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This is to certify that the
thesis entitled

CLASSICAL MIXING APPROACHES TO DETERMINE
EFFECTIVE PERMITTIVITY AND PERMEABILITY OF A
TWO-PHASE MIXTURE

presented by

DANIEL STEVEN KILLIPS

has been accepted towards fulfillment
of the requirements for the

MS degree in ELECTRICAL ENGINEERING

 

 

@/

Major Professor’s Signature

mm

Date

 

MSU is an Affinnative Action/Equal Opportunity Institution

 

 

 

LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN Box to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

 

 

 

DATE DUE DATE DUE DATE DUE
- AU?
8 2 6:: g 2005
596% (I 2008

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c:/CIRC/DateDue.p65-p.15

CLASSICAL MIXING APPROACHES TO DETERMINE EFFECTIVE
PERMITTIVITY AND PERMEABILITY OF A TWO-PHASE MIXTURE

By

Daniel Steven Killips

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Electrical and Computer Engineering

2004

ABSTRACT

CLASSICAL MIXING APPROACHES TO DETERMINE EFFECTIVE
PERMITTIVITY AND PERMEABILITY OF A TWO-PHASE MIXTURE

By

Daniel Steven Killips

Presented in this thesis is a discussion of four classical mixing approaches to predicting
the relative permittivity of a two-phase mixture. These formulations are: Maxwell
Gamett, Clausius Mosotti, Bruggeman and Coherent Potential. All four of these will be
discussed in close detail and then compared to actual measured data from a mixture
comprised of small hexafen'ite spherical particles randomly dispersed throughout a non-
magnetic polymer background. A description will then be given on the accuracy of these
predictions and why some work better than others.

Based on the concept of duality, the electrostatic and magnetostatic formulations
obey the same conditions for a given geometry subject to appropriate boundary
conditions. Therefore the mixing approaches derived for permittivity should apply just
the same to permeability. These formulations are again calculated and graphed with
measured data in order to show how the relative permeability of a mixture is not as easily
computed as was the permittivity due to the interaction between particles that is evident
in magnetism.

Finally, an application of these predictions is presented along with conclusions

and future work.

To my wife Shana

iii

ACKNOWLEDGEMENTS

First I would like to thank all those in my life who have helped me get through the thesis
writing process with what I can only hope is my sanity. Special thanks goes to Dr. Leo
Kempel for giving me the opportunity to further my education and for the invaluable
experience I receive. Thanks also to Dr. Edward Rothwell for his excellent teachings and
his availability for questions and advice. I also wish to thank Dr. Dennis Nyquist for
participating on my committee and for helping with the measurement technique.

Much appreciation goes to Dr. Steve Schneider from AFRL/SNRR, Mr. Emil
Martisnek from AFRL/SNZ, and Dr. Jeffery Berrie from MRC for their sponsorship and
advice throughout my research. Special recognition goes to Dr. Mark Scott from MRC
whose knowledge in magnetism has been more than a great help with my work. Also,
thanks goes to Andrew Bogle for instructing me on the stripline and its measurement
technique.

Finally, the most important acknowledgment goes to my lovely wife Shana,

without her I would not be the person I am today.

iv

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. vi
LIST OF FIGURES .......................................................................................................... vii
CHAPTER 1 - Introduction ................................................................................................ 1
1.1 — Introduction ............................................................................................................ 1
1.2 — Permittivity ............................................................................................................ 1
1.3 — Permeability ........................................................................................................... 3
1.4 - Overview ................................................................................................................ 3
CHAPTER 2 - Dielectric and Magnetic Properties ........................................................... 6
2.1 - Permittivity ............................................................................................................. 6
2.2 - Permeability ............................................................................................................ 9
2.3 — Domains ............................................................................................................... 14
2.4 - Magnetic Materials ............................................................................................... 16
CHAPTER 3 — Mixture and Measurement ....................................................................... 18
3.1 - COD Composites ................................................................................................... 18
3.2 - Stripline Procedure ............................................................................................... 19
CHAPTER 4 - Mixing Formulations ............................................................................... 22
4.1 - Maxwell Gamett Permittivity ............................................................................... 22
4.2 - Clausius Mosotti Permittivity ............................................................................... 27
4.3 — Bruggeman Permittivity ....................................................................................... 35
4.4 - Coherent Potential Permittivity ............................................................................ 41
4.5 - Maxwell Gamett Permeability .............................................................................. 47
4.6 — Bruggeman Permeability ..................................................................................... 50
4.7 - Coherent Potential Permeability ........................................................................... 55
4.8 - Onsager Permeability ........................................................................................... 59
CHAPTER 5 - Applications .............................................................................................. 65
5.1 - Square Patch Antenna Bandwidth ........................................................................ 65
CHAPTER 6 — Conclusions and Future Work ................................................................. 71
6.1 - Conclusion ............................................................................................................ 71
6.2 - Future Work .......................................................................................................... 73

LIST OF TABLES

Table 1: Summary of major mixing formulas [22]. ........................................................... 5

vi

LIST OF FIGURES

Figure 1: Description of dielectric mixture with spherical inclusions placed in an
environment. 6,- is the permittivity of inclusions, ee is the permittivity of the
environment. u,- is the permeability of inclusions, It... is the permeability of the
environment. ............................................................................................................... 2

Figure 2.1.1: Description of polarization mechanisms [6]. ............................................... 7

Figure 2.2.1: Model of a magnetic field due to an electron moving along its orbital
around the nucleus. ..................................................................................................... 9

Figure 2.2.2: Illustration of orbital angular momentum in the z-direction for a d-orbital.
................................................................................................................................... 11

Figure 2.3.1: Bloch wall transition for adjacent domains with anti-parallel
magnetizations [25] ................................................................................................... 15

Figure 2.4.1: (a) magnetic moments randomly aligned in absence of magnetic field, (b)

magnetic moments exposed to applied magnetic field. ............................................ 17
Figure 3.1.1: Types of composites for magnetic mixing [1] ............................................ 19
Figure 3.2.1: Description of stripline used in measurements [8]. .................................... 19

Figure 3.2.2: Description of three cascaded two port networks; region a, region b, and
region s. ..................................................................................................................... 20

Figure 4.1.1: Description of dielectric mixture with spherical inclusions placed in an
environment. e,- is the permittivity of inclusions, 69 is the permittivity of the

environment. ............................................................................................................. 22

Figure 4.1.2: Plot of effective permittivity vs. changing volume fraction. ee = 3.5 and e,- =
16 ............................................................................................................................... 24

Figure 4.1.3: Maxwell Garnett calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=O.4. ............................................... 26

Figure 4.1.4: Maxwell Garnett calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=0.25. ............................................. 26

Figure 4.1.5: Maxwell Garnett calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=0.1. ............................................... 27

vii

Figure 4.2.1: Electric fields by a spherical inclusion. E is the applied field, and E is
the field inside the particle. ....................................................................................... 28

Figure 4.2.2: Clausius Mosotti calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=O.4. ............................................... 31

Figure 4.2.3: Clausius Mosotti calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=0.25. ............................................. 31

Figure 4.2.4: Clausius Mosotti calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=0.1. ............................................... 32

Figure 4.2.5: Comparison of Clausius Mosotti to Maxwell Garnett and Experimental data
for f=O.4 .................................................................................................................... 33

Figure 4.2.6: Comparison of Clausius Mosotti to Maxwell Garnett and Experimental data
for f=0.25. ................................................................................................................. 33

Figure 4.2.7: Comparison of Clausius Mosotti to Maxwell Garnett and Experimental data
for EOJ. ................................................................................................................... 34

Figure 4.3.1 : Bruggeman calculated permittivity graphed with measured permittivity of a
sample with volume fraction f=O.4. .......................................................................... 37

Figure 4.3.2: Bruggeman calculated permittivity graphed with measured permittivity of a
sample with volume fraction f=0.25. ........................................................................ 37

Figure 4.3.3: Bruggeman calculated permittivity graphed with measured permittivity of a
sample with volume fraction f—=O. 1. .......................................................................... 38

Figure 4.3.4: Maxwell Garnett, Bruggeman, and Experimental data graphed for
comparison for f=O.4. ................................................................................................ 39

Figure 4.3.5: Maxwell Gamett, Bruggeman, and Experimental data graphed for
comparison for f=0.25 ............................................................................................... 39

Figure 4.3.6: Maxwell Garnett, Bruggeman, and Experimental data graphed for
comparison for f=0.1 ................................................................................................. 40

Figure 4.4.1: Coherent Potential calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=O.4. ............................................... 43

Figure 4.4.2: Coherent Potential calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=0.25. ............................................. 44

viii

Figure 4.4.3: Coherent Potential calculated permittivity graphed with measured
permittivity of a sample with volume fraction f=0. 1. ............................................... 44

Figure 4.4.4: Maxwell Garnett, Bruggeman, Coherent Potential, and Experimental data
graphed for comparison for f=O.4. ............................................................................ 45

Figure 4.4.5: Maxwell Garnett, Bruggeman, Coherent Potential, and Experimental data
graphed for comparison for f=0.25. .......................................................................... 46

Figure 4.4.6: Maxwell Garnett, Bruggeman, Coherent Potential, and Experimental data
graphed for comparison for f=0.1. ............................................................................ 46

Figure 4.5.1: Maxwell Garnett calculated permeability graphed with measured
permittivity of a sample with volume fraction f=O.4. ............................................... 48

Figure 4.5.2: Maxwell Garnett calculated permeability graphed with measured
permittivity of a sample with volume fraction f=0.25. ............................................. 48

Figure 4.5.3: Maxwell Garnett calculated permeability graphed with measured
permittivity of a sample with volume fraction f=0. l. ............................................... 49

Figure 4.6.1: Bruggeman calculated permeability graphed with measured permeability of
the sample with volume fraction f=O.4. .................................................................... 51

Figure 4.6.2: Bruggeman calculated permeability graphed with measured permeability of
the sample with volume fraction f=O.25. .................................................................. 51

Figure 4.6.3: Bruggeman calculated permeability graphed with measured permeability of
the sample with volume fraction f=0. 1. .................................................................... 52

Figure 4.6.4: Maxwell Garnett, Bruggeman, and Experimental data graphed for
comparison for f=O.4. ................................................................................................ 53

Figure 4.6.5: Maxwell Garnett, Bruggeman, and Experimental data graphed for
comparison for f=0.25 ............................................................................................... 53

Figure 4.6.6: Maxwell Garnett, Bruggeman, and Experimental data graphed for
comparison for f=0.1 ................................................................................................. 54

Figure 4.7.1: Coherent Potential calculated permeability graphed with measured
permittivity of a sample with volume fraction f=O.4. ............................................... 55

