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dissertation entitled

COMPOSITE MATERIAL DESIGN AND

CHARACTERIZATION FOR RF APPLICATIONS

presented by

DANIEL STEVEN KILLIPS

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Composite Material Design and Characterization for RF
Applications

By

Daniel Steven Killips

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Electrical Engineering

2007

ABSTRACT

Composite Material Design and Characterization for RF Applications
By

Daniel Steven Killips

As radio frequency technology continues to evolve, understanding of the electro-
magnetic behavior of materials is increasingly significant. Applications of such ma-
terials can range from radomes in which controlled transmission is required, to radar
absorbing materials in which high loss and low reflection is a dominant factor. This
is the motivation for creating materials in which the permittivity and permeability
can be controlled depending upon the application.

The design of such materials, which will ultimately determine the effective permit-
tivity and permeability, will depend on such factors as: geometry, volume fraction,
thickness, and permittivity and permeability of the individual materials being com-
bined. Composite design using classic mixing formulae, such as Maxwell-Garnet and
Bruggeman formulations, are not valid for dense composites or ones possessing signif-
icant coupling between inclusions. Hence, more accurate homogenization techniques
are needed. For layered composites, alternating dielectric and magnetic layers sim-
ulated using the method of wave matrices for both isotropic and anisotropic layers
is both useful and efficient. For composites consisting of rod shaped inclusions, an
integral equation based formulation is utilized.

In addition to material design, care must be taken in characterization of these
materials such that the effective material properties are valid. Various characteriza—
tion methods and fixtures are used in this work to determine effective properties for
a variety of composite materials. These composites were fabricated both at Michigan

State University and at partner universities over the past several years.

For Shana and Ella

iii

ACKNOWLEDGMENTS

There are many who deserve thanks and appreciation. I must first extend my
gratitude Dr. Leo Kempel for extending to me the opportunity to continue my edu-
cation with Michigan State University. This experience has been invaluable to both
my development as a researcher and as a person who enjoyed and cherished the advice
given to me. Another special thanks is extended to Dr. Ed Rothwell, who has contin-
ued to spark my interest in electromagnetics since my sophomore year. Dr. Rothwell
has also been available countless times to answer questions in times of confusion for
a wide range of topics. To Dr. Shanker Balasubraniam, much appreciation for his
consultation on the integral equation formulation including the use of the periodic
Green’s function. Also, his classes have given me a better understanding of some
of the more complex topics in theoretical electromagnetics. Thanks to Dr. Martin
Hawley for agreeing to participate in my graduate committee and for his excellent
recommendations on how to improve my dissertation. Recognition must be extended
to Dr. Charles Lee from AFOSR for this work performed under grants F-496-2002-
10196 and FA9550-06-1-0023 as well as the NSF Career grant, ECS-0134236. Drs.
Mark Scott and Jeff. Berrie have provided very beneficial conversations on the topics
of magnetics and composite materials. Also, Dr. Stephen W. Schneider deserves
appreciation for his thoughts contributing to the motivation for this work. Finally, I
would like to thank my wife, Shana, for her love and support and for putting up with
me throughout this process. Not to mention, she gave me a beautiful daughter who

was able to help me keep smiling through even the most stressful times.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................. viii

LIST OF FIGURES ................................ ix

KEY TO SYMBOLS AND ABBREVIATIONS ................. xii
CHAPTER 1

Introduction ..................................... 1

1.1 Permittivity and Permeability ...................... 2

1.2 Composites ................................ 3

1.3 Material Characterization ........................ 5
CHAPTER 2

Composite Background and Characterization .................. 6

2.1 Motivation ................................. 6

2.2 Homogenization .............................. 7

2.3 Composite Synthesis ........................... 8

2.4 Composite Characterization ....................... 9

2.4.1 Dielectric Test Fixture and Impedance Analyzer ........ 9

2.4.2 Magnetic Test Fixture and Impedance Analyzer ........ 11

2.4.3 Stripline Field Applicator .................... 12

2.5 Previous Work: Spherical Inclusions and Classical Mixing Rules . . . 14

2.5.1 Maxwell Garnett ......................... 15

2.5.2 Bruggeman ............................ 16

2.5.3 Coherent Potential ........................ 16

2.5.4 Conclusions ............................ 17
CHAPTER 3

Isotropic Layered Composites ........................... 31

3.1 Wave Matrix ............................... 32

3.1.1 Reflection and fiansmission Coefficients ............ 33

3.1.2 Wave Matrix Method ....................... 35

3.1.3 Scattering Matrix ......................... 39

3.2 Analysis and Results ........................... 4O

CHAPTER 4

Anisotropic Layered Composites .......................... 52

4.1 Background and Derivation ....................... 53

4.1.1 Geometry ............................. 53

4.1.2 State Vector and State Equation Formulation ......... 53

4.1.3 Transition, Transmission, and Reflection Matrices ....... 58

4.1.4 Layered Media and Local Transition Matrix .......... 61

4.2 Verification of Anisotropic Formulation ................. 63

4.2.1 Transmission through Free Space ................ 63

4.2.2 Reflection from PEC space .................... 64

4.2.3 Comparison with Isotropic Layers ................ 64

4.3 Ferrimagnetic Materials with Anisotropic Permeability ........ 65

4.3.1 Permeability Tensor ....................... 65

4.3.2 Demagnetization Factor ..................... 66

4.3.3 Bias Transverse to Direction of Propagation .......... 67

4.4 Layered Composite of Ferrimagnetic and Dielectric Materials ..... 69

4.5 Simulation of Double Negative (DNG) Material ............ 71

4.5.1 Plasma as ENG Layer ...................... 71

4.5.2 YIG as MNG Layer ........................ 74

4.5.3 DNG Layered Composite ..................... 74
CHAPTER 5

Composites with Rod Shaped Inclusions ..................... 94

5.1 Background and Derivation ....................... 95

5.1.1 Geometry ............................. 95

5.1.2 Integral Equation ......................... 96

5.1.3 Periodic Green’s Function .................... 102

5.2 Verification of IE Code .......................... 106

5.2.1 Penetrable Cylinder ........................ 106

5.2.2 Infinite Slab ............................ 107

5.3 Analysis of Composite .......................... 108

5.3.1 Geometries ............................ 109

5.3.2 Increasing Layers ......................... 110

5.3.3 Volume Ftaction ......................... 111

5.3.4 Free-Space Inclusions ....................... 113

5.4 Conclusions ................................ 114
CHAPTER 6

Conclusions and Future Work ........................... 139

6.1 Material Characterization ........................ 139

vi

6.2 Composites and Unique Contributions ................. 140

6.2.1 Composites with Spherical Inclusions .............. 140

6.2.2 Composites of Alternating Layers ................ 140

6.2.3 Composites with Cylindrical Inclusions ............. 141

6.3 Future Work ................................ 142
BIBLIOGRAPHY ................................. 145

vii

LIST OF TABLES

Table 3.1 Tunable Effective Permittivity range summary for isotropic layered
composites. .............................. 51

Table 4.1 Tunable Effective Permittivity range summary for anisotropic lay-
ered composites. ........................... 88

viii

Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8

Figure 2.9

Figure 2.10
Figure 2.11

Figure 2.12

Figure 3.1
Figure 3.2

Figure 3.3

Figure 3.4
Figure 3.5

Figure 3.6
Figure 3.7
Figure 3.8
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6

LIST OF FIGURES

Possible Geometries for Composite Material Design ........ 19
Homogenization Illustration ..................... 20
Agilent 16453A Dielectric Test Fixture [6] .............. 21

Measured permittivity of acrylic using 16453A Dielectric test fixture. 22
Agilent 16454A Magnetic Test Fixture [6]. ............. 23
Measured permeability of acrylic using 16454A Magnetic test fixture. 24
Sideview of the stripline field applicator [5]. ............ 25
Field lines within the stripline: a.) Dynamic Electric Field, b.)

Dynamic Magnetic Field. ...................... 26
Measured permittivity and permeability of acrylic using stripline. 27
Composite with spherical inclusions ................ 28
Comparison of 3 classical mixing laws for dielectric contrast ratio

of 52' /€e = 2. ............................. 29
Comparison of 3 classical mixing laws for dielectric contrast ratio

of si/ee = 25. ............................ 30
Geometry for layered materials. ................... 43
Illustration of the analysis on layered materials when layers A and

B are swapped. ............................ 44
Incident, reflected, and transmitted waves for an interface between

2 isotropic materials .......................... 45
Reflection and transmission of plane wave amplitude coefficients. . 46
Reflection and transmission of plane wave amplitude coefficients

for multiple layers. .......................... 47
Case 1: Layer A has volume fraction 0.75. ............. 48
Case 2: Layer A has volume fraction 0.50. ............. 49
Case 3: Layer A has volume fraction 0.25. ............. 50
Geometry for anisotropic layered material derivation. ....... 76
Transmission through free space geometry. ............. 77
Scattering parameters for free space case ............... 78
Reflection from PEC in region 3. .................. 79
Scattering parameters for PEC case. ................ 80
Comparison of calculated S-parameters using the isotropic and

anisotropic formulations ........................ 81

ix

Figure 4.7

Figure 4.8
Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13
Figure 4.14

Figure 4.15
Figure 4.16

Figure 4.17

Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9

Figure 5.10
Figure 5.11

Comparison of extracted permeability and permittivity for both
isotropic and anisotropic methods.

Demagnetization Factors for various geometries ........... 83

Effective permeability for extraordinary wave with the illustration
of ferrimagnetic resonance (FMR). ................. 84

Effective wavenumber for extraordinary wave assuming negligible
loss. .................................. 85

Effective permeability for layered anisotropic composite consist-
ing of biased ferrite and dielectric materials for different volume
fractions, where subscript ”F” denotes the volume fraction of the
ferrite. Also, F-D-F implies that the ferrimagnetic material is the
outer layer while D-F-D implies that the dielectric layer is on the
outside ................................. 86

Effective permittivity for layered anisotropic composite consisting
of biased ferrite and dielectric materials for different volume frac-
tions, where subscript ”F” denotes the volume fraction of the ferrite. 87

Effective permittivity for plasma immersed in a magnetic bias field. 89

Ferrite effective permeability and plasma effective permittivity

with negative values. ......................... 90
Layered composite for DN G simulation. .............. 91
Negative permeability and permittivity for layered composite in

Figure 4.15a. ............................. 92
Negative permeability and permittivity for layered composite in

Figure 4.15b. ............................. 93
Geometry and mesh for core—shell model used in IE formulation. . 116
Pyramidal basis function for node n [53, 54, 55] .......... 117
Convergence of the hankel function .................. 118
Comparison of the exact and IE surface fields for dielectric rod. . 119
Comparison of the exact and IE surface fields for magnetic rod. . 120
Comparison of the exact and IE surface fields for lossy rod. . . . . 121

Comparison of the exact and IE scattering parameters for lossy slab.122
Extracted permittivity and permeability for a lossy slab. ..... 123

Comparison of effective permeability of anisotropic to isotropic fer-
rite. .................................. 124

Geometry and unit cell for the core-shell model with one cylinder. 125

Geometry and unit cell for the core-shell model with three cylinders.126

Figure 5.12
Figure 5.13

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20

Figure 5.21

Figure 5.22

Figure 5.23

Geometry and unit cell for the core-shell model with five cylinders. 127

Geometry and unit cell for the alternating core-shell model with
three cylinders. ............................

Geometry and unit cell for the alternating core-shell model with
five cylinders. .............................

Effect of increasing layers on homogenized permittivity and perme—
ability for rods in a row [a,b] and alternating [c,d] with a volume
fraction of 0.45 .............................

Effect of increasing volume fraction on homogenized permittivity
and permeability ............................

Effect of increasing volume fraction on homogenized permittivity
for composite consisting of alumina and teflon ............

Geometry for analysis on effects of air gaps on permittivity and
permeability.

Effective permittivity and permeability of composite with holes for
volume fraction of 0.45 with YIG as cylindrical inclusion ......

Effective permittivity and permeability of composite with holes for
volume fraction of 0.18 with YIG as cylindrical inclusion ......

Geometry for analysis on effects of air gaps on permittivity and
permeability.

Effective permittivity and permeability of composite with holes for
volume fraction of 0.45 with Teflon as cylindrical inclusion. . . . .

Effective permittivity and permeability of composite with holes for
volume fraction of 0.18 with Teflon as cylindrical inclusion. . . . .

xi

128

129

130

131

132

133

134

135

136

137

138

KEY TO SYMBOLS AND ABBREVIATIONS

&: Unit vector

Z: Column Vector

Rank 2 Tensor

fill

71-): Spatial Vector

MUT: Material Under Test

TEM: Transverse Electromagnetic
FMR: Ferrimagnetic Resonance

TM: Transverse Magnetic

IE: Integral Equation

EFIE: Electric Field Integral Equation

PEC: Perfect Electric Conductor

xii

CHAPTER 1

INTRODUCTION

The motivation in this dissertation is to be capable of understanding how different
geometries of composite mixtures can effect the electromagnetic properties of materi-
als, called permittivity and permeability. Typically, an RF designer is constrained to
certain electromagnetic characteristics of materials which are available in bulk form.
A potential solution to this problem is to add an additional degree of freedom in the
design of RF devices, namely the capability to design a composite material that best
fits the application. This could range from circumstances requiring total transmis-
sion such as in the case of a radome structure generally used to protect an antenna
from the elements. Ideally, the permittivity and permeability would be designed to
be equal to each in order to provide a better impedance match to free space which
minimizes reflection. In addition, the product of the permittivity and permeability
would be designed to be as low as possible in order to reduce the index of refraction.
Another potential application would be used for antenna loading. Typically, designers
use dielectric loadings in antennas in order to reduce the overall size of the antenna.
However, the addition of dielectric materials increases the capacitance of the antenna
substrate and subsequently results in a degradation in the bandwidth. It has been
proposed by Hansen [1] that an antenna with a magneto-dielectric substrate can fur-
ther reduce the surface are of a patch antenna but also increase the bandwidth. In
this situation, the product of the permittivity and permeability would need to be as
high as possible in order to fully decrease the size of the antenna. This is just a subset
of the potential benefits of having the capability to design a composite material with
appropriate values of permittivity and permeability.

The analysis performed in this dissertation will form a foundation for the un-

derstanding of the electromagnetic properties of two—phase composite materials for
three different geometries. These geometries include; composites with spherical inclu—
sions analyzed using classical mixing laws, layered composites using method of wave
matrices for both isotropic and anisotropic layers, and composites with cylindrical
inclusions using the method of moments. It will be demonstrated how the volume
fraction and differences in permittivity and permeability between the two—phases can
be used to tune the effective permittivity and permeability of the composites. An
additional degree of freedom in the layered composites is provided based on the num-
ber of layers in the direction of propagation. The unique contribution in this work
involves analysis of the composites consisting of layers and columnar inclusions based
on the effective permittivity and permeability. In this stage of development, the
analysis is purely theoretical, however experimental verification is a crucial portion
of future work in order to further the design methodology.

This chapter provides an introduction to the concept of permittivity and perme-
ability, composites and methods of characterizing these materials. It should form a
good basis for main concepts covered in this dissertation while chapter 2 will go into

greater detail about the motivation and composite structures.

1.1 Permittivity and Permeability

The interaction of electromagnetic fields with their surrounding mediums can be de-
scribed with various concepts including polarization, susceptibility, and even conduc-
tivity to describe one type of loss mechanism. This work focuses on the parameters
permittivity and permeability which signify the dielectric and magnetic properties of
materials, respectively. Mathematically, this can be seen in (1.1), where both permit-

tivity, E and permeability, fl are in their most general form and the double overlines

indicate that these quantities are rank 2 tensors.

(1.1a)

ml DI
II
”ill

ll
til
:1 ml

(1.1s)

The vector field quantites will be referred to as follows: B and D are the magnetic
and electric f lux densities, respectively, while ff and E are the magnetic and electric
field intensities, respectively. For time-harmonic fields, many materials can have
both 5 and it represented using complex scalar quantities, where p0=47r*10-7(H / m)

and £0 28.854*10—12(F / m) are the permeability and permittivity of free space.

I - II
H = How - Jur) (12a)

8=€o(€$~-j€§~’) (1.2m

+jwt is assumed for all the work done in this dissertation. The

The time convention 6
real portion of these values represents the stored power while the imaginary portion
represents the dissipated power and the ratio of imaginary to real is referred to as
the loss tangent. It should be noted, that the quantities in paranthesis in (1.2), are
relative values and therefore are dimensionless.

The motivation of this work is to accurately calculate the permittivity and per-

meability of materials with an ultimate goal of composite design in which both it and

5 can be adjusted to better fit a particular application.

1 .2 Composites

The majority of the analysis in this work consists of two—phase composite materials,
involving dielectric-dielectric or dielectric-magnetic mixtures. These composites are

clearly inhomogeneous, meaning the values of permeability and permittivity vary with

position within the material. Therefore, homogenization of the composites is a term
that will be introduced which implies that an effective permeability and permittivity
will be used to represent an inhomogeneous mixture. The three different geometries
to be investigated are: spherical particles mixed in a background, alternating layered
structures, and finally rod shaped inclusions mixed in a background material [2]. In
the following chapters, all three of these composite geometries will be analyzed using
different methods.

The composite consisting of spherical particles has been previously analyzed [3]
using classical mixing formulae [4] and compared to experimentally obtained data.
It was shown that these mixing laws become less reliable to accurately predict the
permittivity as the volume of the particles relative to the total volume of the composite
is increased. These mixing laws are even more unrelieable in the ability to predict
permeability for even small volume fractions. One reason for this inaccuracy is due
to the fact that these classical mixing formulae do not account for particle to particle
interaction.

Next, layered composites will be analyzed using the method of wave matrices for
both isotropic and anisotropic layers. The reason for modeling anisotropic layers
is to account for the tensor permeability that is common with many ferrimagnetic
materials. The effect of layering will be investigated for both dielectric-dielectric and
dielectric-magnetic composites. It can be seen that as the volume fraction, thickness,
and constrast ratio of both permittivity and permeability are varied the homogenized
values can be predicted and controlled. This provides a good foundation for design of
composite materials for which a layered structure compliments the given application.

