a“ a —d
120(41): 0ch ”26; ) ...a=x,z (7.16)
afiZy(_d) _ . ~ . ... . ..
___ay—' "’ JwEZZcfl2y(—d)—J§”2x(_d)—1472.22 (_d) (717)

163

where the dielectric contrast and intrinsic impedance are given by

 

1112:52— , Zc=(1+j) “”0 (7.18)
81 20".

Equation (7.12) is the radiation condition, equations (7.13)-(7.15) are the boundary
conditions at the cover/film interface and equations (7.16)-(7.17) are the impedance
boundary conditions at the surface of the imperfect (but good) conductor. As anticipated,
there are 12 boundary conditions required to uniquely compute the 12 unknown spectral
coefficients.

Enforcement of the radiation condition leads to the following result when applied

to equation (7.8)
fi1a(y—)oo)=0 :> W1;=0 ...a=x,y,z (7.19)

since Re{ p1} > 0. Thus, the transform-domain Hertz potential in region 1 consists of a

principal wave and only an up-going reflected wave, that is
fila = VJe-p‘y + WIZC—p‘y ...y' < y < oo (7.20)
771a = Vgep'y +WIZe'p'y ...0 < y <y’ (7.21)

The tangential spectral coefficients can be computed as follows. First, insertion

of (7.10) into (7.16) leads to

Wzgepz" + Wz‘ae‘Pzd = —L;—(W;;,epzd - Wz’ae‘l’z") ...a = x,z (7.22)
0C C

and upon solving for W2“; gives

 

 

W73, = -R,e‘2P2dW2‘a ...a = x, z (7.23)
R2 : 0'ch ”p2 = 1—p2 /0'ch ’ 0' Z = (1+1) (01“00'0 (7.24)
Uch+p2 1+p2/O'CZC c c 2

164

Next, substitution of (7.21) and (7.10) into (7.13) and (7.14) produces the result
V; + 141,; = N2(W,:, + 1112;) ...a = x, y,z (7.25)
p104: —W1;>=—p2N2(W2t. -W1‘..) ...a=x,z (7.26)

Solving the above equations in terms of W]; and using (7.23) gives

m; = —V,; + N2(1— Rze—szd )W,;, ...a = x, z (7.27)
111,; = V; — 32— N2(l + Rze'zpzd )Wz'a ...a = x, z (7.28)
P1

The spectral coefficient W2}, is determined by equating (7.27) and (7.28), leading to the

following expression
W2; =7",Vc; ...a=x,z (7.29)

where the tangential transmission and interfacial reflection coefficients at the cover/film

interface are

 

2 _
7; ___ 2 Pl 4}) d R1 = P1 P2 (730)
N (P1+P2)(1—R1Rze 2 ) P1+P2

Substitution of (7.29) into (7.23) and (7.27) or (7.28) identifies the remaining tangential

spectral coefficients, that is
W2; = —R27}e_2p2dVa" ...a = x, z (7.31)
W]; =R,V0: ...a=x,z (7.32)
where the tangential reflection coefficient is

R_m-&€%d
t _ ‘2P2d
1 — R1 R28

 

(7.33)

It is noted in passing that the above spectral coefficients reduce to the expected perfectly-

165

conducting result [45] in the limit as ac —) 00.

The tangential components of the spectral-domain Hertz potential in region 1 can

now be mathematically/physically identified using (7.7), (7.19) and (7.32). The result is

 

-p1|y-y'| -p1(y+y') ” “ r
fila =fill; +5101 = I 5—2——+R, e 2 Jag/1”” )dy' ...a =x,z (7.34)
y, P1 P1 1051

The first term in (7.34) represents the principal-wave contribution that travels directly
from the source point y' to the observation point y. The second term in (7.34)
represents the reflected—wave contribution that travels downward from the source, gets
reflected from the cover/film interface and travels upwards to the point of observation.
Note, a closer examination of R, in (7.33) reveals that the reflected-wave contribution is
actually comprised of a primary reflection from the cover/film interface and a secondary
reflection from the film/conductor interface. A similar picture holds for the normal
component of the Hertz potential.

The nonnal-component spectral coefficients are determined in the following

manner. First, substitution of (7.10) into (7.17) leads to

‘172 (W239 pzd — Wz-yewzd) = ngzzc (Wi‘yepzd 1‘ W 2_ye—p 2d )

(7.35)
—J'5(W2:ep2d + szce—pzd) - 14(W223p2d 1' Wie-pzd)
Solving (7.35) in terms of W+ produces the result (after some algebra)
2y
ny = C,,e‘2P2dW,‘, - R'ze‘zl’zdwz‘y + Cfle'zpdeZ‘, (7.36)
where the normal interfacial reflection and coupling coefficients are
1‘52 = fwgzzc " P2 = _ 1 ’ngzzc /P2 (737)

 

 

1.608226 +p2 1+jngZc /P2

166

 

. . l—Rz 1—112
ny=J§C a Cyz:}CC 9 C— — (738)

— 1503221: + P2 - P2(1+ ngzzc / P2)
Next, the boundary condition result of (7.25) is used to express W1"; in terms of W23},
namely
_ 2 _
W]; = —Vy + N (ny + Wzy) (7.39)
Substitution of (7.36) into (7.39) gives
W]; = _V; + nyN2e_2p2dW2} + N2(1— 11212—243" )Wz‘y + CWN2e72p2dW2; (7.40)

The final boundary condition is enforced by substituting (7.21) and (7.10) into (7.15),

leading to
1210/; — W1;)+ mm; — W2".)=(1— N2)[14(W5; + Wz‘.) + j€(W2+z + W29] (7.41)

Solving (7.41) for W]; and using (7.23) and (7.36) gives

 

p2C1.e‘2”2d 4411- N2)(1— Rze—ZPZd)]W’

+ _ _
”/ly—Vy +|: P1 21:

(7.42)

— 'p—Z' (1 '1‘ Eze_2p2d )W2_y '1'
P1

 

Ipzcfle‘ZPZ" -1'6(1—N2)(1-R1e‘2"2d)I _
p W22
1

Using (7.29) and equating (7 .40) and (7.42) produces the following result for W23, (after
a little algebraic effort)

W23, = TnxV; + Tny V; + T: V; (7.43)
where the normal transmission coefficients are

(11N2 -p2)C,.e‘2P2d + 7411- N2)(1 — 121-2121)

Tnx=_Tt 2 -- -2 d
(PIN +P2)(1"R1R29 p2)

 

(7.44)

167

2p
Ty = 1 _ _ (7.45)
n (PrN2 + P2 )(1 - R1R23_2p2d)

 

T. _ T (1211112 -p2)C..e“2"2" +1411-N2)(1-R2e‘2P2d)
——t — — —
(p1N2 +122 )(1—R1R2e 1’2")

 

(7.46)

n

2—
R,=————p‘N2 1’2 (7.47)
p1N +112

Substitution of (7.43) into (7.36) and (7.40) or (7 .42) determines, with the aid of (7.29),

the remaining normal-component spectral coefficients

W2”; = (CW7; — R,Tn")e‘2P2dV; — 11262102" TnyV; + (CWT, — R,T,Z)e'2P2dV,‘ (7.48)

W1; = my; + 1851/; + RgV; (7.49)
where
R; = N2 [nye'zi’zdz + (1 - 1528—21214 )T;] (7.50)
R; = N2(1— 11212—242" )Tny -1 (7.51)
R; = N2 [Cyze‘zpzdz + (1 — Eze'2pz" )T,f] (7.52)

The normal component of Hertz potential in the spectral domain for region I can

therefore be identified using (7.7), (7.19) and (7.49). The result is

 

-P1|y-y'| 1 -p1(y+y') 2 r 1
7in :51]; +5.3: I yi——+R,,e .J(f1’y)dy' (7.53)
,1 2P1 2P1 1061 _
where
11,, = 5611; + 9115+ 2R; , .7: 1.}, + y‘)", + 2] (7.54)

The first term in (7.53) is the principal-wave contribution and the second term is the

reflected-wave contribution. Note, equations (7.34) and (7.53) state that the normal

168

component of current only couples into the normal component of potential, whereas the
tangential components of current (upon interaction with the interface) couple into both

the tangential and normal components of Hertz potential.

7.3.3 Identification of the Hertz Potential Dyadic Green’s Function
The principal part of the spectral-domain Hertz potential in region I can be easily

deduced from section 7.3.1 or 7.3.2, that is

~ ~ _' r j I) ' I
III; = IGp(/l,y|y )Mdy ...a = x,y,z (7.55)
. 16081
y
where
,_ 1 ~ g e-p1Iy-y'l
Gp(1,J’|)")=GP(/1,y-)") :-_2p— (7-56)
I

The principal part of the spatial-domain Hertz-potential Green’s function immediately

follows by taking the inverse transform of (7.55), resulting in

{2,(7)

 

. .. 1.0') .
_(27: )24) I I ”game/’1 1122: VIGP(rr)| —111ij1 (7.57)

where

 

GP(F|F’)=GP(F—F')= I I GP(2 y— y)ejMF ”(122 (7.58)

(2 )21...

Thus, the principal Hertz-potential dyadic Green’s function is identified as

ifflf): IGP(F|F')-1.(L)-dV' , GP(F|F')=IGP(F|F') (7.59)
V1 16051

In a similar manner, the reflected Hertz-potential dyadic Green’s function is

identified by

169

ii, (F) = I G’(F|F’)-1@—dV' (7.60)
V.

1031
where
6' = 205+ 90;,“ 90:0,; + pop + 20,32 (7.61)
r 1 0° 00 ~ .° I '7- F—F'
005 = ——-— I IG;fl(/l,y|y )ej’u )dZ/l ...a,,B = x,y,z (7.62)
(27r) _oom
,, e-pl(y+y')
050, = R, —— ...a = x,z (7.63)
2P1
~ e’PiU‘l‘y')
G;a = R: —— ...a = x,y,z (7.64)
2P1

The total Hertz-potential dyadic Green’s function in region 1 is therefore, by

superposition
6(7)?) = 6P0)? ) + 6' (7| F') (7.65)
and the total Hertz potential is
771(7) = I dawn-glam (7.66)
V. [0’51

7.3.4 Identification of the Electric-Field Dyadic Green’s Function
The electric-field dyadic Green’s function in region I is identified by using the

following computation

E = 1:377, + VV . if, = I Ge (7|? ) - .7(F')dV' (7.67)
V.