Figure 4.7.2: Coherent Potential calculated permeability graphed with measured
permittivity of a sample with volume fraction f=0.25. ............................................. 56

ix

Figure 4.7.3: Coherent Potential calculated permeability graphed with measured
permittivity of a sample with volume fraction f=0.1. ............................................... 56

Figure 4.7.4: Maxwell Garnett, Bruggeman, Coherent Potential and Experimental data

graphed for comparison when f=O.4. ........................................................................ 57
Figure 4.7.5: Maxwell Garnett, Bruggeman, Coherent Potential and Experimental data

graphed for comparison when f=0.25. ...................................................................... 58
Figure 4.7.6: Maxwell Garnett, Bruggeman, Coherent Potential and Experimental data

graphed for comparison when f=0.1. ........................................................................ 58
Figure 5.1.1: Bandwidth for constant permittivity with permeability varying from 1 to

10 ............................................................................................................................... 66
Figure 5.1.2: Bandwidth for constant permeability with permittivity varying from 1 to

10 ............................................................................................................................... 67
Figure 5.1.3: Bandwidth plot for permittivity of 2.2 and permeability of 2. ................... 68
Figure 5.1.4: Bandwidth plot for permittivity of 5.1 and permeability of 1.5. ................ 69
Figure 5.1.5: Bandwidth plot for permittivity of 4.4 and permeability of 1. ................... 70
Figure 6.1.1: Comparison of volume percent to weight percent ...................................... 72
Figure 6.2.1: Types of composites. .................................................................................. 73

CHAPTER 1 - Introduction

1.1 - Introduction

Materials inherently have dielectric and magnetic properties which describe how they
interact with electromagnetic energy. Knowing these properties allows for a wide range
of uses for many materials. However, sometimes it is desirable to mix or combine

different materials in order to achieve the desired parameters. When this happens, it

becomes increasingly difficult to predict the values of the effective permeability (peff)

and effective permittivity (gefl). This has been the focus of research and study for over a

hundred years. Dielectric mixture theory was developed as far back as 1891 when
Maxwell tackled the problem of obtaining the electromagnetic characteristics of simple
magnetic mixtures [22]. The method for obtaining the dielectric properties has been the
bulk of the research over the years because most materials have an effective permeability
equal to one, unless dealing specifically with magnetism. Also, the mathematics

involved with solving for 8

617

is significantly less difficult and its behavior is more easily

predictable.

1.2 — Permittivity

Many classical approaches are available for determining the effective permittivity of a
mixture. Table 1 shows many of these approaches; however the four reviewed in this
thesis are Maxwell-Gamett, Clausius Mosotti, Bruggeman, and Coherent Potential [20].
The first two of these formulations, Maxwell Garnett and Clausius Mosotti, focus on the

inclusions affecting the mixture by being dispersed throughout a background medium

while the latter two cases approach the effective permittivity as an equal contribution
from the background as from the inclusions based on their respective volume fractions.
These models involve small dielectric spherical particles dispersed in a background with

a different permittivity. This is illustrated in figure 1.2.1.

 

8m-

 

ge’fle

 

 

 

 

Figure 1: Description of dielectric mixture with spherical inclusions placed in an environment. 6,- is the
permittivity of inclusions, 6e is the permittivity of the environment. (I,- is the permeability of inclusions,

Me is the permeability of the environment.

The ratio of inclusion volume to volume of the environment is called the volume fraction
(f). It will be shown how Maxwell-Gamett formulation, which looks at the effective
permittivity as an averaged value, relying heavily on volume fraction as well as the
perrnittivities of the inclusions and environment. Clausius Mosotti takes a different
approach by replacing each inclusion with a dipole moment as a result of the
polarizability of each spherical particle. These are two common and very basic mixing
principles and will be derived then compared to measured data. Bruggeman theory is
also analyzed for permittivity which looks at the effective permittivity as a result of the
inclusions and environment equally as much. Another well-known formula which is

relevant in the theoretical studies of wave propagation in random media is the so-called

Coherent Potential formula [20]. These latter two principles are also compared to the
measured data. The basis from which these concepts arise is electrostatics. A static
electric field can be defined simply as an electric field that does not change with time [4].
These assumptions can be made because the particle size is much smaller than a

wavelength. The maximum diameter a particle can be in order to make this

approximation is dmax and is directly proportional to wavelength [11].

d

III

2.
max 5 (1.2.1)

1.3 - Permeability

By the concept of duality, the electrostatic and magnetostatic problems are dual for any
given geometry [12]. Therefore the permeability would seem to be found by the same
methods as was done for permittivity simply by exchanging p. for e. The permeability of
the hexaferrite sample is compared to actual measured data using the classical mixing
models: Maxwell Garnett, Bruggeman, and Coherent Potential. Also, an important
mixing rule for effective permeability derived using Onsager’s field relations is

demonstrated for ferromagnetic materials [5, 19].

1.4 - Overview

One application of predicting the values for lie/f and ee/fis to find various engineering

design parameters such as the bandwidth of a square patch antenna with the mixed

material as the substrate background [7]. The material chosen for use in this thesis is a

hexaferrite spherical inclusion dispersed in a non-magnetic background. A closer look at

the geometry of this mixture can be seen in Figure 1.

In magnetism, hexaferrite is considered to be a ferrimagnetic material and was chosen

because of its low conductivity, 10"4 to 1 [18]. Another important characteristic is its
ability to maintain magnetic non-trivial properties at higher frequencies as compared to
ferromagnetic materials. Mixing becomes desirable when the properties of the material
alone do not allow for the necessary interaction with EM fields. For example,
ferromagnetic materials are often mixed in order to obtain a lower conductivity, yet retain
its desirable magnetic behavior. Ferrimagnetic materials which are used in this thesis are
mixed in order to obtain a relative permittivity that is lower, while retaining its magnetic
properties. If permittivity is lowered enough to be the same as permeability then
desirable matching characteristics are obtained in a square patch antenna or other devices.
This is because the reflection coefficient of the material goes to zero when the
permeability and permittivity are equal to each other. Also, the weight can be reduced if

mixed with a less dense material which is a very desirable characteristic.

The format of this thesis is as follows: First, a background on dielectric and magnetic
properties of materials is given, followed by a description of the measurement techniques
used to find the actual permeability and permittivity of the material. Then a few classical
methods are analyzed for permittivity, and later applied to permeability. Next will be a
description of one application which is that of bandwidth for a patch antenna, and finally

conclusions and suggested future work.

 

Description

Mixture Formula

 

Maxwell, 1891, spheres, 2-phase
Maxwell Garnett formula [15]

 

3Vi£e
Heirs.-+25.)/<e.-a.)I-V,.

 

Rayleigh, 1892, spheres, 2-
phase [14]

 

 

Wiener, 1912, arbitrary shape, 2-phase; u=form
factor; u = 288 for spheres; u = 281. for disks;

u =1/2[g, — 3g ) for needles [23]
1 e

 

 

 

Bruggeman, 1935 spheres,
2-phase [3]

 

 

Bottcher, 1945, spheres, 2-
phase [2]

 

 

E —a 1 a
z e e

5—8 a —a
e: r e

38 1 8i+28

 

Polder, van Santen, 1946
ellipsoids, multiphase [17]

 

 

Looyenga, 1965 spheres,
2-phase [13]

3(13
r 1 r h

 

 

Sihvola, Kong, 1988
ellipsoids, random
orientation [1 l]

 

3 Vial. —£hl£a +Ni(£_£h)i

gzgh+i§1 3[£a+Ni(£i-£h)]

WhCl‘C gazgh+a(8-8h) OR

 

 

 

 

 

Table 1: Summary of major mixing formulas [22].

 

CHAPTER 2 — Dielectric and Magnetic Properties
2.1 - Permittivity

In a microscopic sense, the electronic properties of matter depend on how tightly the
electrons are bound [6]. Materials with loosely bound electrons are called conductors.
When a conductor is placed in an electric field the electrons move in the opposite
direction with respect to the applied field. However, materials with tightly bound
electrons are called dielectrics. A dielectric material is a nonconductor of electric charge
in which an applied electric field causes a displacement of charge but not a flow of
charge [9]. This displacement involves moving the positive charges in the direction of
the applied field, and the negative charges in the direction opposing the field which

results in an electric dipole with its moment in the direction of the applied field. The

collection of such dipoles is represented by an electric polarization (P ).

The phenomenon of electronic polarization is caused by the displacement of the charge
center of the electron cloud with respect to the nucleus [20]. Again, at the microscopic
level, the polarization can be described with three mechanisms: electronic, atomic, and
orientational polarization [6, 20]. This is illustrated in Figure 2.1.1. Atomic, or ionic,
polarization occurs in materials where the molecules are formed by atoms which are
bound together with ionic bonds. The dipole formation occurs when an electric field

distorts the molecular orbit so that a net negative and positive charge center is created [6].

 

Mechanism No applied field Applied field

llilém

 

Orientational

Polarization /' T

2
\/

 

 

 

 

Atomic
Polarization _.
mc
+
Electronic *“
Polarization ® inc

 

 

 

Figure 2.1.1: Description of polarization mechanisms [6].

Orientational polarization occurs in polar materials that already have dipoles even in the
absence of an applied field. However, those dipoles are aligned randomly and have zero

net polarization until an electric field is applied causing the already existing dipoles at

align with the field. The electronic polarization vector P can be described as the sum of
dipole moments per unit volume, and can be represented by equation (2.1.1) where the

strength of a dipole moment p can be represented by equation (2.1.2).

P = N? (2.1.1)

I) : Qdfi (2.1.2)
N is the number density of dipoles, Q is the magnitude of a charge, of is the separation
between the positive and negative charges, and 12 is the alignment of the dipole from the

negative to positive charge.

These dipole effects divide dielectric materials into four groups based on their
polarization mechanisms: nonpolar, polar, electret, and ferroelectric [6]. Nonpolar
materials do not have any dipole moments in the absence of an applied field, but does
have a net polarization when a field is applied. Polar materials, on the other hand, do
have dipole moments when there is no impressed field but are aligned randomly and
cancel out so there is no net polarization. However, polar materials do form a net
polarization in an applied field. Electret materials do possess a net polarization in the
absence of an electric field, and ferroelectric materials have a nonlinear reaction to an

applied field, and can have a remnant polarization when the applied field is removed [6].

Electric flux density 5 can be represented by the electric flux density in a vacuum 50

plus the polarization P of the medium. This is shown in equation (2.1.3).