The final geometry to be analyzed will involve an integral equation formulation
to solve for the scattered fields due to periodically arranged cylindrical rods in a
background material. Volume fraction is shown to be the dominant factor for elec-

tromagnetic design of this composite type.

1.3 Material Characterization

Material characterization is the process of measuring some particular property of a
sample and using that value to calculate or extract the material properties. Many
characterization techniques are employed at Michigan State University, however only
two of such techniques will be discussed here as they are suflicient for the needs of
this work. The first technique involves a stripline field applicator [5] and the HP8510
Network Analyzer. A sample is inserted into the stripline and the scattering parame-
ters are measured using the Network Analyzer. The permittivity and permeability
are then extracted from the measured S-parameters using a traditional root searching
algorithm. This structure is particularly useful due to its broadband nature and the
capability to clamp the MUT (Material Under Test) into place which minimizes error
due to air gaps. Samples can be characterized using this method in the frequency
range 1GHz-18GHz.

The latter of these characterization techniques involves the E4991A Impedance
Analyzer and both dielectric and magnetic test fixtures [6]. The permittivity and
permeability are calculated from measured capacitance and inductance, respectively.
These test fixtures are operational in the frequency range 1MHz-1GHz.

These characterization techniques allow for experimental verification of composites
in a wide range of frequencies. This is a crucial contribution to the development of a

solid design method for composites used in electromagnetic applications.

CHAPTER 2

COMPOSITE BACKGROUND AND CHARACTERIZATION

2. 1 Motivation

Understanding of the electromagnetic behavior of materials continues to gain impor-
tance as radio frequency technology continues to grow [7]-[16]. These materials are
used in a wide range of applications from radomes in which controlled transmission
is required, to radar absorbing materials in which high loss and low reflection is a
dominant factor. Within this range, applications such as antenna substrates are used
to reduce antenna surface area, increase bandwidth, or modify the input impedance.
Many times, either the material with the most desirable properties is unavailable in
its bulk form or the technology is designed around a readily available material. It is
therefore advantageous to have the capability to choose the properties of the material
such that it best fits the particular application. One possible method to achieve this
design is through the use of composite materials.

Electromagnetic omposites are mixtures of two or more dissimilar materials, either
dielectric or magnetic, combined in a matter in order to control the effective permit-
tivity and permeability. The effort in determining a solid composite design can be
split into three basic activities. The first activity involves material synthesis. This
centers around creating the materials being simulated and understanding the binding
and dispersion processes involved when attempting to combine multiple materials.
The author’s contribution to this activity was to consult with the chemical engineers,
material scientists, and chemists on the tOpic of electromagnetics. However, section
2.3 has been devoted to this topic in a very broad sense to assist in the understanding
of composite design process. The next activity incorporates the computional effort

ivolved. A suitable simulation of the composite is needed for a complete understand-

ing of how the different mixtures can be modified to adjust the effective permittivity
and permeability. The simulation methods used will include classical mixing formu-
lae, analytical solutions, and computational electromagnetic techiniques. Experimen-
tal verification of these simulations is also of great importance and is the final activity
of composite design. Numerous techniques are available for characterizing materials
[17]-[26], two of them will be discussed in section 2.4. The remainder of this work is
on the analysis of three different geometries of composites illustrated in Figure 2.1.
The first of these composites consisting of spherical inclusions is discussed in the last
section of this chapter. Layered materials are investigated for isotropic materials in
Chapter 3 and anisotropic layers in Chapter 4. The final geometry with rod shaped
inclusions is examined in Chapter 5.

It has been suggested here that a composite material would be highly useful for
situations where a bulk material is not commonly found with the desired electromag—
netic properties. However, even if the desired properties are found in a bulk material,
composites could also be valuable in situations where the flexibility or conformability
of the material is of importance. Perhaps the weight of the material is an issue and
a composite material can be shown to have the same electromagnetic properties but
exhibit less density. Finally, with proper understanding of electromagnetic material
properties and composite simulation, a design method could be formulated for which

an appropriate mixture is devised that best fits the application.

2.2 Homogenization

Consider a material occupying a volume region V enclosed by a surface S as illustrated
in Figure 2.2 where the left body shows an inhomogenous material and the right body

shows an effectively homogeneous material. To a wave, the homogenized material

”looks” like it is a material occupying region V with the effective material properties.

> (2.1a)
> (2.1s)

<B> =eeff<
(79’) =ueff<

El Cal

Equation (2.1) gives a mathematical representation of this concept where () is a
spatial average of the fields and the subscript I e f f’ stands for effective. The effec-
tive permittivity and permeability are the design goals for the composites discussed

throughout this dissertation.

2.3 Composite Synthesis

As mentioned above, composites consist of two or more dissimilar materials combined
in order to achieve some property, physical or chemical, which cannot be found in
those materials in their bulk form. The materials within a composite can be specified
by one of two categories, matrix and reinforcement [27]. The matrix phase is the
material which surrounds and supports the reinforcement phase keeping the geom-
etry fixed in position. The reinforcement phase is needed to create the necessary
parameters for the application in which the composite will be utilized. The matrix
phase is commonly a form of plastic or polymer, while the reinforcement phase can
be anything from carbon or magnetic powders and fibers to metallic rods of flakes
depending on the desired end product. Various methods can be employed in order to
process and manufacture composites. Only a few are introduced in this section and
are only be briefly summarized since the focus of this work is on the electromagnetics
of composites. The first method is hand lay-up molding where the reinforcement is
put into position and the matrix is painted layer by layer until the desired thickness
has been reached [28]. The advantages of this time consuming process is the ability

to align the fiber reinforcement materials as needed and to accomodate for irregular

shaped objects. A less time consuming process called spray-up molding uses a cutter
to chop up the fiber reinforcement which is sprayed onto a mold with a combination
of resin mist and catalyst. The mixture cures on the mold at room temperature and
is finished. An advantage besides lower time consumption is the ability to handle
large, complex objects due to the spraying operation. Compression molding uses a
hydraulic press to compress the mixture or layers created by hand lay-up at an ele-
vated temperature and help until the composite has cured. The compression allows
the mixture to be distributed properly over the entire mold giving it the ability to
mold large and fairly intricate products. Finally, injection molding involves heating
the material into a molten state then injecting it into a mold at very high pressure
where it cools to solidification. A few polymer processing techniques have been in-
troduced with a brief description in order to include all aspects of composite design.
The focus of this work however, does not involve material production and thus that
topic will be left for this section alone. Also, it should be noted that the terminology,
matrix and reinforcement, in this section was based on [27],[28] and will not be used
anywhere else in this dissertation. The corresponding terminology for matrix is either

background or environment, and inclusions for reinforcement.

2.4 Composite Characterization

Both permittivity and permeability are not values that are easily measured through
direct methods. However, it is convenient to measure other quantities such as reflec-
tion, transmission and impedance from which the permittivity and permeability can

be extracted.
2.4.1 Dielectric Test Fixture and Impedance Analyzer

The first method involves the Agilent E4991A Impedance Analyzer and 16453A Di-
electric Test Fixture [6]. The dielectric test fixture measures the admittance of the

material under test (MUT) between two electrodes within the fixture as seen in Fig-

ure 2.3 from 1MHz to 1GHz. Figure 2.3a and Figure 2.3b show the test fixture and a
close up look at the electrodes. Figure 2.3c illustrates the electric field lines between
the electrodes, including the fringing effects.

Prior to taking this measurement, a three part calibration process must be per-
formed. This includes an open measurement with the electrodes separated by a fixed
distance, a shorted measurement with the electrodes in contact with each other, and
the final measurement with a known load, e.g. Teflon. Once the calibration is com-

pleted, the measured complex admittance can be represented by (2.2).

The real part, G, is the conductance and represents dielectric loss. The imaginary
part, Cp, is the capacitance between the parallel plate electrodes and w is the angular
frequency. The effective relative permittivity of the MUT can then be calculated by

(2.3) where CO is the capacitance of an empty test fixture.

_@_.G

5r - CO JJéU (2-3)

Figure 2.4 shows the measured real and imaginary values of permittivity for a sample
of acrylic which has a known relative permittivity of e = 2.6 - 30. Also included in
the figure is a device uncertainty bar that illustrates the potential range of variation
in measurement results. This uncertainty is calculated using (2.4) where t is the
thickness of the sample in mm, f is the frequency in GHz, and Eirm is the real part

of the measured permittivity.

10

 

 

 

I /
A 0.1 t 100
fim = :t 5 + (10 + ———)—,—— + 0. 255”” + [%] (2.4)
5rm f Erm t

 

 

I
f Erm

b

The complete calibration to measurment procedure is repeated five times to ensure
consistency and accuracy of the measurements and to check that the data is well

within the uncertainty bars.
2.4.2 Magnetic Test Fixture and Impedance Analyzer

The next method involves the Agilent E4991A Impedance Analyzer and 16454A Mag-
netic Test Fixture [6]. This test fixture measures the inductance in a toroidal shaped
sample as seen in Figure 2.5 from 1MHz to 1GHz. The inductance within the fix—
ture is created from current flowing upwards through the center conductor and then
outward and down the walls of the fixture. This current loop forms a magnetic flux,
which by the right hand rule, is in the direction normal to the surface created by
that loop. Mathematically, the relationship between the inductance” and magnetic
flux density is given by (2.5), where B is the magnetic flux density vector and I is

the magnitude of the current.

”i/B‘T: i/ohO/ep Triad” (2'5)

As shown in Figure 2.5, no is the height of the fixture, a is the diameter of the
inner conductor, and e is the diameter of the fixture. All of these distances are in

measured in mm,and the self inductance of an empty fixture is given by (2.6) and is

11

an important contribution to the calibration procedure.

H_0

L =
83 2’”

6

Before connecting the fixture to the test head a four part calibration process is per-
formed. This includes an open, short, 509 load, and a low-loss capacitor. Once the
test head has been properly calibrated, the fixture is attached and the self-inductance
is measured and saved completing the calibration. Now the complex impedance Z-m

of the material can be measured.
Zm = R3 + ijS (2.7)

Finally, the complex permeability can be calculated by (2.8).

_ 2’7er — jQJLss
”T — jwpohlng

 

(2.8)

The inner and outer radii of the MUT are given in mm by b and c, respectively and
w = 27r f where f is the frequency in GHz. Figure 2.6 shows the measured real and
imaginary values of permittivity for a sample of acrylic which has a known relative
permeability of m- = 1 - 30. As with the dielectric measurements, Figure 2.6 has a

device uncertainty bar which is calculated using (2.9).

 

 

 

 

Allin, 0.02( 25 ) c , ( 15 )2 2
2;]; 4 hl - 1

lim [+ f h1n(C/b)uf~m + “(film +hln(C/blfli‘m f W

(2.9

2.4.3 Stripline Field Applicator

The stripline is a waveguiding structure consisting of two parallel conducting plates
with a thin conducting strip down the center. Figure 2.7 shows a side view of the

stripline and placement of the material under test. The stripline supports a Transverse

12

Electromagnetic (TEM) wave with zero frequency cutoff and the design described
herein allows for measurements over a wide range of frequencies, namely 1-18GHz.
One advantage the stripline has over similar measurement techniques using coaxial
and rectangular waveguides is the ability to clamp the material in place and minimize
air gaps. Also, the fields within the stripline are concentrated to the area surrounding
the center conductor and are well confined [5] as illustrated in Figure 2.8.

Unlike the characterization procedures discussed in 2.4.1 and 2.4.2 where permit-
tivity and permeability were determined through measured impedance, the stripline
uses measured transmission and reflection. There are three sets of measurements
needed to characterize these materials. The first set involves measuring transmission
through and reflection from an empty stripline. The next step measures the reflec-
tion from a short placed in three different locations within the stripline and the final
measurement is done with a sample inside the stripline.

One method for obtaining both permittivity and permeability involves comparing
the measured and calculated propagation constant within the material region, 6, and
reflection coefficient from the interface of the material, F, until they are identical
using a Newton’s two-dimensional complex root-searching algorithm. The subscript

”m” denotes measured values.

fi(e,ii,w) — flm(w) = 0 (2.10a)

F(e,ii,w) — I‘m(w) = 0 (2.10b)

Both [3 and I‘ can be related to permittivity and permeability using (2.11).

5 = (fix/lira: (2.11a)

r: Fm: (2.11b)

 

13

A technique developed by Nicholson, Ross, and Weir is used to calculate 13m and

I‘m from the measured 311 and $21 at the sample plane [5].

 

Fm (1— 22)
Sfl = 2 (2.12a)
1 — (FmZ)
.. Z (1 - Pin)
52, = ——-———-§ (2.121;)
1 — (FTnZ)
Z = 8Xp(—j/37nt3) (2.12C)

The subscript / superscript ” 3” denotes sample plane S—parameters which can be found
from the measured S-parameters at the stripline terminals using a de-embedding
technique. The sample thickness is given as t3 and the complex permittivity and

permeability are given by (2.13b).

_ E l‘_1"
Er — ’60 (1+ P) (2.138.)
__ ,3 1+r

Finally, measured permittivity and permeability for acylic is shown in Figure 2.9 and

is in agreement with previous measurements.

2.5 Previous Work: Spherical Inclusions and Classical Mixing Rules

Composites consisting of dielectric spherical particles mixed throughout a dielectric
background have been studied for over a hundred years providing many classical
mixing laws to predict effective permittivity [4]. Several of these classical mixing
laws have been summarized and tested against experimental data using the stripline
method discussed in section 2.4.3 for magneto-dielectric composites to check for accu-
racy in predicting BOTH effective permittivity and permeability [3, 29]. A summary

of three of these classical mixing laws as well as general conclusions are provided

14

in this section. The three classical mixing laws, Maxwell-Garnett, Bruggeman, and
Coherent Potential, were chosen because of the difference in their basic philosphy
on homogenization. Figure 2.10 is a geometrical representation of a composite with
spherical inclusions where subscripts i and 6 denote inclusion and environment phases,
respectively. A necessary assumption in the derivation of the classical mixing laws is
that the spherical particles are small enough in diameter such that they approximately
obey the rules of electrostatics and magnetostatics. The maximum diameter, as de-
fined by Sihvola and Kong [30] is given by (2.14), and is appoximately proportional

to wavelength.

A
dmax e 5; (2.14)

2.5.1 Maxwell Garnett

The first and probably most widely known mixing law to be discussed is the [Maxwell
Garnett formulation for effective permittivity shown in (2.15), where f is the volume

fraction of the inclusions [31].

(52‘ - 56)
82' + 258 — f(€i — Ee)

 

Eeff=€e+3fEe (2.15)

This equation is based on the polarizability of a dielectric sphere, and algebraic ma-
nipulation of the spatial averaged electric fields. This formula is assymetric in that
the inclusion phase affects the effective permittivity of the composite diflerently' than
the environment phase. This implies that the inclusion phase is considered a guest
dispersed throughout the environment which is considered a host. The guest versus
host philosophy is the foundation for the Maxwell Garnett formulation in that the

inclusions are compared against the environment.

15

2.5.2 Bruggeman

The next mixing rule is the Bruggeman formulation, (2.16), for effective pern'iittivity

of a two-phase composite [32].

(ho—EVE” +f————Ei_5€ff =

Ee+2€eff €i+2€eff (2.16)

The major difference from Maxwell-Garnett is the symmetry of the Bruggeman for-
mula which no longer employs a host versus guest philosophy. Instead, both the
inclusions and environment phases are evaluated equally against the effective proper-
ties of the homogenized composite. This allows the environment to potentially provide
the same contribution to the effective permittivity of the composite as the inclusions,
unlike Maxwell-Garnett. Essentially, both the inclusions and environment are treated
as inclusions within the entire medium. The Bruggeman formula is known by sev-
eral names; Polder van Santen, ale Loor, Béttcherr, and effective medium model
[414321436]-

2.5.3 Coherent Potential

The final mixing rule discussed in this section is the Coherent Potential formula [37]

which is seen in its most convenient form in (2.17) for spherical inclusions.

356” (2.17)

€eff =5€+fIEi _Ee)3eeff+(1-f)(5i -€e)

Unlike the previous two mixing rules which analyze the problem of a single scatterer
approximation weighted by the volume fraction of the materials, the Coherent Po-
tential formulation uses the Green’s function to represent the polarization density of

the effective medium [4].

16

2.5.4 Conclusions

All three of the classical mixing laws discussed in this section tend to predict the
same effective permittivity as the volume fraction gets very small (f § 0.3) and can

all be approximated by (2.18) [4].

Ei—Ee

— 2.1
51- + 253 ( 8)

5eff mee+3fee

Another important factor in calculating the effective permittivity is the difference in
the dielectric constant of the inclusion and environment phases. As the contrast gets
small the mixing rules tend to agree and also predict the permittivity more accurately.
The cause for this is due to the fact that the classical mixing laws do not take into
account particle to particle interaction, thus as the volume fraction is increased and
the particles become more tightly packed, these formulas will tend to deviate from
each other and from the effective permittivity. See Figure 2.11 and Figure 2.12 for a
graphical illustration of the effects of volume fraction and dielectric contrast. These
figures compare the effective permittivity of the three classical mixing laws described
in this chapter as a function of volume fraction.

When the volume fraction is low (f § 0.3), and there is not a great dielectric
contrast, all three of the classical mixing laws discussed here can be used to predict
the effective permittivity of a composite with spherical particle inclusions fairly ac-
curately. On the other hand, using these same classical mixing laws in attempting to
predict permeability through the concept of duality between the electric and magnetic
fields, the formulas were unable to yield accurate results and hence could not be used
as a tool for composite simulation and design [3]. Another important observation
in [3] and through extensive experience measuring materials with spherical magnetic
particles, is that this geometry doesn’t allow for a significant magnetization within

the composite and thus the permeability is approximately unity and therefore is es-

17

sentially non-magnetic. This is partially due to the fact that spherical ferrimagnetic
particles have a demagnetization factor of 1 / 3 and hence these particles need to be
very tightly packed in order to achieve an appreciable permeability. This difficulty
is the motivation to progress to a new geometry for composite design. This new
geometry is a layered structure and is discussed in chapters 3 and 4 for isotropic
and anisotropic materials, respectively. The reason for layered composites is that it

provides a controlled geometry which can be modeled using exact methods.