Of course, the field will contain a principle and reflected part, that is

E = EP + 13" (7.68)

170

where

ii" = k,2i:°{’ +vv.;z‘{’ = I GeP(F|F')-.7(F')dV' (7.69)
V!
E’ = klzfif +VV-7'r’ I G” (rIF') J(r )dV’ (7.70)

The principal part of the electric-field dyadic Green’s function was previously
investigated in Chapter 5 and will therefore not be repeated here. Application of (7.70)
leads to the following expression for the reflected part of the electric-field dyadic Green’s

function

G” = xGe'x + x0” y + xGex;2 +

 

 

 

 

 

 

 

(7.71)
yG;;x+ynyy+yG;z 2+2Gerx+2Ge zyy+2szr2
2 r 620’
G§;=—[k1216’+a 62+ yx] (7.72)
16031 6x 6x6y
620’ 620' 2 r
G§C=J— 22 , G:;= .1 ”+6622 (7.73)
1608, 6x6y 16081 6x6y 6x62
626' 2 r
G;;_— — 1620;.» 22" + a G” (7.74)
jaw] 6y ayax
2 r 2 r
a G 2 r
G$=.—1—— 6262+ 2” , G”— kEG;,+ {2 + a G“ (7.75)
16081 jwal 6y Byaz

 

 

 

2 r 2 r 2 r
G” — 1 6 G +6 G er 1 6 C” (7.76)
jaw] azax 626y “Ty: jaw] azay
2 r 620'
G” = —k 2G;+ a 6232 + a a; (7.77)
11081 2

171

620;, 1 °°°° ~, -~.;.;
6x2 r0702 I I’ZG”em W

 

 

—oo-oo

 

aZGr co co aG~r .~ « .,
ya _ 1 I If: ya eJA-(r—r)d2/1 ...a=x,y,z
—oo

 

My 12202 5.. 5y

aGya :‘legipflz—R—g-e-pl(y+y') oua=xryaz
6y 2P1 2

626;, _ 1 °°

 

w '° —- w
lééégaej’”"’ )dzll ...a =x,z

-oo

6va — _ (27;)2 _w

 

aZGra 1 <1) <1) 2 Z(‘ ) 2

y _ "r ' ~r—F' _

ay2 —(2fl)2 I IpleaeJ d ,1 ...a—x,y,z
—oo—ao

 

 

 

 

2 r 00 co “’ -
a an = 1 l 11,660,; ejA.(F-F')d21
Mac (2202 ...... 5y
2 co "' s
a 62;, = 1 00 11,46ng ejl-(F-F')d2’1
(3)262 (27!)2 _oom 6y
6G;a =_ IR e-P1(y+y) =_&e_p‘(y+y') ...a =x 2
8y I 2P1 2 ,
620’ °° °° 652' -* - ~
ya : l 2 I [jg—yie-I’I'v—r )dz/l. ...a =X,y,z
azay (27!) _GHD 6y
620;, _ 1 °°

m 7' - —-v
ICZngej/l-(r-r )dzl

—CD

 

622 (27:)2 m

7.4 Development of the Lossy Microstrip EFIE

(7.78)

(7.79)

(7.80)

(7.81)

(7.82)

(7.83)

(7.84)

(7.85)

(7.86)

(7.87)

The EFIE for the discrete-mode propagation spectrum of the microstrip field

applicator (shown in Figure 7.2) is obtained by following an analogous set of steps as in

172

Chapter 6. The resulting coupled EFIE’s (upon suppressing the limit notation as was

 

 

 

 

 

 

 

done in Chapter 6) are
m . a - I
I dgelé" I [an,(x') + C,,k,(x')]e‘122 dx' — Zskx(x) = 0 (7.88)
-.., _a
I dgefi" I an,(x') + C,,k,(x')]e'/¢X'dx' — Zskz (x) = 0 (7.89)
where

e‘p‘y A k2 — 2 —

Cxx(é,y,é) = j27rwg [ ( 1D" g )+§2C] (7.90)
l
e"P1y A _
sz(69y9;)=czx(€9y’g)=_€._;_—[_h—C) (791)
127mg] D

e_p'y A k2 — 2 —

022694) = j27rwg [ ( 1D,); )+§2C] (7.92)
l
A(§,;) =sinh p2d+ 1’; cosh pzd (7.93)
C C
B(§,{) = p,A(N2 -1) [cosh pzd + j 0822“ sinh pzd] (7.94)
P2
— B + / 0' Z
C(56) = (p53), 2 C) (7.95)
p2 p
D”(§,;) = [p1 + 2- Jsinh p2d+ p2 [1+ ‘ Icosh pzd (7.96)
0026 CZC
2

De(§,;) = (plNz + jcogch )cosh pzd +[p2 + jW—Jsinh pzd (7.97)

P2

Note that ansz are even in 4‘ and sz,sz are odd functions of 6. In

173

addition, examination of (7.90)-(7.97) reveals that these coefficients C0,), are even with
respect to p2. Hence, the branch point k2 (and associated branch cut) in p2 is a
removable singularity. The branch point k] in p1 is not a removable singularity since the

coefficients are not even in p].

Conductor (OS,ZS)

\k— 23 ——’l Regionl (81,110)

y=0 _ )

 

Region 2 (829p0)

 

y=-d
Conductor (0C,ZC)

Figure 7.2 Cross-sectional geometry of the lossy microstrip field applicator.

7.5 MoM Solution for the Lossy Microstrip EFIE
The dual EFIE’s in (7.88) and (7.89) can be solved using a MoM technique.
Similar to Chapter 6, application of expansion and testing functions leads to the following

general formulation

N, N2
2 AJ’cnxnaxn + Z Agnazn : O

'7 17‘ ) ...m =1,...,N, + N2 (7.98)
2 Aznaxn + Z Agnazn : O

n=l n=l )

 

174

where

afl =y11m lcapfamgflnd5+0$5ap ..a ,3: x 2

form =fam(€)= Itam(x)ej;xdx ...a=x,z

—a

a
an = gan(é) = 1351,1056?”ng dx, ...a = x,z

—a

1...a=,6

13“": —Z I tam(x)ea,,(x)dx , 5afl=IO (”fl

'0

(7.99)

(7.100)

(7.101)

(7.102)

Since the microstrip transmission line studied in this chapter exhibits symmetry,

the following analysis (as in Chapter 6) prevails

Nil Amnaxn'l' ~23] Agna C102”:

n=0 n=0 L

Nx-l Nz—l
run run
2 Azx axn + 2 A2: 0
n=0

n=0

...m = O,...,Nx + N2. -1

 

 

Cn(€,€)=. ' [’“k' “+62CI

1275051 Dh

C..<:.6>=C..(:,c>=—— ‘24 —(i—5I

1 27mm] Dh

 

2 2
C..<64>= ‘ I’M"; N425]

j27f€08l Dh

where, for even modes,

Amn_ " 80 1C3“; €)Ifxm (ga)1gxn(§a)d§—Z s Ixmn

175

(7.103)

(7.104)

(7.105)

(7.106)

(7.107)

A2" = 18a 1C... (5,0152... (501:2, (more:
0

A3" = —18a Ica(:,;>12m(:a)1;m<5a>d:
0

A3.” = 8a 1sz (15.01;, (5a)1;,,,(ga)d§ - 231m
0

and for odd modes

AS." = 8a JCM(:,c>i;m(éa)i;m<éa>d:4.11m
0

A3" = —1'8a I C12 (6,022,, (éafigzneaw:
0

A3" = jsa 1C... (5,072,, (507;, (and:
0

A2?" = 8a Isz(€,C)1~?zm(§a)i§zn(§a)d<§-Zsizmn
0

(7.108)

(7.109)

(7.110)

(7.111)

(7.112)

(7.113)

(7.114)

Thus, the analytical procedure for determining the propagation constant and surface

current distribution is identical to Chapter 6, except that the factors, C a , are different.

7.6 Numerical Results

In this section, the numerical results for the microstrip propagation constant

5 = ,6 - ja and surface currents kx and k2 will be discussed. It will be assumed that

n1 = Jar] =1 and n2 = Jan = 3.13 , unless specified differently. Since the convergence

rates for the microstrip and stripline propagation constant and surface currents are

176

similar, no convergence plots will be provided. In addition, since the non-Galerkin and
Galerkin microstrip MoM techniques produced identical results for the lossless scenario,
no comparison plots will be supplied (as was done in Chapter 6 for the stripline field
applicator).

Figure 7.3 shows that the full-wave theory developed in Chapter 7 compares
reasonably well with the known perturbation theory result [3], [18]. Figures 7.4-7.7 show
the effects that ground-plane and strip conductivity have on the principal-mode
propagation constant. Similar to Chapter 6, the phase and attenuation constant for the
microstrip increase as conductivity decreases and strip losses are more prominent, as
expected. Figures 7.8-7.9 demonstrate the effects that strip conductivity has on surface
current distribution. The longitudinal current is not significantly influenced by conductor
loss and surprisingly, the transverse surface current decreases as conductivity decreases
(this is opposite to the behavior of the transverse current for the stripline device).
However, since the microstrip supports hybrid modes, this behavior may be anticipated.
Finally, Figures 7.10-7.11 reveal the effect of frequency on the transverse and

longitudinal surface current densities.

177

Attenuation Constant - Theory Comparison
0' =os=1.e6 S/m, w=1.5 mm, t=.635 mm

 

     
 

 

 

 

 

 

7 ~ g

E 6 ’ ............

ca. ’ .........

E 5 1 ...........

:3 . .......

E 4 _ ..........

m L ,,,,

s .....

U 3 - ...... — Full-Wave Theory

"a , ------------- Perturbation Theory

2 2

c:

a) .

3 0

<1: 1 .
O l A L r l I
1.00 8.25 15.50 22.75 30.00

Frequency (GHz)

Figure 7.3 Full-wave and perturbation theory comparison for attenuation constant a.

178

Phase Constant [3/k0

Phase Constant - Ground Plane Conductivity Profile
os=1.e200 S/m, w=1.5 mm, t=.635 mm

   

 

   

 

 

 

 

 

3.05 '
2.95 L ”..g?”
l ....... g=1.e200 S/m
--------- og=1.e7 S/m

------------ og=l.e6 S/m

2.85 ------- og=l.e5 S/m

2.75 ' ‘ I ' ‘ '

1.00 8.25 15.50 22.75 30.00

Frequency (GHz)

Figure 7.4 Effect of ground plane conductivity on phase constant ,6.