D =DO+P (2.1.3)

Where DO and P can be represented
0 : 80E (2.1.4)
P=%LE (us

where 50 is the permittivity of free space, E is the applied electric field, Xe = 8r —1 is
the electric susceptibility and Sr is the relative permittivity of the medium. The

permittivity is a bulk property of a dielectric material and can be used to describe its
reaction to an applied electric field. The relative permittivity of a mixture is one of the

desired characteristics that will be discussed in this thesis.

2.2 - Permeability

Unlike the dielectric properties of matter previously described, the magnetic behavior
needs to be analyzed more rigorously in order to be accurate, making it much more
challenging. The magnetic effects can be the result of one or all of the following three
things: net nuclear spin, asymmetric electron orbital, and net electron spin [6]. Generally
the nuclear spin contribution is much less than the contributions fiom the electron orbital
and spin. According to quantum theory, electrons can have a spin-up or a spin-down
state and an orbital is allowed to have only one of each spin state. A net magnetic field
will be produced if an orbital has only one electron due to its spin. A magnetic field will
also be produced by the movement of that electron along its orbital. These are the two

major contributions to the magnetic behavior due to its electrons.

The magnetic field due to the electron traveling along its orbital can be modeled as a
magnetic moment. A current is produced by the moving charge, and since the current is
traveling in a loop a magnetic field is produced according to the right hand rule. This is

illustrated in Figure 2.2.1.

Magnetic field due
to traveling charge

N)

Current due to
moving charge

  

Traveling

/ electron

Figure 2.2.1: Model of a magnetic field due to an electron moving along its orbital around the nucleus.

This field can be represented as the magnetic dipole moment due to the circling charge.

in- : 217212 (2.2.1)
Where 771‘ is the magnetic dipole moment, I is the current due to the electron and is
multiplied by the surface area of the orbital [18]. This model can be taken into further
consideration because these electrons can only have specific energy states and the orbitals
have specific positions. The principal quantum number n is the value that determines the
energy of a particular shell or orbit [16]. This number gives us the orbital angular

momentum quantum number I.

l = 0,1,2, ..... ,(n — 1) (2.2.2)

M=h [1+1 (22.3)

M is the total angular momentum of an electron due to its orbital motion, and h is

Planck’s constant. The magnetic quantum number m determines the component of the

l
orbital momentum along the direction of the applied magnetic field [16]. Therefore the
angular momentum in the z-direction can be determined by equation (2.2.4). This is

illustrated in Figure 2.2.2 for a d-orbital (1:2).

M = m,h (2.2.4)

10

 

 

Figure 2.2.2: Illustration of orbital angular momentum in the z-direction for a d-orbital.

Next, the intrinsic angular momentum is caused by the electron spinning about an internal

axis. The spin quantum number (ms) determines the component of the spin (3) along the

direction of the applied field [16]. Unlike the orbital angular momentum, this number is
restricted to i l / 2 . Thus the contribution of the spin angular momentum in the i z-

direction is restricted to i h / 2.

S = m 71 (2.2.5)

Z S
By quantum mechanical theory the total angular momentum .7}? of an atom containing a
number of electrons is equal to the sum of the total orbital angular momentum A771 and

the total spin angular momentum Sh [21].

J = M + S (2.2.6)
The orbital and spin magnetic moments that result from these values is seen in equations

(2.2.7) and (2.2.8).

11

 

71
mL = ”J M (M +1)(E§‘;‘j (2.2.7)

,u eh
ms = ‘x/ SIS + lip—20;) (2.2.8)

Where 8 is the charge of an electron and m is the mass of an electron. Since the charge of
an electron is negative, the magnetic moments are directed opposite to the vector sums of

their angular momentum [21]. The effective magnetic moment m1 of the atom is

directed along the same direction as .7 and is shown in equation (2.2.9).

 

,u eh
m1 2‘8 J J+1[ 20m) (2.2.9)

J(J+1)+S(S+1)—M(M —1)
2J(J+1) (2.2.10)

 

g=1+

g is known as the Lande g-factor and it determines the splitting of the energy levels in the
presence of a weak external magnetic field [21]. When an atom with a nonzero net
magnetic moment is exposed to an externally applied magnetic field its total angular

momentum precesses, moves with a gyrating fashion, about the magnetic field lines.

This happens because the electrons are fixed to their orbitals, therefore .7 precesses about
the field lines and not the electron itself. When there is more than one atom present, the
predictions become more complicated because of coupling between the different atoms
[6]. The orbital contributions to the net magnetic moment tend to cancel due to their

alignment in the crystalline phase of the transition metals [21]. The coupling that occurs

12

between adjacent atoms is due to their spins and can be represented by exchange

 

energyWex.
W... = -2JeS,- '5,- (2.2.11)
J _ 3ch
e ZZS(S+1) (2.2.12)

In equation (2.2.12) k is Boltzmann’s constant, Tc is the Curie temperature, and Z is the

number of nearest neighbors. Si and Sj are the total spin angular momentums of the

atoms located at i and j respectively. Je is known as the exchange integral which

expresses the difference in Coulomb interaction energy of the system when the spins are

parallel or anti-parallel [21]. For atoms with spins aligned parallel to each other, Je is

positive and the result is ferromagnetism and there is a net magnetic moment due to those

atoms. However, if the spins are anti-parallel Je is negative and the result is anti-

ferromagnetism with a net magnetic moment of zero due to those atoms.

When looking at a magnetic material in a macroscopic sense, there exists a magnetic

polarization vector similar to the electric polarization vector discussed in the dielectric
material section. The magnetic flux density E that results from an applied field I? is the

sum of the flux density in a vacuum E0 and the magnetic polarization vector of the

material 117 , commonly referred to as magnetization.

B— : 30 + 1‘7 (2.2.13)

13

where

E0 = .UoH (2.2.14)

M = floZmH (2.2.15)

17 is the static applied field, ,uo is the permeability of free space, gm = ,u, ——1 is the

susceptibility, and p, is the bulk relative permeability of the material. Equation (2.2.13)

can now be written in terms of the static applied field and the permeability.

B :2 ,uoyrH = ,uH (2.2.16)

Permeability is a very important quantity in electromagnetics and is part of the focus of

this thesis.

2.3 — Domains

In section 2.2, the magnetic moment was described for a single atom. When adjacent
atoms have parallel moments they form a domain within the material. The domain will
have a net magnetization in the direction of the aligned moments within it. In an un-
magnetized material the domains randomly align yielding a net magnetization of zero.
How these domains react to an externally applied field determines its type of magnetic
material. The transition layer between adjacent domains magnetized in different
directions is called a Bloch wall. The thickness of this wall is found from the exchange
energy, which tends to increase its thickness, and the anisotropy energy, which tends to

decrease its thickness [21]. For example the wall thickness for iron with domains whose

14

magnetization is aligned anti-parallel is about 10‘7 meters [21]. Figure 2.3.1 illustrates

this transition region.

 
   
 
  
  
   
 
 

  

." y WSW
is: ass-ta. {‘4’
o a ' \ .1

g‘tt fi‘.‘ 5‘?“ -‘- I

a: * or

"137'? . .
I’lu ’45". 9,?" .

 

 
     

4,?

1. #5 f ‘
‘c
D

J

‘ <=f*d

 
 

Figure 2.3.1: Bloch wall transition for adjacent domains with anti-parallel magnetizations [2 5].

The mixture discussed in this thesis is assumed to have magnetic inclusions that are small
enough that each particle is one single domain. When trying to find the effective
permittivity and permeability of a mixture it is easier if the particles each represent only
one net magnetization, whereas a multiple domain particle would have different
magnetization directions within itself. The particles also need to be far enough away
from adjacent particles in order to not interact with one another which can change the
ability to be single domain. When this happens the material no longer exhibits the

characteristics that were desired by a mixture of single domain inclusions.

2.4 - Magnetic Materials

The sum of all magnetic moments in a volume is the magnetic polarization vectorIW .
This is also referred to as the saturation magnetization when all the magnetic dipoles in a
volume are aligned along the applied magnetic field [6]. Depending on whether the sum
of moments adds to or subtracts from an applied field, magnetic materials can be split
into two categories. The first involves materials with relative permeabilities less than
one, and whose magnetization vector acts against an applied field. This type of material
is called a diamagnetic material. In the absence of an applied field, the orbital and spin
moments cancel, and there is no net magnetic moment on each atom [18]. The second
category involves those materials whose net moments align with an applied field. This
type of material is called a paramagnetic material, and in the absence of an external field
the spin moments are greater than the orbital moments leaving individual atoms with a
net permanent magnetic moment. A common example of a paramagnetic material is that
which is found in kitchen magnets. In either case, the density of magnetic moments A7
is zero in the absence of an applied field [18]. In the second category, there are three sub-

categories; ferromagnetic, anti-ferromagnetic, and ferrimagnetic materials.

Ferromagnetic materials, such as nickel and iron, have all their magnetic moments
aligned and parallel to each other. Due to this alignment, ferromagnetic materials have a
very high polarizability, otherwise known as magnetizability in terms of magnetism, and

add to an applied field easily [20]. Regions where these aligned moments create a

nonzero magnetization 1T4— in the absence of an external field are called magnetic

domains.

16

 

 

 

 

 

 

 

4‘”, W i;

 

 

 

 

 

 

(al I (bl I

Figure 2.4.1: (a) magnetic moments randomly aligned in absence of magnetic field, (b) magnetic moments
exposed to applied magnetic field.

In the macroscopic sense, these domains all orient in a way that the net magnetization is
zero. Anti-ferromagnetic materials have magnetic moments which are aligned anti-
parallel to each other canceling out any net magnetization. Because of this orientation
the polarizability of anti-ferromagnetic materials is very low. Finally, ferrimagnetic
materials have moments that also align anti-parallel to each other but they are not equal

in magnitude resulting in a net magnetic field.

Ferrites are the material focused on in this thesis. Ferrites are considered a very useful
group of ferrimagnetic materials. Developed in the 1940’s by researchers at the Phillip
Laboratories as low-loss magnetic media for supporting electromagnetic waves, ferrites
have very low conductivity, very high relative permeability and relative permittivities in
the range 10-15 [18]. The ability to maintain magnetic characteristics at higher
frequencies is another benefit of fen'imagnetic materials. Also the saturation

magnetization tends to be much higher than that of ferromagnetic materials.