18

 

 

 

 

 

 

 

 

 

c.)

Figure 2.1. Possible Geometries for Composite Material Design

19

 

Figure 2.2. Homogenization Illustration

20

      

T
_ U
M

   

Diameter - 10 mm

Upper electrode

Diameter - 7 mm

.5).
2;; 4.1 h. .5
“czar“ a". 3.. an

o' a 3 .0 3%.:

gzar 6. 1
tion 1'4”“

 

               

Figure 2.3. Agilent 16453A Dielectric Test Fixture [6].

21

4 = . a ,
3 5; DeVIce

' -. ..... fl Uncertainty“
3 _ ”In...nunnun-unon-unnuuuucnunun I... a.-nun-announu-nnu

2.5» 7r - - - _,
2? ..................... ' .../. ................................. 7

 

 

 

 

1.5:: r -
1;. -
0.[5 sf _
0].. ......... >1 ............................... .
as 1 1

 

100 200 300 400 500 600 700 800 900 1000
Frequency [MHz]

Figure 2.4. Measured permittivity of acrylic using 16453A Dielectric test fixture.

22

Agilent 16454A

 

 

 

 

 

 

 

 

 

 

Figure 2.5. Agilent 16454A Magnetic Test Fixture [6].

23

 

 

Device ..........
E Uncertainty ,,,,,,,,
1.5g / ........... _
....... in ...........
1 /7
, .............. i ......
0-5l‘ ............ ‘
. fl: ............
o L ...... A-----------------_--------------::
-0.5

 

 

 

100 200 300 400 500 600 700 800 900 1000
Frequency [MHz]

Figure 2.6. Measured permeability of acrylic using 16454A Magnetic test fixture.

24

ground plane

      
 
 

center conductor

 

 

    

ground plane

sample
material , I
//
tuning
wedges

Figure 2.7. Sideview of the stripline field applicator [5].

25

 

 

 

 

a.) b.)

 

 

 

 

 

 

 

 

 

 

 

Figure 2.8. Field lines within the stripline: a.) Dynamic Electric Field, b.) Dynamic
Magnetic Field.

26

 

 

 

 

 

 

 

2 5 ~ —“' T — “f — — ‘ ~
s;/
2 ~ _
1.5 ~ [1; ~
1 ~‘ \ -_
0.5 — 8» [an ~
0 ----------—-_.> ———ml—.------I-P-ul"'[
'0'5 2 4 6 8 1o 12 14 16 18

Frequency [GHz]

Figure 2.9. Measured permittivity and permeability of acrylic using stripline.

27

8.

 

 

 

000 cc
Q Q O 0

0’0

Iue, 5e

 

 

Figure 2.10. Composite with spherical inclusions

28

2 I 1 1 f

 

 

 

 

 

 

 

 

1.9_ ”"Maxwell Garnett _
- Bruggeman
> 1.8~ ——-Coherent Potential 5
:5 1.7L -
3'3
E 1.6~ “
L-
: 1.5 i
,‘z’ 1.4 —
5
o , - 4
E 1 3 gi/ge = 2
1.2— ‘
1.1~ “
1 l l l 1
O 0.2 0.4 0.6 0.8 1

Volume Fraction

Figure 2.11. Comparison of 3 classical mixing laws for dielectric contrast ratio of
cr/ee = 2.
i

29

 

25 1 r . .

 

 

 

 

 

 

 

- ,-_ Maxwell Garnett
20 Bruggeman
a. F ——- Coherent Potential -
IE
5
E 15 ~
5
m
n.
“>’
'5 10~ ~
0
o
3:
LL!
5 s
8,. /se = 25
0 0.2 0.4 0.6 0.8 1

Volume Fraction

Figure 2.12. Comparison of 3 classical mixing laws for dielectric contrast ratio of
Ei/Ee = 25.

30

CHAPTER 3

ISOTROPIC LAYERED COMPOSITES

This chapter focuses on composites consisting of layered isotropic materials with a
_}
geometry given by Figure 3.1. Isotropic materials are those in which D is related to

—> —-> —->
E and B to H by complex scalar quantities such that. the field direction in each pair

is aligned [38].

(3.1a)

ml at
H

l'
mi ml

(3.11))

The focus of this chapter will be limited to the simulation of isotropic dielectric mate-
rials and the impact of layering on the effective permittivity of the composite. Mag-
netic materials are discussed in chapter 4. The analysis of this structure will be done
using the method of wave matrices for planar layered dielectric materials assuming a
plane wave normally incident on the interface. Although the wave matrix method is
fully capable of handling oblique incidence, normal incidence is assumed here in order
to accurately model the characterization procedure within the stripline described in
section 2.4.3. Doing so will allow the effective permittivity to be extracted from the
calculated scattering paramters using the same program as for the stripline. Also,
with normal incidence, phase matching can be achieved at each interface. Section
3.1 derives this method in detail and follows directly from Collin [39]. Using wave
matrices, a straightforward calculation of the reflection and transmission coefficients
for a layered structure can be used in the extraction of complex effective permittivity
as demonstrated in section 2.4.3 for the stripline material characterization technique.

In the design of a composite with tunable effective permittivity, there are three

main parameters of interest. The number of layers, the difference in the dielectric

31

constant of the two materials (dielectric contrast ratio), and the volume fraction of
the layers will form a basis for the following analysis. To simplify the problem, the
material layers are extended to infinity in the directions parallel to the interfaces but
is finite in length in the direction perpendicular. Therefore, the analysis will proceed
for several different dielectric contrast ratios and volume fractions. The layering effect
will be examined by first holding the total thickness, d, fixed in length, while second
increasing the number of layers within the composite. Hence, the volume fraction
is kept constant. This is illustrated in Figure 3.2 where the first case involves ma-
terial A on the outside, followed by interchanging the materials such that material
B is on the outside. It will be demonstrated later that the effective permittivity is
heavily persuaded by the outer layer for composites with a small number of layers.
The permittivities of adjacent layers, A and B, are based on the dielectric contrast
ratio and are therefore real numbers independent of frequency. Since the layers are
non-dispersive, a frequency analysis is not performed here because the effective per-
mittivity would be a constant value over any frequency range in which the sample
thickness is not equal to or greater than a half wavelength. In that case, possible

resonances could potentially cause problems in the characterization process.

3.1 Wave Matrix

The wave matrix method is an exceptionally useful and simple tool for the analysis
of layered dielectrics. In the case of a normally incident plane wave, each layer can
be completely described by its material properties (8, p) and thickness. This section
begins with a derivation of reflection and transmission coefficients in terms of per-
mittivity and permeability for one interface. This is followed by the implementation

of multiple interfaces and finally the method of wave matrices.

32

3.1.1 Reflection and Transmission Coefficients

This study begins with the investigation of the reflection and transmission of a plane
wave normally incident at the interface of a planar dielectric in the same plane as the
transverse component of the wave. Figure 3.3 illustrates that when an incident wave
arrives from the left (z<0) at the dielectric interface, part of that wave is reflected and
part is transmitted. The incident and transmitted waves are shown traveling in the
positive z-direction while the reflected wave is traveling in the negative z-direction.
For this work, the time convention eth is assumed and suppressed which results in

the following representation for the fields.

73”: TifiEe ‘J'klz (3.2a)
'13": 327$ij (3.2b)
it: 11733.2 ‘ijZ (3.2c)

The wavenumber, k, in (3.2) is dependent on frequency, permittivity and permeability
and can be represented as kz- = tum-E? where i could be either 1 or 2, and the vector
amplitude coefficients for the transverse components of the electric fields are given
by TIfE. The transverse components of the magnetic field can be written in terms of

the electric field amplitude coefficients and impedance matrices in each region.

1139: 71 AWE (3.3a)
——> 22—1 —>
~11}, = Z1 411’}; (3.3b)
——> =—1 ——+
xiii], = —22 will]? (3.3c)

The inverse of the impedance matrix (admittance matrix) is given by 7; =
051(3):? — 533)) where i is given by either 1 or 2 and the wave impedances by

77., = i/I‘i/Ei' Assuming there are no currents or charge buildup along the inter-

33

face, boundary conditions imply that the tangential fields must be continuous.

a

Z(2:=0)+E)T(2:=0)=Et(z=0) (3.4a)

e1.

2(.«:=i))+?i”‘(z=(i)=Eii(z=o) (3.4s)

These boundary conditions can also be represented by the transverse components
of the fields since these are tangent to the interface. Also, in the z = 0 plane, the

exponentials in (3.2) becom unity.

11733 + T112; = T171]; (3.5a)
—>i -—>,r. _ -—>t
\IIH+\IIH—\IJH (3.51))

Using (3.3), the magnetic field arnplitued coefficients can be written in terms of the

transverse electric field components and (35b) becomes

=—1 81-4 51—)7. 812—1 —-)t
ZO .( _71_1pi _ Lip )= _T'_ZO .11; (3.6)
”T1 E #71 E ”7'1 E

Furthermore, the reflection coefficient, R1, into region 1 from region 2 at z = 0
is defined as the ratio of the reflected field to that of the incident field. Also, the
transmission coefficient, T21, into region 2 from region 1 is given as the ratio of the

transmitted field to the incident field.

T21 = i (3.7b)

34

Using (3.7), both (3.5a) and (3.6) can be written in terms of the incident electric

i
field, \IIE.

(1+R1)\1135 = TglxiflE (3.8a)
1[ 571 En ] i 1 571 i
— ,/—— —R \IJ = — ——T 111 (3.8b)
770 M1 M1 1 E 710 M1 21 E

Dividing out \I/iE, this becomes two equations and two unknowns for which the re-

flection and transmission coefficients can now be solved.

R = M (3.9)
"T2 + 777']
2
T21 = ——31‘—2—— (3.10)
7772 + ”T1

Now that the reflection and transmission coefficients have been derived for a plane
wave normally incident on the interface between two isotropic materials, this discus-

sion can be extended to multiple interfaces and the method of wave matrices.
3.1.2 Wave Matrix Method

The derivation of wave matrices begins with a description of the problem seen in
Figure 3.4. A plane wave is incident from the left in the region z<0 with an amplitude
of a1 and another wave with an amplitude of b2 incident from the right in the region
z>0. For this analysis, all waves traveling in the positive z-direction will be denoted
by a, while the waves traveling in the negative z-direction will be denoted by b. The
backward traveling wave in region 1 is comprised of the reflected portion of a1 and the
transmitted portion of ()2 as seen in (3.11a). The same is true of a forward traveling

wave in region 2 given by (3.11b).

01 = R1a1+ T1202 (3.11a)

a2 = R202 + T2101 (3.11b)

35

Solving for a1 in (3.11b) and inserting it in (3.11a), the backward traveling wave in
region 1, bl, can be written terms of the forward and backward traveling waves in

region 2. The same can be done to solve for al.

a2 R2132
a1 = —-—-——— (3.12a)
T21 T21
R1132 31
bl = (T1 ————) b2+—a (3.12b)
2 T21 T212

On the left side of (3.12) are the fields in region 1, while on the right side are the fields
in region 2 as well as the reflection and transmission coefficients which are functions of
the material properties in both regions. These equations can now be put into matrix
format as shown in (3.13).

a1 1 1 432 a2 A11 A21 a2

:_T__ = (3.13)
b1 21 R1T21T12”R1R2 b2 A12 A22 52

The matrix H shown above is commonly referred to as the wave-transmission chain
matrix and it relates the amplitudes of the forward and backward propagating waves
in region 1 to those in region 2. Symmetry of the problem under consideration Yields
T12 = T21 and R1 = —R2 for reflection and transmission [on either side of the
interface. Remembering also that T21 2 1+ R1 and T12 = 1 + R2 the elements in H

can be simplified into the following forms,

36

 

 

-R2 R1

A = = .

12 T21 T21 (314b)
R

A21 = 311 (3.14c)
T T —R R

A22 — 21 12 1 2:; (3.14.1)

T21 T21

which can be put back into a simpler matrix form.

0’1 = TI— 1 R1 “2 (3.15)

b1 21 R1 1 b2
Once multiple interfaces are incorporated, the length of each region must be taken
into consideration since the phase of the waves will change depending on the prop-
erties of each region and the distance each wave travels within the regions. This is
done by considering first a forward traveling wave ale-”fl“z and then a backward
traveling wave blejkz . For 220, the waves are simply represented by their amplitude
coefficients, however the waves at a distance 2 = z1 from the z = 0 plane are given as
a2 = ale—jkzl and b2 = blejkzl. Finally, relating the waves in region 1 to region 2

gives the following equations where kzl is the electric thickness 01 as represented in

Figure 3.5 which accounts for the phase progression of the wave in that layer.

a1 = agejkzz1=a26j01 (3.16)

bl -.= er—J'kzz1=b2e“j91 (3.17)

37

This phase relationship can also be put into matrix form.

al = 636 0 02 (318)

Combining both (3.15) and (3.18) both reflection and transmission at. a single interface
as well as propagation through the layer have been included. Finally, after carrying

out the matrix multiplication, (3.20) gives the wave transmission matrix for one layer.

'9
a 1 1 R e] 0 a
01 21 R1 1 0 63—J6 02
a1 = L 616 Hie—fl) “2 (3 20)

This gives a straightfoward method for calculating the fields at the z = 0 in terms of
material properties, layer thickness and the fields at the interface located at z = 21.
For the case of an arbitrary number of layers, (3.21) can be formed by cascading each
wave transmission matrix until finally having the fields at z = 0 in terms of the fields

at z = Zn+11

ejgi Ric-jail

a1 = fi 1 an+1 (3 21)
. T- . 1'0- —1'0- '
()1 2:1 '1. Rze 2 8 ’I, bn+1
where Ti and R7; are given by the following equations.
R,- = ————7” — "H (3.22)
772' + tit—1
2 .
11,-: —32—— (3.23)
771' + 77i—1

38

3.1 .3 Scattering Matrix

This wave transmission matrix is exceptionally useful because the matrices can simply
be cascaded for each additional layer. Another useful property is the ability to convert
this final matrix into other useful quantities, such as the scattering matrix, impedance
matrix, and admittance matrix [40]. The scattering matrix is of particular interest
to this work because it relates the backward traveling waves to the forward traveling
waves as seen in (3.24). Besides, scattering parameters are the measured quantity
at practice. With the calculated S-parameters, the effective permittivity can be ex-
tracted using the same procedure as in section 2.4.3 for the stripline characterization

technique.
bl = 511 512 a1 (3 24)
b2 321 322 (12

The scattering matrix is calculated from the wave matrix using (3.25) [40].

511 = 411 + 412/770 - 421110 - A22 (3253)
411+ A12/770 + A2100 + A22
512 = 2(411422 - 412421) (3251))
411+ A12/770 + A21710 + A22
2
S : 3.25C
21 A11+412/170 +A21770 +422 ( )

-411 + A12/770 - A21710 + A22
A11+ Alz/no + A21770 + A22

 

 

 

 

(3.25.1)

For the two-port network used in this model the wave transmission matrix for each

layer can be reduced to the form seen in (3.26) [40].

A11,i = cos(62-) (3.26a)
4122' = inosinlt’z') (3263b)
2121,,- = jnO-lsinwi) (3.26c)
A21,- = cos(6z-) (3.26d)

39

3.2 Analysis and Results

Using the methods described in the previous section, the effective permittivity of
a composite comprised of layered dielectrics is investigated. More specifically, the
composition consists of two dissimilar dielectrics with differing relative permittivities
alternating such that the first layer is the same as the final layer. This last requirement
allows the system to be symmetric, a necessity of the inversion method used herein.

As the number of layers is increased the total volume of the mixture being sim-
ulated is held constant, as is the volume of each of the two dielectrics. Therefore,
different volume fractions, dielectric contrast ratios and number of layers will be a11-
alyzed in order to interpret their consequence on the effective permittivity of the
composite. The fixed thickness of the composite being simulated is )1 / 50 and is suffi-
ciently small enough to realize homogenization.

Figure 3.6 gives the effective permittivity for the first case in which the volume
fraction of dielectric A is 0.75 and has the lower value of permittivity. As seen in
Figure 3.2, the scattering parameters are calculated when dielectric A is on the outside
(e.g., facing the impinging waves), then again when dielectric B is on the outside.
For the situation when dielectric A is on the outside, the eflective permittivity is
seen to start at a lower value and begin to rise as the number of layers is increased
and eventually saturate to the dashed line calculated using the linear law which is
discussed below. The later situation when dielectric B is the outer later results in the
effective permittivity beginning at a higher value and lowering to the same dashed
line. For all cases in which layer A has the lower permittivity in the dielectric contrast
ratio, this trend is repeated. Figure 3.7 is the same analysis as in the previous case,
however the volume fraction of material A is 0.5, and finally Figure 3.8 illustrates the

same effects with the volume fraction of material A equal to 0.25. In each of these

40

cases, six different dielectric contrast ratios are calculated ranging from 1 : 4 up to
1 : 49.

As mentioned earlier, the dashed lines in each figure are calculated using the linear
law by (3.27). This is the simplest formula for determining the effective permittivity
of a two—phase mixture where subscript A implies dielectric A, and subscript B implies

dielectric B, while the volume fraction of material A is denoted by f A, etc.

Eeff=fA€A+fBEB (3.27)

This straightforward equation assumes total homogenization and the effective per-
mittivity is purely a function of the dielectric constants and their volume fractions.
Whether the outer layer is the high or low dielectric contrast the effective permit-
tivity will eventually merge towards the value given by (3.27) as the number or layers
is increased. This implies that the effects of the layering is negligible once the layers
become extremely thin. Figure 3.6 shows that the effective permittivity can be tuned
to any value between 1.65 and 16.6, and Figure 3.8 yields an effective permittivity
range from 2.38 to 48.6. Table 3.1 summarizes these ranges for each of the dielec-
tric contrast ratios and volume fractions. These results allow for a potential design
platform for planar layered composites, and demonstrate how the different number
of layers within a composite can be used to control or tune the effective permittivity
to better fit some particular electromagnetic application. A possible reason for the
change in permittivity as the number of layers is altered is due to multiple reflections
within the composite. A material with multiple interfaces has the potential to trap
more power and hence have a greater effective permittivity at the outer edges. Or
perhaps a composite material can be designed with a certain number of layers as to
allow for greater propagation through the material or to reduce trapped power and

lower the effective permittivity. The analysis performed in this chapter for lossless

41

dielectric composites based on contrast ratios and volume fractions has demonstrated
the effects of layering. However, in real-world applications the materials being used
will have some form of loss associated with the permittivity, permeability or both.
Just as the number of layers and multiple reflections can help control the effective
permittivity, they can also effect the loss of the composite. As a wave propagates
through a lossy material and is bounced back and forth, the wave will have greater
attenuation after traveling through the entire composite. This results in a greater
negative imaginary portion of permittivity. N ow that the effects of layering are bet-
ter understood, it is imperative that the model be extended to include layers of actual
materials since many magnetic materials have a tensor permeability and the goal is
ultimately to design and control a composite’s permittivity and permeability. This is

done in Chapter 4 and includes loss in the analysis.