179

Attenuation Constant a (Np/m)

Attenuation Constant - Ground Plane Conductivity Profile
os=1.e200 S/m, w=1.5 mm, t=.635 mm

 

 

 

 

 

 

 

7.5 -
6.5 ~ ............
5.5 - ............
4'5 — ........... ___... og=l.e200 S/m
L ------------- - O‘g=l.e7 S/m
3'5 b """"""""" Gg=1.e6 S/m
---------- og=l.e5 S/m
2.5 - .......
1.5 L— ..0' .....................................
0-5 .............................................................................
'O.5 ‘ l a 1 1 - 1
1-00 8.25 15.50 22.75 30.00

Frequency (GHz)

Figure 7.5 Effect of ground plane conductivity on attenuation constant a.

180

Phase Constant B/kO

3.05

.N
co
0:

2.85

2.75

I

 

Phase Constant - Strip Conductivity Profile
og=1.e200 S/m, w=1.5 mm, t=.635 m

I
.....
0".
.......
IIIII
D. I'
.....
.......
....
"""""
v' N
.....
O a.
.....
O .0
a
‘0
O
s
I.
0
Q
0
0
I
I
C
‘I
I
‘I
0
I

.a
o
o
'o
a
o
a
o
a
o
9'
o

......
g 0
O I.
O o
0' V
. Q

  

Ԥ
..
Q

 

   
  

 

 

 

 

.......... — os=1.e200 S/m
------------- - cs=l.e7 S/m
---------------- os=l .e6 S/m
------- os=1.e5 S/m
8.25 15.50 22.75 30.00
Frequency (GHz)

Figure 7.6 Effect of strip conductivity on phase constant ,6.

181

Attenuation Constant a (Np/m)

Attenuation Constant - Strip Conductivity Profile
og=1.e200 S/m, w=1.5 mm, t=.635 mm

 

 

 

 

 

 

 

1o — .......
9 p .........................

8 .- ..................

7 ~ .............. — 651.6200 SM
67 ITIII .... 2:12:32

5 k """" °s=1.e5 S/m

4 — o.

3 - .........................................................
2 1' .......................................

1 ___________________________________________________________________________________
o >’ """""

Frequency (GHz)

Figure 7.7 Effect of strip conductivity on attenuation constant a.

182

Surface Current Kx - Strip Conductivity Profile
og=1.e200 S/m, w=1.5 mm, t=.635 mm, f=10 GHz

 

 

 

 

0.010 .

0.009 r

’ — 05:1.e200 S/m

:2 0003 f ------- - os=l.e7 S/m
€45 0.007 L """"" 65:1.86 SA“
0 1 ------- 65:].65 S/m
“c _
g “006 1
g 0.005 ~ ......................
q.) 0004 _ ..... ~. ”3:;
.2 . ‘. .. '...:
E 0.003 -
O ..a '..
'3‘ 0.002 -

0.001 ~

0.000 . J A l 1 1 L A 1 A l g L r l A l

 

 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Transverse Strip Distance x/w

Figure 7.8 Effect of strip conductivity on surface current kx.

183

Relative Amplitude of K2

1.0 '

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

og=1.e200 S/m, w=1.5 mm, t=.635 mm, f=10 GHz

Surface Current Kz - Strip Conductivity Profile

 

—O'

 

O'
0'
O’

S
S
S
S

=1.e200 S/m
=1.e7 S/m
=1.e6 S/m
=1.e5 S/m

 

 

 

’...-......
-.-..._.. ...-..I-C-ICCCCC.
........-.-....--.....- Ill-ll

 

'
..
‘o
....‘C
-
...

...-nuulll--.‘-:n.-
. ... .I ........ u ..

 

 

 

0.3 0.4

0.5

0.6 0.7

Transverse Strip Distance x/w

Figure 7.9 Effect of strip conductivity on surface current k2 .

184

Surface Current Kx - Frequency Profile

 

 

 

 

 

0025 _ og=os=1.e6 S/m, w=1.5 mm, F635 m

g 0.020 r .............................

t...

O

S _

L37. 0015 — f=1 GHz

2‘ """"" - #10 GHz

<1: 0 010 _ --------- {=30 GHz

.2 '

E

a) ‘.

ad _____________________
0005 ~ ............................ ..
0.000 “A” .... ‘ E 4 4 a

 

 

 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Transverse Strip Distance x/w

Figure 7.10 Effect of frequency on surface current kx.

185

Relative Amplitude of K2

I

1.0

0.9

l

0.8
0.7 '

I

 

0'6 — f=1 GHz

 

0.5 7 ------- - f=10 GHz
--------- #30 GHz

 

 

 

0.4
0.3

 

 

0.2
0.1 "

 

l . l l . l 1 J

0.0 . l 1 l l l l .
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Transverse Strip Distance x/w

186

Chapter 8

CONCLUSION

This dissertation has provided several methods and schemes, beyond the NRW
technique, for accommodating errors in electromagnetic material characterization
measurements. Chapter 2 discussed two methods, the direct and deembed techniques, for
measuring and characterizing samples located within multi-layered materials.
Experimental rectangular waveguide measurements were provided to verify the
theoretical analysis. Although both methods are based on the wave-matrix approach, it
was shown that the direct method must be used if sample homogeneity is to be accurately
monitored.

Chapter 3 provided a theoretical technique for accommodating sample-to-wall
gaps in rectangular waveguide measurements. The gaps were handled by regarding the
waveguide as inhomogeneously filled in the cross-sectional plane with LSM and LSE
propagation modes supported in the sample/gap region. Characteristic equations for the

corresponding propagation constants were derived and solved numerically to determine
shifts from the ideal T E10 propagation constant of a uniformly-filled guide. A modal

analysis was utilized to obtain, under small gap conditions, approximate expressions for

the wave impedance and interfacial reflection and transmission coefficients. This was
done by considering a single T E10 mode incident upon, and reflected from, the sample

and only a single LSM or LSE mode inside the sample region. Approximate

expressions for the scattering parameters were also obtained using wave matrices. Gap

187

analysis measurements for a non-magnetic Alumina (A1203) sample were performed to
substantiate the theoretical analysis. It was shown, as expected, that top/bottom gaps had
a much more significant impact on constitutive parameter errors (5%) than left/right gaps
(<1%) due to the electric field configuration for the TE“) rectangular waveguide mode.

In Chapter 4, a coupled-mode perturbation theory based on an impedance
boundary condition was used to account for wall loss due to imperfectly-conducting
boundaries in rectangular waveguide field applicators. The coupled-mode perturbation
theory was specialized to single-mode operation and a complex correction to the ideal
TElo mode propagation constant was subsequently identified. This complex correction
was compared with the standard power-loss method, which provides attenuation
correction only. Rectangular waveguide measurements showed that obtaining the
additional phase correction using the coupled-mode perturbation theory was crucial to
accurately accommodating for the imperfectly-conducting walls. In addition, a high-
dielectric resistive-card sample was measured to reveal the sensitivity that constitutive
parameter computations have when changes in conductivity occur.

The main objective of Chapter 5 was to develop the dyadic Green’s function for
the electric field maintained by a general 3D current source immersed in a lossy stripline
background environment. This goal was achieved by employing an intermediary electric-
type Hertz potential in the Fourier transform domain and satisfying appropriate boundary
conditions. The electric-field dyadic Green’s function was formulated using two
methodologies. The first method involved the analysis of a symmetric slab waveguide in
which Hertz potential boundary conditions were invoked at the dielectric interfaces, the

outer regions were allowed to become good conductors and the electric field was

188

carefully calculated using relations in Appendix A. The source-point singularity was
discussed and appropriately handled using Leibnitz’s rule. Although this singularity had
no bearing on the results of Chapter 6, a clear understanding is vital for thick-strip
analysis and applications. The second method utilized Hertzian-potential impedance
boundary conditions to ultimately compute the electric-field dyadic Green’s function. It
was shown that both methods produced identical results, but the second method required
considerably less effort. The magnetic-field dyadic Green’s function, which was not
required in this dissertation, was also computed for completeness.

Chapter 6 involved the analysis and numerical solution of a lossy stripline
structure. These objectives were accomplished through the following sequence of steps.
First, the general 3D current source of chapter 5 was specialized to an infinitely-long strip
surface current symmetrically located between imperfectly-conducting plates. Next, an
EFIE was developed by satisfying an impedance boundary condition on the strip
conductor in the axial Fourier transform domain. As a final step, the EF IE was solved
using a MoM technique and the principal-mode propagation constant and corresponding
surface current distribution were subsequently identified. In regards to the MoM solution
for the lossy stripline, a non-Galerkin’s MoM technique using Chebyshev polynomials of
the first and second kind had to be devised since the well-known Galerkin’s method
produced divergent integrals. The resulting integrals from the non-Galerkin’s technique
were identified and computed in closed form with the aid of Appendix E.

The results of the full-wave lossy stripline analysis and corresponding MoM
solution revealed several expected phenomena. First, it was shown that the full-wave

analysis lead to a complex correction to the propagation constant, which is vital in the

189

material characterization process. Next, it was observed that the MoM solution
converged rapidly since the chosen Chebyshev polynomials closely modeled the real
physical behavior of the surface currents. Indeed, it was shown that even one expansion
term lead to highly accurate results for the surface currents and propagation constant. It
was also demonstrated that the Galean and non-Galerkin MoM solutions produced
identical results under lossless conditions. The full-wave theory compared closely with
the standard power-loss perturbation method. Effects of ground and strip conductivity
and frequency on the propagation constant and surface currents were examined and
discussed. Finally, the influences of strip width and refractive index were reported.

A lossy microstrip transmission line was investigated in Chapter 7 using a
procedure similar to Chapters 5 and 6. First, the dyadic Green’s function was developed
for the electric field maintained by a general 3D current source immersed in a lossy
microstrip background environment. The general 3D current was then specialized to an
infinitely-long strip surface current and boundary conditions were invoked on the
imperfect strip conductor, leading to an EFIE. In the final step, the same non-Galerkin
MoM technique of Chapter 6 was employed for solution of the lossy microstrip EFIE.
The full-wave theory for the microstrip, as was the case for the stripline, compared
closely with the power-loss perturbation method. Effects of ground-plane conductivity,
strip conductivity and frequency on the principal-mode propagation constant and surface
current were also examined.

This dissertation has also provided a basis for future investigations. For example,
strip and microstrip transmission lines having thick, imperfect center conductors can now

be investigated using the general electric-field dyadic Green’s functions developed in

190

Chapters 5 and 7, respectively. The edge-singularity behavior will be different than the
infinitesimally-thin conductors of Chapters 6 and 7, thus different expansion and testing
functions will be required. A detailed study of the edge singularity reveals that
Gegenbauer polynomials are the appropriate choice. Future investigations should also
include coupled lossy strip and microstrip field applicators. Finally, higher-order modes

of lossy strip and microstrip devices should be explored.