17

CHAPTER 3 — Mixture and Measurement
3.1 - COD Composites

COD composites are mixtures with magnetic inclusions, often spheroids and ellipsoids,
embedded in a nonmagnetic medium [1]. The volume fraction (f) of this material is the
volume of the inclusions to the volume of the background and varies between 0 and 1.
For example, when f =0.25 then 25% of the total volume is the inclusions while 75% is
the surrounding matrix. In this thesis, one type of material has spherical particles with a
radius of approximately 2 microns and has been mixed with volume fractions of 0. 1,
0.25, and 0.4. According to Sihvola and Kong [1 1], the maximum diameter of the
spherical inclusions allowed in order to apply the rules of electrostatics and

magnetostatics is given by equation 3.1.1.

ah...x E— (3.1.1)

Here, A is the wavelength of the incident field. This is a necessary assumption for
application of the mixing principles. There has been a significant amount of work
dedicated to determining the effective permeability and effective permittivity of such a
mixture. Some of the classical methods will be looked at in this paper such as: Maxwell
Garnett, Claussius Mosotti, etc. Figure 3.1.1 shows the COD composite as well as C1D
and C2D which represent respectively magnetic wires and magnetic thin films embedded
in the background. The C1D and C2D composites will be discussed later in the section

on future work.

18

C2D C1D COD

 

 

 

 

 

 

OD 000
o _. o oo 00
we 0 o o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.1.1: Types of composites for magnetic mixing [1].

3.2 - Stripline Procedure

The testing procedure for extraction of permeability and permittivity was done using a
stripline waveguide and a network analyzer which is used to measure the S—parameters.
Using the measured S-parameters and a root search program written in Matlab, we are
able to obtain the complex permeability and permittivity of the sample over a wide
bandwidth if necessary. Figure 3.2.1 shows the setup for the stripline waveguide and
placement of the sample material. The tuning wedges are grounded, conducting,

triangular structures which are used to maintain a nearly uniform width-to-height ratio.

k... "we—w

    
  

center conductor

 

 

  

ground plane
material

tuning
wedges

Figure 3.2.1: Description of stripline used in measurements [8].

The stripline is then split into three cascaded two port networks. This is illustrated in
figure 3.2.2. The three regions are described as the sample region ( s ) and the

surrounding air filled regions ( a ) and ( b ).

 

 

 

 

 

[sa] [8.] [Sb]

 

 

 

 

 

 

 

 

 

 

 

Figure 3.2.2: Description of three cascaded two port networks; region a, region b, and region 3.

By equating the measured S—parameters with the following expressions, the interfacial

reflection coefficient (F) and the propagation factor ( B ) is found.

SS Fl—z2
11 =
1_ (1.2)2 (3.2.1)
zl—F2
y =
2‘ 1_ (1.2)2 (3.2.2)
was
z=e an)

a)
,3 = k = :x/grflr (3.2.4)

20

Now using the values for ,6 and F the relative permittivity and permeability can be

determined using equations (3.2.5) and (3.2.6) respectively.

_.3 1-F
a, ‘7‘; 1+1— (3.2.5)

_,6 1+1“
(4“,; l—F (3.2.6)

These equations are an example of how the complex relative permittivity and

permeability of a sample placed in a waveguide can be determined. However, the

stripline does not allow for measurements directly near the sample. Therefore, the SSH

and 8521 must be extracted from the S-parameters measured at the ends and the structure.

This can be accomplished using a de-embedding technique. For more information on this

technique, see Infante [8].

The stripline is very useful because it supports a transverse electromagnetic (TEM) wave,
which has a zero frequency cutoff allowing for very low frequency measurements. Also,
when the stripline is designed properly, the TEM wave will allow for a large bandwidth
of 100 MHz up to 18 GHz [8]. The characteristic impedance of the structure is
determined directly from its height and width, which allows for ease in the transition

from the stripline to the coaxial cable.

21

CHAPTER 4 - Mixing Formulations
4.1 - Maxwell-Garnett Permittivity
One of the more classical approaches to finding the permittivity of a mixture is the
Maxwell-Garnett formula. This relation looks at a mixture of dielectric materials from a
macroscopic viewpoint using volume-averaged fields. Consider Figure 4.1.1 in which
spherical inclusions are randomly dispersed throughout a medium, called the

environment or matrix.

8.41.-

8e’lue

 

Figure 4.1.1: Description of dielectric mixture with spherical inclusions placed in an environment. 6,- is the
permittivity of inclusions, 6,. is the pemiittivity of the environment.

The spherical particles are allowed to be randomly positioned, where f is the volume
fraction of inclusions to environment. Therefore, 1 - f is the volume fraction of the
environment to the inclusions. Relating the macroscopic values of the electric flux

density to electric field quantity gives the following:

<5>=8eff<E> (4.1.1)

<5>=éi5dV (4.1.1a)
V

22

where 8eff is the permittivity of the entire mixture and the symbol < > denotes the

volume average represented by equation (4.1.1a). Defining the macroscopic fields in

terms of their volume fractions gives:

<13 > = f 81E: +(1"f)5eEe (4.1.2)
<§>=fE,+(1—f)Ee (4.1.3)

Where Ei is the magnitude of the field within the inclusion and Ee is the magnitude of

the field surrounding the inclusion and throughout the environment. Inserting (4.1.2) and

(4.1.3) into equation (4.1.1) and solving fora

eff:

8 _f£iEi+(l_f)5eEe
W 1E.+(1—f)E.

 

(4.1.4)

For an inclusion that is much smaller than a wavelength allowing for use of the

electrostatic field solution and Rayleigh scattering by a spherical object [10], Ez’ and Ee

can be replaced by the ratio Ee / Ei .

 

E 38
e = 9 (4.1.5)
El. 6,. + 286

Formally dividing (4.1.4) by Ei and utilizing (4.1.5) yields the Maxwell Garnett mixing

formula.

23

_ fa. +(1—f)e.(3a. ((8.- + 24.))

 

 

8 — (4.1.6)
9” f +(1—fX38. /(6.- + 28.))
Equation (4.1.6) can be rewritten in the more commonly seen form [20]:
(81' — 8e )
eefl = ge + 3 fee (4.1.7)

81' +28e—f(8i —ge)

This equation is considered to have a solid foundation in quasi-static cases where the size

of the inclusions is much smaller than wavelength. Looking at a plot of relative

permittivity found using equation (4.1.7) vs. the volume fraction one can look at the two

extreme cases for a mixture purely made of inclusion and a mixture with no inclusions.

This is seen in Figure 4.1.2.

Effective Relative Permittivity

Maxwell Garnett Effective Permittivity vs. Volume Fraction

18

A
O
A

 

 

 

— Real 6

 

O
1

a:
L

‘1
l

 

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Volume Fraction

Figure 4.1.2: Plot of effective permittivity vs. changing volume fraction. 66 = 3.5 and E,- = 16.

24

For a volume fraction of 0, corresponding to no inclusions, the equation reduces to

gejj’ 2 Se which is just the permittivity of the environment alone. Also, for a volume
fraction of 1, equation (4.1.7) becomes 8eff = 8i which is the permittivity of the

inclusions. However, the latter case is not physically realizable because of the spherical
shape of the inclusions it would be too difficult to get a volume fraction of 1 since there is
always some interstitial volume. But this does show that the Maxwell Garnett formula

should be accurate at least in the regions closer to the extreme cases.

Equation (4.1.7) has been graphed over a frequency range of 2 GHz to 12 GHz for three
different volume fractions; 0.1, 0.25, and 0.4. The inclusions are hexaferrite spherical
particles of permittivity 16 with a radius of 1.91 pm dispersed in a background with a
relative permittivity of 3.5. Although the calculated effective permittivity is independent
of frequency it is easier to compare it to the measured data if it is plotted as constant over
this range. Measured data is taken from our strip-line measurements. A detailed
description of this measurement technique can be seen in section 3.2. Figures 4.1.3, 4.1.4

and 4.1.5 show these comparisons with volume fractions 0.4, 0.25, and 0.1 respectively.

25

Maxwell Garnett Permittivity [ f=O.4]

 

 

4 — Ep Real [Exp]
-— Ep Real [MG]

 

 

 

Relative Permittivity

   

 

4% 3 4 5 6 7 8 9101112
Frequency (GHz)

Figure 4.1.3: Maxwell Garnett calculated permittivity graphed with measured permittivity of a sample with
volume fraction f=O.4.

Maxwell Garnett Permittivity [ f=0.25]

 

 

— Ep Real [Exp]
— Ep Real [MG]

 

 

 

Relative Permittivity
&

 

 

 

3 -1

2 u

12

O mmmlz.11:n11111111111111!111115.11111?11mmunmmhmnmmn rmwmm

42% 3 4 5 6 7 8 9101112
Frequency (GHz)

Figure 4.1.4: Maxwell Garnett calculated permittivity graphed with measured permittivity of a sample with
volume fraction f=0.25.

26

Maxwell Garnett Permittivity [ f=0.1 ]

 

4 —- Ep Real [Exp]
3 4 — ' ‘7 ‘ -—“— Ep Real [MG]

 

 

 

Relative Permittivity

 

 

-1 3456789101112
Frequency (GHz)

Figure 4.1.5: Maxwell Garnett calculated permittivity graphed with measured permittivity of a sample with
volume fraction #0. 1.

These figures reveal that as volume fraction is reduced, the accuracy of Maxwell Garnett
formulation becomes more accurate. This occurs because equation (4.1.7) does not take
into consideration the effect of particle-particle interaction. As the volume fi'action

becomes smaller, the separation between adjacent inclusions is increased allowing for

less, if any, interaction.

4.2 - Clausius Mosotti Permittivity

The Clausius Mosotti relation involves polarizability of the individual spherical

inclusions. This polarization is averaged by replacing the scatterers with dipole moments

[20].

27

The dipole moment 5 of a particle can be defined by:

I3 = 055' ++(4.2.1)

where a is the polarizability, and E‘ is the local electric field. I? can be represented by

the Mosotti field [18].

E: E + L (4.2.2)
380

Where E is the applied electric field, 7’ represents the macroscopic dipole moment
density, also called the Electric Polarization vector, and F/ 380 is the field resulting from

the polarized molecules and is found outside the particle often called the “Lorentz field”.

 

lTTTTTTTF

Figure 4.2.1: Electric fields by a spherical inclusion. E is the applied field, and F is the field inside the
particle.

Using equation (4.2.2), the macroscopic dipole moment density I? can be found by:

F = Naif': Na[1'~:' + i] (4.2.3)
380

28

where N is the number density of dipoles. Solving for I—3

F = (M); (4.2.4)
380 — Na

This is the macroscopic dipole moment density found due to the applied field which is

propagating in free space and can be represented I? = 1860177 .

Z __ 3Na
e _——380—Na (4.2.5)

This is the susceptibility and can be represented le 2 5r —1. Using the susceptibility to

solve for permittivity

 

3 + 2Na / 80
8 = 808, = 80 (4.2.6)
3 — N a / 80
and rearranging to solve for polarizability
380 (8 — 1)
a = —-—5——— 4.2.7
N (a, + 2) ( )

gives the Clausius Mosotti formula for polarizability [18]. Equations (4.2.6) and (4.2.7)
can be manipulated to describe a spherical inclusion surrounded by a dielectric medium.

