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MSU IcAn Affirmative Action/Equal Opportunity imtituion
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CHARACTERIZATION OF MICROSTRIP WITH SUPERSTRATE

USING HERTZIAN WAVE MATRICES

By

Boutheina Kzadn’

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1994

ABSTRACT
CHARACTERIZATION OF MICROSTRIP WITH SUPERSTRATE
USING HERTZIAN WAVE MATRICES

By

Boutheina Kzadri

Dyadic Green’s function for the EM field maintained by electric currents immersed in
a planar layered environment are constructed through electric Hertz potentials using
wave matrices to determine the spectral amplitudes in their Sommerfeld-integral rep-
resentations. Such Green’s functions are appropriate for the analysis of contemporary
integrated electronic and optical circuits operating at micro/mm/optical wavelengths,
where the circuit components are located adjacent to a layered surround environment.
This systematic new formulation accommodates general volume currents and removes
any uncertainty regarding completeness of the field representation, while naturally
accommodating the source-point singularity.

A full-wave analysis of a microstrip transmission line with superstrate is devel-
oped based on an integral equation description of the microstrip circuit structure. In
order to obtain a complete circuit model of the microstrip transmission line with
superstrate, its dispersion characteristics, current distributions and characteristic im-
pedance are investigated via the rigorous full-wave integral equation approach. It is
found that the principal mode of the microstrip with superstratc remains always a

bound mode since it never leaks. Moreover, the microstrip line with superstrate

behaves similarly to the conventional microstrip in the sense that the current distribu-
tions for the principal and higher-order modes behave in the same manner.

A full-wave analysis of the characteristic impedance for the conventional
microstrip using both voltage-current and power—current definitions revealed that both
methods give results very close to each other. It is also established that the character-
istic impedance, for the conventional microstrip and the microstrip with a superstrate
layer and a substrate having the largest permittivity, will always increase with fre-

quency.

DHJICAIED 10 MY HUM, AIEF
Am

TOMYSONIDCOMELAM

iv

ACKNOWLEDGEMENTS

I would like to express my sincere thanks and appreciation to professor Dennis
Nyquist for his guidance, support, and inspiration throughout the course of this
research. His valuable time, kind consideration, and understanding have been an
incentive for the completion of this dissertation.

Sincere gratitude is extended to the other members of my guidance committee:
professor Chen, professor Rothwell, and professor Lappan for their interest and
criticism.

Last, but not least, I would like to express my special gratitude to my husband
Atef for his patience, love, eare, and encouragement during every step I made

towards the completion of this research.

TABLE OF CONTENTS

LIST OF TABLES ...................................... ix
LIST OF FIGURES ..................................... x
CHAPTER 1 INTRODUCTION ......................... 1
CHAPTER 2 LITERATURE REVIEW .................... 6
2.1 GREEN’S FUNCTION FOR PLANAR LAYERED MEDIA ...... 6
2.2 MICROSTRIP CIRCUITS ........................... 7
2.3 CHARACTERISTIC IMPEDANCE ..................... 11
CHAPTER 3 DYADIC GREEN ’8 FUNCTIONS FOR THE EM

FIELD IN PLANAR LAYERED MEDIA BASED
UPON WAVE MATRICES FOR ELECTRIC HERTZ

POTENTIAL ............................ 15
3.1 INTRODUCTION ................................ 15
3.2 CONFIGURATION ............................... 16
3.3 HERTZ POTENTIAL AND ELECTRIC FIELD ............. 18
3.4 WAVE MATRICES FOR SPECTRAL HERTZ POTENTIAL ..... 20
3.5 DYADIC GREEN ’8 FUNCTION FOR LAYERED MEDIA ...... 24
3.5.1. Potential in Source Layer ..................... 24
A. Generalized tangential reflection coefficient ........ 25

B. Generalized normal reflection and coupling
coefficients ........................... 26
C. Potential in the source region ................. 29
3.5.2 Potential in region 1 due to currents in region i ....... 36
Tangential and normal components ............... 36
Coupling component ........................ 38
3 .6 SUMMARY ......... ' .......................... 42

vi

CHAPTER 4 APPLICATION TO INTEGRATED-CIRCUIT
ENVIRONNIENT: SIMPLE MICROSTRIP

STRUCTURE ........................... 46

4.1 INTRODUCTION ................................ 46

4.2 FIELDS IN THE COVER LAYER ..................... 46

4.3 FIELDS IN THE FILM REGION ...................... 52

4.4 INTEGRAL EQUATION DESCRIPTION OF MICROSTRIP

TRANSMISSION LINE ............................ 55

4.5 MOMENT METHOD SOLUTION ..................... 61

4.6 NUMERICAL RESULTS ........................... 63

4.7 SUMMARY ................................... 64

CHAPTER 5 COMPUTATION OF CHARACTERISTIC

IMPEDANCE OF MICROSTRIP TRANSMISSION

LINE ................................. 69

5.1 INTRODUCTION ................................ 69

5.2 VOLTAGE-CURRENT METHOD ..................... 70

5.3 POWER-CURRENT METHOD ....................... 74

5.3.1 Power in the Cover Region .................... 77

5.3.2 Power in the Film Region ..................... 80

5.4 NUMERICAL RESULTS ........................... 83

5.5 SUMMARY ................................... 85

CHAPTER 6 ANALYSIS OF MICROSTRIP STRUCTURE WITH

SUPERSTRATE LAYER .................... 91

6.1 INTRODUCTION ................................ 91

6.2 FIELDS IN THE SUPERSTRATE LAYER ................ 94

6.3 FIELDS IN THE SUBSTRATE LAYER ................. 100

6.4 INTEGRAL EQUATION DESCRIPTION OF MICROSTRIP ..... 105

6.5 MOMENT METI-IOD SOLUTION ..................... 110

6.6 NUMERICAL RESULTS ........................... 114

6.7 SUMMARY ................................... 117

vii

CHAPTER 7 CHARACTERISTIC IMPEDANCE OF
MCROSTRIP TRANSMISSION LINE WITH

SUPERSTRATE LAYER .................... 131
7.1 INTRODUCTION ................................ 131
7.2 FORMULATION OF THE PROBLEM ................... 131
7.3 NUMERICAL RESULTS ........................... 136
7.4 SUMMARY ................................... 138

CHAPTER 8 EM CHARACTERIZATION OF MATERIALS IN A

MICROSTRIP SUPERSTRATE ENVIRONMENT . . . . 143

8.1 INTRODUCTION ................................ 143

8.2 CHARACTERIZATION OF FIELD APPLICATOR:
MEASUREMENT OF TRANSITION REGION S PARAMETERS . . . 144

8.2.1 Transition region "a" ....................... ‘ 144
8.2.2 Transition region "b" ....................... 146
8.3 DE-EMBEDDING OF SAMPLE-REGION S PARAMETERS FROM

MEASURED TERMINAL S PARAMETERS ................ 148

8.4 DETERMINATION OF MATERIAL CONSTITUTIVE PARAMETERS
FROM MEASURED SAMPLE-REGION S PARAMETERS ...... 149
8.5 NUMERICAL RESULTS ........................... 151
CHAPTER 9 SUMMARY AND CONCLUSIONS .............. 155

APPENDIX A CANONICAL TRANSFORM DOMAIN SOLUTION
TO THE PLANAR INTERFACE REFLECTION

PROBLEVI ............................. 159
APPENDIX B ELECTRIC DYADIC GREEN’S FUNCTIONS ...... 168
BIBLIOGRAPHY ...................................... 173

viii

 

LIST OF TABLES

Table 4.1: Convergence of the propagation constant upon the number of
basis functions used in the current expansion. ............. 65

ix

 

Figure 1.1:

Figure 2.1:

Figure 3.1:
Figure 3.2:

Figure 3.3:

Figure 4.1:

Figure 4.2:

Figure 4.3:

Figure 4.4:

Figure 4.5:

Figure 4.6:

Figure 5.1:

Figure 5.2:

Figure 5.3:

Figure 5.4:

Figure 5.5:

LIST OF FIGURES

Microstrip line with superstrate.

Some transmission line structure suited to microwave circuit
integration. ..................................

Configuration of planar layered media. .................

Normal and tangential wave amplitudes.

Coupled wave amplitudes. .........................
Configuration of general open microstrip integrated circuit ......

Typical background environment in a microstrip circuit.

General configuration of an open microstrip transmission line . . .

Dispersion characteristics of the principal mode for the
configuration of Fig. 4.3 with w = 3.04 m, t = 3.17 mm,
and m=3.42 .................................
Frequency-dependent characteristics of normalized longitudinal
current distribution, relevant to the example of Fig. 4.4. ......
Frequency-dependent characteristics of normalized transverse
current distribution, relevant to the example of Fig. 4.4. ......
Characteristic impedance versus frequency using the voltage-
current method (w = 1.5 m, t = 0.635 mm, and e, = 9.8). . . .

Characteristic impedance versus frequency for two different
methods (w = 1.5 m, t = 0.635 mm, and e, = 9.8). .......

Characteristic impedance as a function of microstrip
width at 10 GHz (t = 0.635 mm, and e, = 9.8). ...........

Comparison of characteristic impedance with published results
(w = 1.52 m, t = 3.17 mm, c, = 11.7). ..............

Comparison of characteristic impedance versus microstrip width
with Pozar’s results at 3 GHz. (t = 1.27 mm, c, = 10.2). .....

17

31

31

47

56

66

67

86

87

88

89

90

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:

Microstrip with a superstrate layer ...................

Background environment of a microstrip with superstrate .....

Configuration of microstrip transmission line with superstrate

Convergence of the normalized propagation constant as a
function of number of terms used in the Chebyshev series

expansion for the current. ........................

Dispersion characteristics of the principal and the first two
higher-order modes of the microstriop line with superstrate.

Dispersion characteristics of the principal mode with

superstrate refractive index as parameter. ...............

Dispersion characteristics of the principal mode with

superstrate thickness as parameter. ...................

Comparison between the longitudinal current distributions
of the principal mode for the conventional microstrip and
those for the microstrip with superstrate operating at two

frequencies. .................................

Comparison between the transverse current distributions
of the principal mode for the conventional microstrip and
those for the microstrip with superstrate operating at two

frequencies. .................................

Dispersion characteristics of the principal and the first
higher-order modes of the microstriop line with superstrate.

Dispersion characteristics of the principal mode with superstrate

refractive index as parameter. ......................

Dispersion characteristics of the principal mode with superstrate

thickness as parameter. ..........................

Longitudinal current distribution of the principal mode for the
microstrip line with superstrate operating at two different

frequencies. .................................

Transverse current distribution of the principal mode for the
microstrip line with superstrate operating at two different

xi

92

93

. 106

118

.119

. 120

121

122

. 123

. 124

125

126

127

 

Figure 6.15:

Figure 6.16:

Figure 7.1:

Figure 7.2:

Figure 7.3:

Figure 7.4:

Figure 8.1:

Figure 8.2:

Figure 8.3:

Figure A.l:

Figure A.2:

frequencies. .................................. 128

Longitudinal current distribution of the first higher-order mode
for the microstrip line with superstrate operating at two different
frequencies. .................................. 129

Transverse current distribution of the first higher-order mode for
the microstrip line with superstrate operating at two different
frequencies. .................................. l 30

Characteristic impedance versus frequency for different
superstrate refractive indices with w = 1.5 mm, t =0.635 mm,
and ds /t = 1. ................................ 139

Characteristic impedance versus frequency for different
superstrate height with w = 1.5 m, t =0.635 mm, and
ns = 2. .................................... 140

Characteristic impedance versus frequency for different ,
superstrate refractive indices with w = 1.5 m, t =0.635 mm,
and ds /t = 1. ................................ 141

Characteristic impedance versus frequency for different
superstrate height with w = 1.5 m, t =0.635 mm, and

ns /t = 3.13. ................................. 142
Transmission-line field applicator with sample inserted in

superstrate. .................................. 152
Equivalent two-part network. ....................... 153
Frequency dependence of complex permittivity for teflon

sample. .................................... 154
Single interface. ............................... 159
Effect of a propagation path length. ................... 166

xii

 

 

CHAPTER 1

INTRODUCTION

The microstrip is a particularly useful transmission line medium for microwave
and millimeter-wave integrated circuit applications. This dissertation presents a rigor-
ous analysis of a new microstrip structure, namely a microstrip with a superstrate
layer. The microstrip device is printed on the substrate layer and resides in the su-
persrtrate region as shown in Fig. 1.1. To our knowledge, this structure has not been
analyzed extensively in the literature. In order to obtain a circuit model of the micro-
strip transmission line with superstrate layer, its propagation characteristics and char-
acteristic impedance will be evaluated. The analysis of characteristic impedance for
this particular structure is believed to be new and has not been discussed in the litera-
ture.

The rigorous analysis of printed circuit elements such as microstrips requires
the use of the dyadic Green’s function associated with the layered background.
Therefore, the dyadic Green’s functions associated with a multi-layered environment
are constructed through the electric Hertz potential in a systematic manner using wave
transmission matrices. Determining the Hertz potential in any region of a planar
layered environment, maintained by general volume currents residing in any other
layer, will be rendered to multiplication of wave matrices. This analysis will be
specialized to obtain the dyadic Green’s function associated with the four layered
background environment of the microstrip with superstrate in Fig. 1.1. We note that

l

 

 

       

MA 0, /7Z¢7////W/II/x/W/V/Vflflyfl'zwy / //7/,//¢///// //////// //// 777/
/ fl / I / l ' I / /-///: ,’/

/// 77/7/

Figure 1.1: Microstrip line with supersrtate.

this otherwise extensive effort will be made relatively easy by using wave transmis-
sion matrices.

The text in this dissertation is divided into nine chapters. Chapter 2 presents a
literature review of the main topics discussed in this thesis, namely, Dyadic Green’s
functions for planar layered environments, microstrip transmission lines and charac-
teristic impedance. In Chapter 3, the dyadic Green’s functions for the EM field
maintained in any region of a planar, layered environment by general electric volume
currents in any other region are constructed through electric Hertz potentials, to deter-
mine the spectral amplitudes in their Sommerfeld-integral representations. The con—
struction of the dyadic Green’s functions exploits wave transmission matrices for the
tangential and normal components of potential maintained by respective currents as
well as coupling matrices which couple tangential currents to normal potential compo-
nents.

In Chapter 4, an electric field integral equation (EFIE) description for general
microstrip circuits is developed, and then applied to microstrip transmission lines.
This work was originally performed by Yuan and Nyquist [l], and is included here
for completeness. However, the dyadic Green’s functions associated with the layered
background of a conventional microstrip environment are constructed using wave ma-
trices as proposed in Chapter 3. Moreover, propagation modes on a single lossless
microstrip transmission line are analyzed. Numerical solution to the homogeneous
EFIE are implemented by the Galerkin’s method of moments. Chebychev polynomi-
als weighted by appropriate edge-condition factors are utilized as basis functions in
the current expansion. The currents are obtained in a convenient quasi-closed form of
rapidly convergent Chebychev polynomial series. Results of dispersion characteristics

3

 

and current distributions for the principal and higher-order modes are presented and
compared to previously published data.

In order to obtain a complete equivalent transmission-line representation for
the microstrip, not only its dispersion characteristics must be evaluated but also the
characteristic impedance. Chapter 5 presents a full-wave analysis for the characteris-
tic impedance of conventional microstrip transmission line. Both voltage-current and
power-current definitions of characteristic impedance are utilized and compared to
each other. Numerical results for both methods are presented and compared to previ-
ously published ones.

Chapter 6 presents a rigorous analysis of a microstrip structure with
superstrate layer. Based on the integral-equation formulation, the microstrip line with
a superstrate is studied by an approach similar to the one presented in Chapter 3 for
the conventional microstrip. A rigorous full-wave solution to the integral .equation is
pursued again using the Galerkin’s method of moments. The dispersion characteris-
tics and current distributions of the guiding structure are analyzed for the principal
and high-order modes. Extensive numerical results are presented.

Chapter 7 completes the circuit modeling of the microstrip transmission line
with superstrate layer by analyzing its characteristic impedance. The voltage-current
method will be used since it is less analytically involved and, as established in Chap-
ter 5, gives results very close to those of the more accurate power-current method.

Chapter 8 exploits both an experimental method and the full-wave analysis for
the dispersion characteristics and characteristic impedance of the microstrip circuit
with superstrate to deduce the constitutive parameters of materials located in the

superstrate layer of a four-layered microstrip background environment.

4

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

 

sion on this research, along with some recommendations for future investigation on

this topic.

 

CHAPTER 2

LITERATURE REVIEW

2.1 GREEN’S FUNCTION FOR PLANAR LAYERED MEDIA

The study of waves and fields in planarly layered media is quite a classic
problem and has been studied by numerous researchers. Many books have been
written on this subject. We have listed the works chronologically [2] - [9], but this
list is by no means complete. The development of dyadic Green’s functions for lay-
ered media is an important subject.

The components of Green’s functions carry complete information regarding the
general characteristics of wave propagation and coupling in a specific multilayer
medium. Due to the importance of this subject, substantial research work has been
published [10] - [15]. Different methods have been proposed to construct the dyadic
Green’s function for layered media.

The construction exposed in Kong [8] and Chew [9] is based upon either the
normal electric or magnetic field components. Another approach is based on the
normal components of electric and magnetic Hertz potentials as in Stoyer [10].

A method of derivation of spectral domain Green’s functions for a multilayer
geometry has been described by Pozar and Das [13] that uses equivalent transmission
line sections to account for several layers. A similar approach is adopted by

Michalski and Zheng [15].

2.2 MICROSTRIP CIRCUITS

Microstrip is a particularly useful transmission line medium for microwave and
millimeter-wave integrated circuit applications. Circuits using microstrip can be
implemented in many radars, some segments of point-to-point radio links, and certain
portions of satellite communication systems [16]. Microstrip transmission lines are
widely used in microwave integrated circuits (MIC’s) for these systems. Most of the
structures are also suitable for various high-speed digital applications. In addition,
monolithic microwave circuits (MMC) have to be interconnected using microstrip
lines and therefore such lines are important structures.

The microstrip falls into the category of the so called inhomogeneous planar
transmission lines. In its basic form, it consists of a conducting strip printed on a
dielectric substrate which is in turn backed by a ground plane as shown in Figure
2.1(a). This is called the open microstrip line. Some other types of planar
transmission lines are shown in Figures 2.1(b)-(t) [17]. The microstrip with a cover
plate and the shielded microstrip are shown in Figures 2.1(b) and (c), respectively.
Figure 2.1(d) illustrates the inverted microstrip structure. The slot line and coplanar
transmission lines shown in Figures 2.1(e) and (0, respectively, are also used in a
number of applications.

Although the microstrip has a very simple geometric structure, the
electromagnetic fields involved are actually complex. It is clear that the microstrip
involves an abrupt dielectric interface between the substrate and the air above it.
Therefore, the microstrip belongs to a family of inhomogeneous transmission lines.

This implies that no simple TEM or waveguide-type TE and TM modes exist

 

independently. An accurate and thorough analysis requires quite elaborate
mathematical treatments. The early work on planar transmission-line structures
was based on quasi-TEM analysis [18]-[Zl]. Most of these papers were directed

toward the evaluation of the static capacitance of the structure, from which the

effective dielectric constant (ed) which determines the propagation constant

(k, =k0i/e—d')’ and the characteristic impedance are subsequently derived. A useful set

of approximate relationships was derived by Wheeler [19]. Yamashita [20] presented
a theoretical method to analyze microstrip lines based on a variational calculation of
the line capacitance in the Fourier—transform domain.

Microstrip cannot support a pure TEM, or any other simple electromagnetic
field mode. Therefore, the quasi-TEM analysis, which is approximate, is inadequate
for estimating the dispersion properties of the line at higher frequencies. Hybrid
mode analysis of the microstrip structure is required. There have been several
approaches to the hybrid mode analysis of microstrip [l], [22]-[34]. Some of these
are briefly discussed below.

Denlinger [22] presented an approximate hybrid mode solution that gives the
frequency dependence of phase velocity and characteristic impedance of an open
microstrip line deposited on either a dielectric or a demagnetized ferrite substrate.
Getsinger [23] reported an interesting approach by proposing an alternative model for
microstrip which is arranged for more straightforward analysis than the microstrip
itself. It is important to note that the model is not physically realizable but it is much
easier to analyze than the real microstrip because it is simply a parallel-plate line.

The most popular method used to analyze the microstrip transmission line is

 

 

(a)

 

 

 

 

 

(C)

{—3—‘——_l

(0)

 

(b)

 

 

(d)

 

 

(0

Figure 2.1: Some transmission line structures suited to microwave circuit
integration.

 

 

the spectral-domain method. In this spectral or Fourier-transform method, the
Green’s function and boundary conditions are formulated in the spectral domain. The
integral equation for the currents on the strip is also solved in the transform domain.
This method is very attractive due mainly to the fact that the Green’s function is
relatively simplified in the spectral domain. This method appears to have originated
in an early paper by Yamashita [20] and was refined by Itoh and Mittra [26]. A more
recent effort by Jansen [28] gives a survey of the spectral~domain approach for
microwave integrated circuits including the shielded-, covered-, and open-type
microstrip. He also discussed the different aspects of this approach and considered its
numerical efficiency.

The dispersion characteristics of microstrip lines have been investigated by a
number of authors using a variety of methods. However, the numerical results shown
in many papers were calculated using a small number of basis functions to save
computation time, and the current distributions were not expressed accurately. This is
the major cause for the significant disparity between computed results, as shown by
Kuester and Chang [27]. The current distributions are fundamental quantities as
sources for the electromagnetic fields of microstrip lines. Therefore, it is crucial to
accurately represent them. In general, it is preferable to take the edge behavior of the
current into account explicitly since this results in greater accuracy with fewer terms
in the current expansion. Kobayashi [30] proposed closed-form expressions for the
current distributions that satisfy the edge singularities. Using these expressions, the
frequency-dependent characteristics for the effective relative permittivity of microstrip
lines were calculated by spectral-domain analysis. In a later paper by Kobayashi [33],
the spectral—domain analysis using Chebyshev polynomials as basis functions is used

10

to obtain the frequency dependence of current distributions and the effective relative

permittivity of an open microstrip line up to h/Ao =1 (normalized substrate thickness).

In a paper by Fache and De Zutter [32], these characteristics were not shown in

frequency ranges higher than h/Ao=0.2.

2.3 CHARACTERISTIC IMPEDANCE

Accurate modeling of interconnections have gained increasing importance due
to their presence in high-spwd electronics and micro/millimeter-wave integrated
circuits. Hence, it is necessary to obtain an equivalent transmission-line model which
represents a circuit description of the microstrip structure so that it can be analyzed
when connected to TEM structures such as loads and drivers. A large amount of
attention was paid to evaluating the dispersion characteristics of microstrip, as seen in
the previous section. In order to obtain a complete equivalent transmission line
representation, not only the dispersion characteristics must be evaluated but also the
characteristic impedance. There have been many different approaches to the dynamic
problem of microstrip characteristic impedance and quite different functions of
frequency have been predicted, extending even to opposing trends [35, 36].