191

APPENDIX A

EM FIELDS AND HERTZIAN POTENTIALS

A.1 Introduction
Appendix A provides a brief overview of Maxwell’s equations, EM fields and

Hertzian potentials. Included in the analysis is the development of the wave equations

for both the EM fields and the Hertzian potentials.

A.2 Maxwell’s Equations and the Wave Equations for E and I?
Maxwell’s equations for a simple medium (linear, homogeneous and isotropic)

and the continuity equation in spectral-domain point form are

v x 13(7) = — M41190") (A.1)

V x 17(7) = ](7) + 0173(7) + @5520) = 7(7) + jwaEm (A.2)
V-E(F) = p(F)/£ (A.3)

v.90) = 0 (A4)

V-j(F)=—ja)p(F) (A5)

where 7(7) is an electric source (i.e., impressed) current and 6‘ =§(l— ja/wé) is the
effective complex permittivity. In general, all the above quantities are also functions of
a) (which has been dropped for notational convenience). If the excitation is time
harmonic (i.e., sinusoidal steady state), then a) is interpreted as a frequency-domain

variable and the peak field quantities in the time domain are recovered by multiplying by

192

eja” and taking the real part. For example, E(F,t) in the time domain will be

E(F,t) = Re{E(F,w)ef“”} (A.6)
If the excitation is non-sinusoidal, then a) must be interpreted as a Fourier-transfonn

variable and the field quantities in the time domain are computed by taking the inverse

transform. For example, E(F,t) in the time domain will be

E(F,t) = E(F,a))ej“"da) (A.7)

_1-
2x

zit—1.8

The wave equation for E is determined by taking the curl of (A.1), substituting
(A.2) into the resulting relation, applying the vector identity VxVx E = V(VoE)—VZE

and using equations (A.3) and (A5). The result is

VZE+k2E =jco,uj—.—1-V(Vo.7) (A8)
1608

where k2 =w25y. The wave equation for 1:] is found by taking the curl of (A.2),
substituting (A.1) into the resulting equation and using VxVx H = V(VoFI)—V21:I and
(AA). The result is

VZH+k2H = —VX.7 (A.9)

A.3 Electric Hertzian Potential
Hertzian potentials are primarily used as an intermediate and simplifying step to

determining electric and magnetic fields. An electric Hertzian potential can be identified
by observing that (A.4) implies that P! can be written as

H=janxif (A.10)

193

since VoV x 7? ¥ 0 by vector identity. The electric field can be determined by substituting

(A. 10) into (A.1) and using the vector identity V x VCD = O. This leads to
E=k2i+VCD (A.11)

The wave equation for if can be identified by substituting (A.10) and (All) into

(A.2), applying the identity Vxinf: V(V-if)—V27? and using the gauge (b=V-fi.

The resulting Helmoltz wave equation is

—.

V27? + kzir‘ = —+1— (A.12)
1a»:

Equation (A.12) can be decomposed into three scalar equations in Cartesian coordinates

(hence reducing mathematical complexity) as follows

Ja

V2750, + k2”, = ——,—— (A.13)
10)::
where a = x, y, 2. Substituting (1) = V-ir’ into (A.11) leads to
E = 1:27? + V(V-7‘r’) (A.14)

Since (A.12) reveals that if is maintained by an electric current, it is called an
electric Hertzian potential (magnetic Hertzian potentials are also used, but will not be
discussed here). A comparison of (A.13) with (A8) or (A.9) demonstrates why Hertzian
potentials are introduced into the mathematical analysis of electromagnetic problems. In
equation (A.13), each component of if is simply and directly related to each component
of .7 . The relationship between E , .7 in (A8) or H , .7 in (A.9) is more complicated and

therefore the solution is, in general, more difficult to determine and more strongly

singular. It’s easier to solve for if first then obtain E and I? using (AM) and (A.10).

194

APPENDIX B

EVALUATION OF THE FOURIER TRANSFORM INVERSION INTEGRAL FOR

THE SPECTRAL-DOMAIN PRINCIPAL-WAVE GREEN’S FUNCTION

B.l Introduction
It was stated in Chapter 5 that the solution to the Fourier transform inversion

integral for the spectral-domain principal-wave Green’s function was

°° MU-y') “Ply-y]
—1— e. . dn= e (3.1)
2” .... (77 + JPX’I - 1p) 2p

 

 

@502)» y')=G§(Z;y-y')=

 

where y is the field point, y’ is the source point, p is the spectral-domain wavenumber

and 77 = 77m (a real axis integration). The purpose of this appendix is to prove the above
result using Cauchy’s integral theorem and formula. Cauchy’s integral theorem states, if
a function f (z) is analytic (differentiable at a point and a neighborhood about that point)

everywhere within and on a closed contour C , then

(f) f(z)dz = 0 (3.2)
C

Actually, it is sufficient to require f (z) to be differentiable strictly inside the contour and
continuous in the closed region bounded by it, including points on the boundary [67],

[68]. Cauchy’s integral formula states, if f (z) is analytic within and on a closed contour

C , then if 20 is interior to C

 

(f) f(z) dz = :1: 1'27: f(zo) (13.3)
C

z—zo

195

depending on whether C is oriented in the counterclockwise/clockwise direction.

The parameter p = p054) is not a fimction of 77 , thus equation (B.1) reveals that
n = ijp are simple poles in the complex n-plane (they are not branch points). Also,
Re{ p} > O has been chosen (as mentioned in Chapter 5), thus 77 = +jp must be located in
the upper-half of the complex n-plane since Re{ p} > O :> Im{jp} > 0. Similarly,
I] = —jp must be located in the lower-half of the complex n-plane since

Re{p} >0 :> Im{—jp} <0.
The integrand, for the original real-axis integration, is a complex function of a
real variable and is therefore not an analytic function since it doesn’t satisfy the Cauchy-

Riemann equations [69]. In order to compute the integral in (8.1) using Cauchy’s
integral theorem, the integrand must be analytically continued off of the 17m -axis
(avoiding the poles :tjp in the process) to establish a region of analyticity [70]. The
above analytic continuation, in this case, is easily accomplished by allowing 7] in (8.1) to
become complex [71], that is, 77 = 77,, + 177:»; . The integrand is now analytic in the entire
complex ”-plane, except at the simple pole singularities 77 = ijp, thus the original
integration contour can be deformed off of the real axis and Cauchy’s integral theorem
invoked.

The specific closed-contour C chosen for application of Cauchy’s integral
theorem is motivated by an examination of the integrand, more specifically, the term
ejrfly—y’). If y—y'>O, then emy'y') =e"”’m(y'y')ej'7"(y_y'), thus 77,-," >0 provides
exponential decay (important for mathematical convergence). This prompts closure of

the real-axis contour in the upper-half plane (making sure to circumvent the pole

196

singularity 17 = jp since Cauchy’s integral theorem states that no singularities can be

contained within or on the closed contour), as shown in Figure 8.]. Similarly, Figure B.2
shows the pr0per closure and contour when y — y' < 0 since ’71»: < 0 provides the

necessary exponential decay for convergence. The evaluation of the integral in equation
(8.1) using Cauchy’s integral theorem and formula will be carried out in detail in the

following sections.

8.2 Evaluation of Cf for y — y' > 0
Figure B] shows the closed contour C = C R + C; + C; used in Cauchy’s integral
theorem for the evaluation of Gzp when y— y' > O. The segment C R is the real-axis

contour, C; is the portion of C that circumvents the pole singularity at 17 = jp and C;

is the semi-circular segment that closes the contour in the upper-half plane. Since

f (n)=ej"(y ’y') /(77+ jp)(n-— jp) is analytic everywhere within and on C, Cauchy’s

integral theorem can be applied, leading to

 

 

gBflmdn= [f(n)dn+<]>f(n)dn+ C] f(n)dn= o (3.4)
where
eery-y' ) R ejn(y-y' )
lm 13.5
A fwd”: «(71+J'PX77-IJ'P) 01:7, R»w_£(n+1p)(njp)dn ( )

Therefore, using (3.4) and (3.5), the following desired result is obtained

°° emu-y )

l..d7-=<l>

4 (77 + JPX'I - 1p)

emU-y )

C.(n+jp)(n- jp)d

eflKy-y’ )

— 77] (B.6)
3(n+jp)(n- jp)d

 

 

 

197

The contour integral along C; can be determined by using the parameterization

77 = Rely, dry = j Rejadfl and Jordan’s lemma [72]. The result is

 

 

 

emo—y') Ir ejRe/‘(y-y') ~19
] , , an: im] .6 . .9 . jReJ d0 —+ 0 (13.7)
C; (n+1p)(n-Jp) R—mo (Re! +jp)(ReJ — jp)
A Im{n}
C-l-
+
C9 .
JP
r t >
’R CR R Re{n}
.x
“JP

 

Figure B.1 Evaluation contour for Cf when y — y' > 0 .

The term ej77(y-y') , as mentioned in the previous section, has an exponentially decaying

behavior that provides the necessary convergence in (B.7). Note, the above result is also

valid for y — y' = 0 (which is not always the case). The integral along the contour C;

can be evaluated using Cauchy’s integral formula. The pole n = jp is interior to C;

while 7] = ~—jp is exterior to it, thus eflKy-y') /(77+ jp) is analytic everywhere within and
on C; . Therefore, application of Cauchy’s integral formula leads to

198

 

ej71()"‘)") ejn(y-y’) e-p(y-y’)
96 . . dn=-J'27r . =—7r— (8.8)
C. (77 + 1an - 1p) (77 + 1p) _ . p

p 'I—JP

 

 

Combining the results of (B6), (8.7) and (B8) leads to the following desired result

6?” Z 1 °° emu-y') d e-p(y-y') O B 9
' ' :— :____— ... _ '2 .
2( ’yly) 27: I(77+jp)(77-jp) 77 2p y y ( )

 

B.3 Evaluation of Cf for y - y’ < 0
The closed contour used for evaluating Cf when y — y' < 0 is shown in Figure

B.2. In this case, emy'y') provides the necessary convergence (i.e., exponential decay)

along the Cg, contour due to the lower-half plane closure. An application of Cauchy’s

integral theorem leads to

01mm}

 

 

 

Figure B.2 Evaluation contour for Cf when y — y' < O.