By replacing £0 with ac for the relative permittivity of the environment in equation

(4.2.6) and writing (4.2.7) in terms of a dielectric sphere

_ 3+2Na/ae
54f - 5e 3—Na/8 (4.2.8)

 

29

Remembering that [5 = a1? , i5 = VP and using the static field solution for a dielectric

sphere immersed in a field (4.2.9)

38
E1. = ——e— E, (4.2.9)
8,. + 28,
the polarizability a can be found to be
8, _ 8 (4.2.10)
a=3QV—4—JL
q+2g

V is the volume of the spherical particles. Equations (4.2.8) and (4.2.10) will give the

effective permittivity of a spherical inclusion sparsely distributed in a dielectric host.

Using the same material as in the Maxwell Garnett section, the Clausius Mosotti formula

has been graphed versus frequency for 3 different volume fractions: 0.4, 0.25, and 0.1.

The particle radius is 1.91 pm. The measured data is as before.

30

Clausius Mosotti Permittivity [ f=O.4]

 

 

7 4 VAN
— Ep Real [Exp]

4 ~ — Ep Real [CM]

 

 

 

Relative Permittivity
01

 

 

O 1.’ ' Ti‘Tllillll'UlHll 1.! .-"lllllllllll’llilllll Jillll llllllllillllll'lllli llllll l-"l"illlll'llllllllnllil'll !'= ' Il'l'lllllll llllllllllllll llll i' ”lili‘i' llTlY'llll W

2 3 4 5 6 7 8 9101112
Frequency (GHz)

Figure 4.2.2: Clausius Mosotti calculated permittivity graphed with measured permittivity of a sample with
volume fraction f=O.4.

Clausius Mosotti Permittivity [ f=0.25]

74

' m l—Ep Real [Exp]
4 ~ —Ep Real [CM]

 

 

 

 

Relative Permittivity
U'I

 

 

O TWWTWTW'TW .1. .: 1.. (WTWUWUWTTWUW

2 3 4 5 6 7 8 9101112
Frequency (GHz)

Figure 4.2.3: C lausius Mosotti calculated permittivity graphed with measured permittivity of a sample with
volume fraction f=0.25.

31

Clausius Mosotti Permittivity [ f=0.1 ]

 

 

 

 

 

 

9-
8..
E 7*
.2 -
E 6
E 54
g 4 - gk‘ __ —EpReal[Exp]
g 3_ ' ' fi “ ' "“‘ ' -‘-—‘-—:— —-EpReal[CM]
a 2*
l! 1...

 

0 W'rrrrrmrmnmnmmmrmmmnmmn ,H‘H Hllllllll ill in Ill , 1. 1

42 3 4 5 6 7 8 9101112
Frequency (GHz)

 

 

Figure 4.2.4: Clausius Mosotti calculated permittivity graphed with measured permittivity of a sample with
volume fraction f=0.1.

Similar to Maxwell Garnett results, the Clausius Mosotti formula seems to be more
accurate as the volume fraction decreases. Although this formulation seems to be a less
averaged approach to modeling effective permittivity as the Maxwell Garnett by
replacing each particle with a dipole, it still does not account for any interaction between
particles. Therefore as the inclusions are more sparsely distributed, the more accurate the
prediction can be. The following three figures show the results from Clausius Mosotti and

Maxwell Garnett together for purposes of comparison.

32

Maxwell Garnett, Clausius Mosotti, and
Experimental Permittivity for f=0.4

 

 

 

- ...-.. Ep Real [MG]
- - - -Ep Real [CM]
— Ep Real [Exp]

 

 

 

IWTWWWWWWTWWWTWmlWIWWW

23456789101112

Frequency

Relative Permittivity
O -3 N on 45 01 O) \l on CD

Figure 4.2.5: Comparison of C lausius Mosotti to Maxwell Garnett and Experimental data for f=0.4

Maxwell Garnett, Claussius Mosotti, and
Experimental Permittivity for f=0.25

64 Exp

 

 

5 W ~--Ep Rea|[MG]

MG, CM
- - - -Ep Real [CM]
3 ~ —Ep Real [Exp]

 

 

 

Relative Permittivity
.p.

 

 

0 Hill. .Hlllllliilllllli rlllldllllllllilll lllllilllll‘ -"||llllli;l‘llllllll]1.11£1 - illlillllllllllnluillilllli lfillillllrmllllllllililllllIlillllillllllllllllllllll

23456789101112

Frequency

Figure 4.2.6: Comparison of Clausius Mosotti to Maxwell Garnett and Experimental data for f=0.25.

33

Relative Permittivity

01

.b

O)

N

—8

C

Maxwell Garnett, Clausius Mosotti, and
Experimental Permittivity for f=0.1

 

P '.u-'—'- ”W'—* -‘w-

 

 

llllliIJlllllz 'llllllI‘ll”l‘ll-llllllli'll'llllh‘llilllli.lllllilllli,liliilllllllliulllll "lllllllllllIllllllslllllllllllllllll - llllllllllllllllE'llllllllllllllllllllll

2 3 4 5 6 78 9101112
Frequency

 

 

:--- Ep Real [MG]
. - . -Ep Real [CM]
_ Ep Real [EXP]

 

Figure 4.2.7: Comparison of Clausius Mosotti to Maxwell Garnett and Experimental data for f=0. 1.

equation.

sz/V

following formula.

3 + 2(f / V)38eV[(8, — 8e)/(8l. + 28e)](1 / 88)

Figures 4.2.5, 4.2.6, and 4.2.7 show that for the geometry used here the Clausius Mosotti
and Maxwell Garnett formulations predict the same effective permittivity. This led to
further investigation into the two equations. When the polarizability is as given in
equation (4.2.10) and the number density N is represented as in equation (4.2.11) the

Clausius Mosotti formula can be shown to be exactly the same as the Maxwell Garnett

(4.2.11)

Inserting (4.2.10) and (4.2.11) into equation (4.2.8) for effective permittivity we get the

 

84 2 5e 3—(f/V)3e.V[(e. —e.)/(8.- + 25,)11/6.)

34

(4.2.12)

 

Dropping out like terms, dividing by 3 and multiplying by l8l. + 286 M81. + 288) we get

equation (4.2.13).

81' +282» _f(£i -8e)+3f(gi —8e)
8428-44-8.) (“13>

 

8eff =89

Realizing the first three terms of the numerator in (4.2.13) are equal to the denominator,
this equation can be further simplified to give equation (4.2.14) which is exactly the same

as equation (4.1.7) defined in the previous section as the Maxwell Garnett formula.

(81" 8e)

8. +28. -f(8.- -8.)

 

8.), = 8.. + 38..f (4.2.14)

This explains why the effective permittivity found using the Maxwell Garnett and the

Clausius Mosotti yield the same results for the dielectric spheres.

4.3 — Bruggeman Permittivity

The Bruggeman formula for the homogenization of a mixture is widely used the
electromagnetics. It is known for different names depending on where it is used; in the
remote sensing community it goes under the names Folder-van Santen and the de Loor
formula. It is also known as the Béttcher formula or in materials science it is often
referred to as the effective medium model [20]. This model analyzes the polarization of
the mixture as a function of the environment equally as much as the inclusions. The
Bruggeman philosophy is represented in equation (4.3.1) where N is the number of
isotropic phases. For the case in this thesis N=2; one is for the environment and the other

for the spherical inclusions.

35

N _.
ijwzo (4.3.1)

Letting j =1 represent the environment phase ee = 61, and the volume fraction f1 = (I - f).

Now for j =2, the inclusion phase permittivity will be e) = 62 and the volume fraction f2==f.

Now writing out the summation for these two terms we get the following formula.

81‘ - gefl

8e _ gaff
(1-f)—+f———=O (4.3.2)
8e + 28,]!— 8,. + 2861,

Bringing the first term in (4.3.2) over to the right side of the equation and multiplying out

the denominators we get equation (4.3.3).

f(8i _ gefl Xge + 2880' ): (.f _1Xge — 861:7 x51 + 2881]) (4°13)
This can be multiplied out and put in the form of a quadratic equation with the following

coefficients.

2
EeflA+SeflB+C= 0

A = —2
B = 2f8,. — 3f8e + f8, + 288 — 81. (43-4)
C = 81.86,

Solving the quadratic formula represented by (4.3.4) and taking the non-negative result
gives the effective permittivity of the mixture. This was done for the material of
hexaferrite inclusions with a relative permittivity of 16, and enviromnent with relative

permittivity of 3.5 for all three volume fractions; 0.4, 0.25, and 0.1.

36

Bruggeman Effective Permittivity [f=O.4]

 

 

 

 

9 _
8 4M
3 7
g 6- W
a 5 4 —-Ep Real [Exp]
4, 4 - -—Ep Real [Brugglj
5 3 4
1!
at 2 “
1-
0 _

 

|l.. . ll llllllllllll l l‘x.l.‘l l lll lllllll 1.41111111111anme

2 3 4 5 6 7 8 9101112
Frequency

Figure 4.3.1: Bruggeman calculated permittivity graphed with measured permittivity of a sample with
volume fraction f=O.4.

Bruggeman Effective Permittivity [f=0.25]

 

 

 

 

7 1.
6
g M
re 5 J
E 4 1 -—Ep Real [Exp]
3 3 ~' l—Ep Real [Brugg]
E 2 ~
e
n: 1]

0 lFWT. (I I]. .l ll.l H ..ll.lllllllTlTTTT'-T"W‘.11Tm‘l..1.. lul'.1 lllll"- ll ””17. llllll '. .1” Wu. ll 1! l mu .ITTTTTWTTU

2 34 56 78 9101112

Frequency

Figure 4.3.2: Bruggeman calculated permittivity graphed with measured permittivity of a sample with
volume fraction f=0.25.

37

 

Bruggeman Effective Permittivity [f=0.1]

h 01 O)
1 1

l

 

 

— Ep Real [Exp]
- Ep Real [Bulge]

 

N
1_

Relative Permittivity
00

 

 

2 3 4 5 6 7 8 9101112
Frequency

Figure 4.3.3: Bruggeman calculated permittivity graphed with measured permittivity of a sample with
volume fraction f=0.1.

The results shown in Figures 4.3.1, 4.3.2, and 4.3.3 seem to be fairly close to the
measured data. The following three figures compare these results using the Bruggeman

formula to the results obtained from the Maxwell Garnett (Clausius Mosotti) equation.