The classical definition of characteristic impedance as the ratio of voltage to
current at any point along a transmission line is meaningless for non-TEM structures.
For a perfect TEM line, the electric field is conservative in the transverse plane,
hence the voltage is uniquely defined as the negative path integral of electric field
from one conductor to another along any path on the same transverse plane.
However, for non-TEM structures such as microstrip lines, the path integral of

electric field mentioned above is dependent on the path of integration. This matter

11

 

 

has been clearly discussed by Getsinger [37]. Voltage cannot be uniquely defined,
and hence the above definition of characteristic impedance is ambiguous. As a result
of the ambiguity in the definition of voltage, there is a wide disagreement among the
microwave community about how the microstrip characteristic impedance should be

defined. As an example, Bianco et al [35], defined impedances in terms of mean

voltage V and center voltage V6 to yield a total of five definitions as follows:

 

 

V
z = _
mm ,
VC
2P
= _. 1
VCVC
2°40) = 2P
_ W*
0,50) - 2P

where * denotes complex conjugation. The complex power P is evaluated using

Poynting’s theorem. These definitions lead to a variety of results. The general

conclusion found by Bianco et a1. is that 20.1 and 20.4 always rise with increasing

frequency whereas the remaining impedances all fall with 203 exhibiting the smallest

variation.

Getsinger [38] developed, using the model described in the previous section,
an expression which has microstrip characteristic impedance varying inversely with
the square root of 6mm- In a more recent paper, Getsinger [37] defines the "apparent
characteristic impedance” on the basis of accurate measurements of the reflection loss

in the transfer of power between the source and the microstrip line. It was found that

12

 

the measured impedance showed a large frequency deviation. Hashimoto [39]
presented rigorous, closed—form expressions for the characteristic impedance of
microstrip given by the ratio of the electromagnetic power flowing along the strip to
the square of the total longitudinal electric current. Pozar and Das [13] used a full-
wave analysis to evaluate the characteristic impedance of several microstrip
configurations by using the ratio of average voltage to the total longitudinal current.
They showed that the characteristic impedance rises with frequency. As shown

above, different authors choose different definitions of characteristic impedance

(namely the voltage-current definition 2 = VII , the power-voltage definition

2 = |V|2/2P and the power-current definition Z =2P/ |I|2). All the three definitions of

characteristic impedance lead to different results due to the ambiguity in the definition
of current and voltage. However, it has been shown by Brews [40] that all three

definitions of characteristic impedance become equivalent if we require the complex
power P to satisfy the relation P=I ‘V/ 2 , which is a natural requirement upon any

transmission line model. Moreover, he required that both the microstrip and the
equivalent transmission line have the same propagation constant. The voltage or the
current is chosen in such a way that one of them can be given a circuit interpretation
but not both. In case of a single microstrip, if the current is selected as the
independent variable, it can be chosen to be the total longitudinal current. If the
voltage is selected as the independent variable, it can be chosen to be the strip center
voltage. It is only in the low-frequency or quasi-static limit that a circuit meaning can
be assigned to both the voltage and the current. Furthermore, Dezutter [41] showed

that power current definition is the most appropriate model as a circuit description

13

 

since it has the most TEM-like character, as its value only starts to increase at higher
frequencies as compared to the other models. Consequently, several authors adopted
this method [42, 42, 44].

A recent paper by Cheng and Everard [45] presented a new method for the
derivation of the characteristic impedance of an Open microstrip line assuming the
quasi-TEM mode of propagation. It is based on the spectral-domain approach with
4 rectangular shaped basis functions. Finally, Slade and Webb [46] used a Finite
Element Method to compute the characteristic impedance for several microstrip

geometries.

14

CHAPTER 3

DYADIC GREEN’S FUNCTIONS FOR THE EM FIELD IN PLANAR
LAYERED MEDIA BASED UPON WAVE MATRICES
FOR ELECTRIC HERTZ POTENTIAL

3.1 INTRODUCTION

In this chapter, dyadic Green’s functions for the EM field maintained in any
region of a planar, layered environment by general electric volume currents in any
other region are constructed through electric Hertz potentials, using wave matrices to
determine the spectral amplitudes in their Sommerfeld-integral representations. The
electromagnetics of planar layered media has received much research attention since
the original treatment by Sommerfeld [2], with relatively recent efforts including
those exposed in Brekhovskikh [3], Wait [4], Felsen and Marcuvitz [5], Stoyer [10],
Kuester [ll], Kong [8], Mosig [l4], Michalski and Zheng [15] and Chew [9].

The construction exposed in this chapter is based strictly upon the electric
Hertz potential, and exploits wave transmission matrices for the tangential and normal
components of potential maintained by respective currents as well as coupling
matrices which couple tangential currents to normal potential components. This
method differs from prior ones where the fields were obtained from generating
functions consisting of either the normal electric and magnetic field components as in
Kong [8] and Chew [9] or the normal components of electric and magnetic Hertz
potentials as in Das and Pozar [l3] and Stoyer [10]. Another approach is taken by
Michalski and Zheng [15] where they use a transmission line analogy to layered

15

media. By using wave transmission matrices here, the Green’s functions are con-

structed systematically.

3.2 CONFIGURATION

Each of the planar layers is assumed to be linear, isotropic and locally

homogeneous, with complex constitutive parameters £521,260 and u, =m,-2 p0 , leading
to wave number k, and intrinsic impedance n, in the i ’th layer. The wavenumber is
k,=to eiufinlmlko and r],= u,/e,=m,n0/n, with komothe free space wavenumber

and intrinsic impedance, respectively. Contrast among the various layers is described

by ”final/"r and M,=mM/m,. A coordinate system is chosen with the y-axis
normal and the x,z axes tangential, respectively, to the planar interfaces at y =y‘. ,
such that the i ’th layer resides within y,<y<y,_l and y, denotes the coordinate of the

boundary between the i ’th and the (i + 1) ’th layer as indicated in Fig. 1. An electric

volume current source is assumed to reside within the i ’th layer. As a result of
multiple reflections at boundaries between the layers, two waves will exist in each of

the regions, with the exception of regions 1 and N. One of these waves propagates
in the positive y direction, and the other in the negative y direction. Wave ampli-

tudes in region I above the source region are referenced to the (I - 1) ’th interface and

denoted by (If while amplitudes in region I below the source region are referenced

to the I ’th interface and denoted by of.

16

yr

3:

yt-z

31-1

yr

ym

YN-z

yN-l

Region 1: (61,111)

Region 2: (62.11,)

 

 

Region (i-l): (94,914)

 

Region i: (app) 0,. 1

a; 16‘. 1 a,‘ I

 

 

 

Region (i+1): (€91,914) 0&1 1 (11:1 I
O
O
0
Region (IV-1): (en-vI-‘N-t) a§-11 aJ-nt
6,; I

interface]

interface 2

interface (i -2)

interface (i -1)

interface i

interface (i +1)

interface (N -2)

interface (N -1)

Figure 3.1: Configuration of Planar Layered Media

17

3.3 HERTZ POTENTIAL AND ELECTRIC FIELD
In this section, final results of the new formulation are exposed to provide an

overview, with the details following in subsequent sections. The electric field in the

I ’th layer can be expressed by generalization of the technique used by Bagby and

Nyquist [47], as E, = (k,2 +VV-) II” where the Hertz potential in the I ’th layer main-

tained by currents in the i ’th layer is

firm = 17m] Girl r’)'.7(i”)dv’ (1)

with G, = Iqu,’ + G; . The first term leads to the principal wave excited in an

unbounded i ’th layer, and has the sommerfeld-integral representation

 

_‘ 31“""I’e rib-3’! 2 (2)
Gf-'-(?|r’) H 20:?!” d).

where p, = 134:, and I = £5 +£C is the 2-D spatial frequency transform variable,

with 12 = £2+C2 and d2}. = dEdC . The branch cut for the square—root in p, must

be chosen to implement Re{p,} >0. This principal Green’s dyad has been shown by

Viola and Nyquist [48] to accommodate the electric field source-point singularity, and
permits identification of Yaghjian’s depolarizing dyad [49]. The scattered dyad has

the form

ogrrlr’) = (we) 6.2+? [(2% +£%]G,L+G;j] <3)

18

with scalar component representations

: .. .. . r [,1
6130!") 31:01? I ) er-(H’)
GMO'IFI) = If SuO’I)’ 94-) W
Grimm -' 31:0 13”,“ p,

d2; , (4)

Clearly 6;, yields tangential potential components due to tangential currents, 0,};
yields the normal potential due to normal current and 6,; couples tangential currents

to normal potential. Scattering coefficients S; are determined through the wave

transmission and coupling matrices, and assume the generic forms

‘ S I.
5:? = Z 311(1)e’°""”’“~f0r q = t.n.c . (5)
k ' l

Expressions for the 3,3, are obtained using wave matrices as described below, and the (bf,

are simple expressions, e.g., of, = p,(y-y,) +p,(y,_l-y’ H) which take special forms

when I =i and where t, represents the thickness of the i ’th layer.
If the differential operator is passed, with due regard (using Leibnitz’s rule)
for the source-point singularity (when I =i) of G, through the superposition integral

in (1), then that field is obtained as
" _ Hi"! 0 -o -o -0 /
5,0) - —f Gu(r|r’)- .7(r’)dv . (6)
It, V
The electric Green’s function is identified, by the method described by Viola and

Nyquist [48], as earl?) = PV(k,2 +vv-)G',,(r|r’) + [saw-r"). Notation pv

l9

indicates that G; should be integrated in a principle-value sense by excluding an

innate "slice" principal volume at y = y’ to accommodate the source-point singularity

of the principal-wave contribution at that point, and I: = 1‘9" is an associated

depolarizing dyad.

3.4 WAVE MATRICES FOR SPECTRAL HERTZ POTENTIAL
Uniformity of the planar, layered media parallel to the x-z plane prompts

Fourrier transformation on those variables, leading to the transform pair
.- r
fade) = ffn,(r)e-I"'dxdz
"' t fora =x,y,z . (7)

 

l -- .. .
11 =— n r. e’x'dzl
.(i) (2101].! .< .y)

The spectral-domain Hertz potential components in the I ’th layer satisfy the trans-

form—domain Helmoltz equation as follows

 

fir - =7 (Ly)

d2 2 [.(AJ) " (8)
— - ~ ‘ = . no. or a =x .

(dy’ ']{112.(1.y)} ”’6' f M

We must distinguish a region I residing above the source region i , where the

incident waves are travelling upward, from a region I below the source region,
where downward travelling incident waves exist. Hence, all quantities associated with

a downward pointing arrow are for a region I below the source region i , while those

associated with an upward pointing arrow are for a region I above region i. For a

20

region I ti underneath the source region, .7“ =0 and appropriate homogeneous
solutions to (8) are

finds) = am) e"’"” + arid) cw") <9)
where the wave amplitudes are referenced to the I ’th interface. For the case of
upward travelling incident waves the source-free Hertz potentials are

13,..(530') = a;.<x)e’*°"'-" + are» "N". <10)

where the wave amplitudes are referenced to the (I - 1) ’th interface. In the I=i

source region, the total potential is expressed as the sum of a principal wave main-

tained by .7“ in an unbounded environment and a homogeneous solution representing

waves scattered from the planar interfaces at y, and yH as follows
11,42.» = rim,» + fruity) . (11)
The principal wave is given by

(12)

 

~ _. , j‘(x,yl) 7113"“
“from = f ‘ . ‘
’r-r 106‘ 2pi

which is a forced solution to eq. (8) as developed by Bagby and Nyquist [47].
The Sommerfeld [2] boundary conditions for electric Hertz potential are
generalized to accommodate arbitrarily polarized potentials at the interfaces between

media having both electric and magnetic contrast. In the spectral domain, the

boundary conditions at the I ’th interface require [50]

21

 

114.05.» = N,M,fr nude)

- 13

w = N26fl1+1,¢(xay) I "for“ "xii ( I
a? ' ay ,

and
fl,,(i°.y) = szfir.1,(7-.y)

(aligns) _ afimfi.»
6y 6y

where the last condition couples normal potential components to tangential ones.

(14)
. -(N3M,2—1)I,-efi,,,,(i.y) amped»)

Applying conditions ( 13) and (14) at any interface leads to relations among the
scattered wave amplitudes which can be described by wave matrices as detailed in
Appendix A.

For a region I below the source region, a downward recursion is derived.

Hence, the tangential amplitudes are related by [“ch = [A,'i][a,,l’a] for a =x,z,

where the 2 x 2 matrix [Aft] is the tangential wave transmission matrix which chains

tangential wave amplitudes in region (1+1) to those in region I . The first entry in
the 2x1 matrix [on] is the incident wave amplitude at the I ’th interface while the
second entry is the reflected wave. Hence the above notation is a shorthand for

II I

aI+l,¢
6
“(+1.41

for a =x,z . (15)

 

 

The boundary conditions couple normal and tangential potential components such that
0‘?_ =[A'n I] ably +[C ’1] j: 01:1,: + j: “1+1; ] (16)
a", dbl; aI:loZ

aI+lJ
22

 

 

For a region I residing above the source region, an upward recursion is

derived. Hence, the wave amplitudes in region I are related to those in region (I - 1)

as follows

a 51 a:
1“, =PHIIQ: + CI-1]j£ I.” + jC ll" , (18)
0H" 01-1.: 01-1,:

In eqs. (16) and (18), [Aft] and [A1311] are normal transmission matrices

and

 

while [(3,1) and [CHI] are the interfacial coupling matrices. PM] chains normal

wave amplitudes in region (1+1) to those in region I . [CA] couples tangential wave

amplitudes in region (I + 1) to normal wave amplitudes in region I . The wave

matrices are identified as

23

-l
l ¢l+l qul ¢l*l
[Afl] = .____ forq = t,"
T‘l q -1
I RI l¢l+l ¢Hl

1 4’1-1 nglt‘bI-ll
[A1311] = —-— ...forqgt,"

7131' R" i “
I-l ¢I-1 ¢I-1 (19)

-l
-2 -2 '4’ - ’4) — I
1-N,-,M,_, ' 1 ’ ‘

 

[941‘ 2,,

 

-1
¢1-1 ¢I-l I
-l
Ntlez-l -¢l+l -¢l*l
[C11] "'

2p _
1 $1 +1 ¢I +11

where ¢r+r = exp(pth) and 4)“ = exp(thH), with tM = |y,-yM| the thick-
ness of the (I + 1) ’th layer. R," and T," are interfacial reflection and transmission

coefficients defined in Appendix A.

3.5 DYADIC GREEN’S FUNCTION FOR LAYERED MEDIA

The wave matrix formalism allows the Green’s function to be determined
explicitly and in a systematic manner. First, the fields in the source layer i are
evaluated, then fields in any source-free region I due to currents in region i are
determined.

3.5.1. Potential in Source Layer (I=I)

Since only outgoing waves exist in regions 1 and N, the wave matrices can be

applied to obtain relationships between the wave amplitudes in those regions and the

source region i . For regions above the source region, an upward recursive scheme

24

is used; for layers below the source region, a downward recursive scheme is used.

A. Generalized tangential reflection coefficient

Downward recursion: The tangential amplitudes at interface I are related to those in
region N as follows

[:53] . [fins] 1 [(23.1 ,0, 0, W. (20)

I,a

Hence, recursion relations for the overall reflection coefficients associated with

downward travelling wave incident upon the I ’th interface are obtained as in Chew

[3]

R'i + at, I ”MM

1 + Rfi 98,11 admin

 

where of. = Sift ai, (o: = x,z). Eq. (21) is obtained from eq. (20) by solving for

the reflected wave in terms of the incident wave in region I . We first considered the

case of two layers, then three, then four regions. In each case, the reflected wave is

written in terms of the incident wave, thus the recursion relation (21) follows. The

generalized reflection coefficient 8th associated with downward incident waves

accounts for reflections from all interfaces below the region I .

Upward recursion: The tangential wave amplitudes in region I at interface (I - 1)

are related to those in region 1 as follows

25

[22:] In M I III

. (-1
Hence, we obtain the generalized reflection coefficient for upward travelling incident

waves

' I I 'ZPi-l‘i-I
$1-11 = RI-l + aI-Z e (23)

l + RIt-lt $;_2I e '2PI-I‘I-l

 

where Ii", = $1.11 6,; (a = x,z). $2.41 accounts for all reflections from all

interfaces above region I .

B. Generalized normal reflection and coupling coefficients

Downward recursion: The normal wave amplitudes at interface I are related to those
in region (I + 1) and are coupled to the incident tangential waves at interface (1+1) as
shown in eq. (16). We want to chain wave amplitudes in region N to those in region

I . Note that the tangential wave amplitudes in region I are now available through eq.

(20). Rewriting eq. (16), we have

,. a'. . a: . a2
= [4: 1][ '."’ + [C11] [15 [ '."'] +16 [ '3‘] ] . (24)
“(+1; “(+1; “1+1;

Now, we write the normal wave amplitudes in region (1+1) in terms of the normal

ale
0:;

 

 

 

and tangential wave amplitudes in region (I + 2) leading to

[2'3” WI

1+1;

aI:2,z
01:2;

 

0'32“ HI
0112,!

 

 

 

 

:3» . [0,.,I](jia ] <25)
01:20

26

Using eq. (25) in eq. (24), we have
11 Panda “1:2,, + [0111(15 t: M" JC [a [In] ]
“(+1; “(+1.1

0,,”
The tangential wave amplitudes in either region (1+1) or (I +2) can be written in

 

 

 

     

 

(26)

01:2,:
0,:

 

 

 

 

terms of those in region I using eq. (20) as follows

 

 

I' - ' _ W
aI+l,a _ I -1 aka
. - I411] .
_al+l,¢‘ 01,.
“1:2,. - “1+1...
‘ = P1111]! I... a=x,z . (27)
Lal+2,¢‘ 01:1.6

 

 

a-
-1 -1 1m
= PM W I

a“ J

Exploiting eq. (27), the normal wave amplitudes in region I as in eq. (26) are now

given as

(:23

'1] WI :.

 

 

     

 

+ {[0,1] 94:11" + M [M MI" W" I

Cu]

“1.:

_ (28)
a

x j: ‘2‘ +1:
aw

 

 

 

Next, the normal wave amplitudes in region (I +2) are written in terms of those in
region (I +3) and are coupled to tangential wave amplitudes in that region. This

process is repeated until normal wave amplitudes in region N are chained to wave

27

amplitudes in region I . Hence, we obtain

1W1] I I3I + [911

1.:

N-
a
’1‘ wt

“1.:

ab =
“A?

If

 

 

 

 

 

I (29)

 

 

 

a
k . a
where the overall coupling matrix [C,1] from interface I to interface (N - 1) is given

in terms of interfacial coupling and normal and tangential wave transmission matrices

asfollows
[911 = [0:1] WI“ + WIIICMIIIAMIVII“
+ Ifiwq I [CN-IIII4§-11]" I f1 I491“ I -

k-N-Z

(30)

Similarly, the normal wave amplitudes at interface (I - l) are related to those in

region 1 and are coupled to the upward incident tangential waves at the same inter-

] (31)

[CI-1'] = [cl-11] [Alt-1'].l * [A];t][C,_zt]I4,‘_2tI'lIA,'_ltI'l
+ + (‘11me I [Clt] WII" I kfizth‘ I .

Application of the above wave matrices leads to the overall normal reflection

face leading to

5:;

“1.:

1.:
at:

6,": ii

.. + II
“A?

 

 

 

 

 

11L W'] I [“30] + [9-11] I}:

*-

 

 

where

(32)

(8641,87 1) and coupling (CHI ,C,l) coefficients at the (I-l)’th and the I ’th inter-

face. For a region I below the source region, we use eq. (29) to solve for the re-

flected normal wave amplitude in region I in terms of the incident waves in the same

28

region and writing the tangential reflected waves as a“. = 9!?! an; ( a =x,z ) leads to
0,; = 91710,; + CllUEal; +jCa,;) . (33)
Similarly, for a region I above the source region, we have
5,; = 37414:, + 9-1'0551; +jca,;) (34)
where 9371 is the overall normal reflection coefficient for downward travelling inci-

dent waves at interface I . The latter coefficient is given by a recursion relation
identical to the one for 93:1 with the superscript t replaced by n in that expression.

The overall coupling coefficients at each interface are

- -a;'1(cmt+stjtcmt) + (Czul+9t;1le)
'mf-IHCHJ-tt +8tI-l'Cer-rt) + (6211-1t +9tI-1'C2244')

~23

(35)

C...1

and Cu J1 , for example is the first entry of the coupling matrix [C,l] given by eq.

(30).

C. Potential in the source region

Referring to Fig. 1.1, the wave amplitudes in the source region are referenced
to both i and (i - l) interfaces. Hence, the scattered wave amplitudes at these inter-

faces are given in terms of the incident wave amplitudes as follows

all“ = filaha+ for “ =x’z (36)
51,: = 3:4 '51.:

and

29

at; = 331...; + Wheaten) (37,
6,; = 87.1%; * 9-1'0551; WI) °

The incident wave amplitudes 5;. and a"; in the source region need to be quantified

in order to find the total potential in the i ’th layer maintained by volume currents in

the i ’th region; they can be obtained with the aid of Figs. 3.2 and 3.3. Direct re-

flection of any potential component is accommodated by Fig. 3.2, while Fig. 3.3 is

applicable to coupling of tangential to normal waves.

Tangential and normal components

Referring to Fig. 3.2, the total upward travelling incident wave amplitude at

interface (i - 1) consists of an upward travelling principal wave, (if; maintained by

source currents, augmented by the incident wave amplitude at interface i , reflected

and then shifted to interface (i - 1) as follows
= a:;+a;.e"~=ar;macaw" <38)

with the understanding that q=t when a =x,z and q=n when a =y. Similarly, the

total downward travelling incident wave amplitude at interface i is given as

0,; =af; Wig-”Hag + ,-,ta,:’,e"‘" . (39)
The cf; and 53’; are the principal waves at interfaces i and (i - 1), respectively,
maintained by source currents in region i . Replacing y by y,,, in expression (12)
and noting that y>y’ at interface (i - 1) leads to the expression for (If; as

30

 

‘3

Region (i - 1)
)‘H 1 I interface (i-l) $1”
Region i . I - 1 at} 614
all aifl . . g
y, mted'ace I St, 1

 

Figure 3.2 : Normal and tangential wave amplitudes.

Region (i-l)

yI-l
Region i T - 1 iii 5,17
Y; a“ a“ interface i 9:31, C,l

 

interface (i-l) Stilt, CHI

 

Figure 3.3: Coupled wave amplitudes.