199

emU-y ) eflKy-y ) emU-y)

d =- d — d B.10
I (n+1'an-jp) 7) C;(n+jp)(r7-jp) 77 iMi‘JPXU-J'P) '7 ( )

 

 

 

-oo

The contribution from C; , similar to C; , vanishes for y — y' S 0 due to proper closure
conditions and Jordan’s lemma, that is

eflKy-y’) '” ejRej9(y-y')

[ , ,dn=1im] .9. .9.
c; (77+JPX77—JP) R->°° 0 (Re! + jp)( Re] — 1p)

 

 

jRejgdB —> 0 (13.11)

An application of Cauchy’s integral formula produces the result

 

 

 

 

emu-y') emu—y') ep(y-y’)
<[>( + , . )d77=j27r—(-——_—) =—7r (3.12)
C; 7) mm 110 77 119 ”91.1, 1)
Therefore, combining (3.10), (3.1 1) and (B.12) produces the result
,4 g 1 °° eJ'IIU-y') ep(y-y')
GP(A;yy')=— . . dn= ...y—y'so (13.13)
2 ' 27r_£(n+1p)(n-Jp) 2p

B.4 General Representation of G2”

The expressions for G; in (B9) and (B.l3) can be combined into a single result

that is valid for all values of y — y' by recognizing that

 

 

, y—y' ...y-y'>0
ly-y|={ , , (3.14)
-(y-y) my-y < 0
This leads to the final desired expression that was to be proved in (B. 1), namely
- p I l .. p Z ) 1 °° emu-y') d e-PIy-y'l (B 15)
G ( ;yy')=G ( ;y—y' =— , , n: .
2 2 27: _0, (77 + Jp)(n-Jp) 2p

200

 

 

APPENDIX C

ELECTRIC-TYPE HERTZIAN-POTENTIAL

BOUNDARY CONDITIONS AT A MATERIAL INTERFACE

 

C.1 Introduction

Electric-type Hertzian-potential boundary conditions at a material interface are

 

developed in Appendix C. These boundary conditions, which are based on the continuity

of tangential electric and magnetic field components, are valid at the interface between

two media having constitutive parameters (81,;11) and (62,;12 ). The analysis contained

in this appendix directly follows the development of Nyquist [26], which is based upon

the work of Sommerfeld [25] and Banos [27].

C.2 Geometry

 

Figure C.1 shows the material interface used in the development of the Hertzian-
potential boundary conditions. The medium in region 1 (y > O) has an effective complex
permittivity and permeability of (81,;11) and the medium in region 2 (y <0) has an

effective complex permittivity and permeability of ( 62, p2 ).

 

if?) y
Interface (EDI—i!) (814‘!)
y=0 f ’
f " x
(EZ’HZ) (329%)

Figure C.l Material interface for Hertzian potential boundary conditions.

201

C.3 Hertzian-Potential Boundary Conditions
The electric-type Hertzian-potential boundary conditions at a material interface

can be identified by invoking continuity on the tangential electric and magnetic field

components, that is (for all values of x and 2 along the y = 0 interface)
o‘z-E,(x,y=0+,z)=d.17:2(x,y=0‘,z) ...a=x,z (C.1)

éolfl(x,y=0+,z)=&-ff2(x,y=0",z) ...a=x,z (C.2)

where (see Appendix A for details)

1?: = kzir’ + V(V-7'r’) (C.3)

I? = 1'an x 77 (C4)

v27? + 185 = ——¥— (C5)
1608

and k2 = (025,11. Note that, in equation (C.2), it has been assumed that no surface current
exists at the material interface.

Substitution of (C.3) into (C.1) and (CA) into (C2) (and dropping the functional
dependence of the x and 2 variables for notational convenience) results in the following

set of relations (for a = x, z )

_ 6 .. a s
E1.(y=0*)=E2.<y=0) => k.’n..+5x—V-n1=k%nzx+5x—V-rrz (0.6)

_ a x a _.
E12(y=0+)=Ezz(y=0) => klzfllz+ng1=k22flzz+gngz (C7)

 

 

67: an
Hlx(y=0+)=H2x(y=O-) => jw31£a;z‘ a;y]=jw£2[agjz- 6?](08)

202

_ , 57!] 67: , 5712 67:
le(y=0+)=H22(y=O) : Jwgl[_L—_—1L]=]w£2[ axy- ajxj(C9)

 

5x 0y
The above boundary conditions appear difficult to implement since all components of if

and its derivatives are implicated. However, because of the uniqueness theorem, it is

 

found that not all components of ii are required to represent the EM field. Individual
components of excitatory current (i.e., Jx,Jy,Jz) are studied to deduce (using the

principle of superposition) which components of if are required to satisfy the above

 

boundary conditions on tangential E, If for a general electric current source .7 .

C.3.1 Horizontal Source .7 = 52.1,

An examination of (C5) reveals that J x maintains Itx only if the current source
was immersed in unbounded space, that is, xx alone maintains all fields (also see
Appendix B for verification of this fact). The presence of the boundary in Figure C]
may cause J x to couple into other components of if. However, as a starting point, the

naive conjecture that all fields can be represented by if = so, only will be made.

Substituting if = far, into (C.7) leads to

=+= =~ 3950:2512);
E120 0) E220 0) 2 62(6):] 62(6):] (0.10)

Since this relation has to hold for all points x, 2 along the y = 0 interface, it is concluded

from (C. 10) that

”Ix =7r2x (C-ll)

l

|

1

Similarly, substitution of if = in, into (C.6) leads to
203

_ 6 672’ 6 67:
Elx(y:0+)=E2x(y=O) => k127r1x+5x“( af)=k22”2x+g[ a?) ((112)

 

 

Inserting (C.11) into the partial derivative terms of (C.12) produces the following result
1:127“, = kgnz, (C.13)
Equation (C.13) is in contradiction with (CH), hence it is concluded that the conjecture
of if = 7277,‘ maintaining all fields was incorrect. Thus, 7? = 5277,, alone is not sufficient to
satisfy boundary conditions for the excitation .7 = 72.1,, .
The above analysis prompts the more educated conjecture that all fields are
maintained by

if = affix +5275, (C.14)
That is, the horizontal source .7 = xe couples into both horizontal and vertical

components of Hertzian potential 7:), and fly due to the presence of the boundary.

Substitution of (C.14) into (C.7) results in

 

 

E,,(y=0+)=E2,(y=0‘) => %V.i,=%V-fiz :> V-fi,=v.iiz (C.15)
67:1,, +aflly = 67:2,: +a7t2y 672'“, _67r2y =6”2x _afllx (C16)

Inserting (C. 14) into (C6) and using the result from (C. 1 5) leads to
E,,(y=0+)=E2,(y=0‘) => k127z'lx =k227r2,r (c.17)
51#17?1x = gzflzflzx (C.13)

Substitution of (C.14) into (C.8) gives

671']

H —0+ -H — " 6””
|x(y— )- 2x(y"0) : 8| 62

62

y

 

 

= 82 (C.19)

204

 

 

$17!”, = Szfl’zy ((2.20)

Substitution of (C. 14) into (09) and using the result from (€20) produces

 

 

67: 67:
le(y=0+) =sz(y= O-) -_-> 61[ ly — afllx]=£2[ 2y — aflzx] ((3.21)

 

 

 

 

 

fix by 0x 5y
67: 67:

=> 81 6”“ -£2 fix— = a] A— 52 2y = 0 (C22)

6y 6y 6x 6x
a, 6”” = 2 97521- (023)

5y 5y
Finally, inserting (C. 1 8) into (C. 1 6) establishes the result
671'”, _ 67:2}, 2 (I _ 82612 ] aflZI (€24)
6y 6y 51/11 6x

No contradictions were encountered in the previous analysis, thus the above educated

conjecture was correct.

C.3.2 Vertical Source .7 = ny

It will be assumed that .7 = ny maintains only a vertical component of Hertzian

potential. Thus, it will be conjectured that all fields can be represented by

if = 97:, (C25)

Substitution of (025) into (C6) and (C7) gives the following result

 

_+_ __ _a_67r1y_267r2y

Ebro/‘0 )-E2x(y-O) 2) 6x[ —"ay j—axL 6y ] ((3.26)
_+_ __ 26x1y_§_6772y

Ele-O )-E22(y-0 ) => 6z[—6y ]-6z {—6)} ) (C27)

205

67: 67:
1y _ 27

C28
6y 6y ( )

 

Note, the derivatives with respect to x and z in (C26) and (C27) can be removed (via
integration) since these relations must hold for all values x,z. However, the derivative
with respect to y cannot be removed since it holds only at a single point, namely, y = 0

(note, integration is an operation that can be performed only if an interval or some

neighborhood about a point exists). Similarly, substitution of (C25) into (C8) and (C9)

 

 

gives
_ 671'] 671' 2
H..(y=o*)=H2.(y=0) => 51——y=52——y (C29)
62 62
6751 6772
H :0“ =H :0 :> a y=£ y C30
12 (y ) 22 (y ) l ax 2 ax ( )
81751}, =€2fl2y (C31)
Note, no contradictions were encountered above, thus the conjecture was correct.
C.3.3 Horizontal Source .7 = 2.],
If the conjecture that all fields can be represented by
7? = 577:), + 27:, (C32)

for the current excitation .7 = 2.12 is correct, then an analogous set of steps as in section

C.3.l can be utilized, leading to the following relations
klzfllz = 1:227:22 : glylfl-lz =82/J27r22 (C33)

8171]), = 8232), (C.34)

206

671']: _ 67722

el——£2

5y 5y

 

67(1), _ aflz)’ =[1_ 82712 J @7722
6y 6y 81711 52

C.3.4 General Source .7 = 12.7 x + ny + 2.1,.

(C35)

(C36)

Based upon the above special cases in sections C.3.1-C.3.3, the boundary

conditions for Hertzian potential can be deduced (by linear superposition) when all

components of .7 are active. A comparison of (C.17) with (C33) leads to

2 2
kl 7[la = k27r2a => allulflla = 82/127121 '"a =x,z

An examination of (C20), (C31) and (C34) reveals, by superposition, that

517’1y =52”2y

Substitution of (C37) into (C6) and (C7) gives

_ 6 - 6 -

Elx(y=0+)=E2x(y:0) 3" EVWFEVWZ

+ _ 6 _. 6 .

E1z(y=0 )=E2.~(y=0) => —V-fl1=—V'flz
62 62

:> V‘fil =V'fi2

677]), _ 0712), _ aflzx _ 671'” + 672.22 _ 67712
6y 5}, 6x 6x 62 62

am, _ 67:2, =[1_ 32p, X0772, + 07:2,]
6y 6y 51/11 5" 52

 

 

 

 

((3.37)

(C.38)

(C.39)

((3.40)

(C41)

(C42)

((143)

since 7:16, = (82,172 / 81710220, from (€37). Finally, substitution of (C38) into (C8) and

(C9) produces the result

207

67:12 67722

 

 

Hlx(y=0+)=H2x(y=0_) 3 51 =52 6y (C44)
_ 67:, 67:Jr
H1.(y=0*)=H2.(y=0) 2 8. ay‘ =82};— (0.45)

Note, the boundary condition in (C43) results in coupling of 7:}, to 7:x and 7:2. Also, if

the interface is along x or z , the above components may be cyclically perrnutated to

obtain the proper relationships.