42

 

 

 

 

 

Figure 3.1. Geometry for layered materials.

43

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i—d—s ie—d—s i——d——i

Figure 3.2. Illustration of the analysis on layered materials when layers A and B are
swapped.

44

(”Olurl 9 gogrl ) (1‘10ng ’ 808r2)

\
\
_.. \
E1W\

W177

 

Film

Region 1 \ Region 2

 

Figure 3.3. Incident, reflected, and transmitted waves for an interface between 2
isotropic materials.

45

 

W
b,[ 1‘11 \ R,b,

WZ&_\/ W b2 2
Region 1 \ Region 2

 

Figure 3.4. Reflection and transmission of plane wave amplitude coefficients.

46

NQ
Q
203

 

l
El

3%

t.

 

NV

___.__"i§-§f

-
---‘----

 

 

 

 

,2.

Figure 3.5. Reflection and transmission of plane wave amplitude coefficients for mul-
tiple layers.

47

   

18

 

 

Linear

  

16
14—

?~
12 ~c~

 

10—

 

 

 

 

3 5 7 9 1 1 13 15 17 19
Number of layers

Figure 3.6. Case 1: Layer A has volume fraction 0.75.

48

 

 

 

 

 

 

5 7 9 11 13 15 17
Number of layers

Figure 3.7. Case 2: Layer A has volume fraction 0.50.

49

 

 

 

 

 

 

 

 

 

:Z: : f:.':i::=::.':.':.':.'nmmmmmusus In Human-rum)
0 -

'53 5 7 9 11 13 15 17 19

Number of layers

 

Figure 3.8. Case 3: Layer A has volume fraction 0.25.

50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dielectric Contrast Ratio
fA 1:4 1:9 1:16 1:25 1:36 1:49 Total
0.75 1.65-1.82 2.5-3.5 3.7-5.8 5.2-8.8 71-124 94-166 1.65-16.6
0.5 2-3 3.7—6.3 605-1095 9.1-16.9 12.95-24 17.5-32.5 2-32.5
0.25 238-42 485-91 8.4-16.1 13-25 1865-358 25.4-48.6 238-486

 

Table 3.1. Tunable Effective Permittivity range summary for isotropic layered com-

posites.

51

 

CHAPTER 4

ANISOTROPIC LAYERED COMPOSITES

The purpose of this chapter is to demonstrate a method of calculating the scatter-
ing parameters of a planar, layered, anisotropic composite material. From these
S-parameters, the complex effective permittivity and permeability of the mixture is
extracted using a complex 2D Newton’s root searching algorithm as in section 2.4.3.
The S-parameters are calculated at the front and back interfaces of the layered com-
posite in order to effectively homogenize the composite. This is accomplished using
state vector and state equation formulations stemming from Maxwell’s equations
[41, 42]. In Chapter 3 the scattering parameters for isotropic dielectric layers were
determined using the method of wave matrices. This demonstrated how the num-
ber of layers, as well as the volume fractions and dielectric constants of each layer,
could be manipulated in order to achieve an effective permittivity that better suits
the application. Also, the greater the dielectric contrast ratio, the wider the range of
tunable effective permittivities will be accomplished.

In this chapter, the layered composite model is extended to include anisotropic
layers such as ferrimagnetic materials which can be comprised of ferrites or magnetic
garnets, for example. The objective is to design a layered mixture that exhibits mag-
netic behavior and achieve a tunable permeability. Ferrites are chosen because unlike
ferromagnetic compounds, ferrites or ferrimagnetic materials do not have high resis-
tivity but has anisotropy induced by an external applied field [38]. In ferrimagnetics,
the individual elements of the permeability tensor can be controlled using a static
magnetic bias field (e.g., the field from a permanent magnet) [40]. The derivation of
the reflection and transmission coefficients are introduced as well as various examples

including: narrow line-width ferrimagnetic composites, wide line-width ferrimagnetic

52

composites, and double negative materials.

4.1 Background and Derivation

4.1 .1 Geometry

The geometry of the composite in this derivation is given by Figure 4.1. Just as in
chapter 3, the material extends to infinity in the x and y directions with a finite
thicknes in z. The configuration consists of three regions where region 2 contains
the anisotropy in the permeability tensor, permittivity tensor, or both. In all but
one case in this chapter, both regions 1 and 3 are considered to be free space. The
interface between regions 1 and 2 is located at z = 0 and between regions 2 and 3
at z = at. First, the reflection and transmission from an anisotropic slab is derived.
Next, the slab is partitioned into layers such that the fields at the n — 1 interface will
be calculated in terms of the fields at the n interface. Finally, this is continued until
the fields at the z = 0 interface are known in terms of the fields at the z = d interface

as seen in the preceeding chapter.
4.1.2 State Vector and State Equation Formulation

To start, the two bulk Maxwell’s equations, more specifically the two curl equations,
Faraday’s law and Ampere’s law, will be separated into their six different components
where both permittivity and permeability are rank 2 tensors. The notation used to
represent a rank 2 tensor, or matrix, will be square brackets. A rank 1 tensor, or
column vector, will be denoted by curly braces. Spatial vectors will remain using the

over-right-arrow notation.

 

v x 75’ = —jw[p]- Ti (4.1a)
5yEz - 5.2133; l #:rx Hwy #5132 Ha:
52572: — (5sz = —jw nyg; Hyy #yz Hy (4.1b)
_ SUEZ! - 69533 _ L #22: My #22 J _ Hz .

 

 

 

 

 

53

v x fi = jw[e]-E (4.2a)

P

 

 

 

 

 

 

_ .1 - 7
63/112 '— 63H?! €173; Exy 5533 E1;
62H]; — (5tz : jw Eyqj Eyy Eyz Ey (42b)
_ 5$Hy - (5ny 3 L 52.1: Ezy Ezz E2 1

These equations can be separated into transverse, (4.3),(4.4), and longitudinal com-

ponents (4.5).

(52132: = —jw (Hyerr + #yyHy) - jwflysz + 5rEz (4-33)

In (4.3), the left side of the equation has the normal derivatives of the transverse
electric fields and the right side contains all three components of the magnetic field
along with the transverse derivatives of the longitudinal components of the electric

field. The same is true of (4.4), except the magnetic and electric fields are transposed.

dsz = jw (nyEx '1' Enyy) ‘l" jWEyzEz + (53sz (4.4a)

63Hy = ~ju) (8513:3131; + EltyEy) '- jWEszz + ($sz (4.4b)

Finally, the longitudinal fields in (4.5) can be inserted into both (4.3) and (4.4)
reducing the six components of Maxwell’s equations into four components in terms

of the transverse electric and magnetic fields only.

—1

 

 

H = , 6 E —6 E —— H + g H 4.5a
2 294122” 9 y 3:) ”22,0122: :1: I1 y y) ( )
1 1
green Ezz

. u -—+ —) —) A —) a .
The transverse electric field, Et( r ) = Eg:( r )x + Ey( r )y, and transverse magnetlc

field, HA?) = Hg;(7’)it + Hy(?)g, can be represented in a form in which the x, y

54

and z dependences are separated.

75? = Eiae‘j’ifl‘jkyy (499

1?: = H;(z)e—jk$$—jkyy (4.6b)

The reprersentation given by (4.6), illustrates how the dependence in the x and y
directions allows phase matching to be achieved at the interfaces and implies that the
transverse fields depend on their position in z. Now (4.3) and (4.4) can be put into

the matrix form seen in (4.7).

d 3i 7
21; _, = [Al' _, (4-7)
Ht Ht

This is called the state equation where the matrix [A] is a 4x4 matrix containing the
operations on the transverse components of the fields. The column vector containing
the longitudinal dependence of the transverse fields can be represented by the state
vector, {V}, in (4.8).

Exfz) '
{V} = 133/(z) (4.8)
1122(2)

L III/(Z) -

The state equation can now be represented in its compact form given by (4.9).

 

 

is {V} = [A1 - {V} (4.9)

The z dependence from the column vector in (4.8) can be separated out and repre-

sented by (4.10).

55

{V}={\Il}e)‘2= y 602 (4.10)

 

 

l. 90 -
Now the z derivative from the state equation in (4.9) can be replaced by A which

yields the eigenvalue equation in (4.11) where [I] is the identity matrix.

([A] — A [11> - {w} = o (4.11)

The calculated eigenvalue in the representation of {V} in (4.10) denotes the propa-
gation phase and attenuation of a plane wave through region 2. The elements in [A]
are given by (4.12) in its most general form. It should be noted that in this analysis,
the plane wave is assumed to have normal incidence upon the layered composite, this
implies that km and Icy in (4.12) are zero reducing the complexity of [A]. This eigen-
value equation will solve for the fields in an anisotropic region assuming plane wave
propagation. The next section derives a solution for the fields at the interfaces of an
anisotropic slab of thickness d. This method is similar to the wave matrix method in
chapter 3 because each layer within region 2 will have a solved matrix that can be
cascaded to the solved matrices of the other layers. A non-trivial solution to (4.11)
can be obtained by setting the determinant det ([A] — /\ [1]) equal to zero and solving
the resulting characteristic polynomial of [A], for the unknown eigenvalues A. Once
the eigenvalues are determined, they can be used in (4.11) to find the corresponding

eigenvectors.

56

 

jkyltyz +jk2°521r
#22 522
‘J'k'yflyz + jkaf€ry

 

 

 

 

 

 

 

 

 

22 522
_ ILL) L z - k k
_Jw#y$+3 ly ,Uccz _] 1: y
#22 01522:
. jwflyzflzy jkg
—]w/.Lyy + —— + -—
#22. 60822
"jkyflarz + jkyEza:
#22 522
jkxflrrz + jky€zy
#22 522
. - 2
. w Jk
lel-m: _ J szflzzr _ y
#22 szz
. jwllsrz/tzy jkyka:
Jammy — —
#22: 00522
. 'we 35; , 'k ,k
JWny—J 3} 1+3 2‘ y
522 “#22
.1 , 2
. JWE z€z k
JWEyy— y 3! _ J :1:
Ezz wflzz
JkyEyZ + jkcr/izx
522 #272
‘jkeryz + jkmfizry
522 #22
. - 2
. CUE €z_ Jk
522 1«41.22
. jWExzez jk k1:
‘JEJJy + y ‘- y
522 wflzz
*jkyEafz + jky/lzrr
522 #22
“£5 + jkyflzy
522 #22

57

(4.12a)
(4.12b)

(4.12e)

(4.12a)
(4.12e)

(4.121)

(4.12g)
(4.12h)
(4.121)
(4.12))
(4.12k)

(4.121)

(4.12m)
(4.12n)
(4.120)

(4.12p)

4.1.3 Transition, 'IYansmission, and Reflection Matrices

The transition matrix is used to describe the relationship between the fields at the
z = 0 interface to the fields at the z = d interface as illustrated in Figure 4.1a.
This matrix is used to describe the transition of the fields through region 2 in terms
of material properties and layer thickness. The transition matrix, [B] is given in

mathematical form by (4.13)

{V(z = 0)} = [B] ' {V(Z = d)} (4-13)

and is a 4x4 matrix with 2x2 submatrices given by [b] in (4.14).

[B] = lbll lbzl (41,)

[b3] PM]

Also, the reflected and transmitted electric fields can be represented by (4.15) which

introduces 2x2 reflection and transmission matrices for the fields at the interfaces

2 = 0 and z = d.
13,? (0) = [R] . 51(0) (4.154)
13,5 (d) = [T] - E§(0) (4.15b)

The subscript ”t” implies that these are the transverse fields in the a: and y direc-
tions, while the superscripts ”r”, ” i”, and ”t” represent the reflected, incident, and
transmitted fields, respectively. The total field in region 1 are given by the superpo-
sition of the incident and reflected fields while the total field in region 3 consist of the

transmitted field only. Therefore the transition matrix equation given in (4.13) can

58

be written by (4.17).

Eif + E; = ["‘(z s 0) (4.163)
E; = Effie; _>_ d) (4.16b)
H; + H; = Htwt(z S 0) (4.16c)
Hf = Hide. 2 d) (4.16d)
_ i+r l - t ]
Ea: (0) Edd)
EHT 0 b b Et d
g () ___ l1] [2] y() (417)
Harm) [’23] Val Had)
. iii/+70) - LHW) _

 

 

 

 

This can be multiplied out of matrix form to give the following two equations for the

electric and magnetic fields.

Ei<o>+Ei<01=ibflfi<d>+ib21f1i<d> (4181
111(0) + 17%) = [b3] . E; (d) + [04] . 111(3) (4.19)

In order to reduce the number of unknowns in the above equations, the transverse
magnetic fields are represented in terms of the corresponding transverse electric field

components through wave impedance matrices.

7(0) = [21] - HAG) (4.20:1)
7(0) = - [le - HAD) (4.201))
Ei(d) = [Z3] - HEW) (4.20c)

59

0 77
[213] = 1’3 (4.21a)
-721,3 0
n
U13 2 21—11% (4.21b)
’ 1,3

The wave impedances, 7713, are functions of both permittivity and permeability of
regions 1 or 3. Using (4.15) and the above impedance matrices, (4.18) can be written
entirely in terms of the incident electric field. The incident field term can be normal-
ized allowing for the solution of the reflection matrix in terms of the transmission,

transition, and impedance matrices.

—+ —)

Ei<o>+iRiEi<o> = [bli-iTi-Ei<o>+ibzi-[zfl-iTi-Eiw) (4.2241)

[R]

lbll - m + 021- [23—1] 4 [Ti — [I] (4.221»)

The same procedure can be aplied to (4.19). Once the incident electric field term has
been normalized, the reflection matrix from above can be inserted leaving only the

transmission matrix as the unknown.

Fifi»- [le—l - 13113730): [Z1]-1 = [’93] - [Tl ' E30) + [174] ' [Tl-Em [331—1
(4.23)

[Tl (“91 + (1.4112314) = 0211" - ([611 - [T] + [b2] 1231-1 - lTl — 111) - 1211—1)

(4.24)

The transmission matrix is given in terms of the impedance and transition matrices

in (4.25).

[T] = 2[Z3l-(lb1l-lz3l + lb2l + [12311211423] + lb4l'lZ1l)_1 (4-25)

60

This transmission matrix is then inserted into the equation for the reflection matrix

in (4.2%).

[R1 =(lb11+ [2421-1231—5214]

- ([1211 1231 + [421 + [b3]- (21] - [231 + [44 «an—1 _(11

Finally, the reflection matrix is given by the following equation in terms of impedance

and transition matrices.

[Rl =(lbll'l23l + lb2l + [193112111231 + [1’41 121])

((1211 .1231 + [421 + [b31121] 1231 + [4411211)‘1

These reflection and transmission matrices have been derived for a homogeneous
slab of anisotropic material as shown in Figure 4.1a, and are functions of impedance
matrices which describe the isotropic surrounding regions and the transition matrix

which describes the slab.
4.1.4 Layered Media and Local Transition Matrix

Reflection and transmission have now been derived for an anisotropic slab in terms
of impedance and transition matrices for which the latter has yet to be solved. The
transition matrix represents the propagation inside region 2 between interfaces at
z --= O and z = d. In this section, region 2 is no longer an anisotropic slab, but a
layered anisotropic material. Therefore a local transition matrix, [Bn] is introduced,
which represents the fields at the interface z = zn_1 to the fields at z = zn within
region 2 where those interfaces surround the nth layer. This is illustrated in Figure

4.1b and mathematically given by (4.26).

{Vn—i} = [Bln ° {Vn} (4.26)

61

This implies that the eigenvalue equation derived in section 4.1.2 will need to be

solved in each individual nth layer.

([472] - Anlll)'{‘1’n} = 0 (427)

Once the eigenvalues, An, and corresponding eigenvectors, {\Iln}, are known for each
layer the state vector is also known and given by the following representation with

coefficients {C n } .

{vn} = {444} 41"an- {Cn} = {‘I’n} -f([D<zn)1> - {on} (4.28)
F eAlzn 0 0 0 l
0 e422" 0 0
f ([077, (anl) = A (429)
0 0 e 32” 0
0 0 0 6442n

 

 

_ d

The function f ([D(zn)]) is an exponential function of the diagonal eigenvalue matrix

above and has the property below.

f ([1971 (Zn—1 + anl) = f([Dn(Zn—1)l)'f(an(Zn)l) (430)

Using this relationship, the fields at the interface 2 = 271—1 can be written by (4.31).

{141-1} ={wn}-f([Dn(zn_1— 40]) - {ail} - {wn}-f<[Dn(zn>1> - {on}
(4.31)
Utilizing (4.28) for {Vn} into (4.31) for {Vn—l} and comparing to (4.26), an expres-

sion for the local transition matrix is obtained.
anl =f‘1’nl'f([Dn(Zn_1 — Zn)l)’{‘1’n}_1 (4-32)

62

Once all of the local transition matrices have been solved for each layer, they can
be cascaded together in order to obtain the total transition matrix which relates the
fields at the z = 0 interface to those at the z = d interface and accounts for the

interaction of the fields within the anisotropic region.

N
[B] = [1an] (4.33)
n=1

where N denotes the number of layers. Using the total transition matrix and the
impedance matrices for the surrounding regions, both reflection and transmission can
be calculated and used to extract the complex permittivity and permeability of the

layered composite as done in section 2.4.3.