38

Maxwell Garnett, Bruggeman, and Experimental
Permittivity for f=0.4

 

 

-Ep Real [MG]
- - - -Ep Real [Brugg]
-— Ep Real [Exp]

Relative Permittivity
O —8 N 00 A 01 O) \l on (D

.1‘11i‘llli'l1illlll‘TillllinW'VIHIWIW lmilll'lll T: "‘

 

“lfllllluml ' , 'mm

23456789101112

Frequency

Figure 4.3.4: Maxwell Garnett. Bruggeman, and Experimental data graphed for comparison for f=0.4.

Maxwell Garnett, Bruggeman, and Experimental
Permittivity for f=0.25

 

“Ep Real[MG] l

l - - - -Ep Real [Brugg]lI
- —Ep Real [Exp] '

Relative Permittivity
O A N w 4> U1 0') \l on

 

.111.1.1.111I 11111.1.1111 1111111]11111.1 11. 1111.11111 r 11.1 111.1. 1.1101111111111111

23456789101112

Frequency

Figure 4.3.5: Maxwell Garnett, Bruggeman, and Experimental data graphed for comparison for f=0.25.

39

Maxwell Garnett, Bruggeman, and Experimental
Permittivity for f=0.1

 

 

 

 

 

 

 

 

5 -
31 Bruce
E 4 “ii—2:;~_=~ _,...__:__. 2 ”M m T
E ' ' sip“ *—
g 3 - -------1-1-Ep Real [MG]
a - - - ~Ep Real [Brugg]
.‘z’ 27 -—-Ep Real [Exp]
5
g 1 1

0 l'llll.:1lllllllll-lllllllllllllli.I:Illllll|llll1'llllllllllllllill.l1lllillllllllli'l.‘lll-‘Illll'lllllllllllllllllhlllllllll lllllllllll lllllllllflllllllllillllmm

2 3 4 5 6 7 8 9 1O 11 12

Frequency

Figure 4.3.6: Maxwell Garnett, Bruggeman, and Experimental data graphed for comparison for f=0.1.

Figures 4.3.4, 4.3.5, and 4.3.6 show that the Bruggeman approximation is a little more
accurate in determining the effective permittivity of a material composed of dielectric

spheres dispersed throughout a matrix.

The interpretation of the Bruggeman formula, equation (4.3.2), is that the formula
balances both mixing components with respect to the unknown effective medium, using
the volume fraction of each component as weight giving it symmetry which is not the
case for Maxwell Garnett and Clausius Mosotti whose focus is on the inclusion more than

the environment [20].

40

4.4 - Coherent Potential Permittivity

Another method for finding the effective permittivity of a mixture with spherical
inclusions is the well known formula called Coherent Potential formula [20]. This can

commonly be seen in the following form.

 

38
8 =8 +f(8.—8) 9’7 (4.4-1)
eff e z e
3881f +(1_fxgi —ge)
This can be rearranged into a quadratic formula with the coefficients
A = 3
B:(1_fX8i —ge)—3f(8i _ge)_3ge (4.42)

C =—8e(1—fX81—59)

The philosophy behind the approaches that led to Coherent potential mixing formulas is
that one should not treat a single scatterer floating in isolation in the environment when
the dipole moment and the local field are calculated. Instead, the Green’s function which
is used to enumerate the field of a given polarization density is taken to be that of the

effective medium, not that of the background [20].

The Coherent Potential formula is derived using the quasicrystalline approximation for
spherical particles embedded in a background medium. In the low frequency limit this

approximation can be reduced to the following [26].

61,031,132): 6; (4.4.3)

Where I)", and )5, are the momentum operators.

41

 

 

 

5 = v05". + flm5[3 1:2 +§1K 0ldrrzlg(r)- 1]] (4.4.4)

and

 

 

f z 1 + j 2 Ka3z

m = "’ 4.4.
1+z/(3K2) 91+z/(3K2) ( 5)

K is the coherent propagation constant of the effective medium, f = nova is the volume

fraction comprised of the number density multiplied by the volume of a spherical

inclusion whose radius is a. The transition operator fm represented by equation (4.4.5) is

inserted into (4.4.4) and gives the following.

 

 

. v02 , 2 3 Z m
C ___ 1+ 2(1— f)/(31<2 )l1 + J 5 Kg 1+ 2(1— f)/(31<2)xl1+ 4m°1i[g(r)—1]ll (4.4.6)

The dispersion relation according to the quasicrystalline approximation is represented by

equation (4.4.7) [26].

K 2 = k2 + n06 (4.4.7)

Again, K is the propagation constant of the effective medium and k is the propagation
constant of the background medium alone. Since this is the low frequency approximation
the imaginary term, which is dependent on the particle size can be neglected and (4.4.6)
reduces to its first term alone [26]. Therefore inserting the first term only of (4.4.6) into
(4.4.7) and using the following representations for K and 2 we get equation (4.4.10) for

the effective permittivity of the mixture.

K 2 = 8eff80a2y0 (4.4.8)

42

2 2 2 2
z=ki —k =0) 210808,. —a) #0808, (4.4.9)

2 2
novo(w [105051-50 flogoge)

2
= +
gefl8oa),u0 60 #08088 1+(0le108051' ‘wzflogoge X1 _f)/(3£eff£0a)#0)

(4.4.10)

Here ee and e,- are the relative permittivities of the background medium and inclusions

respectively. Dropping out like terms and simplifying this equation can be fiirther
reduced to be the same as equation (4.4.1) shown above. For more detailed background

Quasicrystalline Approximation and Coherent Potential please refer to Tsang, Kong, and

Shin [26].

Using the same material as in the previous three sections, this effective permittivity is
graphed at three volume fractions: 0.4, 0.25, and 0.1, along with the actual measured

data. The following three graphs illustrate these comparisons.

Coherent Potential Permittivity [ f=0.4]

 

 

 

— Ep Real [Exp]
4 —1 —Ep Real [CP]

 

 

 

Relative Permittivity
U1

 

 

1 :WWWWWWWWWWWW

2 3 4 5 6 7 8 9101112
Frequency (GHz)

Figure 4.4. 1: Coherent Potential calculated permittivity graphed with measured permittivity of a sample
with volume fraction f=0.4.

43

Coherent Potential Permittivity [ f=0.25]

74

 

 

 

WW

— Ep Real [Em]

 

 

Relative Permittivity
U'l

 

 

4 2 —Ep Real [CP]
3 -
2 .1
1 ..
o MWWWWW .. WWW...
2 3 4 5 6 7 8 9 1O 11 12

Frequency (GHz)

Figure 4.4.2: Coherent Potential calculated permittivity graphed with measured permittivity of a sample
with volume fraction f=0.25.

Coherent Potential Permittivity [ f=0.1 ]

 

4 q —Ep Real [Em]
3 d F " - '_ —-Ep Real [CP]

 

 

 

 

Relative Permittivity

0 ”TWTTTWTTTTTUH TTUTTTTTTTTTTTTTTTT WWTTWWYTWWWWWWTWW

412 3 4 5 6 7 8 9101112
Frequency (GHz)

 

Figure 4.4.3: Coherent Potential calculated permittivity graphed with measured permittivity of a sample
with volume fraction f=0.1.

44

 

A comparison of these three methods; Maxwell Garnett, Bruggeman, and Coherent

Potential is shown in the following three figures along with the experimental data.

Maxwell Garnett, Bruggeman, Coherent Potential,
and Experimental Permittivity for f=0.4

 

 

 

 

 

 

92‘

.2

E 6 4 ’ " MG ' --~-~Ep Real [MG]
E53 ----EpReal[Brugg]
2 4 a — — Ep Real [CP]
3; 3 1 ——Ep Real [Em]
11E

 

0 ‘.1111111=1111r17'11111117*n111111111mwmm.11 lllll .mnmmmnmmmnmnnmmm.l lllllllll mmmmmm

23456789101112

Frequency

Figure 4.4.4: Maxwell Garnett, Bruggeman, Coherent Potential, and Experimental data graphed for
comparison for f=0.4.

45

Maxwell Garnett, Bruggeman, Coherent Potential,
and Experimental Permittivity for f=0.25

 

 

 

 

 

 

 

8 a
E" 7 i
:2 6 “M E83 _ C
E 5 -1" 7'" T:,":-;'::".::.A'::"‘:: LLMB'UEQ
g MG 1-— Ep Real [MG]
0. 4 7
o 3 _ - - - -Ep Real [Brugg]
.‘g‘ 2 - — Ep Real [CP]
g 1 —Ep Real [Em]____

C
.__J___ 1

 

llllllllllllll lllll ‘llllllllllllllll llllllllll'lllllllllll‘1lll.lll Ilillll lllllllllllllllllllll llllll ‘Iill ['1 "l ll‘.Illsllllllllllllllll||||l|lllllllll

3 4 5 678 9101112
Frequency

N

Figure 4.4.5: Maxwell Garnett, Bruggeman, Coherent Potential, and Experimental data graphed for
comparison for f=0.25.

Maxwell Garnett, Bruggeman, Coherent Potential,
and Experimental Permittivity for $0.4

 

 

 

 

 

 

 

9 _.
E 8
.2 7 1
3g 6 - -- 1 — Ep Real [MG]
g 5 - - - oEp Real [Brugg]
2 4 '1 — — Ep Real [CP]
'43 3 ‘ —Ep Real [Em]
a 2

0 111111111111111111111111111111111111111111111111111111111...............11.1'. ........... .11111UUUUTTWTTTTTITTTTTIWTWWTTWWTTTITTTTWTH

23456789101112
Frequency

Figure 4.4.6: Maxwell Garnett, Bruggeman, Coherent Potential, and Experimental data graphed for
comparison for f=0.1.

46

Figures 4.4.4, 4.4.5, and 4.4.6 show that although the other two methods for determining
effective permittivity seemed to be relatively accurate, the Coherent Potential equation

seems to be most accurate for all three volume fractions of the hexaferrite mixture.

4.5 - Maxwell Garnett Permeability

By duality, the electrostatic and magnetostatic formulations obey the same conditions for
a given geometry [12] subject to appropriate boundary conditions. Therefore, given the
spherical geometry of the inclusions, the permeability could be found using Maxwell
Garnett formulation. The derivation would be the same as for permittivity [10], except
rather than the electric displacement being related to the electric field as in (4.5.1), the

flux density is related to the magnetic field intensity as in (4.5.2).

(12>: 63215 > (45-11

<§> = flefl<H> (4.5.2)

The resulting equation is the same as for permittivity with 8 replaced with y.

(m-w)
11. +271. -f(#. we)

 

Il’leff : #8 + 3ffle (4.5.3)

Equation (4.5.3) is graphed using the same hexaferrite sample as previous sections. The

relative permeability of the inclusions [L] is 10, and the relative permeability of the non-

magnetic environment 118 is 1. This data is graphed against the measured values for

permeability at volume fractions 0.4, 0.25, and 0.1 respectively.