31

 

., + '° - l-l J¢(X 6 5”
a3, = figment) = e ”H I; jet): ‘2; dy’. (40)
I i I

Similarly, to obtain the expressions for af; , we replace y by y, in expression (12)

and note that y< y’ at interface i , leading to

 

... ”-1 J (XJ’) e'Pd’
up = HP (2: ’ ) = ep‘y‘f a. I . (41)
" u y‘ ,, me. 2p. dy

Solving for 0,; and 6;. from eqs. (38) and (39) leads to the unknown total incident-

wave amplitudes in terms of the known principal-wave amplitudes as

 

 

a- = a3; + Hie Mag:

La 0‘ ‘7’ - (42)
6+ = difa + ”-11: ‘aa

L6 D‘

with 0" =1 - a: 1 91111 . With the above construction, we finally obtain the scattered

potential in the source layer i due to reflection only as

fi' .(X‘y)= 0‘. *Heph’1)+a “WI-l)

(43)
=- 8131a,,“ "0 ’9 + 81‘,’_,16,;e"°""‘).

Substituting the wave amplitudes a"; and (if, given by eq. (42) into the above Hertz

potential expression and using eqs. (40) and (41) for the principal waves in region i

yields

32

 

ll

 

4 41:
I141.» = 2 331(1) I 1.7“ ”2 ‘ pdy’ (44)
hi

jwe,
where the 8,3, are expressed in terms of the generalized reflection coefficients as

follows

 

Dq I...forq=t’n (45)

“a:
a; a
ll
5‘1.

 

and the phase shift factors of, are

«I. = p.(-y-y’+2y,-,)

ti = p.0-y’+2)'.-r2y1)
«bi = Pt(-y+y’+2y,-.-2y.)
$14: = p10+yl'2)’3) -

(46)

Coupling components

Referring to Fig. 3.3, the normal component of the incident wave amplitude
due to coupling at interface (i - 1) consists of a normal reflection of the coupled com-
ponent augmented by a coupling of tangential components of the incident wave ampli-
tude at interface i with everything shifted to interface (i - l). Mathematically, this is

written as

a; = (124331 + q1F3')c"" . (47)

33

 

A similar argument leads to the coupled normal component at interface i as

“to = (5837-11 + (Ii-1'13”)!”i (48)

where F," and F," are defined as

Fa" = 150.; + Kai; ; F,“ = jea; + jcag, . (49)
Solving eqs. (47) and (48) simultaneously, the coupled wave amplitudes are obtained

in terms of total tangential components as follows

.. _ Ff‘c,_,tst;‘1c'z’" + F,“c,1e""

 

 

“to:
D" (50)
- Ff‘qtst:_,tc"’" + F,‘*c,_,te""
a“ = D“ .

Since the tangential wave amplitudes are written in terms of the known principal
wave amplitudes as in eq. (42), we substitute the expressions for 0"}, at.“ a"; and a,;

in eq. (50) to obtain

at“. = Bite""<i£a£; + jcaa‘) + Bifse'zmtifiafl' + m3?) (5,)
6.; = Bée'z”‘U€d£ + mg) + Biew'tizai; + m5)

where the 8,; (k

1,---4) are identified in terms of the overall reflection and cou-

pling coefficients as follows

34

l

 

 

 

 

Bite! = D"D‘ {glaring-1'94," + CHI}
1
813 = My {CHIWI + 931-1191} (52)
c _ 1
Bus ’ D'D' {9137.11 * ailCl-l”
l -2,
3;, = D'D‘ {cHtstftstjte "‘ + qt}.

The normal scattered potential component must be augmented by the coupled compo-

nent given by
End.» = dice‘p’o-y“) + spew-’9 . (53)

Now, we substitute eqs. (51) into the above expression and use eqs. (40) and (41) for
the principal wave amplitudes. Consequently, the expression for the coupled compo-

nent of the Hertz potential becomes

4
flux.» = 2 3121(1)

l-l

 

f-J'EJ",(X.y’) waldo) e*””'*’ g] (s4)
.. jue, 2”:

where ¢f1(y|y’,i) are defined as in eq. (46).

We finally obtain the total transform domain Hertz potential in region i ,

which is the sum of scattered and principal waves at any point, where the principal

wave is given by eq. (12). Hence, we have

find.» = 112.6.» + 133.03.?) + Games) a = x.y.z (55)

Next, inverse transforming the Hertz potential as in relation (7) and writting .7423!)

in terms of space domain currents as

35

 

1036') = IIJ,(?’)e"fi" dS’ (56)

leads to expression (1), where the 8,; and of, in the Green’s dyad components are

identified as in relations (45), (46) and (52). Once the potential in the source region
is known, the potential in the I ’th layer can be deduced from it by wave matrix

multiplication as discussed in the next subsection.

3.5.2 Potential in Region I due to currents in region i
The potential in any source-free layer I can be constructed in terms of the
now known wave amplitudes in the source region using wave matrices. We assume

that the region I under consideration resides below the source region i such that a

downward recurSion is used. The Hertz potential in region I consists of only scat-

tered waves with tangential, normal and coupling components as follows

fin..(xay) = fiIL¢(X’y) + bayfi’a;(xfiy) a = xaysz 0 (57)
We want to write the wave amplitudes a; and oi, in region I in terms of the wave
amplitudes in the source region i .

Tangential and normal components:

In this subsection, we assume that the normal components of potential are due

to only normal component of primary current in region i . Referring to Fig. 3.1, the

tangential and normal wave amplitudes at interface I are related to those at interface i

as follows

36

a“

a
“Ln = [qul ] a

 

 

 

TI for a =x,y,z (58)
La

where the expression for {D} l] is defined as

I

[bit] .-. k1:1_[.4:t]-1I ‘

h

’ = 59
61311 dlqul I forq t,n . ( )

 

 

. i." 4211‘ .
We then obtain the unknown tangential wave amplitudes in region I in terms of the

known wave amplitudes of. as follows

01:. _._ dnqul 01:0 + (1142.110ch (60)
01; = €121,110); + dIngaL‘a '

The scattered potential in region I due to reflection only is consequently written in
terms of the wave amplitudes in region i . By using eq. (60) in eq. (9) we obtain

fiaafi‘y) =(dl‘11J1aL'. + (113.11 of“) €930")

- , _ _ (61)
«detect. + «Lia...» "0 "’
Substituting the wave amplitudes at. given by eq. (42) into the above expression
leads to
filed.) = 311': gm") 51’; + 3326 "’""""’ (if; (62)

+Blge’10’YIIOf; +31?“ 710-190;; .
The coefficients 8,3, ( k=1,...,4) are expressed in terms of generalized reflection

37

 

 

coefficients and the entries of the matrix ID,"1I as follows

81-11(‘11141 +d1‘12419til)

 

 

 

 

B"=
111 0‘
Bq_i-11'(d211*d2qu‘9m)
112' D4 (63)
B, (df’undfutstft)
D!
114 D" °

Finally, we use eqs. (40) and (41) for the principal waves of: in eq. (62) to obtain

4 wiob’l)
fii..(i.y)=£ 8110) f j “a.” dy’ <64)
k-l

jwe, p,

 

where the phase shift factors are

4)) mom a") 710-») W.
¢i=p.<y.-i-Y’)+Pz(v-yi)+Pa (65,
W: = 19.6.? ’) 12,6 1,)

<13 = 7.01 1') +1010 3)) -

Coupling component:

In this subsection, the normal components of Hertz potential in region I due
to coupling to tangential currents in region 1 are obtained. The normal wave ampli-
tudes at interface i are related to those in region I and are coupled to the tangential
amplitudes at interface i as in eq. (29) with, region I replaced by region i and

region N replaced by I , in that equation, leading to

38

0.
+11 ‘1‘
“1.:

“A?

”1]
where the overall coupling matrix at interface i is

[CH] = [Cd] M" + MGMIIWI'M"

   

 

 

 

 

   

[Cal] (1'5: ] (66)

 

     

1-2 -1 (67)
+(,I_I,l4ill][cz-HJW-dl (311M ]
Isolating the normal wave amplitudes in region I leads to
L4 11W] MAW" - [911(jez: +1: 3} ]. <68)
u e

 

 

 

 

   

 

 

Multiplying the above matrix equation by the product of inverse transmission matrices

“‘3‘ + 1c “"3 . (‘9)
“u “1.:

[Afillr [Aflrl from the left provides

' 40:11

O

 

 

 

 

 

 

0,; ,
+ F

where [0:1] is defined as in eq. (59) and [Fat] is defined as

‘ [”1"][9‘]

sw‘ fnu‘ (7°)

[Fu‘]

ml fzual

Hence, the normal wave amplitudes in region I are obtained from eq. (69) as

06=dfiilae+d£flaa$ +(fxu‘ *fMWWiaa—N‘W (71)
as=dmd+datas+<rmt veimiuviawflaa -

We want to find the normal wave amplitudes in region I due only to coupling with

39

tangential currents in the source region 1 . Using the fact that the normal wave
components 0:, in the source region consist of normal components maintained by

normal currents augmented by coupling components due to tangential currents. Hence

the normal wave amplitudes in region I due to coupling with tangential currents in

the source region i are

0L2=dfillaé+dfiylaé +(fllJIl +flwlfitil)(iEaL-, +jCaL-z)
aé=$11103;+d£flaé+(fzwl+fm,19t:1)(i£ag;+jca‘;) .

(72)

Referring to Fig. 3.3, the upward travelling coupled wave amplitude 01:, at interface

i consists of the incident coupled wave amplitude 6,; at interface (i - l) shifted to

interface i as follows

a,; = a; e“ (73)

The coupled wave amplitudes 0,; and 6,; were obtained previously in eq. (50) in

terms of the total tangential wave amplitudes in the source region. Exploiting eqs.
(50) and (73) in expression (72) leads to

:-

F
- -2 ‘
“L: ={dfiquwt‘,'.,1e p"+clldl;.ll+(f11£1...-t'lulaIIIDu};

I
3+

- - F
«as» cm ”‘+d{5.:1C.-n9tf1e P's—g
D p" (74)
a;={d2'},lc,18tf.,te'2”‘+C,1d£,l +(I211:l 422.111 My}?
F"
- u - 1
*{dzix‘cu-i" ph+d£JlCl-lt81 1‘ m};

40

where F,“ are defined in eq. (49).

Since the tangential wave amplitudes in region 1 are written in terms of the

known principal wave amplitudes as in eq. (42), we substitute the expressions for

a,;,a,:z,d,; and 5,; in eq. (74) to obtain

at;- =8,er Tho-E 55 *jC (it?) + 31305“; +jC aft.)

(75)
a; =B,:,e‘”*(iea,f; +1165) + 35.0105 +1605)

where the B; coefficients are defined below

8:1 3 [Cll(d&'ll +d1L18tf_lte'2”‘) +(f‘ul +f1ulfltz)D"]$:_,t +(31-11641;1 #134191?)

 

 

 

 

0'0"
as. [94% +datimr.ne"”')+(fmx meow] 91.1! «2.. «at» aw?!)
D‘D"
Be = €119,111! +dfiJlaI-it‘-m9*(fii.ul +f1ulfitf)D"+C,_lte'”‘(dl'ul +d131‘8719ti1)
B D'D"
Ba“: 916123.11 +d5'lla?’lte-2p’j+(f2|fi1 +fmlai)Dn+Cl-1Ie.ph(d2:fl+d£J1$f13t:I)
0‘0"

(76)

The Hertz potential due to coupling is written in terms of the spectral ampli-

tudes for the normal component coupled wave as

fads?) = aéep‘o.” + an}: 1910-») . (77)

Using the same procedure as for tangential potentials, the Hertz potential due to cou-

pling is given as

41

 

fajfijxdy’) +j(.'i,(53,y’) (”My“) / (78)

X B A
fiLJJF Z M) jwe, 2?: dy

k-l

where the phase factors tbf, are defined in eq. (65).

Applying an inverse transformation on spectral potentials given by eqs. (64)

and (78) leads to the space domain Hertz potential in region I as expressed by rela-

tion (1) where the scattering coefficients 5,? in the expressions for the Green’s func-

tions are identified in terms of the above B); coefficients and phase factors 4%,. For

a region I above the source region, the analysis is similar to the above with the

exception that an upward recursion is used to find the Hertz potential.

3.6 SUMMARY

Dyadic Green’s functions for the EM field in a planar layered environment are
formulated strictly in terms of the electric Hertz vector. Spectral amplitudes in their
Sommerfeld-integral representations are obtained using wave matrices. The formu-
lation based on the electric Hertz vector is complicated due to coupling of normal and
tangential potential components by the boundary conditions. Wave matrices are intro-
duced to handle the coupling in a systematic manner. This method accommodates
general electric volume currents immersed in any region of planar layered environ-
ment. This approach removes any uncertainty regarding completeness of the field

representation and naturally accommodates the source-point singularity.

The electric field in the I ’th layer can be expressed in terms of the Hertz

vector as E, = (k,2 +VV-) fin where the Hertz potential in the I ’th layer maintained by

42

currents in the i ’th layer is

fix?) = %f Guam- jaw (79)
i v
with G, = 13qu + C3,: . The first term leads to the principal wave excited in an

unbounded i ’th layer, and has the Sommerfeld-integral representation

 

G P017”) = If 8110-1") e flab-1| d2). (80)
‘ 2(2u)2 °
.. P;

The scattered dyad has the form

(gm?) = (weaagery (Jr-£— +2§)G,;+G;,y] (81)

 

with scalar component representations

012W”) " SiO'Iy’Jt)
6.2.01?) = ff army’s)
6.1m?) -~ SIICYIY’sl)

6111"")
2(23)2P5

d2). . (82)

 

Scattering coefficients 3,} were determined through the wave transmission and cou-

pling matrices, and assume the generic forms

‘ l I.
s: = *2 317:0” WW'M-u for q = t,n,c . (83)
-r

Expressions for the Bu coefficients and the phase shift factors 4‘1: were obtained

using wave matrices as described above.

For potential in the source region, the corresponding 8,, coefficients and

phase shift factors ¢u are

43

 

 

 

 

 

 

Bug] = ”‘1‘
D!
£4 = ”‘1‘”!
m Dq ifor q = t," (84)
Ba”: = 8:2
3‘; = E
D?
B‘ = 1 {Q18’ 18' 162"" + c, I}
111 0'0: 1-1 1-1 --1
3;, = 1 {gnaw + awe,”
DIIDI’ (85)
“ - 1 lat” t 9?: 1
Big ’ Dubt [Cl 1-1 + I CI-l }
3;, - l {C‘_119tfl$§1e'2”‘ + qt}

- D"D‘
and
Q: = PK‘Y'y/+ZYr-r)
2
¢u = no -y’+2yi-r2yr) (86)

(bi = p,(-y+y’+2ys-r2y,)
(>3 = p.0+y’-2y.) .

The general reflection coefficients in region 1’ are given in terms of interfacial refle-

ction coefficients (defined in Appendix A) as follows

R'ql + 8‘4”! e’ZPM‘M

a!) =
1 + K}: 9:1,: e‘z’m‘m

 

q a. -2p ‘ ...for q=t’n. (87)
RH1 + 1-2' e ”H

l ... Rig-rt 311-2! e'zh-r‘r-r

 

”ti-1' ‘

The overall coupling coefficients at interfaces 1' and (i - 1) are, respectively

- -st;'1(cm1+stjlcm1) + (Cmnstfilcwl)
'37-1'(C11J-1I +3I-1'Cm-rt) + (Cm—11+9tI-1'Cmql)

5.?
l

(88)

9-.)

where Cll J1 is the first entry of the overall coupling matrix [Q1]. The coupling

matrices at interfaces 1 and (i -1) are written in terms of interfacial wave matrices

 

as follows
[Cd] = [Cd] W" + wile...1(4.i.1]“}4:1]"
N-2 -r f r t -r (39)
+... , (II,P‘*"‘] ] [CN_,1]W_,1] ”.112 Pm] ]
[Cl-1t] = [Cr-rt] [AI-11F + [AAI][C,_21]P4,'_21]'1P{_11]-!
(90)

+ + ( 121 PM ] [C11] PM" ( sfizwtl-l ] .

t - H
The interfacial wave matrices are defined by eq. (19). Consequently, finding the

Hertz potential in a planar layered environment is rendered to multiplication of wave

matrices.

45

ll

CHAPTER 4

APPLICATION TO INTEGRATED-CIRCUIT ENVIRONNIENT:
SIMPLE MICROSTRIP STRUCTURE

4.1 INTRODUCTION

In this chapter, we apply the results obtained in chapter three to derive the
electric dyadic Green’s function associated with the layered background of a typical
microstrip environment consisting of conductor/film/cover layers. The microstrip
device is printed on the film layer and resides in the cover region as shown in Fig.
4.1. We begin by specializing the results from chapter three to obtain the EM fields
in the cover layer of the tri-layered environment of Fig. 4.1, then the EM fields in
the film region are derived. In a latter section, a general electric field integral equa-

tion (EFIE) description of the microstrip circuit is developed.

4.2 FIELDS IN THE COVER LAYER

The tri-layered conductor/film/cover structure, typical of the background envi-
ronment in a microstrip circuit, is depicted in Fig. 4.2. The electric current source is
embedded in the cover layer. The electric field in the cover region is given as in eq.

(3.6) as follows

 

_ 1'11. ~. .
3.09 - kc fy mer’) .7(?’)dv’ (1)

where the electric Green’s dyad is identified in terms of Hertz potential Green’s dyad

as

46

 

 

Figure 4.1: Configuration of general open microstrip integrated circuit.

47

\JH
f—t

 

...............................................

..........................................

.......................................

..........................................

............................................
...........................................

........................................................................

    

.......................................................................
............................................................................

..............................................................................
..............................................................................

_ y_4

.....................................................................

 

.............................................

   

  

            

; ..m;;7;,;f/y,7//7

,; Vi.'.«,r/,,¢://,3/:‘;‘/,,t’12:111'fl%228"{:f3:335}???’2// -. / , '7’ 9c/€:///2’ IZ’V/ 31/ I I
I. ,1, 4 C ’ .3 . I. ‘ I, '. / 4;; [71,5515]; .(‘,//,.’/~;.,;2 :1"; x24“)?! ”/4/ ;////// /

Figure 4.2: Typical background environment in a microstrip circuit.

 

48

 

He

CC!

 

EX:

 

G”;(?I?) = PV(k.’+vv)<’i..(r|r) + £50279 . (2)
The Hertz potential Green’s dyad decomposes into a principal part and a scattered

part as follows
(3.: iG:+é;. (3)

The principal Green’s function is given by eq. (3.2) while the scattered Green’s dyad

is given by eq. (3.3). The scattered dyad has the following scalar components

6;.(FIF’) - So'co'ly’J)

 

11-(r-r’)
6;,(?|r’) g If Séblyfl) 2‘2 )2 d2). . (4)
65ch 7’) "' Sibly’m ( 1‘ pt

The scattering coefficients Sc: were determined through the wave transmission

and coupling matrices, and assume the form

4 k .
S0: = ,2 3&1): 4,01%.»me q = t.n.c . (5)
- 1

Hence, finding the components of the Green’s dyad reduces to obtaining the Bee

coefficients and the phase shift factors 4)“ . Since there is no interface above the

source region, all quantities related to upward recursion vanish i.e. 91H! = CHI =0.
Exploiting eq. (3.45) the B“ coefficients are specialized as
8" =8“ :8 =
“I qch “3 ...for q=t,n . (6)
Ba‘ = a?!

For coupling components, we use eq. (3.52) to obtain

49

Bcir = BciZ = Roi-3 = 0

(7)
3;, = Ccl .
Consequently, the scattering coefficients have the special form
Sc: =B§,(A)e 4LOIy’4) for q =t,n,c (8)
where the phase shift factor ¢:c(y|y’,}.) is defined from eq. (3.46) with y, =0 as
my.» =p.(y+y) . <9)
Now, we need to determine the general reflection coefficient $31 and the
overall coupling coefficient Ccl . From eq. (3.21), we have
$31: anstfie'zm ---forq=t,n (10)

 

1+Rc‘lat}le'2”’

where tf=t is the thickness of the film region. The overall reflection coefficient 91}!

reduces to the interfacial reflection coefficient since there is no interface below the
film/conductor boundary. From Appendix A, the interfacial reflection coefficients at

the cover interface are

prc -pf

2
Mcpc pf . ch1= 2 (11)

Kg! = 2 ,
Mcpc+pf Ncpc+pf
where M3 =mlemc2 and N: =nf2/n,2 . Since interface f is adjacent to a perfect con-

ductor, we have

Rf'l=-l ; Rf"1=1. (12)
Hence, the overall reflection coefficients in eq. (10) reduce to

50

01

SL

 

Us

 

ge:

 

_ Rc‘l-enz”

 

 

at: , ,1,
1 -RC 1e (13)
RC”! ”'2”
$21 = -
l ”(fie-2”
The overall coupling coefficient Cc! is given by eq. (3.35) as follows
Cc1 = -a:l(cll,cl +afrlclzcl) + (C21;1+$::1C22.c1) (14)
where Cl l".1 is the first entry of the overall coupling matrix [C01]. The coupling ma-

trix is given by eq. (3.30) and is specialized to this structure as
[C1]=[Cc1] [A;1]"+[A:1] [Cfl] [.4,‘1]" [43]" . (15)
Since interface f is a perfect conductor, the product of matrices
. t 'I t '1 . .
[Ac 1] [Cfl] [Afl] [Acl] reduces to the null matrrx. Hence, usrng eq. (3.19) , the
overall coupling matrix is specialized as
[Ccl]=[Ccl] [41]"

_ 1 -N;’M;’ ‘1 '1
2P c l 1

(16)

Substituting the entries of the overall coupling matrix in the expression for the overall

coupling coefficient in eq. (14) leads to

_ 1- Ni»! ‘2

6* ‘(l+$21)(1+821)- <17)

C

Using eq. (11) in (13) and after some manipulations, the final expression for the

generalized reflection coefficients are

51

 

2*(1) ‘ 2'01)

where
z I“(21) = pr. +p,cothp,¢
Z ‘01) =N3p. +pfianhpfi -

(l9)

Substitution of eq. ( 18) in eq. (17) leads to the final expression for the overall cou-

pling coefficient as

= 21:414fo - 1)
z'mzhm '

Cl

C

 

(20)

Finally, the scattered Green’s function components are simplified as follows

Ge’fllfl - 8:10.)
G.L.(?I?’) = ff 9221(1)
agar) “' C30.)

ejI-(f-P’) e -p.0+y’)
2(21t)2 c

d2). . (21)

 

Eq. (21) is a well known result [48]. Pole singularities within the integral representa-

tions of the Green’s function components, as implicated in the reflection and coupling

coefficients 8121(1), 8121(1) and C610.) , lead to the surface waves propagating in

the tri-layered structure of Fig. 4.2. In fact, Z ‘0.) =0 and Z I‘0.) =0 lead to the ein-

genvalue equation for IE and 7M surface-wave modes, respectively, supported by

the layered structure of Fig. 4.2.