4.2 Verification of Anisotropic Formulation

The solution presented here for reflection and transmission from a planar anisotropic
material is an analytical solution. However it is still useful to check the computer
program for errors either in the formulation or in the coding process. The following
three cases will verify the code and solution by comparison to propagation through
free space, reflection from PEC, and scattering from an iostrpic layered composite by

which it can be compared to the code from chapter 3.
4.2.1 Transmission through Free Space

The first check of the layered anisotropic formulation will be computed for the case
shown in Figure 4.2 where regions 1, 2, and 3 are all free space. Therefore #2- : p0
and 52- = 50 for i=1, 2, or 3. As seen in Figure 4.3a the result is that the magnitude of
ISlll is O and |521| is 1, which implies total transmission as expected for free space.
Phase for 5'21 at z=d is given in Figure 4.2b and demonstrates phase wrap-around
occuring at 6GHz. This observation will become useful for verifcation in the following

C888.

63

4.2.2 Reflection from PEC space

The next check is performed for free space in regions 1 and 2 and perfect electric
conductor (PEC) in region 3 as seen in Figure 4.4. The magnitude of the scattering
parameters is shown in Figure 4.5a and illustrates total reflection, ISHI = 1 and
ISZII = 0 as expected from PEC. Also, Figure 4.5b shows the phase of 311 at z = 0
after the wave has traveled to z = d and back again. Phase wrap around occurs at
3GHz which is half the frequency location for the phase of .321 in the previous section
for the case of free space. As distance traveled is increased, the phase will take on
steeper slopes. In this situation, the wave has traveled twice the distance to get from
the PEC and back as opposed to the previous case and hence the slope of 5'11 is
twice as steep as for 321 in free space and therefore wrap—around occurse at half the
frequency. This helps to further verify that the code and formulation are working
properly.

4.2.3 Comparison with Isotropic Layers

The final check in the verification process is to simulate a composite of layered
isotropic materials and compare the results to the same simulation using the wave
matrix method described in chapter 3. The composite has 15 layers with alternating
A and B materials each having a thickness of 100nm. Material A has properties
of free space while material B is non-magnetic (#r = 1) with relative permittivity
of 25. Figure 4.6 shows the magnitude and phase of the scattering parameters for
both the isotropic and anisotropic analysis methods, while Figure 4.7 illustrates the
extracted permittivity and permeability values. Both methods yield the same val—
ues for S-parameters and constitutive parameters providing further verification of the

anisotropic formulation.

64

4.3 Ferrimagnetic Materials with Anisotropic Permeability
4.3.1 Permeability Tensor

The purpose for including anisotropic layers is to include ferrimagnetic materials in
order to tune the effective permeability as well as permittivity. Narrow line-width
ferrites provide a good low-loss magnetic material for which the anisotropy and perme-
ability tensor elements can be controlled using a bias magnetic field. The permeability

tensor is given by (4.34) for a bias field in the x-direction [40].

 

 

' 1
no 0 0
[,u] = 0 I) j); i -— bias (4.34)
O ——jn p

The elements in the permeability tensor are calculated by (4.35) where w = 27r f is

the operating frequency and p0 = 47r X 10—7 is the permeability of free space.

w (A)
u = #0 (1+ ——20—Ln-§) (4.35a)
(4)0 — w
ww
(4)0 "" (4)

Also, “’0 = #OVHO and cum = rim/Ms where H0 is the strength of the internal bias
field in oersteds and M3 = 47rM3 is the saturation magnetization in gauss. The ratio
of the spin magnetic moment to the spin angular momentum for an electron is called
the gyromagnetic ratio and is given by 7 = 1.759 x 1011 C/ Kg [40]. The larmor
frequency can be calculated by f0 = Lao/2w = (2.8M H z/oersted)(H0 oersted) and
fm = wm/27r = (2.8M H z/ oersted)(47rM3 gauss). One more thing to include in the
calculation of larmor frequency is the linewidth, AH in order to account for magnetic

losses inherent in magnetic materials. This is done by using (4.36), giving a complex

65

valued larmor frequency.

.44 ’7AH _ 2.84MH2 oersted AH oersted

 

(4.36)

4.3.2 Demagnetization Factor

The internal bias field, H 0 is not only dependent on the strength of the external static
bias field Ha, but also the direction of the field due to boundary conditions at the
surface of the ferrite [40]. Continuity of the tangential magnetic fields is an example
of such boundary conditions. For a thin plate or layer, an internal bias field will be
equal to the external bias field if it is in the direction parallel to the face of the layer.
However, if Ha is in the direction normal to that face, then the internal bias field is

reduced by a designated fraction of saturation magnetization.

H0 = Ha — NMS (4.37)

This fraction is called the demagnetization factor, N. For thin plates, the demagneti-
zation factor of the normal bias field is 1, and hence the internal field is reduced by
the full saturation magnetization. The demagnetization factor can be represented by
three components, N55, Ny, and N 2, such that their sum is equal to 1. These three

components represent the demagnetization factors in the x, y, and z-directions.

N3; + Ny + NZ = 1 (4.38)

The components in (4.38) are illustrated in Figure 4.8 for thin plates, thin rods, and

spheres. The three components of the internal magnetic bias field can be written as

in (4.39).

66

H01. = Hag; — NxMS (4.398.)

H02: 1' Hag — szl’fs (4.39C)

The choice in this analysis for a bias field in the x-dircction was motivated by the
fact that the demagnetization factor is zero in this direction and the internal bias

field is unaffected by the saturation magnetization.
4.3.3 Bias Transverse to Direction of Propagation

Assume the geometry shown in Figure 4.1a where region 2 is a ferrimagnetic material
with no variance in the x and y directions and a x—directed bias field. The incident
wave is a plane wave traveling in the zrdirection given by (4.40) where both EU and

__§
H 0 are constant vectors describing the transverse dependence of the fields.

~ Ora—jkzz (4.40a)

ml D11

73’ =
if = 043—ij2 (4.40b)

Then the six components of the two curl Maxwell’s equations are given by the

following:

67

jszy = —jprHa; (4.41a)

—jszx —_- —jw(pHy+jnHz) (4.41b)
0 = —jw(—jrcHy+/4Hz) (4.41c)
ijHy = jwsEg; (4.41d)
—jsz$ = jwsEy (4.41e)
0 = jwsEz (4.411?)

Solving (4.41d,e) for H3; and Hy and inserting them into (4.41a,b,c), then (4.41a,b)

can be written in terms of Em and Ey only as given by (4.42).

‘ I435, = wzposEy (4.42a)

,u (kg — (4)2415) Ea; = —w2552Ex (4.42b)

It is evident that there are two possible solutions for the wavenumber, kg, in (4.42).

The first solution from (4.42a) which can only be obtained assuming E3; = O, is given

by (4.43).
kg = am (4.43)

This wavenumber represents that of an ordinary wave since it is unaffected by the
magnetization. Therefore the permeability is that of free space and exhibits no mag-
netic behavior. This is due to the fact that the magnetic field components perpendic-
ular to the direction of bias are zero, hence Hy = H z = O for x directed bias fields.
The other solution for k; occurs when Ey = O for (4.42b) and is given by (4.44),

where #6 is given by (4.45).
kg = (AA/[158 (4.44)

68

x4e = —— (4.45)

This wavenumber and effective permeability are representative of the
extraordinary wave which is affected by the magnetization and exhibits mag-
netic behavior. This permeability is dependent on the values, w, “’0’ and cum
introduced earlier in section 4.3.1. Thus if a wave has electric polarization in the
y direction the wave is ordinary, while an x-directed electric polarized wave will
be extraordinary. This effect is termed birefringence [40]. Figure 4.9 illustrates
a plot of normalized effective permeability for an extraordinary wave as shown in
(4.45). In Figure 4.9 the wavey line indicates the ferrimagnetic resonance (FMR)

frequency which separates the region where the real part of permeability is positive

2

(a > 142) from the region where the real part of permeability is negative (#2 < 142).

Thus the effective wavenumber for an extraordinary wave, kg = (42W, will show
propagating modes for frequencies less than FMR. However, assuming negligible
loss, the effective wavenumber will become purely imaginary for frequencies above
FMR, and the waves will become evanescent. This is illustrated in Figure 4.10 for
the effective wavenumber normalized by the free space wavenumber. A final note on
F MR, as the static bias field is increased or decreased, the F MR frequency will also
increase or decrease. This is a very useful property of ferrimagnetic materials since

the permeability can be controlled.

4.4 Layered Composite of Ferrimagnetic and Dielectric Materials

This section analyzes the effects of layering on a composite consisting of alternating
biased ferrimagnetic materials with anisotropic permeability and isotropic dielectric
layers. The analysis follows the same approach as in section 3.2. The effective per-
mittivity and permeability will be calculated for three different volume fractions and

plotted versus number of layers in order to observe the effects of increased layering

69

and volume fraction. The ferrimagnetic material is a magnetic garnet (Yttrium Iron
Garnet, TYIG), with 47rM3 = 2050 gauss, AH = 5 0e, Er = 14 — 30.002 and is
biased by a static magnetic field in the x-direction with a field strength of 250 Oe.
The YIG material properties are taken from a Trans-tech data sheet . The dielectric
layers are Teflon with the properties 5r = 2.08 — 30.001 and hr =2 1. The frequency
was chosen to be lGHz such that it is below F MR with a real permeability of approx-
imately 10 and the material thickness is fixed to 2mm while the number of layers is
increased for three different volume fractions. The choice of thickness was to ensure
the composite was much smaller than a wavelength for homogenization. Using the
formulation above, the reflection and transmission coefficients are calculated for the
layered anisotropic region so that the effective permittivity and permeability may be
extracted. Figure 4.11 illustrates the effective permeability of the layered composite
for ferrite volume fractions f F = 0.75, 0.5, and 0.25, for the case when the ferrite
is the outside layer and also when the dielectric is the outside layer. Figure 4.12
illustrates the same analysis for the effective permittivity. As observed in 3.2, the
permittivity and permeability can be controlled by the number of layers and volume
fractions such that the tunable permeability range is 2.5 to 9.4 and 3.8 to 12.2 for
tunable permittivity. This is summarized in Table 4.1. In conclusion, the effects of
layering demonstrated in Chapter 3 for tunable permittivity based on volume fraction
and contrast ratio have been illustrated here for both permittivity and permeability
for physical materials that can be realizably manufactured into this form. Finally, a
method for analyzing a structure with anisotropic materials has been presented and
confirmed. This tool is used in a further analysis in the simulation of a layered comp—
soite which yields negative effective permittivity and permeability. This is described

in the following section.

70

4.5 Simulation of Double Negative (DNG) Material

Double Negative (DNG) materials offer the potential for realizing interesting effects
when used in a variety of applications. A DNG material is a composite material,
or mixture, whether ordered or unordered, of two or more constitutive materials
yielding effective properties significantly different than the individual components.
The majority of these materials to date are made by embedding periodic metallic
inclusions in a polymer matrix (sometimes literally by hand assembly using laminated
boards [43]). There is some controversy regarding the behavior of such materials,
especially on the scale of the inclusions themselves; however, there is clear macroscopic
experimental evidence that some interesting behavior can be predicted if the two
constitutive properties are simultaneously negative. One of the main subjects of
controversy involves the use of periodic inclusions and whether such a composite
material really can be interpreted as being DNG [44].

In this section, the use of non-periodic solutions to achieve both negative permit-
tivity and permeability is discussed using the method derived in this chapter. The
hypothetical composite will be formed by a bilayer stack comprised of alternating
layers of narrow linewidth ferrite and suitable plasma tubes. The miniature plasma
tubes can be created using microwave applicators [45] and can exhibit the behavior
of negative permittivity for an operating frequency below the plasma frequency. Fi-
nally, the dimensions of the layers are key to homogenization: the effective materials

properties must result from a large number of layers within a single wavelength.
4.5.1 Plasma as ENG Layer

Plasmas operating below the plasma frequency (w < wp) will exhibit a negative

effective permittivity as is evident by (4.46) for the effective permittivity of a plasma.

w?)
Eeff=€0 l-E (4.46)

71

Typically, the plasma frequency can be calculated by (4.47), which is a function of
number density, n, as well as charge of an electron, e = 1.6 x 10_19, mass of an

electron, m = 9.31 x 10‘31, and permittivity of free space.

.62 i
42,, = (___) (4.47)

60m

Once plasma is immersed in a bias field, Ha, both the plasma frequency and effective
permittivity will be altered. The plasma frequency is replaced with the upper hybrid
frequency, 4.2),, which is a function of both plasma frequency and cyclotron frequency
[46]. The cyclotron frequency is dependent on the bias magnetic flux strength, B =

HOHa where the permeability of the plasma is taken to be that of free space.

B
we = e— (4.48)
m

Using (4.48) along with the plasma frequency the upper hybrid frequency is calculated

and inserted into the equation for effective permittivity of plasma biased by a magnetic

field.

. 2 2 2

0%
Eeff = 50 (I — 7) (4.491))

(4)

(4.49a)

This effectively negative dielectric constant is illustrated in Figure 4.13 for two differ-
ent number densities and two different bias field strengths. The upper hybrid plasma
frequencies are also displayed for the different cases from left to right as vertical dashed
lines. The effective permittivity is negative for frequencies to left of the dashed lines
for the corresponding cases. As seen in Figure 4.13, increasing the number density

or bias field strength will shift wh higher in frequency. This will be useful for con-

72

trolling the plasma permittivity when trying to simulate a double negative material.
The effective permittivity of biased plasma shown in (4.49b) and Figure 4.13 is an
approximation that allows representation of the plasma as an isotropic medium. This
is useful for quickly determining the number density and bias field strength necessary
to achieve the desired plasma properties. However, the permittivity of magnetized
plasma is anisotropic with the permittivity tensor shown in (4.50) for a bias field in

the x-direction.

 

 

53 0 0
[E] = 0 51 3'52 If; - bias (4.50)
0 —j52 51

The elements of the permitivity tensor are given by (4.51).

2
81 = 80 (I—(jg—C-fi) (4.513)

w w 2 we no
82 = 80 (( 1p: ()wc(/w)/2 )) (4.51b)

w?»
53 = 80 —;§ (4.510)

Ideally the plasma layer would be solely plasma, however this is not always practical.

 

Therefore, the plasma layer can be implemented using an array of tightly packed
plasma tubes or one wide plasma tube [45]. Preferably the tube or tubes would be
rectangular in shape to improve packing. The tubes are generally made with quartz,
which is used in this analysis and when referring to the plasma layer it implies that
each layer is surrounded by a thin layer of quartz. Within each plasma layer, the
quartz between adjacent plasma tubes has been neglected since it represents only

a small portion of the layer volume and contributes little to the effective material

73

properties.
4.5.2 YIG as MNG Layer

The other material in this double negative composite is a narrow linewidth magnetic
garnet operated near ferrimagnetic resonance (FMR). Pfequencies immediately above
F MR will exhibit a negative effective permeability over a narrow frequency band as

can be seen in Figure 4.9.
4.5.3 DNG Layered Composite

Now that the individual plasma and ferrite layers have been introduced, the lay-
ered composite with negative effective permittivity and permeability can be mod-
eled through the use of the methods described in section 4.1. One layer con-
sists of the tightly packed rectangular plasma tubes with electron number density
n = 8 x 1017 (m-3) . The thickness of each plasma layer is 0.5mm and surrounding
the plasma layer is 0.1mm layers of quartz on both sides. YIG is again chosen as the
narrow line-width ferrimagnetic material with layer thickness of 0.4mm and material
properties described in section 4.3 The x-directed static bias field is chosen to be
1000e such that the plasma permittivity and the ferrite permeability will be simul-
taneously negative as shown in Figure 4.14. These values were calculated using the
effective permeability and permittivity equations given in (4.45) and (4.49b). Figure
4.15a illustrates the configuration for the first case to be analyzed. This case has the
ferrimagnetic layer first followed by the quartz-plasma-quartz layer next and contin-
ues for a total number of seven plasma and ferrimagnetic layers. As expected, this
composite yields effective constitutive parameters of a double negative material shown
in Figure 4.16. Another observation is the non-trivial loss this material exhibits, ren-
dering it so lossy it will not be of very much practical use except as a narrow-band
absorber. The next case interchanges the ferrimagnetic and plasma layers as shown

in Figure 4.15b and the permittivity and permeability are displayed in Figure 4.17.

74

This also yields double negative material properties along with the resulting high
loss. An interesting result in this latter case is how the real and imaginary parts of
permittivity and permeability are approximately the same around 2-2.1GHz. This
implies that this material has all the same effects as the double negative composite
however the wave impedance which is the square root of the ratio of permeability to
permittivity is nearly 1, which is that of free space.

This section discussed using the formulation in section 4.1 to synthesize a DN G
material through the use of multi-layer anisotropic structures. By using a layered
material with alternating narrow linewidth ferrimagnetic and plasma sub-wavelength
layers emerged in a static bias magnetic field, an effective double negative material
can be achieved if operated at the proper frequencies. This multilayer structure
avoided the use of transverse periodic arrays typically used to form a DN G material.
Use of wave matrices allowed for determination of the propagator matrix, which was
used to calculate the reflection and transmission matrices. This gave the 521 and
311 from which the complex permittivity and permeability of the composite could be
extracted using a two dimensional traditional root searching algorithm and was shown
to exhibit negative effective homogenized permittivity and permeability. There has
been much research into DNG materials and their applications [47, 48, 49, 50, 51, 52].
The purpose of this section is to look at alternative physical materials in the making
of such a material. It has been shown the layered composite does exhibit the effective
properties of a DNG material, although the material exhibitshigh dielectric and
magnetic loss as well as narrow bandwidth thereby limiting its practical applications.
The fundamental explanation for this behavior is the fact that a resonator is used to
achieve a negative effective permittivity (e.g. the ferrite acts like a lossy capacitive

material).

75

d d Region 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Region 1 _
Region 2 Region 3 "=0 ":1 "=2 ":3 n=N n=N+1
E ' ‘i
A r: = W
“2,82 ‘ . 0.000000 ‘
am WW ‘1 ‘Q’VV m
E r Anisotropic Region E E r E1
,8 6'
fire] ”3' 3 ”1’ 1 413,53
2:0 =d 20:0 "l Z2 23 2N4 ZN
a.) b.)