47

Relative Permeability

Maxwell Garnett Permeability [ f=0.4 ]

 

 

 

 

 

 

 

 

5 1
4 1
3 i —Mu Real [Em]
2 , —Mu Real [MG]
1 - _—_—_‘_—, VA
0 TWWWWWWTWWW . 1 .- 1 WW

2 3 4 5 6 7 8 9 1o 11 12

Frequency (GHz)

Figure 4.5.1: Maxwell Garnett calculated permeability graphed with measured permittivity of a sample

Relative Permeability

with volume fraction f=0.4.

Maxwell Garnett Permeability [ f=0.25]

00 -h 01
u_ J _._J

N
EL

 

— Mu Real [Em]
-— Mu Real [MG]

 

 

 

 

 

A
I
'1
l
l

 

0 ‘lrrmmtmmnmmnmmm || '1111lll11111 l lll ll 1 1 | 1 1 T71 llllllllll.l1|llll 'llllllll.l lllllll llllllll

 

2 3 4 5 6 7 8 9 1O 11 12
Frequency (GHz)

Figure 4.5.2: Maxwell Garnett calculated permeability graphed with measured permittivity of a sample

with volume fraction f=0.25.

48

Maxwell Garnett Permeability [ f=0.1 ]

 

 

 

 

 

 

 

 

5 -
g
a 4 ‘
N
E 3 1
a, --—Mu Real [Em]
2; 2 _ —-Mu Real [MG]
.5
.9; nip—I:— A
o 1 ~1 —‘ -
n:

O ‘WWWWWWWWWWWWT 11 . 1: :1 l lTTiTFlTTTTTlTTTTTTTTlTTITT‘lTl

2 3 4 5 6 7 8 9 1O 11 12
Frequency (GHz)

Figure 4.5.3: Maxwell Garnett calculated permeability graphed with measured permittivity of a sample
with volume fraction f=0.1.

Inspecting the measured values, it is seen that the real permeability goes to 1 at
frequencies above 6 GHz and becomes non-magnetic. The hexaferrite sample used in
this thesis was chosen because ferrimagnetic materials can stay magnetic up to higher
frequencies than other magnetic materials. However, Maxwell Garnett’s theory is based
on permittivities and lower frequencies and did not need to take this change into account.
Therefore comparison of the calculated data and the measured data should be focused on
the lower frequencies. It is not entirely accurate in finding the effective permeability,
which is not surprising because of the simplicity of Maxwell Gamett’s equation and the
complexity of magnetism. However, upon comparison with the other classical methods
to be seen in the next few sections, this approach is much more accurate than Bruggeman

and Coherent Potential for finding the effective permeability of our hexaferrite sample.

49

4.6 - Bruggeman Permeability

The Clausius Mosotti formula was found to be the same as Maxwell Garnett using this
geometry for permittivity and therefore will not be demonstrated for permeability since it
will only yield the same results for permeability as the previous section. However,
applying the concept of duality and attempting to replace the permittivity entries with
permeability in the Bruggeman equation derived in section 4.3, we get the following
quadratic equation. Again, this is for a two-phase mixture, with the inclusion phase being

that of magnetic spherical particles dispersed in a non-magnetic medium.

pgA+pr+C=O

A=—2 men
B = 2f#. - 3m. + f“. + 2y. - #1-
C : :ul'fle

Solving (4.6.1) and taking the non—negative result will give the Bruggeman effective
permeability for a mixture of this geometry. This was done for the mixture with an
environment of permeability 1, and the hexaferrite particle with a permeability of 10 and

graphed with measured data for all three volume fractions; 0.4, 0.25, and 0.1.

50

Bruggeman Effective Permeability [f=0.4]

 

A
___J,,_____1_ 1

 

—- Mu Real [Exp]

NW —Mu Real [Brugg]
1 2

 

 

 

Relative Permittivity
N

 

 

Frequency

Figure 4.6.1: Bruggeman calculated permeability graphed with measured permeability of the sample with
volume fraction f=0.4.

Bruggeman Effective Permeability [f=0.25]

 

2 — Mu Real [Em] i
-— Mu Real [Brugg]

 

M 1‘
1 1

Relative Permittivity

 

O innmmmmmnmmmmrr"... .1 1

 

1 .1 . WTTWWUWYUWW

. 3 4 5 6 7 8 9 1011 12
-14

Frequency

Figure 4.6.2: Bruggeman calculated permeability graphed with measured permeability of the sample with
volume fraction f=0.25.

51

Bruggeman Effective Permeability [f=0.1]

 

— Mu Real [Em]

— Mu Real [Brugg]
1 f .— ____ - ‘___ g

0 _ll-ll lll lllllll .l ‘ 'WWWIWUIWWWWWW

3 4 56 78 9101112

 

 

 

 

 

 

Relative Permittivity
N

-1J

Frequency

Figure 4.6.3: Bruggeman calculated permeability graphed with measured permeability of the sample with
volume fraction f=0.1.

This prediction does not seem to be very accurate until the volume fraction is down to 0.1
where it is hardly exhibiting any magnetic characteristics. For comparison, this data will

be graphed again with Maxwell Garnett and experimental data to see which is a better

model for permeability.

52

Maxwell Garnett, Bruggeman, and Experimental

 

 

 

 

 

 

 

 

 

Permeability for f=0.4

5 1
8 4~
3
8 3 ~ -------------------------------
E Brugg . g - 'm Mu Real [MG]
& 21 L 1 MG 7 I i "i ----MuReal[Brugg]
0 N
.5 1 1 Exp '____A__v A Mu Real [Em]
5.1
g 0 1111111111111111111:1111:11.1111111.-1.111>1::.1111..1111111111111WWWWWW

112 3 4 5 6 7 8 9101112

Frequency

Figure 4.6.4: Maxwell Garnett, Bruggeman, and Experimental data graphed for comparison for f=0.4.

Maxwell Garnett, Bruggeman, and Experimental
Permeability for f=0.25

 

 

 

 

 

 

 

 

 

E 4 r
'1‘:

3 _
E ...... Mu Real [MG]
3 2 .:.:.:.. :....:..'..:..:. 'Qm‘gg """" . .1.:...:..:. .T. '.. ' ' - NH Real [BrUQg]
g 5" '7 .- MG _ Mu Real [Em]
a. 1 4 Exp _
2
g 0 T llllllllllll'..l‘lTllIIlll'llllllllll-11l'l!.l||!!lilll 111111.1111111111111.11| 11111111111 .

1 3 4 5 6 7 8

 

Frequency

Figure 4.6.5: Maxwell Garnett, Bruggeman, and Experimental data graphed for comparison for f=0.25.

53

Maxwell Garnett, Bruggeman, and Experimental
Permeability for f=0.1

 

 

 

 

 

 

 

 

 

5 1
g
s 4.
8
E 31 ~-~--MuReal[MG]
6’; - - - -Mu Real [Bmgg]
g 2 1 Mu Real [Em]
2o?- .. BWEQ. MG _
g 1 _W::-—~- - - ___________::'--
a E...

0 l.'l'lllllllll|l1.l'lllrlllllllillllllllill1llIlllllillllllllllli':.-ll.lllll'llllllllll 11111111 - 11111 llll' .......... lTTiTTTlTITTTTTlTllllTTTTlTTllTTTTTTiTlTITTTlTlTl

2 3 4 5 6 7 8 9 1O 11 12
Frequency

Figure 4.6.6: Maxwell Garnett, Bruggeman, and Experimental data graphed for comparison for f=0.1.

Figures 4.6.4, 4.6.5, and 4.6.6 make it apparent that the Maxwell Garnett formula is a
little closer to the measured values as the volume fraction is increased. This is most
likely because the Maxwell Garnett equation focuses more on the inclusion being the
result of effective permeability, whereas the Bruggeman formula is symmetric and places
equal importance on the inclusions as it does the environment based on its volume
fractions. In magnetism, the particle to particle interaction needs to be handled more
carefully and therefore the inclusions dispersed in a non-magnetic medium definitely

dominate how the mixture’s magnetic characteristics will turn out.

54

4. 7 - Coherent Potential Permeability

Again, using duality to try to find the permeability of a mixture using the coherent
potential formula for effective permittivity described earlier we have the following plots

for mixtures of volume fractions 0.4, 0.25, and 0.1 respectively.

3,11,,
#617 : #9 +f(ll’li -#e) fl (4'7'1)

Coherent Potential Permeability [ f=0.4]

 

 

 

 

 

 

 

5 _
33
3'3 4 “
N
0
E 3 "1 —Mu Real [Exp]
% 2 - l—Mu Real [CP]
.5 N
I!
a 1 ~ -—++ -
tr

2 3 4 5 6 7 8 9101112
Frequency (GI-iz)

Figure 4.7.1: Coherent Potential calculated permeability graphed with measured permittivity of a sample
with volume fraction f=0.4.

55

Coherent Potential Permeability [ f=0.25]

01

h
1

 

(D
L

— Mu Real [Exp]
— Mu Real [CP]

 

 

 

N
l

 

A
1
‘1
l
l
|

.1

Relative Permeability

 

O

i
‘lll. i lllll in I IITTTTTTH lillll I H, II ml 1 :. . :i. ”ll” .H‘lvllll ”lllllllllll ..”l.vl

2 3 4 5 6 7 8 9101112
Frequency (GHz)

 

Figure 4.7.2: Coherent Potential calculated permeability graphed with measured permittivity of a sample
with volume fraction f=0.25.

Coherent Potential Permeability [ f=0.1 ]

 

— Mu Real [Exp]
2 ,- -— Mu Real [CP]

 

 

 

Relative Permeability
(A)

 

 

2 3 4 5 6 7 8 9 1O 11 12
Frequency (GHz)

Figure 4.7.3: Coherent Potential calculated permeability graphed with measured permittivity of a sample
with volume fraction f=0.1.

56

Similar to the Bruggeman equation, the Coherent Potential formulation seems to be close
when the mixture is barely magnetic with a relative permeability of one, but does not

seem to be very close for higher volume fractions than 0.1.

Finally, the results from Figures 4.7.1, 4.7.2, and 4.7.3 are graphed with the results from
the permeability calculation of Maxwell Garnett, and Clausius Mosotti for purposes of

comparison.