4.3 FIELDS IN THE FILM REGION

The electric field in the film region is given as in eq. (3.6) as follows

52

 

 

-jflc " c -o /
2,0) = k. I. Gfi(r|?’)- ](F’)dv (22)

where the electric Green’s dyad is identified in terms of Hertz potential Green’s dyad

8.8
alarm = (k}+vv-)c‘ifi(?|r’) . (23)

The Hertz potential Green’s dyad consists of only a scattered part given by eq. (3.3),
with the following scalar components
Gail?) . sgoly’m

JI-(r-r’)
Git-(fl?) = H SIOIy’J) 2:2 )2 dzi. (24)
01m?) "' Sfib'ly’J) " p.

 

The scattering coefficients S}: are written in terms of the 3; coefficients and

the phase shift factors 4)}... Hence, these coefficients must be determined in order to

quantify the Green’s function components. The results in the previous section will be

used. Exploiting eq. (3.63) the tangential and normal components are

B;I=B;2=o
Bjé=dfl ,1 +d1‘2f19131 for q=t,n . (25)
81:34:”: aging:

Similarly, we specialize eq. (3.76) to obtain the coupling components as follows

B;,=f,m1 musty +d,';,1cci (26)
31:44”: +fm9tgl+¢g,1cci .

The corresponding phase shift factors are a special case of eq. (3.65) with y, =0 and

y,= -t, yielding

S3

 

3 __ /_
Q: 'pcy/ pfly +0 (27)
4),. =m +150”) .

The matrix [Df‘l] is given by eq. (3.59) as

e.” -Rc'1e-”

 

 

 

-2 -2
[0911143146 ”:
l-Rcl -R‘lepf ePf
- ‘ (23)
‘1’! _ " ‘PJ‘
-2 -2 e RC 1e
[wear = "c ”:
l-Rcl _Rcu‘ep,r e”
and the matrix [Ffil] is given by eq. (3.70) as
[chl] = -[D;1] [Ccl] . (29)

The overall coupling matrix is given in the previous section by eq. (16). Using eqs.

(16) and (28), eq. (29) becomes

 

[F I] = (NC-Mc-z‘Nc-IIQ‘chl) "I” "W (30)
fi 2pc(1 -Rc"l) e” e”

 

 

Now that the matrices [qul] and [F151] are quantified, we substitute their

entries in eqs. (25) and (26) and use the expressions for the overall reflection and

coupling coefficients from the previous section to obtain the final form of the tan-

gential Bfi coefficients as

54

Bl __ Nc-zpc

fi-i‘ . ,,
Sinhpr (A)

31;. ‘Ncgzpc
sinhpfrz’m

 

(31)

 

and the normal 3; coefficients as

Fe
coshpftZ‘O.)
3" = p‘ .
fl" coshpftZ'O.)

 

81:3 =
(32)

 

Similarly, the coupling components of the Bit coefficients are
-2 2 2 -
Be =p¢Nc (NcMc ' l)(1 -e ”’7‘”
2sinhpjtcoshpftl ‘(A)Z I‘0.)

3‘ _ pflc"2(N3M3 -1)(1 +e "Me ""
2sinhp,tcoshp,¢l¢(i)2*(i)

 

(33)

 

where 2"(1) and Z ‘(l) are defined by eq. (19).

4.4 INTEGRAL EQUATION DESCRIPTION OF MICROSTRIP TRANSMIS-
SION LINE

The general configuration of an open microstrip transmission line is depicted

in Fig. 4.3. The conducting strip is embedded in the cover layer adjacent to the film/

cover interface of the tri-layered conductor/film/cover environment. The y axis is

normal and the x and z axes are tangential to the film/cover interface.

If excitation is provided by an incident field 2%) maintained by an impressed

current, a surface current 12(7) is induced on perfectly conducting device surface S,

55

 

 

 

 

 

 

 

7
///5’ / // 'l
//
/ ' / //

Figure 4.3: General configuration of an open microstrip transmission line

56

 

't

 

producing a scattered field E". The boundary condition for the total tangential elec-
tric field at the conducting surface S requires that I-(E'+E") =0, where I is a unit

tangent vector at any point on surface S. Expressing the scattered field E" in the
form of equation (3.6) leads to the following EFIE for the unknown induced current

I? on the conducting strip surface

I-ISG‘GIF’)-I?(f)dS’=%3I-E'(f) v 763 . (34)

The conducting strip extends infinetly along the wave-guiding z -axis, and the

system is therefore 2 ~invariant. Consequently, the axial integral is convolutional and

Fourier transformation on that axial variable is suggested. Hence, EFIE (34) is axial-

ly-transformed, using the convolution and differentiation theorems, leading to

. .. -°k .
"I. §‘(i5 It's :0 was .C)dl’=—:-I—‘t'é“(b°.0 v :5 so (35)

c
where C is the'axially invariant boundary contour of the strip conductor in the trans-
verse plane; 5 =21: +9y is the 2-D transverse position vector and C is the transform
variable corresponding to 1. Lower case fields and currents are transform domain
quantifies.

The transform-domain electric dyadic Green’s function is given by

£‘(b‘lb‘0 = 9116'(§|'p”;z)l (36)
= PV(k3+W-)§(filis’;o + testis-5’)

where V=V,+i j C with V,=£6l&x +98/6y the transverse operator. The depolarizing

57

dyad [=99 is never required since only tangential current components are present.

The transform-domain Hertzian potential Green’s dyad g in eq. (36) is expressed in

terms of Sommerfeld—type integrals as follows

 

am la a) =73’(f5 Ii .0 +§‘(fi Iii x) (37)
where
N5 II)"; = (be +22)g.’(i>° la :0 +i(Vrg:(ii I5 :0 +3.16 I5 :09) <38)
and
g’(5l6’;1)=f‘jw—j:p:cb ’ IdE (39)

(40)

 

g..’(b‘li5;0 = f swat)

8r(I5|I5;C) .. [381(1) } eJEOt'I3e'Pco‘7’)
d
33mm) " CW)

411pc

with VT=£aldx+£jC. Coefficients 91,133 and C0,)! are the same as those given in

expressions (l7) and (18), and are functions of C through A. Note that the subscript

c in the scattering coefficients and in the Green’s functions, referring to the cover re-

/

gion, is dropped for the sake of simplicity.

Singularities in the spectral integral representation of the Green’s functions
lead to similar singularities in the spectral domain microstrip current REC). These

singularities in the complex C —plane consist of simple pole singularities which corre-
spond to discrete propagation modes and square-root branch-point singularities which
lead the radiation field with a continuous spectrum.

Pole singularities of the current in the axial Fourier transform domain corres-
pond to discrete propagation modes. For C near a discrete propagation-mode pole

58

 

eigenvalue C, , the transform domain current can be approximated as [34]

1215.0 - M (41)

where 5.;(6) is the eigenmode current of a wave propagating in :2. direction on the

strip. It can be shown that I =1 [34], so the poles are simple. Substituting (41) in

EFIE (35) leads to

 

c ; -k* -° (ma—‘4.“ 3 ...v-' c, (42)

Since the impressed field €‘(§,C) is regular at C = IC , the integral in the above

equation must vanish at C= th to provide an indeterminate form [34]. Therefore If

must satisfy the following homogeneous EFIE
f-fcs'tiila ;C)-E;:(a)dt’=o v sec (43)
with nontrivial solution only for C = $9. This EFIE consequently defines the discrete

propagation modes and associated propagation constant r C p.

Inverse transforming the spectral current l? leads to the space-domain current

130) as
Rm =-i- f 76(5) clad: . <44)
21: __
From expression (41), and for a single discrete mode, the space-domain eigenmode
current is

59

 

1?,(5.C)=a; U(:z)l?;(a)e"‘r‘. (45)
The latter expression clearly demonstrates that this a wave propagating in :z direc-

tion along the strip with propagation phase constant C p and current distribution

E;(p‘) .The electromagnetic fields have the same 2 dependence through the common

phase factor e ‘1“. From here on, we assume the case for +2. traveling waves only.

Consider a strip conductor of infinitesimal thickness and a width of 2w as

shown in Fig. 4.3. In this case, the EFIE is simplified as

lim I-f §‘(x,y|x’,y’=0;C)°Fp(x’)dx’= 0 ---for -wsxsw . (46)
"0 C

The surface current for a strip of vanishing thickness has only tangential components
as

E(5)=xk.(x)+zk,(x). (47)

Substituting eq. (47) in (46) and exploiting expression (36) for the electric Green’s

dyad yields a pair of integral equations by letting I =2 and I =2 ; they are
Hm I g.;(x.y Ix ’.0;C)k,(x’) + g;(x.ylx’.o;ok,(x) dx’ =0

r0
w (48)

133,1 f 8;.(xJIx’.0;C)k,(x’) + 85(xsylx’.0;C)k,(x’) dx’=0

for -wsxsw .

The scalar components of the Green’s functions are given in Appendix B as

3:,(xylx’.0 71;] c:,<:.oe/‘“-*’>e ”J d: (49)

...for a’pa-x’z

 

 

where the coefficients are functions of C and C as

N2M2-1 2 M2 k2_ 2
C;(€:C)=(c c )Epc+ c(c E)
Z I‘(MZ ”(2.) Z *0)
NzMz-l M2
C;(:.ci=c;,(r,c).( . . )56P._ .zc
’ Z'IMZ'O.) 2*(1)
(”3”? '1)? . + M3<kZ-c2)

(50)

C;(E.C) =

Exchanging the spatial integration with the spectral one in eq. (48) results in the fol-

lowing
um fdte’” f {6;(£.ok,(x') +C;(E.C)k,(x’)}e"“""dx’=0
r0 .. -..

(51)
13:? dee’JI {6;(:.C)k,(x)+ ;(£.C)k,(x)}e1‘“‘*°dx'=o

for -wsxsw .

A moment method solution of the above equation is pursued in the next section.

4.5 MONIENT METHOD SOLUTION

The transverse and longitudinal current components are expanded in series of

Chebychev polynomials weighted by appropriate edge-condition factors as follows

N
kx(x) = 2 an en(x)
1| '0 (52)

N
k,(x) = 2 an 83(1)

n -0
with

e...(x) =T.(x/w) (1 «ac/w): for -wsxsw (53,
e..(x) = T.(x/w) N 1 - (ac/w)5 .

where Tact/w) is a Chebyshev polynomial of order n of the first kind and an and am

61

 

are unknown expansion coefficients.
Following Galerkin’s method, the same basis functions are used as testing
functions. Using the above current expansions in (51), and after many manipulations

[50], The EFIE is finally rendered into the following matrix equation

 

 

.0”,
an
M rut .
1:. A: a," = 0. for m,n=0,1,...,N (54)
a a aw
.“zN.

where the matrix elements are given as

A$=li$ f e‘P"C,,(A)g..(t)f.,(:)de for (2,0 =x,z (55)
y ..
with
8...(5)= f e...(x)e"’dx for a =x,z (56)
and
f...(E) = fe,,(x’)e‘1"’dx for p =x,z. (57)

One advantage of using Chebyshev polynomials as basis functions is that it allows the
spatial integrals in (56) and (57) to be evaluated in closed form.

To obtain a nontrivial solution of the matrix equation, the determinant of the

[A] matrix must vanish. Since the elements of A are functions of C , this require-

ment yields the propagation constant C p. The corresponding expansion coefficients

62

 

 

are then evaluated and the eigenmode currents are obtained in quasi-closed Chebyshev
polynomial series. An additional advantage of using Chebyshev polynomials is that

only few terms in the series are required to accuretly represent the current.

4.6 NUMERICAL RESULTS

Table 4.1 shows a comparison of normalized propagation constants (C/ko)( for
a microstrip line with parameters 2w/t= 1, ef=8 and uf= 1) obtained by Kobayashi

et a1. [33] and the present method. The influence of the number of basis functions N
on the current components is also included. Table 4.1 shows that fast convergence to

the exact values of C lkois obtained even for the cases of N =2. The results obtained

with the present method show that the results in [33] are accurate within 8 percent to
1 percent for higher frequencies. This is mainly due to the fact that in the present
method the basis functions are Chebyshev polynomials of the first kind, whereas in
[33] Chebyshev polynomials of the first and second kind are used as basis functions in

the expansion for the current, and the resulting convergence rates differ.

Fig. 4.4 shows the value of the nowalized prOpagation constant (C / It“) as a

function of frequency up to 15 GHzn We have compared the results with those ob-
tained in [32]. The latter authors used triangular basis functions with a point-match-
ing technique. The respective results differ by less than 2 percent.

The longitudinal current and the transverse current on the strip are shown in
Figs. 4.5 and 4.6, respectively. As the longitudinal current is symmetric with respect
to the center of the strip and as the transverse current is antisymmetric, only the

results for the right half of the strip are displayed. The amplitude of the transverse

63

current changes more rapidly as a function of frequency than the longitudinal current.

Moreover, the magnitude of the former is quite small compared to the latter.

4.7 SUMMARY

The conventional microstrip is analyzed as an example of an integrated circuit
located adjacent to a layered surround. The dyadic Green’s functions associated with
the layered background of the microstrip environment are constructed using wave
matrices as discussed in the previous chapter. The fields in the film region will be
used in the next chapter to determine the characteristic impedance of the conventional
microstrip. A Fourier transform-domain electric field integral equation (EFIE) de-
scription of general microstrip is developed, and then applied to the conventional
microstrip line. The currents on the strip are needed in the analysis of characteristic

impedance as discussed in the next chapter.

 

 

Table 4.1: Convergence of the propagation constant upon the number of basis func-

tions used in the current expansion.

 

 

 

t / x, 0.005 0.05 0.1 0.2 0.3 0.4
{/ko N=l 2.3379 2.474 2.5973 2.7146 2.7623 2.7849
[33] N=2 2.3383 2.4753 2.5995 2.7202 2.7675 2.7897
N=3 2.3383 2.4753 2.5995 2.7202 2.7675 2.7897
N=4 2.3383 2.4753 2.5995 2.7202 2.7675 2.7897
we, N=1 2.417 2.5772 2.6844 2.7662 2.7944 2.8056
...... N=2 2.417 2.5773 2.6845 2.7663 2.7945 2.8069
W N=3 2.417 2.5773 2.6845 2.7663 2.7945 2.8069
N=4 2.417 2.5773 2.6845 2.7663 2.7945 , 2.8069

 

 

 

 

 

 

 

 

65

 

 

Q9999 present method
tmr results from [32]

s» 91 s» 9‘
O -5 N (:1

Normalized phase constant {/ko
N
in

 

I"
m

 

207[ITIITFITIIIIIIIIIIIIIIIIIIIII]

0.0 5.0 10.0 15.0
Frequency (GHz)

Figure 4.4: Dispersion characteristics of the principal mode for the configuration
of Fig. 4.3 with w = 3.04 mm, t = 3.17 mm, and nf = 3.42.

66

 

 

um f=1 GHz
‘l:l'.:l:l-_-l- f=7 GHz
1,2 1212999 f=15 GHz

9 .0 7‘
o m o

O
.p
\

Normalized longitudinal current
0
10

0.0
0.0 0.2 0.4 0.8 0.8 1 .0

Position x/w

Figure 4.5: Frequency-dependent characteristics of normalized longitudinal current
distribution, relevant to the example of Fig. 4.4.

67

 

*****f=1 GHZ
tlzl‘ir f=7 GHZ

 

 

0.02 _ 043.9213 f=15 GHZ

E I

I
g 0.02 :
3 q
0 _.
q) 2
m q
L _
0.) ..
a ..
C 0.01 :
U ..
L -
y -
'0 I
0’ -
.5 -
E 0.01 E
L 4
o ..
Z I

0.00 “l ____________ i 1 I r i ]
0.0 0.4 0.8 1.2

Position x/w

Figure 4.6: Frequency-dependent characteristics of normalized transverse current
distribution, relevant to the example of Fig. 4.4.

68

CHAPTER 5

COMPUTATION OF CHARACTERISTIC IMPEDANCE OF
MICROSTRIP TRANSMISSION LINE

5.1 INTRODUCTION

The dispersion characteristics of an open microstrip were evaluated in the
previous chapter. In order to obtain a complete equivalent transmission line represen-
tation for the microstrip, not only the dispersion characteristics must be evaluated but
also the characteristic impedance. As discussed in chapter two, due to the ambiguity
in the definition of voltage for non-TEM structures such as the microstrip, the classi-
cal definition of characteristic impedance as the ratio of voltage to current is also am-
biguous. However, it has been seen [13] that for a microstrip line, if the voltage is
computed as a path integral from the ground plane to the microstrip and subsequently
averaged as discussed below, this classical definition of characteristic impedance gives
quite good results and it agrees with a more rigorous definition of characteristic im-
pedance, namely the power-current method (to be discussed later). The voltage-cur-
rent method is much more convenient and is less analytically involved.

The other method is based on the power-current definition of characteristic
impedance (Z =2P/12). In this power—current method, the power is computed as an

integral of the poynting vector across the transverse section of the microstrip line.

69

5.2 VOLTAGE-CURRENT METHOD
This method has been tried for conducting planar transmission lines [13].
Consider an open microstrip line with only the principal mode propagating along the

+z direction with propagation constant C0. The voltage is calculated by integrating

the y component of electric field along the y-axis from the ground plane to the strip

conductor as follows

0 o
V(x)=f‘€fl(x.y)dy = -fefl(x,y)dy . (1)
8? -t

The above voltage being a function of x, its average over the microstrip region can be

defined as

W

I ”(x)” (2)

v=""

2w

or, alternatively as the weighted average

f v(x) 1:0) dx
v = ‘w (3)

‘7‘ k,'(x) d:

 

where the current density 190:) is obtained in eqs. (4.52) and (4.53) from the MOM

solution to the integral equation.

The spectral domain electric field in the film region is given as

-jnc w c ‘" -.
k. _f 2,(x.ylx’.o;-c.)-k(p)dx’ <4)

 

2,05) =

70

With reference to Appendix B, the electric Green’s dyad has the following scalar

components

853(xiylx’,0;C) = 2—13— ] 3;,(5,c,y)eiee-r1d5 f0, mpg”. (5)

Only the y-component of electric field is needed and is written as

e (9‘) = :J—n-E (8,;(xsylx’.0;-Co)k,(x’) + 8;(xylx’.0;-Co)kz(x’)}dx’ (6)
.0 kc -

 

 

 

 

where
B. -_ flip, coshp,0+t) jttk}‘+p})(N;M;-l) coshp,(y+n
#69:”) " 2 . + 2
Nfil’fl) smhl’x‘ Nfiz"(1)z'(2) coshpft (7)
B.“ 1.)) . 11p, cos ,0+t) + J'CUc} +p})(N}.M;-1) coshpf(y+r)
’2 ' Nfiz’xi) sinhpjt N;z*(i)z'(i) 603th '

Exploiting eq.(5) in relation (6) leads to

f [18;(£.-Co.y)k,(x’) +B,;(:,-co,y)k,(x') } e“"""d£dx’ ,(8)

-w -.

 

- 3 =1").
3149) “kc

2
Interchanging the spatial integration with the spectral one and recognizing that the
spatial integral over the current can be interpreted as a Fourier transform on the vari-

able x as follows

fk,(x’)e""'dx’ = 911,05} = 4(a)
“' (9)

fume-fwd): = 90.0)) =f,(€)

we have

71

W = $118.50,- (mama ,§(E.-Co.y)f,(5)}e“‘d£. (to)

The current in eq. (9) is expressed as Chebyshev polynomial series given by eqs.
(4.52) and (4.53). Hence, the integral in (9) can be evaluated in closed form. Refer-

ring to section 4.3, we have

" N
ft“) = f*t(x)e""'dx’ =20..f..(£) ---forB =x.z . (11)
-w u-l

where for“) are evaluated in terms of Bessel functions as follows

fa“) = ( l)"1tWJ2,.(EW)

(12)
f...(£)= -j(- 01—” J-;....(EW)+12...”..(5w)+ll

Jw...-.,..(£ w.)

Substituting the expression for the electric field into eq. (1), the average volt-

age in eq. (3) becomes

 

w 0
- f f 9,03) 140:) dydx
V” = -w -1 w
_I 1;...

(13)

I.“ f dx fdy f 18,..(10 4.3112(5)”) ,;<5.-c..y)r.(£)ieivdit;(x)

=c2rtk

 

jgmm
Interchanging the first spatial integration with the spectral one and using the fact that

f (canard: = 2‘05) (14)

72

the average voltage becomes

,- fdyle;(5.- c.m<01;,;(1)+8 (e- Codi/(0f. (5)14:
v =1). " "' . (15)

“' 2nkc
[Mme

 

Now, the characteristic impedance is defined by

24% (16)

where the total longitudinal current I is given by

1 = f k,(x)dx . (17)

Exploiting eqs. (15) and (17), the characteristic impedance is

fdyf18(:.-c..y)r(5)f.(£)+ ,;(£.- Cwflfifzflndfi
z=j""‘ . (18)

211k:
f 143:) dx f k;(x) dx

 

The spatial integration in eq. (18) can be performed analytically to yield

0 ° ' 2 2 2
l - C C + 2)(NcMc '1)
Innis-mm = 717°- - 1 00:: f’
-r ch (3.) ch 0021102!

= ..iCOB '(5.‘C& (l9)

0
fags-coma) = flats-c.)

tanhp

 

where

73

WHO): 1 _(k}+p})(~3M§-1)
NEZ’m Nfz"<z)z'(x)p,

 

 

tanhp} . (20)

Using the fact that 150:) is an even function of x and the orthogonality of Chebyshev
polynomials, integral (18) can be readily evaluated. We obtain

1 = nwaw (21)

where “w is the first expansion coefficient in the series for the z-component of cur-

rent.
Utilizing eqs. (19) and (20) in eq. (18), the final form of characteristic imped-

ance is found as

21:3]ch2 law I2

 

[[COB‘(£.-c,)1;<of;(t)-:B‘(£,-c,)f,(a)1;‘(e)]dg . (22)

The integrand in the above expression is found to be an even function of E. Hence

eq. (22) becomes

 

n " ' . .
Z= ‘ B' .- z - B‘ ,- x , d . (23)
than“, {[c, (a comma) a (a C&f(£)f(5)]£

The spectral integration in eq. (23) is evaluated numerically.

5.3 POWER-CURRENT METHOD
The characteristic impedance is defined in the power-current method, by

z=£%'., (24)

The total longitudinal current I is available in eq. (20). The average power trans-

ported by the principal mode propagating along the microstrip is defined by the inte—

gral its poynting’s vector over the transverse cross section as follows

Pa = 11’- Re g(gxfi.).ids

1 . ‘ (25)
.. Effasxn, -E,H,)dxdy
x)

where E. and H. (a =x,y) are the spatial domain fields excited by the eigenmode

current. The electric fields in the cover and film layer are written in terms of the
Hertz potential there as

Em = (k,2 +vv-)fl,(r) for l=cf. (26)
The magnetic field is found from the electric field using a Maxwell equation as

1” infi) for I =c,f.
z

 

31(7) =

(0

(27)

The space-domain electric field is written as an inverse transform as follows
Em = -1— ] annemdc (28>
2n__

where the axially transformed field is given by

 

mm = inc Lanai: ;o 4&5 .cm’ for t=c.r. (29)

The electric Green’s function is obtained from the transform domain Hertz potential

Green’s function as follows

gimp”. = (kf+W-)§.(5I5’;o . (3o)

75

The integration in eq. (28) is performed by deforming the real line contour and apply-
ing Cauchy’s theorem for contour integral. The discrete propagation-mode

contribution associated with pole singularities is found as
Em = j; animal‘s. (31)
For the single principal mode propagating in the +z direction with propagation con-
stant Co , we have
3,0) = jume'flf. (32)

The transverse components of the magnetic field are obtained from eqs. (25) and (30)

 

 

 

as
. - 58' " ' ' .
H3 (f) = l. ((p) + 1C0. e, (“0:
~“°" 6’ w“ - (33)
r_. ‘ &O.
11;“) a flu ¢;(5)+-L- 2 (Kat.
.W‘ «w‘ 6".