Figure 4.1. Geometry for anisotropic layered material derivation.

76

 

 

W

Beam—1 Beginaz Raglan}
Ei

W

m
E' _4 m E’

S1 1‘“> 8
#0, 80 #0, O #0380

 

 

 

:0 Z:

Figure 4.2. Transmission through free space geometry.

77

1.2 I I 1 T I T I n r r 200
1 150 ~
0.8» 100 ~
50 .
0.6 .
0
0.4»
-50.
0.2» J
-100~
0 450—
02 L A 1 1 J 1 i L 1 1 _2
1 2 3 4 5 6 7 8 9 10 11 12 00
Frequency x 109
a.)

821 8: S11 - Magnitude

 

 

 

 

 

 

Figure 4.3. Scattering parameters for free space case.

78

 

 

y-

 

.—

821 - Phase

T j

 

Y

 

 

b.)

 

/

\\\\

x

 

 

W
Ei
\/\/\/\>
W
E SH“)
#0980
#0280 ‘\ S21
Z=0

 

Figure 4.4. Reflection from PEC in region 3.

 

08>

0.6»

0.4 ~

0.2 >

821 & S11 - Magnitude

 

V I I

 

 

 

 

 

 

 

 

a.)

Figure 4.5.

 

 

 

S11 - Phase
T V Y 200 Y r 1 r T T l v T 1’
\\ k‘
1 50 ‘\ \‘ ‘
\‘ \
100 » x \ -
. \ \\
s‘ \
50 . ‘ \ -4
4 \ \
\\ \
0 ~ \‘ a
- \\\
-50 i \‘ '1
\‘ \\
.100» \ ~ 4
\\
.150 . \‘ .
‘4
1 r 1 .200 l A 1 1 1 1 m 1
9 10 11 12 1 2 3 4 5 6 7 10 1 1 12
x109 le’emy x 109
b.)

Scattering parameters for PEC case.

80

 
 

Magnitude Comparison

 

I T T T T

Phase Comparison

 

      
     
 

 

o ISOtI’OPTC $21
——Anisotropic S

   

21

   

0.7 5°"
0.6»
0.5r o.
0.4»

  

 

 

 

     
  
  

A Isotropic S

l l
_Anisotropic S

.4

ll

0 Isotropic 521 ~
—Anisotropic S2]

 

 

 

0.3 '
A Isotropic 5‘“ .50»
O. ——Anisotroplc S1 1 .
0.1 .
O _L a; 1 l r L i L i 1 -1m
1 2 3 4 5 6 7 8 9 10 11 12 1
Frequency x109

a.)

3 4 5 6 7 8 9 10 11 12
F
requency x109

b.)

Figure 4.6. Comparison of calculated S-parameters using the isotropic and anisotropic

formulations.

81

Permeability & Permittivity Comparison
14

T I I 7 U T Y I' I 1
1 —1

A Isotropic er

 

10 - —-Anisotropic er .
8 L .
6 ~ 4
4 i -

Isotropic ,ur

 

   

 

 

2 . _Anisotropic ,ur

<

o 1 l t I 1 i 1 I 1 l

1 2 3 4 5 6 7 8 9 10 11 12
Frequency “09

Figure 4.7. Comparison of extracted permeability and permittivity for both isotropic

and anisotropic methods.

82

 

[ / Thin Plate
/ Nx=0Ny=0 Nz=1
x

 

 

x1 ' Thin Rod
Z 1 1
( / ()~ N,=— N,=— NZ=O
/ 2 2
y
2
Sphere
x Nle Ny:l szl
3 3 3

Figure 4.8. Demagnetization Factors for various geometries.

83

Magnetic Positive Magnetic Negative (MNG)

 

 

 

 

 

 

 

30 1 . T . a
cD ' ~
<3 _ lie/IUD
<3 --- Iii/flu ‘
<D Jv‘ FMR ‘
C .
<3
5» D a
CD
0 ---------------- ~~ c
\‘ CD”
'5* ‘1 DI
. 1‘3
-15- 5?:
-20- K}
I:
_25 1 1 1 1|
05 1 15 2 2.5 3 3.5 4 4.5
Frequency 9

Figure 4.9. Effective permeability for extraordinary wave with the illustration of
ferrimagnetic resonance (FMR).

84

 

 

 

 

 

20 Magnetic Positive Magnetic Negative (MNG)
- — — k: / k0
JV‘ FMR
1
I .
I
1
i
1
1
\
\
\
\ _
\\‘
0.5 1 1.5 2 2.5 3 3.5 4 4.5
Frequency

Figure 4.10. Effective wavenumber for extraordinary wave assuming negligible loss.

85

 

10

 

 

 

 

 

 

 

 

 

 

' " fr = 0 5
8 7‘ D F ' D ........ fl: = 0 25
7 “-‘-.-_ -------- q
6 ......:......I:..
”eff ‘—‘— -----------------

5 b". 1’ ‘

Pa) ..............
4 _ .................................................................. 1
3 — _____ :

f .....
2 .- —4
1 I 1 , 1

 

 

 

3 5 7 9 11 13 15 17 19
Number of Layers

Figure 4.11. Effective permeability for layered anisotropic composite consisting of
biased ferrite and dielectric materials for different volume fractions, where subscript
”F” denotes the volume fraction of the ferrite. Also, F-D-F implies that the ferri—
magnetic material is the outer layer while D-F-D implies that the dielectric layer is
on the outside

86

13 . , , , . . .

 

 

 

 

 

 

 

 

 

 

 

11 b ' - - fF = 0.5
>/ D F D ........ fr = 0 25
9 _ ‘~-_--------- _
gefl 7 fa, o““—'---- _,
at.
V’ .......................
5” ...............IIIIIIIIIIIIIII.‘.'.'.'.'.'.'.'.'.':::::::::.‘
r ................
31 q
1 l 1 l 1 1 J 1
3 5 7 9 11 13 15 17 19
Number of Layers

Figure 4.12. Effective permittivity for layered anisotropic composite consisting of
biased ferrite and dielectric materials for different volume fractions, where subscript
”F” denotes the volume fraction of the ferrite.

87

 

fF 0.25 0.5 0.75 [[ Total
”efj 2.5-5.2 4.5-7.8 7.2-9.4 [[ 2.5-9.4

Eeff 3.8-6.8 6.1-10 9.2-12.1 [[ 3.8-12.1

 

 

 

 

 

 

 

 

Table 4.1. Tunable Effective Permittivity range summary for anisotropic layered
composites.

88

 

 

 

 

 

 

0 _
l l
.2 . [ _
1 +4.) I: =1x10", Ho = 100
geff .4 . l —+—B.) n = 1x10", H0 = 400
—I—C.) n = 4x10”, Ho =100
+0.) n = 4x10", Ho = 400
‘6 _ ~ — Plasma Frequencies
l l
-3 . i -
[A B C [D
-10 1 1 [ 1 1 1 1 1 11
1 2 3 4 5 6 7 8 9 10
Frequency x 10

Figure 4.13. Effective permittivity for plasma immersed in a magnetic bias field.

89

 

 

   

 

   

-10

I

+ Ferrite [1e

_._ Ferrite ”e

+ Plum 691

0.5 1 1.5 2 2.5 3 3.5 4
Frequency x109

 

 

 

 

 

 

 

Figure 4.14. Ferrite effective permeability and plasma effective permittivity with
negative values.

90

 

 

 

 

 

 

 

 

 

 

0 ~ , ‘
.14.
.2 . .
-3 ,
+ Mu'
- , —-l— MU”
-4 - . '« l + 59’ 4
1. 2 4 Ep.
-5 1 1 1
1.8 1.9 2 2.1 2.2
Frequency x 109

Figure 4.16. Negative permeability and permittivity for layered composite in Figure
4.15a.

92

 

 

 

 

 

 

 

 

 

-2 _ ’ _
+ Mu'
—i— Mu"
_3 _ .1 —. —-I— 59' _
1. . + Ep"
-4 1 1 1
1.8 1.9 2 2.1 2.2
Frequency x 101

Figure 4.17. Negative permeability and permittivity for layered composite in Figure
4.15b.

93

 

CHAPTER 5

COMPOSITES WITH ROD SHAPED INCLUSIONS

The purpose of this chapter is to gain insight into the possibility of a composite
material with tunable permittivity and permeability using a geometry consisting of
rods (or wires) evenly dispersed throughout a background medium. As was done in
the previous chapters, the scattering parameters are calculated at the front and back
interfaces of the composite through which the complex permittivity and permeability
can be extracted [5]. A domain integral equation formulation is employed to calculate
the unknown electric fields within the material region [53, 54, 55]. Once these fields
are known, the scattering parameters can be determined by taking the ratios of the
reflected and transmitted fields to that of the incident field. The integral equation
is solved using the method of moments with pyramidal basis functions and point
matching over a mesh of triangular cells [58, 54, 55]. Although the composite is
infinite in the y-direction, use of a periodic green’s function allows the problem to
be reduced to a unit cell. This unit cell is then replicated infinitely in the :l:y-
direction through a Floquet series [54]. However, the periodic Green’s function is
slowly converging when performing the necessary summation in the spatial domain.
Therefore a method for accelerating this calculation using the Poisson summation
formula is employed to convert the spatial summation into a spectral summation
which is more rapidly convergent [56, 57].

Once the formulation has been completed and verified against exact solutions,
the analysis will proceed using the same two materials as in the previous chapter.
These are Teflon and a narrow line-width garnet, YIG, which is biased by a static
magnetic field perpendicular to the direction of propagation. The bias field allows the

permeability of the YIG to be controlled at the desired frequency of operation, see

94

 

section 4.3. Although YIG is an anisotropic medium, the permeability tensor can be
accurately simulated using a scalar quantity which predicts the effective permeability
and allows the ferrimagnetic material to be modeled as an isotropic material [40].
The effects on the homogenized permittivity and permeability of the composite will
be investigated by altering the volume fractions of the two materials and increasing
the number of cylinders in the direction of propagation either in a row or stagered
between adjacent layers. Finally, the effects of inserting holes, or pockets of free space,

into the composite will be examined.

5.1 Background and Derivation

5.1.1 Geometry

The geometry of the unit cell for a core—shell cylinder inside a square background is
illustrated in Figure 5.1. The two-dimensional surface, which lies in the x-y plane, is
discretized into small triangles, called facets. The composite is assumed to be finite in
the x-direction and to extend to ioo in the z-direetion with no z-variation (6/62: = 0).
Also, since the unit cell is replicated to :too in the y—direetion, the edges at the top
and bottom of the unit cell are periodic boundaries. As will be explained further in
section 5.1.2, the vertices of each triangle are the locations of the unknown electric
field expansion functions called nodes. The fundamental geometry is designated such
that concentric cylinders are placed in a rectangular region. The radius of the outer
cylinder is chosen based on the desired volume fraction for that region. Thus for the
example given by Figure 5.1, the outer radius gives a cylinder whose volume fraction
is 0.5 implying that it occupies half of the total volume of the composite. Finally, the
radius of the inner cylinder is always chosen as half the radius of the outer cylinder

for convenience.

95

5.1.2 Integral Equation

The process of calculating the scattering parameters for this type of composite is
done using an integral equation formulation for which the unknown electric field is
found using the method of moments [53, 58]. An integral equation is one in which the
unknown appears within the integral. This particular approach uses an electric field
integral equation, E F I E , to model a two-dimensional penetrable scatterer assuming a
normally incident transverse magnetic, TM, plane wave traveling in the +x-direction.
The term penetrable is used to indicate that the unit cell is inhomogeneous with
varying permittivity and permeability for which the incident wave can penetrate.
This implies that the integral equation unknowns must be calculated throughout the
entire domain as opposed to just along a particular surface, as is the case for a PEC
scatterer or homogeneous region. In the example shown in Figure 5.1, the composite
could be broken into three regions in which each region has a different permittivity
and permeability. For instance, one region could be the area exterior to the outer
cylinder, a second would be the ring-shaped area between the concentric cylinders,
while the third and final area would be the innermost cylinder.

As illustrated in Figure 5.1, the unit cell is discretized into nodes, edges, and
facets. The nodes represent the unknowns that are to be calculated and the patches
”are regions of constant permittivity and permeability while the edges separate these
regions. In many cases, the unknowns are electric or magnetic currents; however,
in the case of a penetrable scatterer it is often more convenient to represent the
unknowns in terms of the electric and / or magnetic fields. In this work, the unknowns
are the z-directed electric field as will be shown below.

Physically speaking, the incident plane wave will strike the composite and set up

polarization currents within the material. These currents then reradiate a field called

96

the scattered field. The superpostion of these two fields gives the total field.
E’tot ___ fine + 1:33ch (51)

One possible manner of representing the electric and magnetic scattered fields is in

terms of their vector potentials. More specifically, the magnetic vector potential, :4),

l

_)
is calculated from electric current density, J , and the electric vector potential, F,

from magnetic current density, 2?, through the relationship given by 5.2.

*

(5.2a)

“:11 :41
11
a1 kl

G
:1: G (5.2b)

The symbol * is used to denote the spatial convolution where G signifies the Green’s
function. For the geometry used in this analysis, G is the two-dimensional free space
Green’s function

G = —:H32)(kir*1> (5.3)

(2)

where H0 is the zero-th order Hankel function of the second kind representing an
outgoing cylindrical wave with free space wavenumber, k = cum. Combining
5.2 and 5.3 and giving the convolution in its integral form, the vector potentials
can be written by 5.4 where 77’ and 3) indicate source and observation locations,

respectively.

>1
11

[[713'15113211417—4’1W’ (5.44)

// R’(r’)4in32)(klr — r’nd‘p‘" (Mb)

"111
11

Using 5.4, the scattered electric and magnetic fields can be calculated by the following

97

equation through a process known as integration- f ollowed—by-di f f erentiation [54].
‘E’sca = —jkn;f — V x F (5.5)

The symbol 7} = ,/ 110/50 is the impedance of free space. Equation 5.5 represents
the scattered electric field in terms of the electric and magnetic current densities.
Using this expression in the equation for the total electric field and rearranging gives

a commonly used form of an integral equation.
it“ + 3‘an + v x 73 = Kim (5.6)

The incident field is placed on the left hand side of this equation since it is a known
quantity and for the upcoming linear algebra operations is the most convenient po—
sition. The unknowns on the right hand side of the equation are the electric and
magnetic current densities as well as the total electric field. Each of these values
are vector quantities which have three components, one for each direction in the
Cartersian coordinate system (at, y, 2).

Since the incident field is TM Z , all six of its electric and magnetic field components
may be written in terms of the z-component of the electric field. Also, a z-directed
electric field incident on a scatterer will induce surface currents that are in the z—
direction and proportional to the total z-directed electric field. Therefore, it would
be most beneficial to represent all the unknowns solely in terms of the z-directed total
electric field E z- Note the subscript ”tot” for the total field has now been suppressed
for ease of notation. The integral equation is now reduced from a vector form to a
scalar form.

E2 + jknAz + 2. v x 75 = E?“ (5.7)

The electric and magnetic vector potentials are functions of magnetic and electric

98

currents which can also be written in terms of the total electric and magnetic fields

through the following relationships.

= jw50(€7~ —- 1)? (5.8a)

Ni kl

= jw#0(#r — 11H (5.81))

where the relative permittivity and permeability are given by Er and pr, respectively.
Now using the following form of Faraday’s law, the magnetic field can be written in
terms of the electric field. This can be used in the expression for the electric vector

potential F through the calculation of the magnetic current density 2?.

—->

V X E = —jwp0prfi (5.9)

“7' _ 1v x "E’ (5.10)
#r

 

I?

Now the integral equation in 5.7 can be written entirely in terms of the total z-directed

electric field, Ez, using 5.10, 5.8b, and 5.2.

 

E2716 = E3 — 1132(51' —1)EZ * G —' 2- V X {(#Tp— 1V X 2E3) * G} (5.11)
7"

Since it is a constant within each triangular region, the relative permeability in the last
portion of the right hand side of 5.11 can be brought outside the curl operations. Also,
since the integral operations will be performed numerically, it is more efficient to move
the differentiation inside of the integral where it can be performed analytically. This
changes the current process to a di f f erentiation- f ollowed—by-integration procedure

[54]. The identity in 5.12 shows how the curl-curl operation in 5.11 can be reduced

99

to a transverse laplacian operation.

_. _. 274’ A _.
VXVX A: (VtXVtX At—Qggg‘i‘fvta—f)+5(3(Vt'At)—V%Az)
(5.12)

Finally, the integral equation can be given entirely in terms of E z in its most compact

form.
[tr -' 1
#r

Egnc .—_ Ez — k2(er —1)Ez * G +

 

17,2132 4 G (5.13)

The integral equation can now be solved for using the method of moments [58]. The
scatterer is discretized into N nodes forming a mesh of triangular facets as demon-
strated in Figure 5.1. The unknowns are expanded in terms of pyramidal basis func-
tions in which the unknown becomes a linear interpolation of its surrounding nodes.
This is illustrated in Figure 5.2 and displayed mathematically in 5.14 for a single
triangular patch with a local numbering system such that local node 1 is the nth

node of interest.

a: —:r +2: — + x —a:
Bn(:c.y)= (21/3 3y2) (92 313) 3(3 2)

(5.14)
(1321/3 - 933142) + 931(1/2 - .113) + y1($3 - 162)

As node n is approached the value of the basis function tends to one, and as the
other nodes are approached the basis function goes to zero. Using this basis function,
the total field within the cylinder can be given by the superposition of the unknown

electric fields multiplied by the basis function.

N
n=1

Applying point matching, and inserting 5.15 into 5.13 and separating out the still

100

 

unknown electric field coefficient, en, the following matrix equation is formed.

r 1 r 1

e1 Eéncfrn. 111)
62 Efzncfir2. 312)

[Z]<. L=< e i (5.16)

 

 

 

 

[ eN [ Efszxjvmv) ,

J

The impedance matrix, Z, is given by 5.17 where (5%” is the Kronecker delta funtion
(i.e. 6;,” = 1 if n = m and is zero otherwise). If the total number of nodes is given by
N, the impedance matrix has N crN elements since for every mth observation node

th

there is a contribution from every n source node. For instance, the entry Z25 is

the contribution of the field at node two due to the source at node five.