C.4 Summary

The boundary conditions for components of electric-type Hertzian potential for a

y -interface material boundary are

 

 

 

 

 

 

 

_ a”la: _ a”2a _
191/117% - 52712752.: , 51 6y - 52 —6y "-0" - x,z
(C46)
7: _g 7: 67:1), _67:2y _ 1_82.U2 aflzx + 5722
E11);- 2 2y ’ 6y 6y — 51.111 6x 52
If ,ul = ,uz , then the above boundary conditions simplify as follows
3171a = 5272a a - x,y,z
a, 6”” 612$ ...a = x, z (047)
0y 07
67:]y_67:2y = 1_2 [aflzx +67%?)
6y 6y 51 6x 62

As mentioned earlier, the functional variables were dropped for notational convenience.
It is therefore noted that 7:16, = 7:16, (x, y = 0", z) and that 7:20, = 7:2a (x, y = 0_,z).

If the boundary conditions hold for all values of x and z (—oo < x,z < oo ), then

the above relations can be Fourier transformed. Consider the generic Fourier transform

208

pair

0000

77(5, y,§') = j [ 7:(x, y, my“ dxdz (C.48)

—CD—<1)

oooo

(271.)2 l l 71:,y,:)ef”d:d: ((3.49)

 

7:(x, y, z) =

where 7: = x5 + 24’ and F = xx + yy + 22. If the above boundary conditions are multiplied

by e‘j’i'F and integrated over x and z from -oo to 00, then (C46) and (C47) become

 

 

 

8 ~ _ ~ afila _ afiZa _
Lula-la -£2#2”2a 9 81 6y “£2 6y "'a-x92
.. .. (C.50)
.. .. 67:, 67:2 8 ,u . .. . .
Elflly =6272-2y a ayy - ayy =[l—fil]; (767:2): +1433)

817510 —€27Tza a x,y,z

316”” “2912.1 ...a=x,z (051)
5y 0y

072,, 6772,

 

=[l -g—2](j§7?2x +j§fi22)
31

where 721a =7’ila((§,y=0+,§’) and 7720, =7EZa(4‘,y=O’,4’).

209

APPENDIX D

HERTZIAN-POTENTIAL IMPEDANCE BOUNDARY CONDITIONS

D.l Introduction

Electric—type Hertzian-potential impedance boundary conditions are developed in
Appendix D. These boundary conditions, which are based on the electric and magnetic
field impedance boundary conditions, are valid at the interface between a material and a
good conductor. The Hertzian-potential boundary conditions at the surface of a perfect

conductor will also be found as a limiting case of the impedance boundary conditions.

D.2 Geometry

Figure D.l shows the material/conductor interface used in the development of the
Hertzian-potential impedance boundary conditions. The upper material region (y > 0)
has an effective complex permittivity and permeability (8,].1). Note that a = E — ja/ a) ,
where E and 0' are the permittivity and conductivity. The lower region (y < O) has a
conductivity and permittivity (awéc ), where ac >> (056 for a good conductor, and
permeability 77c. The effective complex permittivity is 6“. = EC — jac / a) z —j0'c /a).
The unit normal vector 79 = 9 points out of the conductor and the fields are maintained by

source excitation J .

210

K?) Ay

//« (8.11)

13>
"<>

Interface

 

——-) II
A
£111
3
v

 

xV

(Eofic) (0c>>w€ouc)

Figure D.l Interface between material and good conductor.

D.3 Hertzian-Potential Impedance Boundary Conditions
The Hertzian-potential impedance boundary conditions are developed from the

approximate electric and magnetic field impedance boundary condition [7 3]-[76]

17:

tan

Z. = (1+ 7), [3% (0.2)

represent the tangential electric and magnetic field

= Zcfix 77,," = 2,12 (13.1)

where

and}?

The vectors E ,0,,

tan

components in the material region at y = O+ , Z, is the intrinsic impedance in the

imperfectly-conducting region y < O , If is the surface current and 73 is the unit normal
vector that points out fiom the conductor and into the material, as shown in Figure DJ.
The justification of the above boundary condition is carried out in detail by Collin
[3], but a brief overview is given here for the benefit of the reader. It is assumed that a
plane EM wave in the material region is incident upon the conductor. Due to the high
contrast between the material and conducting region, the wave penetrates normally into

the conducting region, independent of the incidence angle (as verified by Snell’s law).

211

The components of the electric and magnetic field inside the conductor are thus purely

tangential to the surface and can be described by the well-known plane wave relationship

... —o

I:Ic=fixEc/Zc :> H , =fixE /Zc or E,anvc=—Zc&x1'-I,an.c. Theunitvectorz? is

the direction of propagation and Zc = ch /5c is the intrinsic impedance of the

conductor. Recall that ac z —j0'c / a) for a good conductor, which implies that the

intrinsic impedance becomes Zc z ,[jwpc lac = (1+ 1'”pr / 206 (since the factor
J; = (1+ j)/J2 ). Note that this is in agreement with (D2). The wave penetrates

normally into the conductor (:> 12 = —fi ), thus the above relation can be written

Em,“ = Z Clix H, Since the tangential components of the electric and magnetic fields

071,6‘ '

must be continuous across the interface, the tangential fields just inside the conductor

( Elam: ’

Hm“) can be replaced by the tangential fields just inside the material region

(E, 1:1,“ ), that is,

an,
Emmy :0) = Zcfixfimmg :0) => E,a,,(y =0+) = zchxfl,,,,(y :0“) (D3)
The current induced in the conductor is confined closely to the surface due to the high

conductivity, therefore 75 x H m"

z 13 (a well-known boundary condition), where I? is the
surface current. Thus, the above relation is written Em = Zcrix Elm = Z6]? and is the
desired result (D.l), where it is understood that (Emmlfmn) are the tangential field

components at y = 0". Note that an arbitrary field can always be decomposed into a sum
of plane waves at various angles, thus the above results remain valid for any general EM

field configuration. Besides, no matter what type of field exists in the material region,

212

the wave penetrates normally into the conducting region and will have field components

that are planar to the interface and can therefore be described by a plane wave.

An examination of Figure D.1 shows that 71 = y , thus equation (D.1) becomes

£15, + 215, = 2,9 x (£17, + 2H,) = Zoom, + 219) :> (D.4)
g=4m=4m mm
E, = «2,17,r = ZCKZ (no)

The first parts of (D5) and (D6) can be arranged as follows

—5—=Z=Z %=-Z=-Zc (D7)

2 x

where Z is the wave impedance in the material region. Essentially, (D.7) states that

continuity of the tangential components of E and H is guaranteed by matching the
impedances at the boundary/ interface.
Now that the origin of (D.1) has been discussed, the electric-type Hertzian

potential impedance boundary conditions are now developed. As mentioned previously,

when ii = y , the first part of (D. 1) in scalar form becomes
E.<y=0*)=Z.H.(y=0+) (D3)

E.(y=0+)=—Z.H.(y=0+> 03.9)

For electric-type Hertz potentials, the following relations prevail (see Appendix A)

E=k27':'+V(V-7'f) (D.10)

F1=joogvxiz (D.11)

V27?+k27? =——,J— (D.12)
1:08

213

where k2 = 072877. Substitution of (D.10) and (D11) into (D8) and (D9) leads to
6 a , 571' 67:
Ex(y = 0+) = ZcH2(y = 0"”) :> [(2713: +gV-7Z' = _[CUEZC [fi—By—a] (13.13)

67:,

—fl’lD14
6y6z(')

 

Ez(y =0+) = —ZcHx(y =0+) :> k27:z +£4.77 = —ja)£Zc[
Z

The above boundary conditions appear difficult to implement since all components of 7?
and its derivatives are implicated. However, because of the uniqueness theorem, it is

found that not all components of 7? are required to represent the EM field. Individual
components of excitatory current (i.e., Jx,Jy,Jz) are studied to deduce (using the

principle of superposition) which components of 7? are required to satisfy the above

boundary conditions on tangential E, If for a general electric current source .7 . An

examination of Appendix C may help the reader better understand the following analysis.

D.3.l Horizontal Source .7 = 72.1,,

It will be conjectured, for the horizontal source .7 = 731,, that all fields can be
represented by

7? = £75): + 527:), (D.15)
That is, .7 = £7, couples into both 7:, and 7:}, (due to the presence of the interface) and

only these components are required to satisfy boundary conditions. Substitution of

(D.15) into (D.14) results in

67:
Ez(y=0+)=—ZcHx(y=O+) :> a—an-7?=ja)£Zc-E:-’- (D.16)

214

V-7? = jareZczy (D.17)

since the relation in (D.16) must hold for all values of x,z. Inserting (D.17) into (D.13)

leads to
. 67: . 67: 67:
E,(y = 0*) = ZcHz(y = 0*) => k2”, +1052, Tax—y = 1052, [EL—73f] (D.18)

__jwaZc 67:, _ Z6 67:,
x k2 67 jam 07

 

(D.19)

D.3.2 Vertical Source .7 = 9.1 y

It will be assumed that .7 = ny maintains only a vertical component of Hertzian
potential. Thus, it is conjectured that all fields can be represented by
7? = 527: (D.20)

y

Substitution of (D.20) into (D. 13) and (D. 14) gives the following result

. . a - . any
Ex(y=0 )=ZcHz(y=O ) :> —V'71'=_]0)£Zc— (D.21)
6x 6x
+ + a .. - any
Ez(y=0 )=—ZcHx(y=O ) :> —V-7:=](08Zc— (D22)
62 62
V-7?=ja)£Zc7:y (D.23)

D.3.3 Horizontal Source .7 = 2.]z

It will be conjectured, for the horizontal source .7 = 2.12 , that all fields can be
represented by
(D24)

7? = 97:, +222

215

Substitution of (D24) into (D. 1 3) results in

+ + a -o - any
Ex(y=0 )=ZcHz(y=O ) :> é;V-7c=)w.szc—5x— (D25)

V-ir’ = jwsZcrry (D26)

Inserting (D26) into (D.14) leads to

 

. 672' . 671' 672'
Ez(y=O+)=—ZCHx(y=O+) :> k27r2+jwachazl=—10)5Zc[ a; -—6—z¥-] (D27)

__jwsZc 67rz _ Zc 67:2
2 k2 0y jaw 6y

 

(D28)

D.3.4 General Source .7 = )Ux + szy + 2.12

Based upon the above special cases in sections D.3.1-D.3.3, the Hertzian-potential

impedance boundary conditions can be deduced (by linear superposition) when all

components of .7 are active. A comparison of (D.17), (D23) and (D26) leads to

 

 

 

67r
v.7? = ngzcny => 6’” + y + 6’9 = 108475, (D29)
6x 6y 62
67:
——y = ngzcny -— 6’“ — 6’” (13.30)
6y 6x 62

Note, the boundary condition in (D.30) results in coupling of 7:

y to fix and 7:2.