 

 

Using eqs. (32) and (33) in expression (28) and noting that top; =k,'n,'., the average

power in each region is written in terms of spectral domain electric field as follows

1 ° 0 . 69;; 6e;
Re —. . (0(eded+efiefl)+1(¢d__ + 61—) M . (34)
gkl T], & 6y

1
PM“

[0

In the next subsection, the power flow in the film and cover regions will be

calculated in detail in order to evaluate the characteristic impedance.

76

 

5.3.1 Power in the Cover Region

In the cover region, the eigenmode electric field is expressed by eq. (29) as

 

follows
m) = "f“ f §.'<x.y|x'.o;-c.>-E<xodx'. (35)

The electric Green’s dyad has the following scalar components

8:3(xxylxlao;C) = i; f C:a(€’C)e'Pe’ejE¢r-x’)d£ ...for “,9 :x’y’z . (36)

where the coefficients are obtained in Appendix B as follows

_ (k3 - 56M? + p,€’(N.’M.’ - 1)

 

 

 

 

 

 

 

 

 

CG
‘" 2" z‘mz'o.)
C. . -j:p.M3 +1: «:3 +pb<~3M3 - 1)
" 2" z'mz'o.)
2 2 2
C; = C; = -CEMc +chC(NcMc-1) (37)
z‘ Z"(A)Z‘(A)
C. = vow.” +jc<k3 +p:)(~3M3 - 1)
" 2" z’mz'o.)
C. = (kE-czw.’ +io.c2(1v.’M§-1)
“ 2" Z"().)Z‘(A) '

Substituting the above Green’s function into eq. (33), and interchanging the spatial

integration with the spectral one, the scalar field components are obtained as follows

77

2771c

 

 

 

e... = Effie.- code/w:
e = 7'11, 1‘ch ,(E.- (0)615“: "’dE (38)
e... = 2““ kcfne- melee-Nd:
where
FAIL-Co) = 033(5) + ewe) fora =x,y,z. (39)

The spatial integration over the current is again denoted

13(5) = fume-few forp =x,z (40)

and has been evaluated previously in section 5.2 as given by eq. (12).

Noticing that ae,‘/ax= -jEe; and ae;/ay= 16:; and exploiting eqs. (36) in

(32), the average power in the cover region is obtained as

 

 

P(av)c= 2011““? Ike |2k¢:' fdx {dy

{.

+[f F,(5.-c,)e1°'e"°’de]( f F,‘<£’.-c,)ei‘“e"5’d5’]

(fF,(e.-co)e"*e"°’d£][fF.‘(£’.-c,)e1"‘e"‘°’d£']

 

(41)

 

[fF,(E.-c.)e1‘*e"°’dz](f £’F,‘(£’.-co)ef"*e""’de’]

vhf f F,(£.-Co)e"‘e""d£]( f fiat-(9W? "5’d5’]} .

78

The integrands are all well behaved. Hence, interchanging the spatial and spectral

integration is permissible and leads to

 

 

. 1 I .. -<p.+p;)y - g- I),
r... W... I, 7:, {I'Icoir F r 1M l‘ «v I» . ..

+ffE’Fsz. dEdE/ fa «Pam-Dd), f6)“- E’ydx (42)

.jpc‘ff [7sz I} “(Po-793$. feflE-E’kdx

.. o --
The spatial integrations can be evaluated analytically using
] W'de = 21:6(x-x’)
(43)

 

[eve pcvw _ .
pc+pc

Hence, the expression for the power in the cover region becomes

I‘d:

 

P = l
( )c .
4" 41! Ike"2 kc "' pc +pc

 

11.]c5(F.F;+F,F;)+£FF;-Ip.“FF } (44,

Examination of the integrand in (44) shows that it is an even function of E , hence the

power in the cover region is finally obtained as

 

PW=__1_Re_fCJ(FF;+FF;+) EFF; -jpc'F,deE (45)
2"! Ike-'2 kc. 0 Pc+Pc

The above integration is performed numerically.

79

 

 

 

5.3.2 Power in the Film Region

The power in the film region is given by eq. (34) as follows

 

 

.0 e c
1 1 . . . a: at
PM, = 512:” . . (0(efiefi+e,,e,,)+1(efi a: + efl—ayé) dxdy. (46)
-.‘I f

The spectral domain electric field in the film region is given by eq. (4). The scalar

components of electric field are obtained as

 

chow) -- in“ f {3;(xylx’.0.-Co) k,(x’) + 3:1(x,y|x’,O,-Co)kz(x’) 1d?" (47)

C ‘W

for a =x,y,z .

The scalar components of the electric Green’s dyad are given by eq. (5). Some of the

coefficients were obtained in section 5.2. In this section, all the coefficients are

needed and expressed as

13:15:.» = '92": “WWW _ ‘5’?!"fo ~1) amp/0.0]
’ NEPm SW} nfzmwm coshp/r

IainE coshppm] + Ie(k}+p,’)(N3M.’-1) coshpwm]

 

 

8‘ 5.: ) = .
’4 ”y NEZ‘UI) WK NZZ’mz'm 003W
-5; sinhwynn _ capgNEME-I) sinhpxyun

 

 

 

B;(E.C )= -— .
'y Nfzm) Slnhpft Nfz’mz'm coshpf (4s)

agar.» = ages.»

10, coshpvjom + I:(k,’+p,’)<NZM3-n weapon]
Nfz'm sinhpf Nfz'Imz'm MW
'92-? sine/0+0] _ C’pANZMf-n mew]
NZZ‘m W} Nfz'mz'm WW

 

8,35,“) =

B;(E.C.y) -
Substituting eq. (5) into expression (46), and interchanging the spatial integra-

80

 

 

 

tion with the spectral one, the scalar field components are obtained as follows

where

e,(x.y) =

ebony) =

gm) =

 

 

 

 

 

 

 

 

 

 

 

 

 

.jn, Prue-Co) + GIL-(0)1 . + u
Zflkci, sinhpf ijl’w "1" d5
'1'". ' 'F(E.-Co) + G(£.-C,)‘ + m
M. if” —L—.....,. We we ..
1"}, ' ”age-co) + GAL-(J . + m
21min Sinhpft coshpjt Jsmh[pf(y 01¢ d5

2- -
FIN) = (’9 £61225) time)
chh
- N’M2-1 ,

03.0 = ‘ c . ’Pfliffi) mam

ch'zh

1505.0 = 15%“: ”03“)

wk

°N2M2-l 2+1; +

(gm) = 1‘ . . W12 fume we]
' ch'zh

_ + 2_

mm = ‘W" 3‘1 (212(5)
N,z*

- N2M2-1 ,

GALC) 3 ( c c )Pgflfifxfl) “(0]

ch'zh

and 5(5) and fz(E) are those given by eqs. (11) and (12).

Similar to the case of the power in the cover region, the fields are well be-

81

(49)

(50)

haved functions, hence interchanging the spectral integration with the spatial ones is
allowed after exploiting eq. (47) in relation (44). Again, the integration with respect

to x gives rise to a o-distribution. The integrals with respect to y fall into six catego-

ries
Y _ }.sinhpf(y+t)sinhpf'(y+t)
l - .
-. sinhpftsinhpft

 

dy

Y _ o sinhpfo+t)sinhpf‘(y+t)
2 “f

—: sinhpjtcoshpft
Y _ j-sinhpéyflhinhpf'o'fl)
3 ‘
-. coshpltcoshpf t

 

4y

 

dy

(51)
0

Z _ [coshpfo'mcoshpf' (y+t)
l -, sinhpftsinhpf't

Z ‘Fcoshp/(yfikoshpf' 0+0
2 - .

 

dy

 

dy

o .
coshp +t)coshp (y+t)
23 -f 10 f

. ‘0 ~
-: sinhpjtooshpf t

 

Each of the above integrals can be evaluated analytically. Hence the power in the
film region is obtained as

Inclz . l 0 o o o o
av = —— Re C FxeY +1:ny +G xY +G‘G Y
‘ V 2(2u)lk,|2 f. kfn;{ ° ‘ 2 ‘F 2 ‘ 3
0 o c o o 52
+17, zl +156, 23+F, 6,2, +G,G,Zz] ( l
+ E[FIF;YI +Fsz.Y2 +Gsz.Y2. +Gsz.Y3]
+jp,‘ [131.32, + 56:23 + afgz; + 0,6,7,” dE .

 

The integrand in eq. (52) is an even function of E. Therefore the power in the film

region is finally obtained as

82

In |2 ' 1 . . . . .

= _c_' Re (0 FxeY +12ny +G‘FJY2 +GxGxY

W (21:)th [ yn;{ 1 ‘ 2 3
+ 1332, + 156,73 + F;G,z; + 6,0;22]
t €[Ffz'Yn +£0.72 + sz'Yz' + 0.0.73]

+pr [P317521 + Igcgz3 + 6sz7; + Gsz°Zz]}dE .

 

(53)

The above integration is again performed numerically.

The characteristic impedance for the principal mode of a single microstrip line

is obtained from eqs. (24), (45) and (53) as

_ 2
Z - nzwza;(P(av)c+P(avY)

(54)

5.4 NUMERICAL RESULTS
Figure 5.1 shows the frequency dependence of the characteristic impedance for

the same open microstrip transmission line studied in Chapter 4. At each frequency,

the characteristic impedance for the microstrip line can be evaluated after a chame-
teristic value of Co has been found. The voltage-current method is adopted. It is

clear from Fig. 5.1 that exploiting both the transverse and longitudinal current com-
ponents in the expression for the characteristic impedance gives results very close to
those obtained using only the longitudinal component of current. Hence, neglecting

the transverse component to evaluate the characteristic impedance is justified.

In Figure 5 .2, the power-current method is compared with the voltage-current
one. It is clear that the power-current method has the most TEM-like character [41],

since the characteristic impedance is relatively constant at low frequencies and only

starts to increase at higher frequencies, as compared with the voltage-current results.

83

They both coincide at very low frequencies as expected.

Figure 5.3 shows the characteristic impedance a function of microstrip width

for both the power-current and the voltage-current method. As expected, the charac-
teristic impedance decreases with an increase in the width 2w of the microstrip. This
is because for the very wide microstrip line, nearly all of the field is confined to the
substrate dielectric, the structure resembles a parallel-plate structure. In the case of

very narrow microstrip, the field is almost equally shared by the cover and the sub-
strate.

Figure 5.4 shows the characteristic impedance as a function of frequency for a

microstrip of width 2w=3.04mm, substrate height t=3.l7mm, and e,= 11.7 . Our

results are compared to those obtained by Dezutter [41] for both the power-current
and the voltage-current method. Our results agree with those in [41] in the sense that
the power-current method demonstrates a TEM-like behavior as the characteristic
impedance only starts to increase at higher frequencies. Our results were obtained
using only two terms in the Chebyshev series expansion for the current whereas in
[41], a large number of triangular basis functions were required. This illustrates the
advantage of using Chebyshev polynomials as basis functions against triangular basis

functions used by the latter authors.

Finally, our results are compared to those presented by Pozar [13] for the

characteristic impedance versus microstrip width for a microstrip line of substrate

thickness t=l.27mm and e,= 10.2. This comparison is shown in Figure 5.5. Our

results differ by 3% from the results in [13]. This is mainly due to inaccurate model-

ing of the currents in the latter effort.

 

5.5 SUMMARY

A full-wave analysis for the characteristic impedance using both voltage-
current and power-current definitions is presented. It is shown that the voltage-
current method gives results very close to those of the more rigorous power-current
method. Hence, the voltage-current method will be adopted later since it is less
analytically involved than the power—current method. The dependence of
characteristic impedance upon frequency was a controversial issue since quite
different functions of frequency have been predicted. It is established in this chapter

that, for the conventional microstrip, the characteristic impedance will always increase

with frequency.

85

°____°°°° only k2 is used
205 ++_;+_+ both kz and k, are used

20.0

19.5

19.0

Characteristic impedance (0)

18.5

 

1800[ITIIIIIII[IITIIITIIIIITIIIIIr]

0.0 5.0 10.0 15.0
Frequency (GHz)

Figure 5.1: Characteristic impedance versus frequency using the voltage-current
method (w = 1.5 mm, t = 0.635 mm, and e, = 9.8).

86

0.92.0.9 voltage—current method

 

23.0 tfiid- power—current method
-/'\
S 22.0 °
0)
o I
c .
-8 21.0 ,,
Q)
0- ,,
OE , ’l
o 20.0 ’1
1.: '1
L 9
1D ,"
a
o 19.0 0’1
L a
a .4
.C
o
18.0
1700 IIIIIIIIIIIIIIIIIIIITIIIIIIIIIIITTIIlllllIllllllTfl
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 5.2: Characteristic impedance versus frequency for two different
methods (w = 1.5 mm, t = 0.635 mm, and e, = 9.8).

87

 

0.0.0529 voltage—current method
100,0 mzr: power—current method

80.0

Characteristic impedance (0)
.3
O

 

 

0'0 [IllIllIIIIIIIIITTIIITTTIIITIIIIIIIIII[I]

0.0 2.0 4.0 6.0 3.0
Microstrip width 2w (mm)

Figure 5.3: Characteristic impedance as a function of microstrip
width at 10 GHz (t = 0.635 mm, and e, = 9.8).

88

 

 

.___ PI
x”XXVI De Zutter 41
90.0 tenet Pl De Zutter 41
/*
A y
X
saaoo ,,
l

8 /
c If
g 70.0 x ,’
0) l
O. x ‘
.§ x
o 60.0 x 1.,
1.: X l
.12 I
:2 ,.
g 50.0 *’*
L *’
o it"
.C
o

40.0

30-0ITTIIIIITIIIllflllTlltlllilllfi

1.0 6.0 11.0 16.0

Frequency (GHz)

Figure 5.4: Comparison of characteristic impedance with published results
(w = 1.52 mm, t = 3.17 mm, 6, =11.7).

89

(2.0.99.9 voltage—current method

 

5M +++++ Pozar's computation [13]
+
A
g 50.0
0)
0
c +
8 40.0
0)
0.
CE +
o 30.0
.3
.‘L’
L.
<0
*5
o 20.0
L
O
.C
o
10.0
0'0 leTllllIIIIIIIIIITIITTT1TTTFI [Tlfllll'l]
0.0 2.0 4.0 6.0 6.0

Microstrip width 2w (mm)

Figure 5.5: Comparison of characteristic impedance versus microstrip width
with Pozar’s results at 3 GHz. (t = 1.27 mm, c, = 10.2).

CHAPTER 6

ANALYSIS OF MICROSTRIP STRUCTURE WITH SUPERSTRATE LAYER

6. 1 INTRODUCTION

The rigorous analysis of printed circuit elements such as microstrips requires
the use of the dyadic Green’s function associated with the layered background. In
this chapter, a microstrip structure with a superstrate layer is analyzed. The
microstrip device is printed on the substrate layer and resides in the superstrate region
as shown in Fig. 6.1. Hence, for the geometry of Fig. 6.1 the dyadic Green’s
function is needed. We begin by specializing the results from chapter three to obtain
the EM fields in the superstrate and in the substrate regions of the layered
environment of Fig. 6.2. We note that this otherwise extensive effort was made
relatively easy by using wave transmission matrices. Moreover, to our knowledge,
the microstrip with a superstrate has not been extensively analyzed in the literature.

Based on the integral-equation formulation, we study the microstrip line with a
superstrate by an approach similar to that used for an open microstrip. A rigorous
full-wave solution to the integral equation is pursued again using the Galerkin’s
method of moments. The dispersion characteristics and current distributions of the

guiding structure will be analyzed.

91

 

 

 

Figure 6.1: Microstrip with a superstrate layer

92

 

x ‘ .. . .,,,,""' if} 3.: 3.3:. -. q -. a , ~ ,3 y I- O

 

if’i/QI‘yfl/f/ / / //

Figure 6.2: Background environment of a microstrip with superstrate.

93

6.2 FIELDS IN THE SUPERSTRATE LAYER

The four-layered conductor/substrate/superstrate/cover structure is depicted in
Fig. 6.2. The superstrate region resides between interface 0 and interface 5. The

electric current source is embedded in the superstrate region. The electric field in the

superstrate layer is formulated as follows

 

in = 1" f y Garm- me (1)

where the electric Green’s dyad is identified in terms of a Hertz potential Green’s
dyad as
650' 7') = PV(k}+vv-) 630176 + 1‘: 6(F-i”) . (2)

The Hertz potential Green’s dyaddecomposes into a principal part and a scattered

part as follows A
G; rope; (3)

The principal Green’s function is written as

ejiar-r’) e -P.|ry’|

 

G’(? 7’) = d2). (4)
‘ | I! 2(211)2 ,
while the scattered Green’s function is formulated as
8 . A 8 A a A a S 3
6.801,") = (”+3763 +y [(ig +z5z']Gnc+ my] ' (5)

The scattered dyad has the following scalar components

94

 

'Gwlrfi‘ .. SLUly’J)‘ mm
16;,(rlr’) t = ff s;(y|y’,1) 1 2‘2 2 421. (6)
,Gémrfi, "' $50041), "" '

 

 

 

The scattering coefficients S: can be determined through wave transmission

and coupling matrices as in chapter three, and assume the form

4 k .
S} = *2 3,1,0.» *‘WI’A’ for q = t,n,c . (7)
-1

Exploiting eq. (3.45), the B” coefficients are specialized as

 

 

 

, - 9‘2"
Bar] '
D4
3, = my
”2 Dq twfor q=n" . (8)
as = as.
a 913"
BM = _
D‘1 J

For coupling components, we use eq. (3.52) to obtain

 

1 -2,

8;! = 080‘ {C’lxiflzie ‘d‘ + Cc”
1

8;, = 1 {C3521 + sepcct}

 

D'D‘
3:1; {Cctstjlstfile'w' + 0,1}

1‘1

where

Dq=1-atjtst’g1e‘2”". (10)

95

The phase shift factors 9:0 |y’,1.) are defined in eq. (3.46) as follows

0}. = p,( -y -y ’+2y.)

0:. = p.0-y’+2y.-2y,) (11)
0.3. = P..(-y +y’+2yc'2)'.)

02. = 9.0 +y ’-2y.) .

Using the fact that interface c is at y =d, and interface 3 is at y =0 , the phase shift

factors become

<11. = p,(-y-y’+2d,)

2 - - +
‘1’: ' ”'0' y, [2‘19 (12)
0. = p,(-y +y +241.)
11>; = p,(v+y’) .

Now the general reflection coefficients 9ch and 91,1 , and the overall coupling

coefficients Cc! and C31 need to be determined. From eq. (3.21), we have

R,‘1+R}1e'2""

91.” = _
1+R,'191}1e 2"!”

for q = t,n (l3)

 

where tft is the thickness of the substrate region. The overall reflection coefficient
91}! reduces to the interfacial reflection coefficient Rf‘l since there is no interface

below the substrate/conductor boundary. Moreover, interface 1' is adjacent to a

perfect conductor leading to
R;l=-1 ; Rf'l=l. (14)

96

Hence, the overall reflection coefficients at interface 3 in eq. (13) reduce to

R'l 1'2”"
91$! = ’
1-R,‘1e"’f’

R,"1+e'2""

 

(15)
91:1:

 

“lifted”:

where the interfacial reflection coefficients are given in Appendix A as

2 2
m. “km
S 2 '

R‘I =
3 9
M310, +10, NJ, +10,

(16)

Similarly, the overall reflection coefficient 9131 reduces to the interfacial

reflection coefficient Rf! since there is no interface above the superstrate/cover

boundary. From Appendix A, the interfacial reflection coefficients at the cover
interface are

R‘t

C

. 2 2
=p3-Mcpc . RcflI=p3_NCpc (17')

M31). +2, 1113?. +12.

The overall coupling coefficients C31 and Get at interface 3 and interface 0,

respectively, are given by eq. (3.35) as follows

q: = -st:1(c,u1+st;1cm1) + (C2,; +81;1le) (18)
Cct = -st:1(cu‘c1+st‘c1cmt) + (cmnstgtcmt)
where Cl l":1 for example, is the first entry of the overall coupling matrix for upward

97

travelling incident wave at interface 0. The overall coupling matrices are given by

eq. (3.32) and are specialized to this structure as

[91110.11 [41]-14.1] [91114114 [4‘11"
[Cc1]=[cc1] [41]“ .

(19)

Since interface I is a perfect conductor, the product of matrices
[Aft] [Cfl] {A} l]'1 [Aflrl reduces to the null matrix. Hence, using eq. (3.19) the
overall coupling matrices are specialized as

[c31]=[c,1] [A;1]"

_ 1- N;2M;2
2p,

 

-1 -1] (20)
l l

and

2 2
Lu ['1 ’1], (21)

I =
[C‘] 2p, 1 1

Substituting the entries of the overall coupling matrices in the expressions for

the overall coupling coefficients in eq. (18) leads to

_ -2 -
C.” (1 N’ M’z) (1+9121)(1+9131)
2p

‘ (22)

2 2_
cc1=W (1+9t‘c1)(1+at:1),
2p

S

 

Exploiting the expressions for the overall reflection coefficients at interface 3 given

by eq. (15) and knowing that the overall reflection coefficient at interface c reduces

98

to the interfacial one, eq. (22) becomes

_ (1-N,‘2M;’)(1+R,‘1)(1+R,’1)(1-e'2’f)(1+e"”)
2p.(1-R.'le'2"')(1+R."le'2”) (23)

_ (NEME- 1)

c ”T

S

 

(1+R;1)(1+R:1) .

Substitution of eq. (16) and (17) in eqs. (15) and (23) leads to the final expressions

for the B” coefficients. For tangential and normal components, we have

3. = (p, - M311.) (pr, +p,corhp,t)
Z II

B . _ (p. - M31!) (M31), -p,cod1p,t)
2.2 ‘ z 1. (24)

 

 

3;: = 33.2
(p, + M312.) (M31). 'pfoothpf)

‘ —
B”,— Z].