 

”'7" $=$miy=ym

Once the impedance matrix has been constructed, the unknown electric field coeffi-
cients, 67;, can be calculated by inverting the impedance matrix and multiplying it

by the incident field column vector.

{en} = 121—1 {E2710} (5.18)

In the process of evaluating the impedance matrix it is important to note that the
terms which perform a derivative on the basis function can be reduced from a two
dimensional convolution of the surface of each triangular patch to a one dimensional

convolution over the patch’s edges using Stoke’s theorem. This is more easily seen in

101

5.11 with the curl-curl operation.

f/v x (v x 2Ez)-d§ = /A (v x 2E2) «17 (5.19)

The triangle symbol under the integrand in the right hand side is used to denote a
closed loop integral around the entire triangular surface. The pyramidal basis function
is now only differentiated once each with respect to x and y, then it is dotted with
the vector tangent to the triangle edge being integrated, d7. The second term on
the right hand side in 5.17 is integrated over the surface of the facet and is performed
using a seven-point quadrature routine.

A final important point involves the use of the pyramidal basis functions. As the
impedance matrix is filled, the fields at node m due to the sources at node n are being
calculated as mentioned earlier. However, this basis function is slightly more complex
than in the case of pulse basis functions since the source at node n has contributions
for all its neighboring facets for which it is a vertex. Therefore, in each entry of the
impedance matrix there is a summation over all adjacent triangles at every nth source
node.

Section 5.2 will illustrate some examples of this formulation for a single circular
cylindrical rod in free space along with comparison to an exact solution of the same

problem for purposes of verification.
5.1.3 Periodic Green’s Function

The geometry being modeled is a slab with dimensions that are infinite in the y and
2 directions and has a finite thickness in the x-direction. The infinite extent in the
z-direction is accounted for by the fact that the problem is implemented in two di-
mensions (x-y plane). However, the infinite y-direetion is realized through boundary
conditions inherent in the periodic Green’s function. These periodic boundary condi-

tions are sometimes referred to as Floquet harmonics. The periodic Green’s function

102

with period b is given by 5.20 where the primary difference from the original Green’s
function in 5.3 is the dependence on the progressive phase shift of the incident field

in the y-direction.

 

oo .
1 2 . -77
044.414.4633 E) H8’(k\/<x—w’>2+<y—y'—zb>2)e My” (5.20)
iz—oo

For the problem presented here, the incident field is propagating along the x-axis
and therefore kg = O and the exponential term at the end is equal to unity. This
periodic Green’s function may be inserted into 5.17 for the two-dimensional Green’s
function allowing for the simulation of a large (ie. infinite) structure to be reduced
to that of a unit cell. The difficulty of this method is in the slow convergence of
the periodic Green’s function which involves a spatial summation of the zero-th order
Hankel function [56, 57]. This is visually demonstrated in Figure 5.3 which shows how
oscillatory the real and imaginary parts of the hankel function can be as the argument
gets larger. The usefulness of the periodic Green’s function is greatly limited by this
slow convergence since it significantly increases the computation time in the method
of moments solution of the integral equation formulation in order to get an accurate
result. Therefore, it is desirable to find a method to accelerate this convergence.
The Green’s function in 5.20 can be seen as a response to a periodic array of phase

shifted point / line sources given by 5.21 [57].

m u .
1(4, y) = 2: 5a — .r’)e—Jky2b6(y — y’ — 15) (5.21)
i=—oo
A method commonly referred to as the Poisson sum transformation can be applied
to this expression for current which involves taking the Fourier transform and then

its inverse [54]. The transform of the periodic array of point / line sources will give an

array of periodic current sheets [57]. Once the inverse transform has been performed,

103

the resultant expression for the periodic current sources is a spectral summation

rather than a spatial summation.

1 00 fl» (1! y')
_ l " " ‘-
J(a‘,y) — fi . 2 6(11: — :c )6 ya (5.22)
22-00
where
27ri

Therefore the response to this expression for current is given by a new form of the
periodic Green’s function which is also a spectral summation and is proportional to

the inverse of the period.

 

(5.24)

where

142—14. ifk2 >142.
km = W 317' (5.25)

—j, Hag, — k2 if 1451. > k2

For the ”off-plane” case (it: 74 :r’), as Ii] is increased, the wavenumber kyi trans-
forms from positive real which represents an outgoing propagating wave to negative
imaginary which gives an evanescent wave. Therefore, as [i] increases, the summation
in 5.24 begins to converge exponetially reducing the computational cost considerably.
However, in the ”on-plane” case (a: = a’), the source and observation point lie on the
same current sheet, and the exponetial term with kg; disappears and the summation
is again slowly converging. There are additional/ alternate approaches to accelerate
the summation and to account for the ”on-plane” situations [54, 56, 57], but as shown

in section 5.2, this method provides sufficient accuracy as long as the summation is

104

given enough terms in the case when a: = :13, .

A final consideration when using the periodic Green’s function is that the nodes
along the periodic boundaries on the top and bottom of the unit cell must be congru-
ent. The reason is that the nodes along opposite boundaries are essentially the same
node since as the unit cell is repeated in one direction or the other, the nodes that
were on the top, become the nodes on the bottom and so on. For the case of a strip
grating where a portion of the unit cell is PEC which has a surface contribution to
the convolution integral, and the other is free space, which is not being integrated,
the congruency can be easily realized. As long as that portion not being integrated
is placed at one of the periodic boundaries there will be no further steps needed.
However, if both boundaries contribute to the convolution, as is the case in this work,
than this problem can not be ignored. If nothing is done, when the operation in 5.18 is
done, the unknowns will be incorrectly calculated because the top and bottom nodes
would have been superimposed yielding inaccurate results. The solution is to create
the impedance matrix, [Z], as normal but before calculation of the unknowns using
5.18 can be performed, the nodes at either the top or bottom must be eliminated after
their contribution has been incorporated into the remaining entries of the impedance
matrix. This is best explained through a simple example, consider the following 3x3

impedance matrix and 3x1 forcing vector.

 

 

 

 

- , - .
411 A12 413 b1

[Z]: A21 A22 A23 1 {v}: 02 (526)
(A31 A32 A33_ _‘3,

The 3x3 impedance matrix is analogous to a mesh containing three nodes. Suppose
node three is the node on the periodic boundary chosen to be eliminated. The im—
pedance matrix and forcing vector would be reduced to a 2x2 matrix and 2x1 column

vector and the contribution from node three is absorbed into the remaining entries of

105

the reduced tensors.

A +A +A +A A +A

[Znew] = 11 31 13 33 12 32 (5.275)
1 A21+A23 A22
bl+b

{Unew} = 3 (5.27b)
b2

 

Even if the situation from the strip grating example were present in which the nodes
along the periodic boundary were free space, this step could be utilized to reduce the
size of the impedance matrix and forcing vector. The reason is that there would be
no contribution from node three if it was free space and those entries would be zero.

Once the integral equation formulation in section 5.1.2 has been verified by com-
parison to the exact solution for the unit cell alone, see section 5.2, the periodic
Green’s function will be inserted in place of the two-dimensional free-space Green’s

function given by 5.3. This will give the infinite structure needed for the analysis.

5.2 Verification of IE Code

If possible, it is important that once computer code is written, it should be checked
against exact solutions to look for errors. If it is not feasible to get an exact solution for
comparison, then possibly one could obtain experimental data or even another code
which has been verified. In this section, the code implemented in this dissertation
will be compared against exact solutions for a circular cylindrical rod extending to
infinity along its axis, and an infinite planar slab. The slab extends to infinity in both

directions perpendicular to the direction of wave propagation.
5.2.1 Penetrable Cylinder

Surface currents on a penetrable cylinder in free space that extends to infinity in the

z—direction along its axis will be compared to an exact solution using the integral

106

equation described in section 5.1.2. A TM 2 plane wave is incident in the x-y plane
along the d) = 0 direction and is assumed to have unit amplitude. Three cases will
be calculated and compared to determine validity. These are for a purely dielectric
rod, purely magnetic rod, and a combination. of lossy dielectric and magnetic material
properties. A derivation of the exact solution for scattering from a cylinder can be
found in the appendix of [55]. The region outside the outer cylinder is set to free space
while the region within the outer cylinder is set to a homogenous material depending
on the rod material being analyzed. In all three cases the radius of the cylinder is
0.6)..

The first case to be investigated is a lossless dielectric cylinder, with relative
permittivity er = 4. Figure 5.4 illustrates the accuracy of the integral equation cal-
culations in both magnitude and phase for the surface currents along the cylinder.
This provides assurance that the geometry and mesh input information are in cor-
rect format as well as the quadrature performing the surface integration is yielding
accurate results. The next case will confirm the line integration is working correctly.
The cylinder is now a purely magnetic rod with relative permeability pr = 3.3 and
Er = 1. Again, the results shown in Figure 5.5 demonstrate good agreement with the
exact solution and verfication in the use of the line integral routine. The final test of
the code is a cylinder with both permittivity and permeability greater than one, as
well as dielectric and magnetic losses. The properties of the rod are Mr = 5.1 — 30.2
and er = 2.2 — 3001. Figure 5.6 shows good agreement between the integral equa-
tion formulation and the exact solution. The results of the previous three tests gives
confidence in this code and its ability to accurately calculate the fields for a variety
of material types. The following section will take the verification process one step

further and ensure that the periodic green’s function is properly implemented.

107

5.2.2 Infinite Slab

This section will check the validity of the periodic Green’s function when inserted
into the integral equation code which was demonstrated to yield accurate results in
section 5.2.1. The periodic green’s function allows in infinite periodic structure to
be modeled using only a unit cell the size of the structure’s period. Verification
of this capability can be accomplished through comparison of scattering parameters
for that of an infinite homogeneous slab. The exact solution for scattering from
a slab has been illustrated in both chapters 3 and 4 for isotropic and anisotropic
materials, respectively. As before, the incoming wave is TM Z and normally incident
on the planar surface. The thickness of the slab is 5mm and the material properties
are the same as those in the preceding example of a lossy magneto-dielectric rod,
pr = 5.1 - 30.2 and Er = 2.2 — 30.01. The scattering parameters, 311 and 321,
are calculated from 1-3GHz. Figure 5.7 shows how accurate the integral equation
formulation is able to calculate the scattering from a structure with infinite one-
dimenstional periodicity.

One further test of this formulation is to extract the complex permittivitiy and
permeability from the scattering parameters of the slab using a two-dimensional N ew-
ton’s root searching algorithm as described section 2.4.3. The permittivity and per-
meability of the slab whose scattering parameters are given by Figure 5.7, are shown
versus frequency in Figure 5.8. The extracted values are exactly the permittivity and
permeability entered in order to calculate the scattering parameters.

The code and formulation has now been sufficiently verified, the next step is to see
the effects of various circumstances on the homogenized permittivity and permeability

of a composite involving cylindrical shaped inclusions.

108

5.3 Analysis of Composite

Now that the formulation and code has been verified as being capable of handling
this analysis, the goal is to understand how this type of geometry can be used to
control and effectively tune the electromagnetic properties of a composite containing
rod shaped inclusions. The investigation will proceed using the same two materials
used in the analysis of layered anisotropic materials in chapter 4 and the frequency
will again be 1GHz. The dielectric material is Teflon with Er = 2.08 — j0.001, pr = 1
and the magnetic material is YIG with a static magnetic bias field directed along
the axis of the cylinder. The choice for this direction of the magnetic field is due
to the fact that the demagnetization factor along this axis is zero and hence does
not require a very strong field to achieve the desired magnetic properties. This is
equivalent to a bias field in the direction transverse to the wave propagation in the
layered material analysis in the previous chapters. The permeability of the magnetic
material is modeled using the extraordinary wave effective permeability from 4.45.
The permeabililty for a slab of anisotropic ferrimagnetic material with a bias field
perpendicular to the direction of propagation is caelulated using methods of chapter
4 and given in Figure 5.9 along with the permeability using 4.45. This reinforces
the possibility of representing the anisotropic magnetic material in this analysis using
a scalar value for simplicity without loss of accuracy. In the following cases, the
thickness of the composite slab is kept constant at 2mm. The effects of layering are
investigated by increasing the layers of rods from one to five layers. By varying the
radius of the cylindrical inclusions, the volume fraction effects can be illustrated. Also,
the effects of placing holes, cylinders of free space, inside the composite are analyzed
and finally for all situations the dielectric and magnetic materials are interchanged

to see those effects as well.

109

5.3.1 Geometries

The mesh created for each geometry begins with a specified volume fraction for the
cylindrical inclusions. The two main choices have been 0.5 and 0.2 which implies that
the volume contained within the outermost cylindrical radius is half and one—fifth
the total composite volume, respectively. However, within each mesh is a second
concentric cylindrical ring at a radius exactly one-half that of the outer radius. This
additional radius gives the option to analyze smaller volume fractions or to attempt
any core-shell models. A consideration of the mesh will show how the actual volume
fractions differ from the specified volume fractions of 0.5 and 0.2 since the cylinders
are broken up into eight triangles as illustrated in Figure 5.1. This actually gives
a volume fraction of 0.45 and 0.18 rather than 0.5 and 0.2 respectively. Finally,
the effects of additional layers of cylinders inline with each other and staggered are
incorporated as well. Figure 5.10 - Figure 5.14 show all of the geometries created and
described in this paragraph where the values of a, b, and h are different depending

on the situation while t = 2mm in all cases.
5.3.2 Increasing Layers

The first case involves the magnetic cylinders with inclusion phase volume fraction
of 0.45 and the number of cylinders are increased from one to five. Figure 5.15 a,b
illustrates the effective permittivity and permeability for the case when the layers
are aligned such that each successive rod is directly behind the other. Figure 5.15
c,d is for the case when the rods are staggered. It is clear that there is no effect
on the composite’s electromagnetic properties when increasing the number of layers
for evenly spaced cylindrical rod-shaped inclusions. Another important observation
is the permittivity and permeability are the same whether the rods are aligned in a
straight row or whether they are staggered. These results could be the consequence

of the electrical sizes of the rods and composite structure. The thickness of the entire

110

composite is only 1/150th of a free-space wavelength and the rod is still smaller
yet. The subwavelength thickness allows the properties of the material to easily
become homogenized as in the case of large number of planar layers. In comparison
to increasing planar layers, as the number of cylinders increases, wave propagation is
not impeded as dramatically since portions of the slab with cylinders are homogeneous
depending on the line of sight which would allow for easier propagation. As the planar
layers are increased the waves experience more reflections or bounces and this creates
different values for permittivity and permeability and increases loss in the case of
greater attenuation. The effect of increasing the number of cylinders was investigated
for other volume fractions and the result was always the same and therefore those
results have been left out of this work to avoid redundancy. All the following analysis
have also been performed for one, three and five layer cases but have also been left

out here for the same reason.
5.3.3 Volume Fraction

The next aspect of design to be investigated involves the effects of volume fraction.
For the case of a one cylinder layer a volume fraction of 0.45 would require an outer
cylinder radius of b = 0.8mm giving an inner radius a = 0.4mm. The volume fraction
for the outer radius of 0.8mm implies that the entire cylinder including the inner
cylinder is part of that inclusion phase. Thus the inner radius yields a volume fraction
of 0.08 for the inclusion phase. In the other case, with a volume fraction of 0.18, the
outer radius is b = 0.5mm and the inner radius becomes a = 0.252mm analogous to
a volume fraction of 0.032. Therefore, with these two meshes, four different volume
fractions can be investigated and then if the materials are interchanged, such that
Teflon becomes the cylindrical inclusion and YIG becomes the environment medium,
another four volume fractions of the magnetic material can be realized which are 0.55,

0.82, 0.92, and 0.968. The effects of volume fraction on homogenized permittivity and

111

permeability have been illustrated in Figure 5.16a for increasing volume fraction. The
loss tangents are given in Figure 5.16b.

As expected, the effective permittivity and permeability increase as the volume
of YIG is increased within the composite. The permittivity increases steadily as the
volume fraction is increased. On the other hand, it seems that volume fraction has
only a slight effect on permeability when that value is below 0.45 and the rod is
YIG with a surrounding dielectric. Once the materials have been interchanged the
permeability takes a large jump from 2 to 4.3 for the a volume fraction of 0.55 then
begins to take a steady increase as the volume fraction is continuously raised. This
phenomenon also occured in Chapter 4 for the case when the YIG was the outer
layer a higher effective permeability was achieved. A possible explanation for the
large jump in permeabililty is that when the YIG is the environment it interacts
with the incoming field directly as opposed to the wave propagating through the
dielectric first and this seems to yield a greater permeability. This is evident in the
case of interchanging the YIG and Teflon material phases. However, having the YIG
as the environment gives it a planar surface, which happens to match the incident
field’s wavefront and this could have an impact on the permeability as well. It should
be noted that greater volume fractions for the cylinder were not attempted because
the cylinder would need to take on non-cylindrical shapes if the volume fraction
were going to increase in order to fit within the unit cell. This analysis illustrates
a tuneable relative permittivity from 2.8-13.9 and relative permeability from 1-10.8
based on the volume fraction of YIG versus that of Teflon. Unfortunately, for the
cylindrical geometry, as the volume fraction for YIG is increased, the magnetic loss
is also increased.