Substitution of (D29) into (D. l 3) and (D. 14) produces the following results

67: 67:
Ex(y=0+)=ZcHZ(y=O+) :> kznx+jw£ZcEL=jw£Zc[—af-—%] (D.31)

__jwaZc 67rx _ Zc 67rx
" k2 6y jaw 6y

 

(D.32)

216

 

67!
Ez(y=0+)=—ZcHx(y=O+) :> kzzrz+jw£ZcEL=—jwaZc[

_fw82c%_ Zc 51.;
z k2 6y jaw 6y

 

 

If the permeabilities of both regions are equal (i.e., p = pc ), then

Zc _ ZCZ;t _wp/O'c_ 1
MW jawZE wfljZS UCZC

 

 

 

therefore, the impedance boundary conditions can be written as

 

 

 

 

 

67: 67r 67:
y - x z
—= (ogZ 72' ———
J c y 6x 62
”x: 1 6713
Gale 5y
1 67:2
7:2:
Gaze 5y

6”: fix) (D.33)
6y 62

(D.34)

(D35)

(D36)

(D37)

(D38)

If 23 = —j/ , then a similar analysis leads to the following impedance boundary conditions

 

 

 

 

 

 

 

when yatpc
7! =‘— ZC aflx 71' =— Zc 6i aflz—ngZfl' _iQ—éfl
x jaway z jwyay 6y ”ax 62

and
67:

”x=_ 1 6n; ’”x=- 1 67:1. , y=-ja)£Zc7r _6_7r£_67rz
(7ch 8y aczc 6y 6y 6x 62

when ,u=,uc.

D.4 Hertzian-Potential Boundary Conditions at a Perfect Conductor

217

(D39)

(D40)

The Hertzian-potential boundary conditions at the surface of a perfect conductor

can be found by considering the limiting case as ac —> 00. Note that

lim Zc= lim (1+j) %—+o (D41)
0'

o". —)oo 0‘. —)00 c

therefore (D32), (D34) and (D30) reduce to

7: 7: ——y=o (D42)

D.5 Summary
The electric-type Hertzian-potential impedance boundary conditions at a

material/conductor interface are

 

 

 

 

fra=i .20 % ...a=x,2

a)

J ,u ...p¢yc,r‘z=i}7 (D.43)
_672y -+ja)gZ 7! ——a”" _67rz

_ c y 6x 62

Ira=i 1 91:1 a=x,2

0'ch 6y . ..
6 ...p =l’c ,nz iy (D.44)
”y _ . Z 672,, 6722
— 4“ W a. "3:

where it is understood that Ira =7ra(x,y=0+,2) for a=x,y,z. The boundary

conditions at the surface of a perfect conductor are

—-) 00 (D45)

218

 

If the boundary conditions hold for all values of x and z (—oo < x,z < oo ), then

the above relations can be Fourier transformed. Consider the generic Fourier transform

 

pair
7'2'(§,y,§)= j jn(x,y,z)e-11'Fdxdz (D.46)
. , = .0 w " , , jj'FdZ/t D47
mm (2”), linmoe ( >

where Ii = 25 + 2;, 2127 = dgd; and 7 = ix + j>y + 22. If the above boundary conditions

are multiplied by if” and integrated over x and z from —oo to 00, then (D.43),

 

 

 

 

(D44) and (D45) become
fia 2: .Zc 6;“ a=x,z
mm A .
67? ...pipc,n=iy (D.48)
61:1t1w82cfiy —J€7zx 1472-2
~a=ialz 6;? a—x,z
C C A A
6’ ...,u=,uc,n=iy (D49)
72

73 if ——y-=o mac—>00 '(D50)

where 72a =7'Ea (5, y = 02;). If the interface is along x or z , the above components

may be cyclically permutated to obtain the proper relationships.

219

APPENDIX E

OVERVIEW OF CHEBYSHEV POLYNOMIALS

E.l Introduction
Appendix B provides a brief overview of Chebyshev polynomials. The overview
will include fundamental expressions, relevant properties and several useful integrals that

are commonly encountered in MoM solutions of integral equations. The primary

references are [39], [50], [77]-[80].

E.2 Chebyshev Polynomial Properties
E.2.1 Fundamental Expressions

The Chebyshev polynomials of the first and second kind of order n are designated
as Tn (x) and U n (x) , respectively. The Chebyshev polynomials, defined on the interval

—1 s x 31 , can be expressed in the following power series representation

Tn(x)=U,,(x)=l ...n=0 (13.1)
”[n/Z]
T,,(x)=E Z amnx"'2’" ...n=l,2,3,... (E2)
m=0
[rt/2]
Un(x)= Z bman'" ...n=1,2,3,... (13.3)
=0
_ _ m n—2m (”_m_1)!
““4 1) 2 m!(n—2m)! (EA)
(n—m)!

bmn = (—1)'" 2““ (13.5)

m!(n—2m)!

220

where [n/2] means the greatest integer Sn/Z ([—1/2]=—1, [3/2]=1, [2]=2 etc.). If
n is even, then [n/ 2] = n/ 2. If n is odd, then [n/ 2] can be equivalently represented by
[n/ 2] = (n — 1)/ 2. Thus, equations (E2) and (B.3) can be written

(n—l)/2

Tn(x) =3; 2 amnx"'2'" ...n = 1,3, 5,... (E6)
=0
n 71/2 2
Tn(x) = 5 Z amnx"_ "' ...n = 2, 4, 6,... (15.7)
=0
(n-l)/2
Un(x) = Z bmnx"‘2’" ...n =1,3,5,... (E.8)
=0
n/2
U, (x) = Z 1),,"me ...n = 2, 4, 6,... (13.9)
m=0

Equations (E.6)-(E.9) are in a form conducive to numerical computation via computer.
The trigonometric representations of the Chebyshev polynomials (defined on the interval

0 S 6 S 72 ), using the c.o.v. (change-of-variable) x = cos 0 , are

Tn (cos (9) = cos n6 (E. 10)
Un(cosfl) =W (13.11)
srn (9

Both the power series and trigonometric representations are useful for proving the various

integrals in section B.3.

E.2.2 Parity and Recurrence Relations
Chebyshev polynomials of even/odd order are even/odd about the origin, thus the

following parity relations prevail

T.(—x> =(—1>"T,.<x) (13.12)

221

U.(—x)=<—1)"U.(x) (13.13)
The Chebyshev polynomials can be generated (in computer programs, for example) using

the following recursive formulae
Tn (x) = 2xTn—1(x) _ Tn—Z (x) (E-14)
U n(x) = 2xUn_l(x)—U,,_2(x) (E.lS)

A few of the Chebyshev polynomials are

T0(x) =1 U0(x) =1
T1(x)=x U1(x)=2x
T2(x) = 2x2 —1 U2(x) = 4x2 —1
3 3 (E.l6)
T3(x)=4x —3x U3(x)=8x —4x
T4(x) =8x“ —8x2 +1 U4(x)=16x4 —12x2 +1
T5(x) = 16x5 -20x3 +5x U5(x) = 32x5 —32x3 +6x

E.3 Integrals Involving Chebyshev Polynomials
E.3.l Preliminary Formulae and Well-Known Relations

The following well-known (and perhaps some not so well-known) relations are
given here for the benefit of the reader and will be used in the following sections for

proving the various integrals involving the Chebyshev polynomials, namely

_[Z —jz
cosz = 51—15— (13.17)
2
72 _ —jz
sin 2 = i—fi— (E. 1 8)
12
sin A sin B = %[cos(A — B) — cos(A + B)] (E. 19)

222

 

 

 

 

 

1 . .. .. ...
S + S 0
Wm sin(ax)dx = ‘ 5‘“ a 2 COS a f' i ; n = 1,3, 5,... (E20)
0 0 a = 0
1 . .. .. ..
Ixmz’" cos(c'ix)dx= S3 sma+S4 cosa ...a¢0 'n=2 4 6 (E21)
0 1/(n—2m+l) 2:0 ’ ’ ’ '
1
[56'4de = —1—— (E22)
0 n—2m+l
~ (n-2m-l)/2 r (n _ 2m)! 1
S1 = S] (m,n, a) = g (— ) ("—2m_2r_1)!52r+2 (13.23)
~ (n-Zm-l)/2 r n _2 m ! 1
52 =Sz(m,n,a)= (-1)*' ( ) ,2 .1 (E24)
r=0 (n-2m—2r)!a ’
_ ~ _ _ r .
S3 _ S3(m, n, a) _ g0 ( 1) (n-2m—2r)152'+1 (E25)
~ (n-2m-2)/2 r (n _ 2m)! 1
S4 = S4(m, n, a) = no (—1) (n _ 2m _ 2r -1)1 62'” (E26)

 

. n fl .. fl' .-
Jn(&)=£T—7’[)— joos(n6)ela°°59d0 => joos(n0)e1"°°59d6=( gnaw) (E27)
0 0 '1

 

_(_j)"" _ 1...n=0
7,, (0) - 7, 0] cos(n0)d19 _ {0 ...n at 0 (E28)
. n 2 ,. 7r 1
[31—1330 J" (a) ~ g cos[a —-2—(n + 3)] (E29)
(E30)

J n (dejm ) = ejnmJn (5) ...m = integer

J._.(a>+J...(a)=%./.(a) 03.31)

Note that (E20) and (E21) are odd and even functions of 6 , respectively.

223

E.3.2 Orthogonality Relations
The Chebyshev polynomials, with respect to their resultant weight functions,

satisfy the following orthogonality integrals

1T T ...matn
de= ”/2 m=n¢0 (E32)

”1 vl-xz 7r :m=n=0

] U mn(x)U (x)\/1— —x dx= {0 "m” (E33)

7r/2 ...m=n

The above relations can be easily proved by using the c.o.v. x = c0519 , the trigonometric
definitions in equations (E.lO)-(E.11) and the well-known orthogonality conditions of the

sine/cosine functions on the interval 0 S 6 S 72.