 

and

(p. - Nip.) (pr, +p,tanhp,0
Z 0

(p, - Nip.) (pr, -p,xanhp,t)
Z O

B"=

 

8:2:

 

(25)
3;, = 3;,
= (p, + 11132) (M31). -p,tanhp,1)

B"
‘“ z

 

where

. _ 2 + 2 _ _ 2 2 _ -2p,d,
Z ~(ch,+p}anhp,t)(p. Neg.) (p, Ncpz,)(N.p; pfanhpfm -2” (26)
Z " = (M. , +p,oothp,¢)(p, + My.) - (p, - M.p,)(M.p, -p/00fl1p,t)e ‘ .

For coupling components, we have

99

BC

.31:

 

z f’z', {(NiMi - 01p. — Mip.)<p. - Nip.)e "P"
+ (NiMi - 1)(Mip. +p,omhp,:)(Nip. +p,tanhp,:)}

B;Z= 2’”
2'21

 

{(NiMi - 1)<p. - Mip.)<p. + Nip.)

+ (NiMi - 1)(Mip. +p,oothp,:)(Nip. -p,mnhp,:)}
2p,
Z ‘Z "

+ (~fo - l)(M3p. -p,omhp,t)(N3p. +p,tanhp,t)}

8;:

(27)
{(NiMi-1)<p.+Mip.)<p.-Nip.) ‘1

 

B;,=

 

 

P
2 7f, {(NiMi - 00. + Mip.)<p. + Nip.)

+ (NiMi - 1)(Mip, -p,oothpp)(Nip. -p,xanhpp)}
Pole singularities within the integral representations of the Green’s function
components are implicated in the scattering coefficients given by eq. (7). In fact,
setting 2 ‘0.) =0 and Z “(1) =0 leads to the eingenvalue equations for m and TE

surface-wave modes, respectively, supported by the layered structure of Fig. 6.2.

6.3 FIELDS IN THE SUBSTRATE LAYER
The electric field in the substrate region maintained by currents in the

superstrate layer is given as follows

 

2,0) = 1" f y (1,371?)- .7(i”)dv’ (28)

where the electric Green’s dyad is again identified in terms of a Hertz potential

Green’s dyad as

100

came) = (k,2+vv-) (31.01 F’) . (29)

The Hertz potential Green’s dyad consists of only a scattered part given by eq. (3.3)
with the following scalar components
6210?) . siou’pu

Gym?) = ff gun/,1)
ego-17’) "' sgmyci)

d2}. , (30)

 

The scattering coefficients 5; are written in terms of the B), coefficients and

the phase shift factors 115, . Hence, these coefficients must be determined in order to

quantify the Green’s function components. The substrate region resides between

interface 3 and interface I . The phase shift factors are a special case of eq. (3.65)

with y,=0, y._l =d

3,

y,= =1 and 0:4: yielding

4)}. = p.(-y’+2d.) 1,0 +0

4):. = p,(-y’+ 2d.) +190 +0 (31)
4). = p.y’-p,(r+t)

<12. = p.y’+p,(.v+t) .

Exploiting eq. (3.63) the tangential and normal components of the B; coefficients are

4.. 9
B, 437 3131

l
3;!‘led2qlfl +d£]1$:l}

101

where D‘ is defined in relation (10) and (1,", fl is the first entry of the 2x2 matrix
[Dfl] given by eq. (3.59) as follows

- - h "1’ -R‘1"f
111.2111,2 e ’ ‘

 

-r
[DI‘HAJ‘] =
l-Rsl -R'le”' eP/
, " (33)

e.” -R_,"le'p"

[wean

 

 

l -R‘.l L-R’nl ePf epj‘

Similarly, the coupling components are obtained by specializing eq. (3.76) as

follows

1

c =
B” D‘D“

 

{ [C,1(d,';,1+d,‘;,rst:1e'2"")+(flmr +flurstfir)n"] 91:1
+Cct(dfi,1 +d,';,191;'1)}
{ [03(er+dzfifrstzte'2’l"-)+(fm1+fmrstgl)0'] 31:.)
+Cc1(d,‘},1 +dzgf1$:1)}
C 1

BA, =B‘_D-;{C'l(d&’l +d1'illst'c'te'2’fl') +(flml +f1u18§1)D"

l
D‘D"

 

3;, =
(34)

+cc1e"*“'(d,",,1+d,';,191:191;1)}

1 {C,l(d£,l+dfillst:te'2"d')+(f2ml+fmlatil)D”

D‘D"

 

c —
Bfi‘

+Ccte"‘d'(d2"v1+d.£,13t:191:1)} .

where f,” is the first entry of the matrix [F131] given by eq. (3.70) as follows

102

[’"p‘l= ‘19!"‘19‘1 ° (35’
The overall coupling matrix was obtained in the previous section and is given by eq.

(19). Exploiting eqs. (19) and (33), eq. (35) becomes

1V.'2(1V;2M;2 - 1) (1 + 11:1) -e ‘Pf .. . ”Pf

2p,(1-R,"1)

 

[F131] = , (36)

 

 

e” e”

Now that the matrices [qu1} and [F151] are quantified, we substitute their

entries in eq. (31) and (34) and use the expressions for the overall coupling and

reflection coefficients as in eq. (14) and (23). The B; coefficients are obtained in

terms of interfacial reflection coefficients as follows

8;! = ch 18;

 

 

q q C (37)
8152 =Rc 181,4 for q =t,n
3.1.: N.'2M;2e”’(1+k.‘u
(1 -R;1e ‘2” -R;1(R,‘1 - e ‘2”): '2“) (38)
3;. = ”’4‘ ”(I ”if” .
(1 +R,"1e -2pf-Rc"1(R,"l +e'2”)e -ZP’d’)
r _ _ r
B!“ - EN (39)

n u
Bflr‘Bfi .

Similarly, the coupling components of the B}, coefficients are

103

3;, ={R;1(1 -N;’M;’)(1 +11: 1)(1 +R,‘ 1)(1 -e '2’/)(1 +th e i“)
+1NiMi-1)<1+R:I)<1+R."1)(1 +R."1)(1 -R.‘1e"'f)}

 

 

 

x N? e 7"
2p.(1 + R,"1e ‘2” - R." 1(R,’ 1. + e “2”» ‘W’xl - R,‘1e '2” - Rc't(R,'l - e ‘2”). 4%) ’
(40)
31;: ={<1-N;2M;’)(1+R."1)<1 +R.'1)(1 -e"”)<1+R.’te""")
«1113114,,2 - 1)(1 +R;1)(1 +Rc"t)(1 +R,"1)(R;1 105529.41 ..
x N? e 1),:
210,0 +R,’1e ‘2” -R,"1(R,’1 +e‘2’l‘)e'W1)(1 4:162” 41; 101:1 -. "-PI‘). 4%) ’
(42)
B}; =3}; . (43)
The scattering coefficients in the expression for the Green’s function
components in eq. (30) can be obtained as follows
4 l
#0001) =2; nae"! for q =t,n,c. (44)

Exploiting eqs. (37) through (43) and after some manipulations, the above expressions

for the scattering coefficients become

3130011) =2B,;.sinhp,<y+:)[p;1 e W‘s-’1 . . w’]
5310 MA) =2333008hpf0+t)[R:1e “”1"" + e 7"] (45)
8.10011) =2coshpfcv+r)[3;.e "W“ wee ‘1'" ] .

104

The final expressions for the 3;,(11 =t,n,c) and 8;, coefficients are obtained by

substituting the relation for the interfacial reflection coefficients at interface 3 in eqs.

(38), (40) and (42). We obtain

 

 

 

 

B. _ N;’p.(Mip. +p.)
133 ' . ,,
“2"?!” <46)
Bu _p’(N€pc+p’)
m -
coshpjr Z '
3,3, = {N;’ (NiMi - 1)p,<p. - Nip.) (p. + Nip. + (p. - Nip.) e ‘2'”)
l
+(N’M’— 1)2p’(M’p +p othp )
c c 3 " 3 1c ft } mpftzezk
c -2 2 2 2 2 2 -2p,d (47)
Bfi={Nl (NJMJ -1)p;(p3+Mcp¢-)(pg+Ncpc+(p;-Ncpc)e ’)
- 1
+<N’M’-1)2p’(M’p -p p )e 3”"
c c " ’ 3 M ft }coshpltzezh

where Z ‘(1) and 2"(1) are defined in expression (26). We have to note that, by

letting the cover and the superstrate have the same permittivity, the Hertz potential
Green’s functions found above reduce to the ones obtained in chapter four.
6.4 INTEGRAL EQUATION DESCRIPTION OF MICROSTRIP
TRANSMISSION LINE

In this section, we will study the microstrip structure with a superstrate layer.
The general configuration of this microstrip transmission line is depicted in Fig. 6.3.
The conducting strip is embedded in the superstrate layer adjacent to the
substrate/superstrate interface of the four layered conductor/ substrate/ superstrate/ cover
environment. The y axis is normal and the x and z axes are tangential to the

substrate/superstrate interface.

105

 

 

' //////////////////

//
///// M

 

Figure 6.3: Configuration of microstrip transmission line with superstrate

A homogeneous transform-domain EFIE for the eigencurrent on the strip was

developed in chapter four, which, for the present case reduces to

lim f-f §‘(x,y|x’,y’=0;C)-E;(x’)dx’= O «for -wsxsw . (48)
"0 C

with nontrivial solutions only for C = :9. 1?; (x) is the current distribution of the

p ’th propagation mode on the strip and 1C, is the associated propagation constant.

The space-domain eingenmode currents are

1?;(p‘p) =12]:(p)e""‘ . (49)

This a current wave propagating in :z direction along the strip. Since the current
distributions for :z directed waves are essentially the same, we restrict our discussion

to the case of +2. directed current.

The transform-domain electric Green’s dyad is given by

§‘(b‘|fi’) = Zlc‘v'wlb’fiz)! (so)
= PV(ki+W-)§<plp ;o + Ems—m .

The transform-domain Green’s dyad g in eq. (48) is expressed in terms of

Sommerfeld-type integrals as follows

8% lb"; )=78’(b‘ l5 :0 +§’(5 I5 :0 (51)
where
£15 Ii3 :0 =06 @335 la :0 +y(v,gi(p Ia x) +gi<p Ip m) (52)

with rifle-6% +2% and

107

" 'w-x') -p.|.v-y’l
8’(‘p’|5’;€)= f ”I 4; d5 . (s3)

3

 

The scattered Green’s dyad has the following components

g:’(5|5/;C) .. S'O’IY’J)

 

: “(1‘35

g.(b’lp';o = f snob/,1) ‘ d5. (54)
.- ~ -- sc | "A 4"”:

8c(p|p;C) 0')” )

The scattering coefficients 5" are functions of E, through 2.2 = £2 + C2 and are defined

in previous sections. We note that the subscripts referring to the superstrate is
dropped for the sake of simplicity.
The surface current for a strip of vanishing thickness has only tangential

components as

12(5) =xk,(x) +2k,(x) . (55)
Due to the symmetry of the problem, there are two categories of modes for the
microstrip, even modes and odd modes, defined according to the symmetry of the z

component current as

k,( -x) = 'k,(1)

kz(-x) a ’50!) } even made

(56)
k3( -x) = k,(x)

k,(-x) 3 4900} odd mode .

Substituting eq. (55) in (48) yields a pair of integral equations by letting f=1? and

i=2; they are

108

W

up: f {g;.(x.ylx'.o;ok.(x6 + g;(x.ylx’.o;ok.(x ’)} dx’=o
y -

W

(57)
133 f {85:03 Ix ’.0;C)k.(x’) + g£(x.ylx’.0;C)k.(x ’)} dx’=0

for -wsxsw .

The scalar components of the electric Green’s functions can be found as in Appendix

Bas

3:.(xalx’,0)=-21n-f Mum!”mama/flaw” d:

 

 

 

 

 

 

 

 

(58)
for «.5 =x.z
where the coefficients are functions of i and C as

C. = (ki - 5’)Mi(p.+Mip.) + ezp. A *(x)

1“ 2‘01) 292* (59)
C- = (ki - e2)Mi<p.-Mip.) _ 52p, A ‘m ,

” 2‘0.) 2'2“

C.= —ECMi(p.+Mip.) + £Cp.A*m

“ 2‘0) 2'2” (60)
(3-: -ECMi<p.-Mip.) _ “AA-(A) ,

‘ 2"(1) 2'2“
C. _ (ki - C’)M3(p,+Mip.) + czp. A to)

“ 2"(1) 2'2“ (61)
C- s (ki — comm-Min) _ c’p. A ‘m

a

2%).) 2'2”

with

109

 

A *(x) = (NiMi - 1) (p. + ip.) [e ”Mo. - Mip.) + (p. + Mip.)]
+ 2pr,e “7”"(N3Mf - 1) (Nip, -p,:anhp,:)

A '(A) =<NiMi-1)<p. -Nip.) [e ’Wwp. - Mip.) + (p. + ip.)]
+ 2Mip.(NiMi - 1) (Nip. +p,xanhp,:) .

(62)

Substituting eq. (58) into (57), and interchanging the spatial and spectral integration,

leads to

lim fdfi f {(C£,e""+C,;e”)kx(x’) + (cge'p’y+C,;e"’)kz(x’)}ej“‘"”dx’=0

y—O 2. 3, (63)

133 de f {(cge'"+c;e”)k,(x') + (cge"»’+c;e”)kz(x6} aim-”aka
-.. w «for -w sxsw .

A moment-method solution of the above coupled equations is pursued next.
6.5 MOMENT METHOD SOLUTION
The transverse and longitudinal current components are expanded in series of

Chebyshev polynomials weighted by appropriate edge-condition factors as follows

N
k,(x) = 2 an em(x)

nil-,0 (64)
190:) = 2 an em(x)
n -0
with

e,,(x) = T,(x/W) N1 - (ac/W)2
where Tact/w) is the Chebyshev polynomial of order n of the first kind and an and

an are unknown expansion coefficrents.

110

Exploiting the current expansions in (64) gives

elude f :({c,;e "’+c,;.e”)éa,r,,(o

N
+ (CQe-"+C,;e”)2 aafm(E)}dx’=0
u-O

r0

" " N (66)
m a: + w, - p
.0 {e1 d: £{(Cue Cue ”)gayfAE)
N
+ (c;e”+c,;e”)2aufm(e)}dx'=o
M0
for -w sxsw
where
fp.(€)= f 9,,(x’1e'jwdx’ for a =x,z. (67)

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

functions. We define the testing operator as
f mum-u} for ma =x,z . (68)

Then applying this operator to (66), we have

13: ismflwfi }( [(CQe'” +Cge”)a,.fm(fi)
5,;(6 e""+C' w¢”)d.,f,,(£)]dx’0

um): [mode f [(cge""+C.; e”)a..f..<e)
+(c,;e ”+cge”)a,fm(e)]dx’=o

for -w sxsw

(69)

where

111

3...,(5) = f e.,.(x)e1‘*dx . (70)

If we let

A343: f(C.‘.e""+C;.e”)g.,.(£)r,,<ode mow =x.z (71)

then eq. (69) becomes

N
a; 9:0” +A;"am) =0

N (72)
gfifan +A:'am)=0.

The above is a system of linear equations for unknown expansion coefficients a”. It

can be written in a convenient matrix form as

M
:n a: a," = 0, ...for "1,1! =O,1’...,N . (73)
U

 

 

To obtain a nontrivial solution of the matrix equation, the determinant of the

[A] matrix must vanish. Since the elements of [A] are functions of C , this
requirement yields the propagation constant (p. The corresponding expansion

coefficients are then evaluated and the eigenmode currents are obtained in quasi-

112

closed Chebyshev polynomial series.

One advantage of using Chebyshev polynomials as basis functions is that it
allows spatial integrals in (67) and (70) to be evaluated in closed form. Chebyshev
polynomials of even order are even functions while those of odd order are odd
functions. Hence, for even modes, the transverse current is represented by
Chebyshev polynomials of odd order while the longitudinal current is represented by
Chebyshev polynomials of even order; vice-versa for the odd mode. Therefore, four r

types of integrals emerge and can be evaluated as follows

 

w Tara/W) nw
—cos(£x)dx = (-l)"—J (£w) (74)
o Jl-(x/w)2 2 h

(51:)ch = (-1)" "W 2...,(esw) (75)

; T2,.,(x/w) sin _
0 J1 -(x/w)2 2

f T.,(x/w)(/1-(x/w)2cos(ax)dx
0

(76)
= (4)"?- J..(£w) .; 2.,.,,((5w) +%J..,-,,(:w)]

f T..(x/w)(/1-(x/w)2sin(£x)dx (77)

0

= (-l)’-§5![J2,.1(EW)+-;-J2(..1,.,(EW)+-:-J2(.-1,..(€W)]

where J.(x) is the Bessel function of the first kind.

The matrix elements in eq.(71) are inverse Fourier transform integrals. For

113

bound propagation modes, the real axis is taken as inverse integration contour in the

complex E-plane. The surface-wave poles implicated in the coefficients 0:3(5)

through Z ‘(5) and Z “(5) , are on or near the imaginary axis in the proper half plane,

and are not implicated in the integrals.
An important issue that merits examination in this MOM solution is the

convergence problem of the integrals representing the MOM matrix. By examining
the asymptotic behavior of the integrands as £ approaches infinity, it was shown that
the integrals converge regardless of the value of y. Hence, interchanging the limit

y-‘O with the integration is justified in eq. (71).

6.6 NUMERICAL RESULTS
Numerical results are obtained for the typical configuration shown in Fig. 6.3

with various physical parameters. We first consider a microstrip line with parameters

2w=3.0mm, t=dJ =0.635mm, nc=1.0, nf=3.13 and n,=l.4. An additional

advantage of using Chebyshev polynomials as basis functions is that only a few terms
in the series are required to accurately represent the current. Figure 6.4 shows the
convergence of the normalized propagation constant C/ku, as a function of the number
of terms used in the series expansion for the current. It can be seen that fast

convergence to the exact value of C/ku is obtained even for the case of N = 2.

Figure 6.5 presents plots of the normalized propagation constant C/ko versus

frequency, obtained by solving eq. (72). As expected, the EH.) mode is purely bound

at all frequencies. The dashed line is the dispersion curve of the TM0 background

114

 

surface-wave mode, that is the solution to Z ‘0.) =0 for A = 1p. For bound modes,

the propagation constant satisfies Ap< C 49. Hence, AP provides a cutoff condition

with corresponding cutoff frequency. Higher-order modes become leaky when the
frequency is below cutoff. Background surface-wave and/or space radiation is then
excited.

Figure 6.6 shows the dispersion curve of the principal mode for various
superstrate permittivities and fixed superstrate height. It can be seen that as the
superstrate permittivity becomes larger so does the propagation constant. This is as

expected, since for bound modes and for this particular structure, the propagation

constant satisfies k,< C <kf . Hence, as the superstrate permittivity grows, the fields

in that region have more influence on the propagation constant.

Figure 6.7 illustrates the dispersion curve for the principal mode with a fixed
superstrate permittivity and normalized superstrate height as parameter. It can be
seen that as the superstrate thickness gets larger, the propagation constant increases.
This can be explained by the fact that as the superstrate thickness grows, the
propagation constant is more influenced by the fields in the substrate than by the
fields in the supersrate.

The longitudinal and the transverse currents on the strip for the principal mode

are shown in Figures 6.8 and 6.9, respectively. Only the results for the right half of
the strip Osx sw are displayed. As expected, the longitudinal current is even with

respect to the center of the strip and the transverse current is odd. In Figures 6.8 and

6.9, the current distributions for the microstrip of Fig. 6.3 is compared with the

115

current distributions of the conventional microstrip structure of Fig. 4.3, both
operating at two different frequencies. It can be seen that for both structures, the
change in longitudinal current is relatively small as a function of frequency, and the
transverse current significantly increases with increasing frequency.

Now, we turn our attention to the microstrip of fig. 6.3 having air as substrate
and cover. Figure 6.10 shows the dispersion curve of the principal EH0 mode and the
next higher-order mode. It can be seen that the principal mode never leaks as the

curve for the propagation constant stays always above the curve for the background
TMO surface-wave pole 1,. Figure 6.11 shows the dispersion curve for the principal
mode with the superstrate refractive index as parameter. As the superstrate refractive

index gets larger, so does the propagation constant. Moreover, the propagation

constant displays a significant change as a function of frequency for larger superstrate

refractive indices ( for n,=2.5 and n,=3.13). The normalized propagation constant

starts from a value very close to the substrate refractive index nf= 1. This is due to

the fact that most of the field is concentrated underneath the strip in the substrate
region. Figure 6.12 illustrates the normalized propagation constant versus frequency
up to 40 GHz with superstrate thickness as parameter. As the superstrate gets
thicker, the propagation constant increases. This can be explained by the fact that the

propagation constant is more influenced by the fields in the superstrate and

consequently approaches its refractive index of nf=3.13.

Figures 6.13 and 6.14 show the longitudinal and transverse current

distributions, respectively, of the principal IEEH0 mode. As expected, the longitudinal

116

current is even with respect to the center of the strip whereas the transverse
component is odd. Moreover, the transverse current is quite small compared to the
longitudinal component. Figures 6.15 and 6.16 show the current distributions of the
first higher-order EHl mode for two different frequencies. The longitudinal current is
now odd with respect to the center of the strip while the transverse current is even.
Unlike the principle mode, the magnitude of the transverse current for the EHl mode
decreases as the frequency increases. Moreover, the relative amplitude of the

transverse components are significant compared with the longitudinal one.

6.7 SUMMARY

A rigorous analysis of a microstrip transmission line with a superstrate layer
has been presented. The dispersion characteristics and current distributions of the
guiding structure were analyzed for the principal and higher-order modes. It is found
that the principal mode of such structure never leaks. It was shown by Oliner et al
[51,52] that a leaky dominant mode is present at higher frequencies on conventional
microstrip line. Hence, the microstrip with superstrate can be used as an

improvement to the conventional microstrip.

117

 

 

 

nc=1 . n.=1 4', nf=3 3
w t=23 , d,/t=1, f=10 GHZ.

2.95
.3?
b. r“
*5 .
.3 2.94
m a
c
o
o
C
o 2.93
.5
0
or
o
o.
E 2.92
a
‘0 A
(u t. a
.5
6

2.91
E
L
0
Z

2'90 |1jTrIIflllllllllIlllIIIIIITII[IITIIIIIIIIIIIIIIIII

o 1 2 3 4 5

Number of terms

Figure 6.4: Convergence of the normalized propagation constant as a
function of number of terms used in the Chebyshev series
expansion for the current.

118

nc=1., n,=1.4, nf=3.13, d./t=1.

 

 

 

 

3.5
{ 0r
-
"E 3.0
o
...:
(D
S
o 2.5
c
.9
...;
o
O] 2.0
o
o.
o
L
o.
'O 1.5
(u
.t‘
B
E 1.0
0
Z

0'5 lfillll[llllllIIIIWIIIIIIITTIIIIIIIIITTTIIIIIIII—lj

0.0 1 0.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 6.5: Dispersion characteristics of the principal and the first two
higher-order modes of the microstriop line with superstrate.

119

 

 

M n.=1.