In many situations, researchers are avoiding the use of magnetics because of the
loss inherent in most magnetic materials. Therefore, much work involves purely low-

loss dielectric composites consisting of some polymer and a ceramic called alumina

112

which has a dielectric constant of 10 [59]-[62]. So as an illustration, a composite
consisting of Teflon and alumina will be analyzed using this geometry over various
volume fractions. The composite will begin with an alumina cylindrical inclusion
and the Teflon background material and once a volume fraction of 0.45 is reached,
the materials will be switched as was done previously. The results are displayed in
Figure 5.17. Only the permittivity has been included in the plot since the composite
is non-magnetic. The effective permittivity for this composite takes a similar trend to
that of the YIG and Teflon previously shown and gives a tunable permittivity range

of 2.3 to 9.8 with very low loss.
5.3.4 Free-Space Inclusions

The final step in analyzing this composite is to see the effects of holes, or pockets of
free space on the permittivity and permeability. The first of four trials is just a YIG
cylinder inside Teflon, followed by replacing the innermost cylinder by Teflon which
gives a dielectric filled cylinder of YIG inside Teflon. The third trial is to remove
the innermost cylinder and replace it with free space giving a hollowed out YIG
cylinder, and finally removing the YIG altogether and filling the entire cylindrical
region with free space surrounded by Teflon. These configurations are given in Figure
5.18 with the extracted values of permittivity and permeability for the two meshes
having volume fractions of 0.45 and 0.18 are displayed in Figure 5.19 and Figure 5.20,
respectively. The permittivity seems to be effected slightly when the core of the YIG
rod is replaced with Teflon and seems unaffected by removing the Teflon and replacing
the core with free space. The final case shows a drastic drop in permittivity down to a
value just slightly less than that of Teflon by itself. This is expected because the high
contrast composite has been replaced with a very low contrast composite once the
YIG is completely removed. The permeability seems to be unaffected by the changes
made until the YIG is completely removed and the composite no longer contains any

magnetic material which gives an expected value of one for permeability.

113

Now the materials are swapped as was done in the case of the volume fraction
analysis. The first of four trials is a Teflon cylinder inside a YIG background, followed
by a YIG cylinder inside the Teflon cylinder. Finally, the last two steps are to have
a hollow Teflon cylinder inside YIG and then removing the Teflon altogether giving
a YIG background with a free space cylinder inside. These geometries are given in
Figure 5.21 while the extracted permittivity and permeability are shown in Figure
5.22 and Figure 5.23 for outer radius inclusion volume fractions of 0.45 and 0.18 re-
spectively. The permeability appears to be unaffected by the different cylinders of
Teflon and free space except for the case of a Teflon rod with a YIG core which causes
a slight increase. The permittivity seems to decrease a small amount as the quantity
of free space is increased. When the YIG is used as the environment, it appears to
dominate the effective electromagnetic properties since it is the higher permittivity
and permeability in this high contrast composite. Therefore the difference in permit-
tivity and permeability for Teflon and free space does not seem to make a significant
difference and therefore could replace the Teflon if weight was an issue in the design

of a composite.

5.4 Conclusions

In this chapter, an integral equation formulation has been shown to accurately evalu-
ate a structure with infinite periodicity in one direction. The use of a periodic Green’s
function allows the mesh to be truncated to that of a unit cell the size of one period
and reduced computation time greatly. The integral equation calculation was shown
to match exact solutions for a cylinder in free space and an infinite slab of a homo—
geneous material with both complex permittivity and permeability. The analysis of
composites with red shaped inclusions proceeded by first looking at the effects of in-
creasing the layers or number of rods within the composite. This showed a negligible

change in the effective permittivity and permeability and thus is not a good design

114

tool for tuning those properties if the columnar inclusions are evenly distributed.
The next mode of analysis involved the effects of increasing volume fraction which
did yield a significant change in the permittivity and permeability and could possibly
be a method for creating a composite with the desired electromagnetic characteristics.
Finally, the last case analyzed was the effects of inserting inclusions of free-space into
the composite and it was shown that the difference was small compared to the effects
of having Teflon cylinders instead. Therefore, unless the weight of the composite is
of great concern, it may just be best to use a polymer with low dielectric constant
such as Teflon.

In conclusion, the best tool for material composite design with tunable electromag—

netic properties using this geometry is to alter the volume fraction of each material.

115

 

 

 

 

 

 

 

Figure 5.1. Geometry and mesh for core-shell model used in IE formulation.

116

   

       
 

-----—---Lo—.---

§ '
I
-----....-..-...- .--.-..--.‘w------.. ---.-------.-.-.
fl

:9

Figure 5.2. Pyramidal basis function for node n [53, 54, 55

117

0.5 fl

 

 

 

 

 

 

 

 

 

 

° [infill/”11111111111

Figure 5.3. Convergence of the hankel function.

118

 

Strface Currents - Magnitude Surface Currents - Phase

 

 

 

 

 

 

 

 

 

 

1.4 ' Y fir v 0 Y T '
IE
0 ExactD
1.3~ -10~
o
1.2* .20.
o
1.1" .30.
1» 43- 4

 

 

 

 

 

 

1 1 1 1 1 '3) l 1 1 1 1
01130 .135 .90 -45 o 45 90 135 130 .130 -135 -90 45 o 45 90 135 180

Figure 5.4. Comparison of the exact and IE surface fields for dielectric rod.

119

Surface Currents - Magnitude

 

1138*

1.06'

1.04‘

 

  

Y

 

 

 

1.02 1
-180 -135

.wL

Surface Currents - Phase

 

 

 

r Y Y V f

IE I
Exact“

 

 

 

 

 

 

45 0 45 90 135 18)

Figure 5.5. Comparison of the exact and IE surface fields for magnetic rod.

120

Suface Currents - Magnitude Surface Currents - Phase
1 .45 l I 1 r v 1 v I 1 v v v

 

 

 

 

 

IE
1.4 * 0 Erect ~

 

 

 

 

 

 

 

 

 

 

 

 

 

-60 1
-180 -135 JD -45 0 45 90 135 1a)

 

Figure 5.6. Comparison of the exact and IE surface fields for lossy rod.

121

Scattering Parameters - Magnitude Scattering Parameters - Phase

 

 

 

 

 

 

 

 

 

 

 

1 r 80 ‘
>
0.8» :55" 32' i
0"" W " 1.: 0 IE
E 0- — Exact
z 0 IE

 

 

 

 

 

 

 

 

 

O ‘ L 1 'm 1 ‘ 1
1 1.5 2 2.5 3 1 1-5 2 2.5 93
Frequency x109 Frequency xlO

a.) b-)

Figure 5.7. Comparison of the exact and IE scattering parameters for lossy slab.

122

Permittivity and Permeability

 

 

 

 

 

 

 

 

 

 

 

 

 

6 i r
52 ¢ 6
+epr
W4” —o—8Pi -
—o—mur
-‘ 3P —a—mui a
5W ' 2~ : x E
Z . ‘.
c,d- ‘
0 2 4;
-1 l 1 1
1 1.5 2 2.5 3
Frequency x109"

Figure 5.8. Extracted permittivity and permeability for a lossy slab.

123

15

10’

 

PERMEABILITY & PERMITTIVITY

V 7 T

P

o Efl'ective
—- Anisolro ' it

A l A A
v v v

 

 

p

 

 

3 4 5 6 7 8 9

Frequency

Figure 5.9. Comparison of effective permeability of anisotropic to isotropic ferrite.

124

i
1
}

 

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4
.',
a

 

3 (swam/(Mlle)

-(£rsgo’ #rBflO)
-(£r-C£o: I‘m/Jo)

 

 

 

 

 

Figure 5.10. Geometry and unit cell for the core—shell model with one cylinder.

125

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A *-f,:,:l:f:.‘...:; .y y i I}: i x

‘I (85480: lurAIllo)
u (green ’ IurB #0)
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Figure 5.11. Geometry and unit cell for the core-shell model with three cylinders.

126

 

    

 

 

 

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0%;
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WV 00 _E:::::::::::::l
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Figure 5.12. Geometry and unit cell for the core-shell model with five cylinders.

127

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Figure 5.13. Geometry and unit cell for the alternating core-shell model with three
cylinders.

128

 

 

 

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Figure 5.14. Geometry and unit cell for the alternating core-shell model with five
cylinders.

129

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8 P r s 3 7
7 l 8r
6 2.5
l m I
4 A ; 1‘ a
3 -
2 fl, ‘ 9.5 g 6d i
1 . A. __-_, J 0 T __ T__,_ m... _ fl
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x10'3

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7 8, i 2.5 l
6 - 1 5 l
5 . \rfm ;
4 1
3 4 0.5 ’
2 yr ‘ + 5 J

i i 1 _u l , __-___l 1 3 5'

1 3 5

Number of Layers Number ofaliyers
c.) -

Figure 5.15. Effect of increasing layers on homogenized permittivity and permeability
for rods in a row [a,b] and alternating [c,d] with a volume fraction of 0.45.

130

 

  
  
       

    
  

 

 

 

14 ' ' ‘7 ' .014 i
12 . Y [C Inclusions 012 [i Y [O Inclusrons
' i
10 ' .01 Ir
8 .008
6 .006 . J
Teflon Inclusions

4 .004 i

Teflon Inclusions '
2 .002

. 6d
0 _ c ,4_.__4-_.,, .1 “__1m -1...___s,, l . u. ,, 0 . ___i #7: ‘;' , .‘WAL__ ‘ fih
0.032 0.08 0.18 0.45 0.55 0.82 0.92 0.968 0.032 0.08 0.18 0.45 0.55 0.82 0.92 0.968
Volume Fraction Volume Fraction
a.) b.)

Figure 5.16. Effect of increasing volume fraction on homogenized permittivity and
permeability.

131

x104

     
 
 

     
 

 

 

 

 

10 ”””” "Fifim 7 [ """T"”MC-”W" “““FW" 1
9 Aluminalnclusions 6 iAluminalnclusions i
.r : l
8 l 5 L i
7 4 ;
2
2
4 Teflon Inclusions
1 l
2 ‘_,m_.__, . i . L N . .,.j 0 l “MW 1W4“--. -._L_,, ,- .1
0.032 0.08 0.18 0.45 0.55 0.82 0.92 0.968 0.032 0.08 0.18 0.45 0.55 0.82 0.92 0.968
Volume Fraction Volume Fraction
a.) b.)

Figure 5.17. Effect of increasing volume fraction on homogenized permittivity for
composite consisting of alumina and teflon.

132

 

Teflon
:1 YIG
[:1 Air

 

 

 

 

 

 

Figure 5.18. Geometry for analysis on effects of air gaps on permittivity and perme-
ability.

133

,7a*~h__ _ . _ 77‘_.7___

x103

 

 

de-hUiCD‘JCD

 

Figure 5.19. Effective permittivity and permeability of composite with holes for
volume fraction of 0.45 with YIG as cylindrical inclusion.

134

 

 

 

 

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a.)

Figure 5.20. Effective permittivity and permeability of composite with holes for
volume fraction of 0.18 with YIG as cylindrical inclusion.

135

 

37:75:11 Teflon
[:3 YIG
D Air
y y
x

t I t | y‘ t I l t |

T 1

ng _l x
1.) 2.) 3.) 4.)

Figure 5.21. Geometry for analysis on effects of air gaps on permittivity and perme-
ability.

 

 

 

 

 

y
r
h
LUr

136

 

 

 

 

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i

_1 3 4
b.)

Figure 5.22. Effective permittivity and permeability of composite with holes for

volume fraction of 0.45 with Teflon as cylindrical inclusion.

137

 

W W . WW... .012 W . W .,.--- W

 

 

 

 

6 1
. .01 m
. 1

1 .008
1‘ 1 .006 .. .
10 l .004 l
1 -~ 1
9 .002 i l
’1 1 1
a ff“; .......... __: 0 l m - iii _ .1
1 2 3 4 1 2 3 4

a) b-)

Figure 5.23. Effective permittivity and permeability of composite with holes for
volume fraction of 0.18 with Teflon as cylindrical inclusion.

138

CHAPTER 6

CONCLUSIONS AND FUTURE WORK

Presented in this work is a foundation for composite material design. Using vari-
ous methods including both analytical and numerical techniques, accurate electro-
magnetic models have been formulated for three geometries of two-phase composite
materials. These geometries include: spherical inclusions, alternating layers, and
cylindrical inclusions. The following sections summarize the work done in the previ-
ous chapters on composite modeling followed by a description of future work in the

field of composite simulation and design.

6.1 Material Characterization

In this dissertation, material characterization involves the process of indirectly mea-
suring the electromagnetic properties of materials. Two of the various material char-
acterization methods employed at Michigan State University have been introduced
in section 2.3. One of the methods uses the E4991A Impedance Analyzer to mea-
sure the capacitance and inductance of materials separately in order to calculate
the complex permittivity and permeability, respectively. This allows for electromag-
netic characterization for disk-shaped materials of fairly small size (diameter< 20mm;
thickness< 3mm) in the frequency range of 100MHz-lGHz. The second method uses
a stripline field applicator and a two-dimensional root-searching algorithm to extract
the complex permittivity and permeability from the measured scattering parameters.
This characterization technique involves a stripline with a center conductor that sup-
ports a TEM wave normally incident on the material being interrogated in the fre-
quency range of 1-18GHz. This is the basis for the electromagnetic models created
in this dissertation for layered composites and composites with cylindrical inclusions

where the scattering parameters are the focus of the calculations. These calculated

139

scattering parameters are used in conjunction with the root-searching algorithm to

determine the effective permittivity and permeability of the composites.

6.2 Composites and Unique Contributions

6.2.1 Composites with Spherical Inclusions

Composites with spherical inclusions have been investigated in [3] and summarized
in this work for sake of entirety. Three of the many classical mixing laws for effective
permittivity and a composite consisting of small dielectric particles dispersed through-
out a dielectric background were summarized. Based on the concept of duality, these
classical mixing laws were used to predict permeability of a magneto-dielectric com-
posite. Through experimental measurements, the author was able to show that the
classical mixing laws were capable of accurately predicting the effective permittivity
of a magneto-dielectric composite for low volume fractions (e.g. f § 0.3). However,
these mixing laws are not dependable for predicting the effective permeability nor
were they capable of determing effective permittivity for composites with high vol-
ume fractions. The reason for this is due to the lack of particle to particle interactions

in the classical mixing law formulations.

6.2.2 Composites of Alternating Layers

Using the method of wave matrices, composites consisting of alternating layers were
analyzed for both isotropic and anisotropic materials. This formulation provided an
analytic solution for the scattering paramters of a layered composite from which the
effective permittivity and permeability can be extracted using the same methods as
the stripline material characterization techniques. This fast and relatively straight-
forward process for calculating the scattering parameters provided a very useful tool
in the analysis of how the electromagnetic properties are effected by various aspects of
layered composites. The author was able to demonstrate three degrees of freedom in

the design of a layered composite with tunable permittivity and permeability. These

140

degrees of design freedom include: the number of layers, the volume fractions of the
materials, and the difference in the constitutive parameters of the materials being
combined. The latter of these is termed the dielectric or magnetic contrast ratio.
Using actual materials, namely YIG and Teflon, the author showed how the effective
permittivity and permeability varied as the number of layers and the volume fraction
were increased. As a final illustration of the usefulness of this formulation, a material
exhibiting both negative permittivity and permeability was demonstrating by inter-
changing the Telfon layers with plasma tubes. By controlling the external bias field,
the permeability tensor of YIG is shown to have negative values at frequencies greater
than ferrimagnetic resonance and the permittivity tensor of the plasma yields nega-
tive effective values at frequencies below the plasma frequency. An important point
in this example is that the coalescence of these materials biased by a static magnetic
field revealed that a double-negative material (i.e. simultaneous negative permittivity

and permeability) is acheivable without the use of periodic metallic structures.
6.2.3 Composites with Cylindrical Inclusions

The final composite analyzed in this dissertation is one involving cylindrically shaped
inclusions. An integral equation formulation was utilized to model a penetrable two-
dimensional geometry. The periodicity of the composite was incorporated in the
periodic Green’s function and it was shown how the Poisson summation formula was
able to accelerate its convergence. Just as in the layered composite analysis, the ma-
terials being used to form the mixture are YIG and Teflon. The first analysis of this
composite involves increasing the number of cylinders in the direction of propagation
of the incident wave. It was shown how the number of cylinders, whether in-line or
staggered, has no change on the effective permittivity or permeability for inclusions
spaced evenly in both the directions transverse and parallel to the direction of propa-
gation. In the final portion of this section, the author demonstrated how the volume

fraction of the YIG was able to yield a tunable range of permittivity and permeabil-

141

ity. An even greater permeability was achieved in the case of a YIG background with
either Teflon or free-space cylindrical inclusions. However, this configuration yielded
loss tangents on the order of 10‘2. In conclusion, the only degrees of freedom in
the design of a composite with cylindrical inclusions are the volume fraction and the

contrast ratios.

6.3 Future Work

The work presented in this dissertation formed a foundation for RF composite design
by analyzing three geometries of two-phase mixtures. In the analysis of layered com-
posite, only the case of normal incidence has been considered. This has allowed the
author to simulate the characterization technique used by the stripline field applicator
in which scattering parameters for normal incidence are used to extract permittivity
and permeability. However, the effects of propagation in the direction not parallel
to the axis of the stack of layers could yield interesting values for permittivity and
permeability. This might involve altering the root searching algorithm used in the
extraction process to account for off-normal incidence. In the calculation of the re-
flection and transmission coefficients, Snell’s law must be obeyed at each interface.
Also, in this dissertation the layered composites have consisted of mixtures of two
dissimilar materials and the effects of layering, volume fraction, and contrast ratio
have demonstrated a tunable permittivity and permeability. The use of multiple ma-
terials in various combinations could present many different results and therefore an
optimization routine for layered composites should be implemented that allows the
user to specify either a desired amount of reflection and transmission or a specific
permittivity and permeability.

In the geometry with rod shaped inclusions aligned along the z-axis, it was shown
how the volume fraction of YIG was able to alter the permittivity and permeability

of the composite. However, rather high volume fractions were a necessity for ap-

142

preciable tunable permeability with the consequence of loss tangents on the order of
10—2. Future analysis of this form of geometry would involve finite length cylindrical
inclusions aligned along the z-axis, but with propagation also along the axis of the
cylinders rather than perpendicular. The initial difficulty in this work would be the
need to overcome the demagnetization factor of the rod for bias fields transverse to
the direction of propagation.

Besides further analysis on the geometries presented in this disseration, other
geometries should be investigated. One such geometry is that of hexagonal flakes
evenly dispersed throughout a background material. These flakes could improve pack-
ing and also allow for greater flexibility in a composite mixture.

Also, characterization of materials is important in the design of composites. Veri-
fication of the models could be obtained through various characterization techniques
already utilized at Michigan State University if the composites discussed could be
manufactured for testing. Finally, applications for these materials should be provided
to demonstrate their usefulness. For example, using a properly designed magneto-
dielectric material to reduce the overall size of a patch antenna while simulataneously

increasing the bandwidth.

143

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