E.3.3 Integrals Involving T" (x)
Some of the common integrals involving Chebyshev polynomials of the first kind
are given below. Proofs of the following integrals are also provided. It is noted that

these types of integrals are typically encountered in MoM solutions.

1 n(n-l)/2

— S ' ~+S " ...":tO
[Tn(x)sin(&x)dx= 2 #0 am"( ‘Sma 2005“) a -n=1,3,5,... (E34)
0 0 ...a=0

(n— —1)/2 7t
sin(ax )dx= ( 1) 2J,,a( a..) ; n=l,3,5,... (E.35)

if?

1
an(x)oos(ax)dx = ; n = 0 (E36)

224

(
"rt/2

, -2- Zamn(S3sin6+S4cos6) ”6:0
lTn(x)c°S(5x)dx =1 :73 ; n = 2, 4, 6,... (E37)
n amn a

 

(2m=0”—2m+1

(—1)"/2(7r/2)J,, (a)...a¢0,n=0,2,4,...
lj— Tx"( x) cos(ax)dx= 72/2. ..a: 0, n: 0 (E38)

2
0‘“ x 0 ...a=0,n=2,4,6,...

Note again, upon trivial inspection of the above integrands, that (E34)-(E35) are odd

functions of 6 and (E36)-(B.3 8) are even functions of 6'.

Proof: (E34)

 

The proof of (E.34) follows immediately by substituting (E.6) into the integral

expression of (E34) and using relation (E20).

Proof: (E35)

 

The relation in (E.35) can be more easily proved by examining the following

integral

[Ti—“'0: )sin(&x )dx ...k=0,1,2,... (1339)

N17

Note that 2k +1 is an odd integer, therefore the substitution of n = 2k +1 will be made at
the end of the derivation to recover the desired result. Using the c.o.v. x =cos(9 and

substitution of (BIG) into (E.39) leads to (since the integrand is even due to the odd

parity 0f T2k+l)

225

 

l l 17:
[Englmwmg [k T“ “(x x2)sin(&x)dx= —]eos(2k+1)6sin(aoos6)d6 (E40)
-120

o l-x \ll-

Equation (E.40) can be computed in closed-form by expanding sin(6 cos 6) with the aid

of (El 8) and then using relation (E27), leading to

1
T2k+l(x) - ~ _ 7: ~ _ _~
J—W s1n(ax)dx———4j(_j)2k+1[Jan-1(0) J2k+l( 0)] (E.41)

2k+l ]-I

Since [ j(— j) =(—1)k and J2k +1(—Zi)=—J2k “(6') from the analytic continuation

relation (E.30) with m = 1, equation (E.41) reduces to

l
T2k+l(x x)
Ol_ W Sin(5x)dx= ( 1) £2J2k+1(a) (13.42)

Letting n = 2k +1 produces the desired result

I
[129% sin(ax)dx =(_1)("-')/212’-J,,(6) ...a e 0, n =1,3,5,... (E43)
() l—x

Since n is a positive odd integer, relation (E28) implies that

 

' T (x)
I n 2 Sin(&X)dx = 0 ...a = O, n :1,3,5,". (E44)
0 l-x

Besides, if 6 = O , then the integrand is identically zero. Hence, integral relation (E.35)

has been proved.

Proof: (E.36)

 

The proof follows immediately since To (x) =1 and cos(dx) =1 if a = 6.

Proof: (E37)

 

226

Substitution of (B7) into the integral expression of (E37) and using (E21) leads

to the desired result.

Proof: (E.3fi

 

The relation in (E.38) is more easily proved by examining

1
JIM cos(fix)dx ...k = 0,1, 2,... (E45)

oJl-x2

The integrand is even due to parity relation (E.12), thus using the c.o.v. x = c0519 and

(BIG) gives

 

Icos(2k6?) cos(c'i cos 6)d6 (E. 46)

‘___x_ a, _l

0 \Il — x2 _1 l—x2 2 0
Expanding cos(6 c030) with the aid of (E.l7), using (E27) and invoking the relation
J2k(—6) = J2k (6) produces [since (—j)'2k = (-1)"]

l
]— TM" —)cos(&x)dx= (— 1) 7202,61) (E47)

()--\/1 x2

Letting n = 2k gives the anticipated result

1
I Tn(x)2 COS(&X)dx =(- l)"/2— n J "(0) ...5 n: O, n = 0,2,4’.” (E48)
0 l—x

 

The use of equation (E28) leads to the remaining relation, that is

l

7,,(x) , 7r/2 ...n=0 ~

——cos ax dx= ...a =0 E49
0”sz ( ) {O...n=2,4,6,... ( )

E.3.4 Integrals Involving Un(x)

227

Some of the common integrals involving Chebyshev polynomials of the second

kind are given below. Proofs that are similar to the previous section will be omitted for

the sake of brevity.
1 (n—l)/2
b s ' “+S " ...~¢0
jun(x)sin(ax)dx= m=0 "'"( ‘8‘“ 2605“) a '71 1,3,5
0 ...a=0
l (n-l)/2 ("+071 - ~
—1 —J 0
IUn(x)\/l—x2sin(6x)dx=( ) 26 Ml“) a¢ ;n=l,3,5,...
0 O ...6=O
‘ ~ sin6/6 ...6¢O
IUn(x) cos(ax)dx = ~ ; n = 0
1 ...a=0
O
rn/Z
1 men(S3sin6+S4cos6) ...6¢O
jU,(x)cos(ax)dx =4 "'7: ; n = 2,4, 6,...
n
Z—fim— ...a=0
im=On—2m+l

 

(—1)"/ 2(n +l)(7r/2)J,,+l(a)/a ...a ¢ 0, n = 0,2, 4,...

1_[U,,(x)\/1—x2 cos(6x)dx= 72/4. ..6: O,n= O
O. ..6: O,=n 2,,,..46

(E50)

(E51)

(E52)

(E53)

(E54)

Note, (E50)-(E51) are odd functions of 6 and (E52)-(E54) are even functions of 6.

Proof: (E51)

 

The relation in (E51) is proved by examining

l
IU2k+l(x)\/1—x2 sin(6x)dx ...k = 0,12,...
0

228

(E55)

 

Since U2k+1 is odd, substitution of (El 1) into (E55), upon using the c.o.v. x = c050,

leads to the following

1 It
[112 ,,,., (x)\/1 — x2 sin(6x)dx =§ Isin{[(2k + 1) + 1119} sin05in(6 cos 6)d6 (E56)
0 0

Expanding sin{ [(2k +1) +l]6}sin 6 using (E.l9) gives

1
IU2k+1(x)\/1— x2 sin(6x)dx =
0

blt—

It
2i jcos[(2k +1)6]sin(6 cos 6)d6 -
0 0

jcos{ [(2k + 1) + 2119} sin(6 cos (9)66

(E57)

Expansion of sin(6 cos 0) with the aid of (E. 1 8) and using (E27) and (E30) leads to

[cos[(2k + 1)6] sin(6 cos 9)d9 = (—1)" rum, (5)
0

jcos{[(2k +1) + 216} sin(6 cos 6)d0 = -(—1)" 7rJ(2k+1)+2(6)
0

Substitution of (E58) into (E57) produces
1
. ~ I: .. -
IU21..1(x)\/1 -x2 sm<ax>dx =(-1)" 3161.100 + 4266602)]
0

Finally, use of relation (E.31) gives the desired result

k ”[(2k + 1) +1]
25

 

l
IU2k+1(x)\/1-x2 sin(ax)dx=(—1) 62166.07)
0

Letting n = 2k +1 leads to the anticipated result

7r(n
2 ..

(E58)

(E59)

(E60)

1

jU,,(x)\/1—x2 sin(6x)dx = (—1)("")/2 —il—)J,,,,(a) ...a ¢ 0, n = 1, 3,5,... (E61)
6

0

If 6 = 0 , then (E.61) becomes indeterminate since J" H (O) = 0. Instead, one must look at

229

the relation in (E59) prior to using (1331), which takes the following form when

n = 2k +1
1111.0)” sin(6x)dx = (—1)(""”/2 %[J,,(6) + J,,+2(6)] ...n =1,3,5,... (E62)
0
Substitution of (E.28) into (E.62) leads to (since n > 0 here)
Elgar/1:? sin(6x)dx = 0 ...a = 0, n =1,3,5,... (E63)
0

This is anticipated, of course, since the integrand is identically zero when 6 = 0.

Proof: (E54)

 

A comparable procedure, which will be left for the reader, leads to

 

1

jU,(x)J1— x2 cos(6x)dx = (—1)"’2 (" :3)” J,,,(a) ...a = 0, n = 0, 2, 4,... (E64)
(1

0

The following equivalent form, upon using (E.31), is best for analyzing the behavior

when 6 = 0 (since the above is indeterminate again), namely
1
[Un(x)\/1- x2 cos(6x)dx = (—1)"’2 -;5[J,,(6) + J,+2(a)] ...a = 0, n = 0,2,4,... (E65)
0

Use of (E.28) reveals that

l ...
/4 =0, =0
IUn(x)\/1—x2 cos(ax)dx={” ,_ a " (E66)
0 O ...a=0, n=2,4,6,...

Therefore, relation (E54) has been proved (although several steps have been left for the

enjoyment of the reader).

230

E35 Asymptotic Form of Integrals Involving Tn (x) and U n (x)

A careful examination of the previous results reveals, with the aid of (13.29), that

the various above integrals have the following asymptotic behavior

6°?“ ...n=1,3 5,..

 

1
_lim ITn(x)sin(6x)dx ~
0%

_g _L
U sin(a'oc)dx~COSPi 512/5“ )] ...n=l,3,5,...

 

lim IT" (x)cos(6x)dx ~ fiiflfl ...n = 0, 2,4,...
6

co S(&X)dx~cos[a “if” 2)] ...n=0,2,4,...

 

lim “3&0

lim 3U" (x)sin(ax)dx~cosa ...n=1,3,5,...

 

 

l _
_lim IUn(x)\/1—xzsin(6x)dx~cos[a ;3£"+% )] ...n=l,3,5,...
a—mo 0

l . ..
lim IUn(x)cos(6x)dx~-SI#I- ...n=0,2,4,...
6—>oo O a

6-%(n+%)] ...n = 0,2,4,...

1 cos
lim IUn (le —- x2 cos(6x)dx ~ [ 3/2
6—>oo O 5

where the following asymptotic behavior for SI - S4 has been utilized

lim 81,54 "L2 , lim S2983 “'1...
-_)a) 6 '—-)® a

231

(E67)

(E68)

(E69)

(E70)

(E71)

(E72)

(E73)

(E74)

(E75)

[1]

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[3]

[4]

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[8]

[9]

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