Q9999 n.=1.4

manna n.=2.5
3.20 M n.=3.

n.=1., n.=3.13, d./t=1.

 

3.10

3.00

2.90

2.00

Normalized propagation constant C/ko

2.70

 

20wjTTIIIIlITIIIIIIIIIIIIIIIIIIIIrIIIrTIIIIIIIIIIIIII]

0.0 10.0 ' 20.0 30.0 40.0 50.0
Frequency (GHz)

Figure 6.6: Dispersion characteristics of the principal mode with
superstrate refractive index as parameter.

120

ooooo d./t=1

um d /t=2.
“0 99999 d:/t=3.

n.=1., n,=3.13, n.=2.

3.05

3.00

2.05

2.00

 

Normalized propagation constant C/ko

2.85

2.50
0.0 ‘I 0.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 6.7: Dispersion characteristics of the principal mode with
superstrate thickness as parameter.

121

nf=3.13
w/t=2. 36 d./t= 1.

 

1.2
Q.Q999f=10 GHz, n.=1.4, nc=
...... (=40 GHz, n.=1.4, n.=
10 00.009 f=10 GHz, n.=nc= .
c ' sud-.4- f=40 GHz, n.=nc1.
.9
...-J
B
'c 0.8
...—o
.9
'O
‘E
a, 0.6
t A
3 .'
—°- 00‘ - ‘ - - 4: -‘; ‘U
c C 3 3 “ 3 3 3 3 3 S ‘3 E C C
:6 G—O—O-O -e 9- G- '
3
:t‘ .
O!
8 0.2
..l
0.0
0.0 0.2 0.4 0.6 0.8 1 .0

Position x/w

Figure 6.8: Comparison between the longitudinal current distributions
of the principal mode for the conventional microstrip and
those for the microstrip with superstrate operating at two frequencies.

122

n,=:5.13, w/t =2.36, d,/t=1.

Ital-u f=4O GHz, n.=nc=1.
0090.0 f=10 GHz, n.=nc=1.
*****f=40 GHz, n,=1.4, nc=1.
0_0_0_0_0f=10 GHz, n.=1.4, n.=1.

 

 

 

0.06-
c d
o .-
2; _
3 _
£0.04“ Y*—*\*
(0 ‘ V \
'6 _ r ’\
.. A '6 y
c A 2/
e - / \
L - /"
3 _ \
o d I \
00.02-4 I" *
* .
5. - // ' ‘ ‘ r
S - I s ’r . 3 C ;\c
r.- - r . = '
' at . " °
0°00llllllll[[TIIIIIITIIIIllI1llTrWITllllllell1llll|
0.0 0.2 0.4 0.6 0.8 1.0

Position x/w

Figure 6.9: Comparison between the transverse current distributions
of the principal mode for the conventional microstrip and
those for the microstrip with superstrate operating at two frequencies.

123

nc=nf=1., w/t =2.36, n.=3.13, d./t=1.

 

 

 

a 3.5
i
av 0-
4...!
C 3.0
o
4.;
(n
c:
o
o 2.5
c
o
.4:
o
8’ 2.0
o.
o
L
a
'o 1.5
a)
.5
3 .
E 10 ——————————
o .
Z )‘p/ko
0‘5 IIll]ITITTTTIIIITITTIITITIIIII]IIIIITIllllll'llIm
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 6.10: Dispersion characteristics of the principal and the first
higher-order modes of the microstriop line with superstrate.

124

an! n.=1.5
1.50 099.09 n.=2.

0999.0 n.=2.5

000g: n.=3.13

 

 

{ n.=n.=1., w/t =2.36, d./t=1.
v 1.40
4.1
C
U
...—0
03
5
0 1.”
C
.9
...o
o
8‘ 1.20
O.
o
L.
‘3- M
3
.5 1.10
B P—b‘fi—F :: i —¢——+—-—¢
E
L
o
2 1.00

0.90

0.0 5.0 10.0 15.0 20.0 25.0

Frequency (GHz)

Figure 6.11: Dispersion characteristics of the principal mode with
superstrate refractive index as parameter.

125

 

2.00 m d./t=1.
099110 d./t=2.
0990.0 d./t=3.

nc=nf=1., w/t =2.36. n.=3.13.

1 .20

 

Normalized propagation constant C/ko

 

0‘” [TIIIIIIIIIIIIIIITIIIIIIIITIIIIIIIIUIIIIIIIITIIIIW

0.0 5.0 10.0 15.0 20.0 25.0
Frequency (GHz)

Figure 6.12: Dispersion characteristics of the principal mode with
superstrate thickness as parameter.

126

.___.... (=40 GHz
00000 f=10 GHz

n.=n.=1., w/t =2.3‘6, n.=3.13, d./t=1.

1.2

1.0

0.8

0.6

\\

Longitudinal current

0.21:::==::'

0.0
0.0 0.2 0.4 0.5 0.8

Position x/w

Figure 6.13: Longitudinal current distribution of the principal mode
for the microstrip line with superstrate operating at two
different frequencies.

127

 

'I

'2

1.0

any: f=10 GHz
099.09 f=40 GHz

nc=nf=1., w/t =2.36, n.=3.13, d./t=1.
0.04

\\
o
1 1
I)
’1

\\
’1

0.03 “

'1

\\
h

I; “

0.02

I] “

0

Transverse current

0.01

U

 

p
8

.olllllllllllllllLllLLlllllllllijlllllLLllI

."H"'l'l"""H'I'H'H'HI'HH'H'I""""'I
0 0.2 0.4 0.6 0.8 1.0

Position x/w

Figure 6.14: Transverse current distribution of the principal mode
for the microstrip line with superstrate operating at two
different frequencies.

128

my f=22 GHZ

Q.Q_Q_QDf=40 GHZ

n.=n,=1., w/t =2.36, n.=3.13, d./t=1.
1.2

Longitudinal current

11111111111111111111111111111

 

 

0.0 0.2 0.4 0.0 0.9 1.0
Position x/w

Figure 6.15: Longitudinal current distribution of the first higher-order
mode for the microstrip line with superstrate operating
at two different frequencies.

129

 

my f=40 GHz
0.9.09.9 f=22 GHz

nc=nf=1., w/t =2.36, n.=3.13, d./t=1.

.0
a
J

P
N

llllJLlJJJLJlLLJLllJlllljlll

Transverse current
0

 

0.2 0.4 0.0 0.9 1.0
Position x/w

P
c

Figure 6.16: Transverse current distribution of the first higher-order
mode for the microstrip line with superstrate operating
at two different frequencies.

130

CHAPTER 7

CHARACTERISTIC IMPEDANCE OF MICROSTRIP TRANSNIISSION LINE
WITH SUPERSTRATE LAYER

7.1 INTRODUCTION

The dispersion characteristics of a microstrip with a superstrate layer were
evaluated in chapter six. To completely analyze this microstrip structure, the charac-
teristic impedance must be evaluated. We have seen in chapter five that the power-
current method yields results very close to those of the voltage-current method. Since
the latter method is much more convenient and is less analytically involved, we
choose it to compute the characteristic impedance. We also have to note that the
power-current method will be very difficult to exploit since the structure now involves

additional layers.

7.2 FORMULATION OF THE PROBLEM
The voltage-current method developed in chapter five will be applied again in
this chapter to obtain the characteristic impedance of the microstrip line with a super-

strate layer. Consider the microstrip line of Fig. 6.3 with only the principal mode
propagating along the +2. direction with propagation constant Co. The characteristic
impedance is defined by

z=_V_.. (1)
1

where the total longitudinal current I is given by

131

I = f kz(x)dx . (2)

The current density 15(1) is obtained in eqs (6.64) and (6.65) from the MOM solution

to the integral equation. The average voltage Va, was defined in chapter five as
follows

f v(x) k.‘(x) dx
v = "' (3)

j. k;(x) dx

 

where the voltage is calculated by integrating the y component of electric field along

the y-axis from the ground plane to the strip as follows

0 0
var) =f -e,,(x.y)dy = —fe,,(x,y)dy . (4)
0 -t

The spectral domain electric field in the substrate region is given as

2.05) =

 

k'“ f r,‘(x.ylx’.o;-(.) 46(510’ . (5)

Only the y-component of electric field is needed and is written as

9,03) = f 1g,.‘.(x.ylx’.o;-c.)k.(x) + 3,;(xalx',o;-c.)k.(x))dx'. (61

With reference to Appendix B, the electric Green’s dyad in the substrate region has

the following scalar components

132

 

 

g,f,(x,y|x’,0;C) = 2% f L:p(5.CJ)€Jw"I)dE mfor (1,0 =x,y,z . (7)

Only L; and L; are nwded and are written as

L;(e.(.y) =11: N."coshp,0 + t)(p.(1 + e "2””) +Mip.(1— e ’i’"))
xluci +pi)(NiMi-1)(p.(1+e"’*") +Nip.(1-e'i’")) + p, }

 

 

coshpjtl ‘Z " sinhpft 2 h
+ j: ooshpp +00} +pi)(NiMi-1)4piMie ""4
coshpr 'Z "

(8)
1.;(5.c.y)=j(N;iooshp,0 +t)(p.(1 +e '2” '1 +M3P.<1'e "2"“)
[02210311134120 ...... 0.2.0.20) ”’ }
x +

 

 

coshpjtZ‘Z" sinhpftl"
+ jCcoshpfo' + 0(ku + pi) (N314: - 1)4p}M}e '2’"
coshpjtZ‘Z "

Exploiting relation (7) in eq. (5) leads to

7'11.
211k,

 

01“” = f f 1113(5.-c0.y)k.(x)age-(0.0000yaw-0.15.10, (9)

Interchanging the spatial integration with the spectral one and recognizing that the
spatial integration over the current can be interpreted as a Fourier transform on the

variable x as follows

ka(x’)e'j“’dx’ = «71,4191 =f;(§)
-.. (10)

fk.(x5e’1"'dx 91190)) =f.(E)

we have

133

 

 

2:3. I ‘ Aka-(050,15) +L£(E.-C..y)f,(£) 1 ejudi . (11)

 

9,05) =

The current in eq. (10) is expressed in terms of Chebyshev polynomial series given

by eqs. (6.64) and (6.65). Hence, the integral in eq. (10) can be evaluated in closed

form. Referring to section 6.5, we have

‘V N
f,(£) = fk,(x0e"€"dx’ = E a," 4,05) for p = x,z (12)
-w n-l

where the fpn(E) are evaluated in terms of Bessel functions as follows

f..(5) = (‘1)"NWJ2.(EW)

(13)
2.0:) = 1(4)"le

 

J2M1(EW) + éJun+1).1(£w) + %J2(,.4)+1(€W) .

Exploiting the expression for the electric field given by eq. (9), the average

voltage in relation (3) becomes

 

 

 

190
- f 19,05) k.‘(x)dydx
v“: -w-: w
(up
.."'o . (14)
j" fdxfdy f 1 40.04.5100 age-(.0105) )ef‘xdemx)
=25; "' 4 --

[gma
Interchanging the first spatial integration with the spectral one and using the fact that

[130.1% =f.’(€) <15)

134

 

 

the average voltage becomes

0 o
j“ fdy f (Lac-(01405.15) +L,;(:.-c..y)r;(£)f;(e))d£

 

 

 

 

- 1
v” 2111:, w ( 6)
fgma
Exploiting eqs. (2) and ( 16), relation (1) for the characteristic impedance
yields
0 ..
. fdyf1 L;(t.-c..y)r..(£)f.‘(0 +L,;(:.-(..y)f.(0f.‘(£)1d£
= 1'“ " " . (17)
21th W V
f 1:200de k,‘(x) (b:
The spatial integration in eq. (17) can be performed analytically to yield
0
f L;(E.-Co.y)dy = -jc.L'(£.—c.)
5‘ (18)
fL;(t.-(..y)dy =j£L'(e.-c.)
where
L'(£.(.)) =N."( .(1 +e "”1 +Mip.(1-e '2’1‘1)
(ti +pi) (NiMi - 1)(p.(1 + e '2”) + Nip.(1- e ‘2’”1)tanhp,: 1
" pf 2.2). "27. (19)
+ tanhpflk} +pf’1(N.2M3 - 1)411:3M..2e 'W’
p, 2'2“ '

Moreover, the principal mode is an even mode and therefore the longitudinal current

is rcpresented by Chebyshev polynomials of even order as follows

135

N Tn w
kz(x) =2 amz—(x/L , (20)
W 1/1 - (ac/w)2
Exploiting the orthogonality of Chebyshev polynomials and eq. (20), expression (2)
can be readily evaluated as in chapter six. We obtain

I = rtwaw (21)
where aw is the first expansion coefficient in the series for the z-component of cur-

rent given in eq. (20).
Substituting eqs. (18) and (21) in eq. (17), the expression for the characteristic

impedance is found as

1" “COL'151'CJf.(E)f;(€)-EL‘(E.-Co)f.(5)f.'(E)]dE . (22)

2 rt3ksw2 law I2 ..

 

It is found that the integrand in the above expression is an even function of C.

Hence, the final form of characteristic impedance is

z = "’ [[coL'(£.-c.)f.(0f.‘(0-£L'(£.-c.)f.(t)f.‘(£)]d£. (23)

“3‘09: law I2 0

 

The spectral integration in eq. (23) is performed numerically.

7.3 NUMERICAL RESULTS
Numerical results are obtained for the typical configuration shown in Fig. 6.3

with various physical parameters. We first consider a microstrip line with parameters

2w=3.0mm, t=0.6353nm, nc=l and nf=3.13. These are the same parameters used

in Fig. 6.6, which illustrates the dispersion curve. Figure 7.1 shows the characteris-

136

tic impedance versus frequency for various superstrate permittivities and fixed
superstrate height. It can be seen that as the superstrate permittivity becomes larger,
the characteristic impedance gets smaller. This is as expected, since as the super-
strate permittivity grows, the relative dominance of substrate and superstrate fields
varies. In this case, the fields in the superstrate grow correspondingly giving rise to a
decrease in the voltage in the substrate and hence the characteristic impedance de-
creases. We also have to note that this microstrip structure behaves like the conven-
tional microstrip since the characteristic impedance increases with frequency.

Figure 7.2 illustrates the characteristic impedance versus frequency for a fixed
superstrate permittivity and normalized superstrate height as parameter. It can be
seen that as the superstrate thickness gets larger, the chararcteristic impedance
slightly decreases. This can be explained by the fact that as the superstrate thickness
grows, less electric field is concentrated underneath the strip and a decrease in the
voltage and the characteristic impedance results.

Now, we turn our attention to the microstrip of Fig. 6.3, having air as sub-
strate and cover. Figure 7.3 shows the characteristic impedance versus frequency
with the superstrate refractive index as parameter. As the superstrate refractive index
gets larger, the characteristic impedance decreases. Moreover, the characterstic impe-

dance displays a significant change as a function of frequency for larger refractive

indices ( for n,=2.5 and n,=3.l3). We also have to note that, for this particular

structure, the characteristic impedance decreases with an increase in frequency. This
is due to the fact that for this structure the relative substrate/superstrate fields are

completely different from the conventional microstrip since the substrate does not

137

 

have the highest refractive index.

Figure 7.4 illustrates the characteristic impedance versus frequency with super-
strate thickness as parameter. As the superstrate gets thicker, the characteristic impe-
dance decreases. Moreover, this decrease is more pronounced than for the structure
of Fig. 7.2. This can be explained by the fact that the field underneath the strip be-
comes smaller as the superstrate thiclaiess increases. Consequently, the average
voltage in the expression for the characteristic impedance (1) will decrease and so

does the characteristic impedance.

7 .4 SUMMARY

Two types of microstrip line with superstrate were analyzed. First, a
microstrip line with substrate layer having the largest permittivity was analyzed. It
was shown that the characteristic impedance for this structure does not deviate much
from that of the conventional microstrip. Second, the results for a microstrip line
having both the substrate and cover as free space displayed a tremendous change as its
characteristic impedance showed a decrease with frequency. The voltage-current
method was adopted in this analysis since it is less analytically involved than the

power-current method.

138

 

##4## n,=1.
09.9.0.0 n.=1.4
0.0m n.=2.5
AAAAA n.=3.

 

23.0 ”FM “(=11 3. d./t=1.

22.0

N
d
O

20.0

19.0

Characteristic impedance (0)

18.0

1 7.0

 

16001UIIIIIIIIIIIITIITIFIITIII[IlllllfifiTIT—llilllllll]

0.0 10.0 20.0 30.0 40.0 50.0
Frequency (Ghz)

Figure 7.1: Characteristic impedance versus frequency for different superstrate
refractive indices with w = 1.5 mm, t =0.635 mm, and ds /t = l.

139

4444* d./t=1
03mm; d./t=2.
AAAAA d./t_—_3.

22.0 nc=1-. “(=34 3. n.=2.

N
—.
O

N
p
O

19.0

Characteristic impedance (0)

18.0

17.0
0.0 ' 10.0 20.0 30.0 40.0 50.0

Frequency (Ghz)

Figure 7.2: Characteristic impedance versus frequency for different superstrate
height with w = 1.5 mm, t =0.635 mm, and n, = 2.

140

***** n.=1.5
ODD—.29 n.=2.
000.00 n.=2.5
0.0.000 n.=3.13

nc=nf=1., , d./t=1.

01
P
o

 

 

E

Characteristic impedance (0)
in" .3 '
O O

30.0

 

25.0

 

20'0 IIIIIIIIIIIIIIIIIIll[IIlllllll'IITITIIIIflWIIIIIT]

0.0 5.0 , 10.0 15.0 20.0 25.0
Frequency (GHz)

Figure 7.3: Characteristic impedance versus frequency for different superstrate
refractive indices with w = 1.5 mm, t =0.635 mm, and ds /t = l.

141

4__***4 d./t= 1
000.00 0 /t=2.
1:1___1:11:1c11:1 d:/t= 3

50.0 nc=nf=1 0: ng=3.13

40.0

or
P
o

20.0

Characteristic impedance (0)

10.0

0.0
0.0 5.0 10.0 1 5.0 20.0 25.0

Frequency (GHz)

Figure 7.4: Characteristic impedance versus frequency for different superstrate
height with w = 1.5 mm, t =0.635 mm, and n, = 3.13.

142

CHAPTER 8

EM CHARACTERIZATION OF MATERIALS IN A MICROSTRIP
SUPERSTRATE ENVIRONMENT

8.1 INTRODUCTION
In this chapter the dielectric (complex permittivity e) and magnetic (complex
permeability 1:) characteristics are deduced for materials located in the superstrate of

a four-layered conductor/substrate/superstrate/cover microstrip environment. A
microstrip field applicator, providing a flexible broadband measurement methodology
to accommodate material samples of various types, has been designed [53] and imple-
mented for use with an automatic network analyzer.

The field applicator shown in Fig. 8.1 consists of a strip conductor located
parallel to a conducting ground plate. Tapered transition regions connect the coaxial
applicator terminals to the uniform strip region in which material samples are placed.
The sample is located as a superstrate above the strip conductor. The transition re-
gions, identified in Fig. 8.1 were designed to minimize ambient reflections. A meth-
od was devised to measure the scattering parameters of those transition regions bet-
ween coaxial terminal ports and the front and back terminal planes of the sample
region. Scattering parameters of the superstrate-loaded microstrip region are de-
embedded from measured S-parameters.

Measured scattering parameters of the sample region are processed to deter-

mine phase constant Bm and interfacial reflection coefficient 1"”. Equating the analyt-

143

ical full-wave solutions for those quantities, available from previous chapters, to their

measured values leads to a pair of complex transcendental equations [3(e,p,(0) - [3,.(01) =0

and I‘(e,p,(0) -I‘~((0) =0, which are solved numerically for the desired constitutive

parameters (5(0)) and 11(0)).

8.2 CHARACTERIZATION OF FIELD APPLICATOR: MEASUREMENT OF
TRANSITION REGION S PARAMETERS

Using an equivalent two-port network of the system, as shown in Fig. 8.2, the

scattering parameters of transition regions "a" and "b" and the sample region will be

quantified. The reflected wave amplitudes are designated as b1 while a, are the inci-

dent wave amplitudes.

8.2.1 Transition region "a"

Referring to Fig. 8.2, the scattering matrix [S] of transition region "a" relates

the amplitudes of the reflected waves bvb2 to the amplitudes of the incident waves

01,02 as follows

 

brl_ Star 5142 [“1] (1)
b2 s."l 5;; 02 °

Solving for b2 and using the fact that 02 =I‘2 b2 , we have

 

521 (2)

b2= a a, .
I'Snrz

The reflection coefficient at terminal port 1 as given as

144

b
“1

Solving for bl in eq. (1) in terms of al and a2 exploiting az=I‘:,'b2 and eq. (2),
expression (3) becomes

5125210

. (4)
1 - 5:21;

 

P1 :31: '1’

Hence, to determine the scattering parameters for transition region "a", we

nwd to measure I? ( c for calibration, and i= 1,2,3 ) with port 2 terminated by three

known terminations. A short placed at three known locations provides If, = -exp( -jkol,)

with Il =0,l2 = -l and I3 =1. Applying expression (4) for three calibration measure-

ments and solving simultaneously for the desired transition—region parameters subse-

quently leads to [53]

 

SC =I£I - 51028201I¢11 (5)
11 1 l - a 2:1
with
a I a
Slazszal =(P12 “141-1) (1 -S22r2 X1 -8226!) (6)
1‘32 41’
and
a K c -1
Sn = —— (7)
Key? _ 1:2
where the calibration constant K ‘ is defined as
3 r 2 1
K. _ I‘i 4'; PS 4‘1 (8)

 

112-11' 03-11"

145

Finally, with the scattering parameters 8;}. known, the reflection coefficient at

terminal port 2 is deduced from that measured at port 1 using eq. (4) as follows

I‘1 ' S 101 (9)

r, = .
(P. -S:’.)s;. +8552".

8.2.2 Transition region "b"

Again, referring to Fig. 8.2, the scattering matrix of the sample region relates

the amplitudes of the reflected waves to the incident waves as follows

b3
b

s :
Sll S12

_ “3] (10)
52’) 32":

a4

 

 

 

 

4

Solving for b4 in terms of a3 and a. and using the fact that a. =I‘4b‘, we have

Si" (11)
1 -s;21‘4

 

bf

The reflection coefficient at terminal port 3 is given as

“2 b3 . (12)

Solving for b3 in eq. (10) in terms of a3 and a4 and exploiting a. =I‘4b4 and eq.(11),

expression (12) becomes

$1,252,113 (13)
1 -1251;

 

172:3!!! +

We have to note that since terminal port 6 is matched, we have

146

 

P4 =51: o (14)
The transmission through the field applicator is described by

321=3=fl213=5532fi, (15)
0! atasas at“:

Exploiting expressions (2), (11) and (14), eq. (15) becomes

321s; 8:1
521 '-

- b . (16)
l-S£P2 l'SéSu

 

It can be seen from eq. ( 16) that we need to measure Sf: and S218; [53]. First, to

measure S; we measure the reflection coefficient 1"4 at terminal port 4 with the

sample region "empty". We consequently obtain

a

r,=_‘ eW-v . (17)
b
4

JP I...”

Using the fact that =12, where the superscript e denotes that the sample re-

gion is ”empty”, and exploiting eq. (14) leads to

st =1; em . <18)
To measure S2182: we use expression (16) with the sample region empty. In this case

we have

sg=o

52", =e 'N' .

(19)

147

Using eq. (19), expression (16) becomes

 

a b
521 = 32‘s" e "‘4'. (20)
l-SéI‘Z
Rearranging eq. (20) leads to
5.15;: =szi(1 4mg”: . <21)

This completes the necessary calibration of the two transition regions.

8.3 DE-EMBEDDING OF SAMPLE-REGION S PARAMETERS FROM
MEASURED TERMINAL S PARAMETERS

We measure I"; and 52', under "test" conditions with sample present, to obtain

 

s s b
P2 = s", + ___S”S’: 5‘; (22)

l 'Szzsll

so 52b S:
s;--- "4‘ 2:5' (23)
l-Szzf‘é l-SzzSu
For the sample region, we have

S73 :51“ (24)

s a
312-321 -

Exploiting eqs. (24) in expressions (22) and (23) and rearranging leads to

s 2 b
I3=Sfi+m (25)

1-5533

52‘! I'SZaZI‘; t
1—5585 5215;:

 

(26)

148

Expressions (25) and (26) can be solved analytically for sf, and 5;, [53]. We have

to note that the reflection coefficient at terminal port 2 is known form measured

reflection coefficient at terminal port 1 as indicated by eq. (9). If we let

I" (27)

 

we can solve for Sf, and S2: from eqs (25) and (26) as follows

__ P-Sfio2
1’ Sb 2
( ”0) (28)
1-s,';P

s’ =o—— .
21 l-(Sll; )2

:
$11

8.4 DETERMINATION OF MATERIAL CONSTITUTIVE PARAMETERS
FROM MEASURED SAMPLE-REGION S PARAMETERS

The sample-region parameters are related to the measured 8 parameters as

g _ 2)
Sl‘l= 1 T

l-II‘:T2 (29)
ski—L“ T.

l -—I‘2T2

The interfacial reflection coefficient 1‘ and the transmission propagation factor T are

defined as

 

:_ c

r=z‘ 2‘ (30)
C
ZN.

149

T = e 1.5” (31)

where Z,“ are the characteristic impedances of empty and sample regions and [3 is

the propagation phase constant of the sample region. Expression (29) can be solved

by the Nicholson-Ross-Wier (NRW) technique [54,55] for l‘ and T to yield
P=K1l/K2—l (32)

(sf. we’ll-r
1 -(s;', +s;,)r

T= (33)

 

where

K= (5,1)” - (32")2 + 1 (34)
25:, °

Equating measured BMW) and P..((0) to the corresponding theoretical values leads to

a pair of equations which can be solved for complex 6 and p. as follows

B(€al-"9Q) - DJ“) =0
1159",“) ‘1‘”(“0 =0 ° (35)

A full-wave theory for the microstrip phase constant and characteristic impedance was
developed in chapters 6 and 7; it renders the above equations transcendental, and they
must be solved numerically by a 2-D root search. The interfacial reflection coeffi-

cient is related to the characteristic impedance as follows

 

I‘= ‘ (36)

where zc is the characteristic impedance normalized to that of the unloaded region.

150

 

8.5 NUMERICAL RESULTS

Measurements have been implemented for frequencies over the band of 300-2000
MHz. It was found that the reflection coefficient 1‘ is very small for this particular

configuration and gets mixed with the noise. This is due to the fact that radiation

exists and is not accounted for in this Nicholson-Ross-Wier (NRW) technique [54,55].
Hence, only results for the complex 6 of a non—magnetic material have been obtained,
and the root search was found to be stable. Figure 8.3 shows the real and imaginary
part of the permittivity of a teflon sample. It can be seen that the real part of e is

around 2 as expected.

151

transition % / transition
region ”a” Z region ”b”

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Terminal 1 '2 Terminal
Plane ”A” Plane ”B"
Sample reference plane

Figure 8.1: Transmission line field applicator with sample inserted in superstrate.

152

 

[3']

 

l‘x‘p/l

__<r
[- A.

 

 

 

 

[8‘1

 

 

 

 

 

153

Figure 8.2: Equivalent two-port network.

 

 

.-.... 83

3.00

'8

ttivity

“'2’

Inory perml
S

\
I\,\'\"'-'“\’V- v

I
d
'8
'——-‘

.1.
'2

Real or imog
I
'8

 

-4.” [IIIIITTIIIIIIIIIIIIITIIIIIIIIITIIIIIIIII

0.00 0.50 1.00 1.50 2.00
Frequency (GHz)

Figure 8.3: Frequency dependence of complex permittivity for teflon sample.

154

CHAPTER 9

SUMNIARY AND CONCLUSIONS

This dissertation was devoted mostly to two main topics, namely, construction
of dyadic Green’s functions in planar layered media based upon wave matrices and

the analysis of microstrip circuits with superstrate layer. The dyadic green’s

 

functions associated with a multi-layered environment were constructed in a
systematic manner such that determining the Hertz potential in any region of a planar
layered environment, maintained by general volume currents residing in any other
layer, was rendered to multiplication of wave matrices. A rigorous full-wave integral
operator formulation for the analysis of the electromagnetic properties of microstrip
transmission line with a superstrate layer has been presented. Broad electromagnetic
phenomena associated with this structure, including its dispersion characteristics,
current distributions and characteristic impedance have been investigated via the
rigorous full-wave integral equation approach.

Dyadic Green’s functions for the EM field maintained by electric volume
currents immersed in a planar layered environment were constructed in Chapter 3
through electric Hertz potentials. Spectral amplitudes in their Sommerfeld-integml
representations were obtained using wave transmission and coupling matrices. Such
Green’s functions are appropriate for the analysis of contemporary integrated
electronic and optical circuits operating at micro/mm/optical wavelengths, where the
circuit components are located adjacent to a layered surround environment. This

155

systematic construction of EM Green’s dyads for general volume currents removes
any uncertainty regarding completeness of the field representation and naturally
accommodates the source-point singularity.

In Chapter 4, the conventional microstrip transmission line was analyzed as an
example of an integrated circuit located adjacent to a layered surround. The dyadic
Green’s functions associated with the layered background of the microstrip
environment were constructed using wave matrices as discussed in Chapter 3. A
Fourier transform-domain electric field integral equation (EFIE) description of general
microstrip circuits was developed, and then applied to the conventional microstrip
transmission line.

The complete propagation-mode spectrum of the microstrip line was identified.
Numerical solutions to the homogeneous EFIE were implemented by the Galerkin’s
method of moments. Chebyshev polynomials weighted by square-root edge factors
were utilized as basis functions in the current series expansion. Taking the edge
behavior of the current into account explicitly enhances accuracy and accelerates the
numerics. Moreover, by using Chebyshev polynomials only a few terms were
required in the series to accurately represent the current.

A complete equivalent transmission-line representation for the open microstrip
is obtained by evaluating its characteristic impedance. In fact, Chapter 5 presented a
full-wave analysis for the characteristic impedance using both voltage-current and
power-current definitions. These two methods were compared to each other. It was
shown that the voltage-current method gives results very close to those of the more
rigorous power-current method. The dependence of characteristic impedance upon

frequency was a controversial issue since quite different functions of frequency have

156

been predicted. It was established in this chapter that, for the conventional
microstrip, the characteristic impedance will always increase with frequency.

Chapter 6 presented a rigorous analysis of a microstrip transmission line with
a superstrate layer. First, the dyadic Green’s function associated with the layered
conductor/substrate/superstrate/cover background environment was established. The
EM fields in the substrate and superstrate regions were then evaluated. We have to
note that this otherwise extensive effort was made relatively easy by using wave
matrices as presented in Chapter 3.

A rigorous full-wave solution to the integral equation was pursued similar to
the one in Chapter 4. The dispersion characteristics and current distributions of the
guiding structure were analyzed for the principal and higher-order modes. It was
found that the principal mode of the microstrip with superstrate never leaks. The
microstrip line with superstrate behaves similarly to the conventional microstrip in the
sense that the current distributions for the principal and higher-order modes behave in
the same manner.

Chapter 7 completes the circuit modeling of the microstrip line with
superstrate layer by determining its characteristic impedance. The voltage-current
method was adopted in this analysis since it is less analytically involved than the
power-current method; and, as was established in Chapter 5 , the former method gives
results very close to those of the latter method. Two types of microstrip line with
superstrate were analyzed. First, a microstrip line with substrate layer having the
largest permittivity was analyzed. It was shown that the characteristic impedance for
this structure does not deviate much from that of the conventional microstrip.

Second, the results for a microstrip line having both the substrate and cover as free

157

space displayed a tremendous change as its characteristic impedance showed a
decrease with frequency.

Finally, the research in this dissertation has reflected some progress on the
analysis and understanding of electromagnetic phenomena associated with microstrip
structures with superstrate. There remain many topics which deserve further research
effort in the future. In fact, Higher-order modes can be investigated in the leaky-
wave (below cutoff) regime. Moreover, the analysis of microstrip line with
superstrate in anisotropic media can be investigated. Furthermore, afull-wave theory
for coupled microstrip transmission lines with superstrate can be studied based upon

the analysis of a single microstrip line as presented in this dissertation.

158

APPENDIX A

APPENDIX A

CANONICAL TRANSFORM DOMAIN SOLUTION
TO THE PLAN AR INTERFACE REFLECTION PROBLEM

A.l Single Interface
Consider the planar interface between medium (1) and medium (2) as shown in
Fig. (A.l). The total transform-domain Hertz potential in medium (1) consists of an incident

component augmented by a reflected part as

taxi.» = 131,42» + flux.» (M)

where a =x,y,z.

Region (1) (spun n; 1 fr: T ,f
9“ ,a inte ace]-

Region (2) (£2,132) mm 1 flat“ I

 

y=0

Figure A.1 Single interface

Similarly, the Hertz potential in medium (2) decomposes into an incident part and a reflected

part as
172.05.» = 11;.(1'3’) + 111,09.) . (A.2)

The incident and reflected components of Hertz potential in each layer satisfy the following

159

transform domain Helmoltz equation

 

d2 2 T114753) ”La” (A3)
3 em = +12. -
1,: ’y 0

Appropriate solutions to eq. (A.3) are

= 0123):”
01:43:?"
52:41)!” ”
= and)!”

.Fl: 51:
a

0
fl

(A.4)

are

where 011(1) and 6;,(1') are complex spectral amplitudes at the y=0 interface. 01:,(1')

and 52:43..) are regarded as known due to impressed currents while 01:41.) and did-X) are

regarded as unknown and to be determined.

The total Hertz potentials in each region satisfy the following boundary conditions

 

[50]
finds?) ‘ N12“: fludyl
afil"(x’y) - N2 afiz.¢(x‘y) } ...for a =x’z (A05)
ay ‘ ay 1
and
find.» = N12 112,53)
(A.6)

6fi1,(i‘.y)_ 8111,53)
6y a,

 

. -(Mfuf-l) [iifiudw + jcfiumfl -

160

A.l.1 Scattering of Tangential Potential Components

Implementing boundary conditions (A.5) on tangential components of Hertz potentials
at y =0 interface leads to

“’2‘ = Rflagv Tfttdgf (A.7)
d2“: = Tllalm + R1102“:

where the tangential reflection coefficients at interface (1) associated with downward and

upward travelling incident waves are defined, respectively, as

‘

 

 

 

 

 

Rtl _ alta
1 " - -.
“la “16:0
l... =
- °‘ x’z (A.8)
Rh = 57"“
52:“ al2¢=0
and are given as follows
M2 -
Rf! = tpr P2
M12P1+P2 (A3)
121‘: = -R,‘l .

The interfacial transmission coefficients at interface 1 are defined similarly and are given as

 

 

 

' 21w2 _ -
Tl'l = ii" = 7L3}. = N12M12(1+R:1)
01.. 523° Mlplwz
(A.lO)
a+ 2N2M2
T3 = ‘f _ = _2‘__‘fl = N}M,’(1+R,‘t) .
52,. 61.630 Mlpl +1’2

 

161

Solving for the wave amplitudes of. in medium (1) in terms of the wave amplitudes

5;. leads to

, J ;for a = x,z (A4“

 

 

where we define the tangential wave-transmission chain matrix at interface (1) which chains

region (2) wave amplitudes to those in region (1) as
1 Rf:
[Afl] = .J‘. , (A.12)
1,1 R,‘l 1

Note that the downgoing arrow in the notation for [A 1'1] illustrates the fact that the incident

wave in region (1) is travelling in - y direction. Similarly, the tangential wave-transmission

matrix associated with upward travelling incident wave is

1 lift
[ Aft] = __1_ (A.l3)
I
73‘ left 1
I .
and where [A1 1] rs defined such that
+ a4“
(17:. "' ["1“] '1' ; or a = x.z. (A44)
d2,1! aim

 

162

 

 

A.1.2 Scattering of Normal Potential Components
The tangential potential components were quantified independently above, and may

therefore be regarded as available for this normal potential calculation. The boundary condi-
tions on normal potential at the y =0 interface couple normal and tangential components as in

eq. (A.6) leading to

af‘ynzl; = Nf(%:® (A.lS)
p, (a;y -a;.,) were; ~51.) - (NEH! ‘1)[iE(62}+52L) +1“: (5.1. we] .

Solving for the normal scattered wave amplitudes in both regions leads to

01:, = Rflaf.y + The", + of, (A.l6)
52.; = Tlnlalzy * Rlnmzty 1' 62F:
where the normal reflection and transmission coefficients associated with either downward

or upward travelling incident waves are defined, similarly to above, as follows

 

 

 

 

R1"! = “—13.
a1; §2J=Ft=0

R,"t = * _
62‘, aIJ=F,=O

T1"! = 3L0 (A.l7)
62:, 013=Ft=0

T101 = &-| *
a” 620=Ft=0

and r, = (NM-1) wage» + IC(42}+62:J]. c, = Mf/(Ilh’pl +p,).
c2 = 11min +p2).

163

 

The normal interfacial reflection and transmission coefficients are given as

N2 -
R151 = —;p1 p2 = 'Rlnt
N1p1+p2
2 -
7:1 = _;_’2_ = N12(1+Rl"1) (A.18)
N1P1+P2
21v2
1,": = Tfi = N,’(1+R;'t).
N1P1+P2

Our goal is to be able to chain region (2) wave amplitudes to those in region (1) and

vice versa. Hence, using equation (A. 16), we solve for the normal wave amplitudes alt, in

region (1) in terms of those 62*,“ (a =x,y,z) in region (2) leading to

 

 

 

 

 

_ -Rl'l _, 1 ~_
arr .. new ,azs-czF.
T1 1 T1 1 (A.l9)
, T,"lT,"l-R;')R,"t a; 12,") 52:, ( Rflczw
01 = + + C - o
‘y T,"l ’y 1,") ' T,"l '

Writing the above in matrix notation and after many simplifications, we have

 

[:0 = W'ilz: *[Cn‘](j€ [2:] +1421]. (A.20)

[A ['1] is the normal wave matrix associated with downward travelling incident wave at inter-

face (1) and is defined as

l R"1
1 ‘ (A.21)

 

[Alli] =

 

T,"l Rf) 1

[C11] is the interfacial coupling matrix associated with downward travelling incident wave

164

 

and is given as

(A.22)

21112-1 -1 -1
[C11] = EL).— ]

2pl l 1

Similarly, for an incident wave travelling upward the normal wave amplitudes in region (2)

are given in terms of wave amplitudes in region (1) as follows

('12:, = ‘t alt? + Cl ' a1; + °C 012]] (A-23)
[as] P“ 1L); [”(fi [012:] 1 LL:

 

where

1 1 R,"t
[Arr

(A.24)

 

 

 

Rh 1

(A.25)

1- N,”M," -1 -1
[Qt] = _— .

2p2 1 1

A.2 Effect of a Propagation Path Length

If region (2) has thickness t2 as shown in Fig. (A.2), we now have two interfaces.

To obtain the wave amplitudes in region (2) and at interface (2), the wave amplitudes at

interface (1) are shifted downward as follows

“’3' = 57E“: (A.26)
02.!) = due

165

 

y Region (1) 01:. 1 01:1: I interfacea)
, 1 +1
Region (2) ’ T a”

y2 a“ 02" 1 interface(2)

 

 

Figure A.2 Effect of a Propagation Path Length

A.2.1 Tangential potential components
The wave matrix formulation is modified to reference waves in region (2) to the loca-

tion of interface (2) as follows

= [An]

 

“1:3
alto

= [as]
(A.27)

 

 

 

where

(A.28)

and the interfacial tangential transmission matrix at interface (1) is redefined as

166

 

[111:1] = i [‘1’]

(A.29)

[An] = i

 

 

A.2.2. Normal potential components

A similar modification of the wave matrix for normal potential components leads to

 

 

 

at" +31%“? +[C,1](j£ [“1“ +1: [‘5‘
“1.x “2.x “2.: “2.:

 

] (A.30)

where the normal transmission and coupling matrices at interface (1) are redefined as

 

‘2 " 'Pz‘z
[Ail] . 1 ‘h R‘ 1‘ (A31)
1,") R," 1e”1 e “’2‘“
C l = N12Ml2—l ’CM 'C-M (A032)
[ 1 ] 2pl eh" e725 '

 

 

167

APPENDIX B

APPENDIX B

ELECTRIC DYADIC GREEN ’8 FUNCTIONS

B.l Fields in the Cover Region
In the cover layer of Fig . 4.2, the electric Green’s dyad in the transform

domain is identified in terms of Hertz potential Green’s dyad as
E‘tb’lb”;€)= PV(k3+W°)§(b°I6 ;o + raw-ta") (M)
where V=V,+£ K with V,=£6/ax +96/6y the transverse operator. The transform-

domain Hertz potential Green’s dyad is expressed as in eqs. (4.37) and (4.38) as

follows
§(filb";C)=73’(filfi’;C)+§’(filb";C) 03-2)
where
E‘tb’ Ib”;C) = (if #:3331115“) +)"(Vrg.’(b‘lb°’;0 +g,;'(rs leech?) . (13.3)

Using eq. (B.2) and (3.3) in expression (3.1) leads to

relate=m§+yygg+ifig§+figg+fia§ (3,4,
+1958; +3723; +928; +293;

where 3:, («,9 =x,y,z) are written in terms of Hertz potential Green’s function

components as follows

168

 

62 62 a
kc +— ‘)+——gc
8x3: (2 W)“; 8 523

e_ 2+_ 32 a __
Err-(kca ay‘j]ax8c +36;in +8?)

8;=J'C— its» 3’) +J'C— axgyg

8.; ‘85;

a2
+ (13.5)
8,;= (’3‘ a,» —]J’Cg.. +jC:y(g. g’)

kcz+2 + +2—
g;=( c)<g, g?) cayg.

 

g,‘,=(k3+-§;](s.+s’)

. . a
3,, =JC-5;(8.*8’)

Exploiting the Hertz potential Green’s function components as in eq. (4.40) leads to

. l ' e -, —
g.,<x.le’.o =5] calmed” “e ” d: (13.6)
for «,0 =x.y,z

where

169

_ (k3 - £er + p.521N3M3 - 1)

 

 

 

 

 

 

C; -
Z " Z ”(1)2 ‘0.)
C. = -j£p.M.’ +11: (k3 +p:)(~3M: - 1)
y 2" 2"(1)z'(1)
. _ . -C£M3 p,CC(N3MZ-1)
Cu - Ca = +
2" 2*(1)z'(1)
C. 3 -ij.M.’ +J'C(k3 +p.’)(~.’M.’ - 1)
’1 h I) e
Z Z Z
2 2 (32.) 2(1) (3.7)
C‘ = (k, 'C2)Mc +PCC2(N.,M¢ '1)
a z“ Z”(A)Z‘(A)
2 2
Ce = MC (kc +p6‘2)
I) Z],
. 2
Ce ..lcpcMc
0 Z],
Ce - -jEpcMc2
1! Z],
B.2 Fields in the Film Region
In the film region of Fig. 4.2, the electric Green’s dyad in the transform
domain is identified in terms of Hertz potential Green’s dyad as
§'(b‘lfi’;C)= (k3+W-)§(b‘lb°’;C) (3+8)

where the Hertz potential Green’s dyad has the same form as in eq. (3.3) and has the

following components

r ‘

ammo . Bl:e’“’“’+8,é.e"’°"

- . +c) - +r)
8.191530 = f l 81:33” +Bfi'Le ”’0

sctplpx) -- Lager/wowfge-wa

_ I
, e"“”’)e "0 ) (B.9)

 

41Cpc

 

 

170

The Bk coefficients are defined in eqs. (4.31), (4.32) and (4.33).
Exploiting eq. (B.3) in eq. (3.8) leads to

E'Gi Ib‘ ; C) =38; +993; +7223; +3981; +9381; ' (3.10)
+2ng + fig; +928; +2983,

where 3:, (11,11 =x.)’,Z) are written in terms of Hertz potential Green’s function

components given by eq. (3.9) as follows

a?) _g: as
1:24»—

.1 ...—2.1
_a_
g ax

 

. 2+ 6“ a _a_g
8’“ (3‘ Re '55"

a _a_g
33=1C;g.+jC—axayg
3.2%;

c_ 2 32 . . a
..-(...g,)...,.

kcz+2 +2—
g;=( C)8,Caay8.

Exploiting eqs. (3.9) and (B.11) in (B.10) leads to

331(xdlx’m0)-—21f83.,(5 1.)):le d: (11.12)
"for arfl =x9y’z

where

171

_k/z £2 sinhpfo'fln

 

 

 

 

 

 

 

 

C’ENZMZ-l) sinh[pf(y+t)]

 

 

 

 

 

 

 

B €.C.y
“(C N 323(1) smhpf: N32"(A)Z'(A) coshpj:
B‘(e,c,y) = jg”! oosh[p,(v+t)] , j5("12*P/5(N3M3 -l) cosh[p,o++t)]
3311.11.» = '5‘ @PN*‘)] _ C‘PflNfMZ-l) sinh[p,1y+1)]
N31 "(D WW N§z*(l)z+(1.) coshpj:
B;(£.C.y) = 35.0553)
B'(E Cy): _jL’! “WWW JC("12*P})(N3M3 -l) comp/0+0]
fl ’ N 3Z"(l) 3m?! Nfz"().)z'(r.) coshpf:
Baggy) = if. flip/0+0] _ CZPKN3M3 '1) sinh[p,(y+t)]
N3Z"(l) MW Nfz*().)z+(1) coup]:
k2+p2 mPfCY+t)]
3‘ ’ = f f
”(C C .y) 2‘0.) cash”
c prf sinh[pf(y+t)]
B , =
..(5 C3) 2'11) sinhpf
. 18p, sinh[p,1y+r)]
B , = .
”(E C .y) 2‘01) cosh”

172

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