Maxwell Garnett, Bruggeman, and Experimental

 

 

 

 

 

 

 

 

Permeability for f=0.4
5 E
E 4 1 C.
.D ____________________
8 3 ~ ------------------------------ Mu Real [MG]
Brugg
g 2 _ .. - __- - .. .M'G- .. ...-...._. . - . - Mu Rea|[Brugg]
% N -— — Mu Real [CP]
{.5 1 ~ Exp 7......“ A, Mu Real [Exp]
'5
a:

 

0 gillimnlll'lll‘lllllll‘llli'W‘Hl.l'lll".l‘lllllllll'lllminl vmnumu um...llilllllllllllllll'lllvlll iiiiiiiiiiiiiiiiii “mm llllllllllllll

Frequency

Figure 4.7.4: Maxwell Garnett, Bruggeman, Coherent Potential and Experimental data graphed for
comparison when f=0.4.

57

Maxwell Garnett, Bruggeman, and Experimental

 

Permeability for f=0.25

 

 

 

 

 

 

4 1
g 3 -
'g ~ Mu Real [MG]
E 2 4:: “:2 7-7:: :2'C;E' ‘:‘.T.*: r: r: z“. 7.7-7 r: ' ' ' -Mu Real [Brugg]
5‘3 -- --—~-— ---——---——-——--—--— 5m —«——-—-——-—-— ——- -- -- -— - Mu Real [CP]

WA
2 , - Mu Real E

‘5 .
a J llllllmmllllllllllxllllll II'HI'HIIH' IIIIIIIIIIIIIII IIHI lllllllTlTleTlem
m . ,.

0 £11m!) llTTTTlllllllllillll..il

5 678 9101112

Frequency

Figure 4.7.5: Maxwell Garnett, Bruggeman, Coherent Potential and Experimental data graphed for

comparison when f=0.25.

Maxwell Garnett, Bruggeman, Coherent Potential
and Experimental Permeability for f=0.1

1W

Bru

'.d"" “ rum-vi.” fl Rig-"n.

Exp'

v—‘v‘vV—vw v

 

-12? 3 4

Relative Permeability

 

0 iln-llI‘I-HHI'EHIl1”HHHIHHll:iHIHIHIIIJHI ‘.:.Hll|llll

 

56789101112

Frequency

 

 

 

~- --—--- Mu Real [MG]
- - - ~Mu Real [Brugg]
«- —— Mu Real [CP]
Mu Real [Exp]

Figure 4.7.6: Maxwell Garnett, Bruggeman, Coherent Potential and Experimental data graphed for

comparison when f=0.1.

58

 

J

 

Figures 4.7.4, 4.7.5, and 4.7.6 show that Maxwell Garnett is the better approximation for
effective permeability for the hexaferrite mixture used in this thesis. Again, this is
probably due to the fact that the Maxwell Garnett formula focuses on how the inclusion
affects the entire mixture whereas Coherent Potential and Bruggeman look at the sample

as a homo genous medium.

4.8 - Onsager Permeability

The Maxwell Garnett, Clausius Mosotti, and other classical mixing approaches are
considered fairly accurate for determining effective permittivity of sparsely distributed
inclusions which are much smaller than a wavelength, allowing for electrostatic
approximations. However, when dealing with magnetic materials where the permeability
of inclusions has a real value greater than one, there is an interaction that must be
realized. The presence of permanent magnetic moments means that the theories like

Maxwell Garnett are inapplicable to the calculation of ’ueff [l9]. Onsager’s theory

describes the local interaction between the magnetic field and the magnetic medium
which will be used to arrive to an averaged description in terms of effective dielectric and

magnetic permeability [5].

One assumption is that the particles are mono-domain, and therefore dipolar interaction is
negligible, and they are small in relation to wavelength. With these assumptions, it will

be valid to use a quasi-static magnetic approximation for the electromagnetic fields and

59

 

their interactions with the particles [19]. Beginning by looking at the magnetization of a

single particle,

(4.8.1)

where m0 is the magnetic moment of the particle, 17 is the unit vector in the direction of
the permanent moment, a is the complex magnetic polarizability, and F is the local

magnetic field. Onsager represents the local field E as a summation of two terms: the

cavity field component (7 and the reaction field component E which are obtained
through boundary conditions [5]. Since the particle has a permanent magnetic moment, a

inhomogeneous field is created which alters the neighboring particles’ magnetic moments
creating the reaction field]? . In order to find an expression for F: , the homogeneous
wave equation Aw = 0 can be solved for a magnetic moment in the center of a sphere of
radius a surrounded by a magnetic medium with permeability p . General solutions to

the wave equation are

 

W1 : :01!" + If; )Pn (c036)

 

 

”:0 (4.8.2)
°° D
= C r" + " P c056
v12 2( .( > (4,,
after applying the boundary conditions equations, (4.8.2) and (4.8.3) become
3m
W1 : 2 (3056 (4.8.4)
(2p + 1)r

6O

1,112 = fleosQ— 2mr(,u-1)

2

(9 (4.8.5)
r a3(2,u +1) COS

 

Using equation (4.8.6) to represent the magnitude of the applied field and substituting

into the solutions to the wave equation (4.8.4) and (4.8.5) gives the two local Onsager

fields F7 and F: representing the fields inside and outside of a sphere.

 

 

 

 

— m\/1+ 3cos2 6
”Hm H = r3 (4.8.6)
1 2 2” +1 for r>a (4.8.7)
— — 2 —1
F 2 = m +(2,1(1#+1)23 m for r<a (4.8.8)

1?1 looks a lot like the equation for the polarizability of a sphere in section 4.2 resulting
from the electrostatic solution to Rayleigh scattering with permeability replacing
permittivity. The resulting equation gives the cavity portion of the local Onsager field
represent in equation (4.8.9). The reaction field I? which is caused by the magnetic
moments within the particles is represented by equation (4.8.10). Therefore the total
Onsager field as a result of the applied field F1— is represented by the summation of

equations (4.8.9) and (4.8.10) [5].

 

2’“ +1 H (4.8.9)

61

 

 

2(,u — 1) 1'17 (4.8.10)

 

F=§+§ man

The complex polarizability a is represented in its complex form by equation (4.8.12)

where the real part is given by (4.8.13) and the imaginary part is (4.8.14) [24].

a .__ (fgjafigqja-n) (4.8.12)

 

§,:_( 3 )[1 3(sinh(2x)—sin(2x)) ] (4.813)

87 — 2x(cosh(2x)- cos(2x))

 

 

_,,__[ 9 )[1_x(sinh(2x)+sin(2x))] (48.14)

a — 167m2 (cosh(2x)-cos(2x))

Ex" is the real part of polarizability, 5" is the imaginary part of the polarizability, a is
the radius of the particle, x = a / 6 , 6 = c/ Zimya) is the skin depth, c is the speed of

light, 0' is the conductivity, and a) is the frequency. Inserting equation (4.8.11) into

equation (4.8.1), we get

 

 

I—fizm 1_87Z'a-(Iu—l) 412+ 3,1162 1_87[§(fl—1)_lI—1 (4H815)
0 3(2#+1) 2p+1 3(2,u+1)

62

 

 

The total magnetization per unit volume is 117 = N (E) , where N is the number of

particles per unit volume. The total magnetization can then be related to the applied field

by

m0(a)3(2p+1)+9pafi
47r 3(2,u+1)—87rZz—(,u—1)

 

(4.8.16)

where m, is the magnetic moment per unit volume and (12) is found by the Onsager’s

approach.

(12>: f ,Lin'047m3fi f
kT[3(2,u+1)—871§(,u—1)] (4.8.17)

 

Here k is Boltzmann’s constant, and T is temperature. The main point to be noted from
the derivation of equation (4.8. 1 7) is that only the cavity-field component C— has any
effect on the moment, and the reaction field 1—2 is always parallel to r71" and therefore can

not exert any torque on the particle [19]. Since relaxation is not instantaneous, (17> needs

to be multiplied by an Debye relaxation time g(a)).

g(a)) : (1_ ij)—1 (4.8.18)

Where I = To exp(K V / kT ) is the orientation relaxation response function, K represents

the anisotropy energy per unit volume, V = 4M3 / 3 is the volume of a single inclusion.

The quantities if and r720 are normalized values per unit volume and therefore will be

multiplied by the volume fraction f of the samples. Multiplying in f and utilizing

63

 

 

equations (4.8.17) and (4.8.18) in equation (4.8.16), the result is an equation for the

effective permeability of the mixture.
(4.8.19)

4m311—23
9171. 24. +1860 ——9- _
”eff—1 jf( fl ) ( ) 3kT + 9fll‘lejfa

47: _ [3(2yefl + 08751111,, —1)]” 3(2yeff +1)—8m7f(,uefl - 1)

 

Equation (4.8.19) can be rewritten as a cubic equation with the following coefficients and

solved for ye ff .

A lie/f3 + Byefl.2 + Cyefl + D = O

A = 4(9+167:2f2552 —2479(C—¥)

B = —247;7"[§(3—47ya)+ 3X g(a))]

C = —3 [32425sz +12nf(Xg(a))+ 35 )+ 9]
D = —[9 + 16an (3 + 47raf)]

(4.8.20)

Where X =(47ra317102 /3kT) , since there are going to be three solutions to this cubic

equation, the physically realizable #617 must be chosen so that Remefl} is positive.

This formulation has been proven accurate for mono-domain ferromagnetic particles by
Gadenne [19], however the ferrite material used for this thesis has a significantly smaller
conductivity, and higher anisotropic constant and this relation fails because of those
constraints. Although it does not work well for the hexaferrite inclusions, it is important

to keep this in mind for future applications involving ferromagnetic particles.

64

CHAPTER 5 - Applications

5. 1 - Square Patch Antenna Bandwidth

A magneto-dielectric material is one with a permeability and permittivity greater than one
and is the case for the hexaferrite mixture discussed throughout this thesis. A paper by
Hansen and Burke [7] analyzes the bandwidth of a magneto-dielectric backed square
patch antenna using its conductance and admittance. They claim that the results obtained
allow one to analyze the effect of varying parameters, p and 3 , on the bandwidth using a
zero-order theory. Also, that this approach will provide simple results that will only be

changed slightly when looked at by more exact theories. The zero order theory states that

the patch length must be as in equation (5.1.1) for resonance.

H

=—---— (5.1.1)

By empirical analysis, the best fit for conductance of this resonant patch is represented by

G 2 1 (5.1.2)

40 ,urer +170,ur8r

 

The characteristic admittance of a wide microstrip line is

A
27711,}

 

Y0 : (5.1.3)

where 11 is wavelength, 7] = 12071 is the intrinsic impedance, and t is the thickness of the

substrate. For a VSWR=2, the zero order percent bandwidth of the patch antenna is

defined as seen in equation (5.1.4).

65

 

BW = ——
JEQ (5.1.4)
where
”Yo
: __ (5.1.5)
Q 4G

Using this formulation one will be able to predict the bandwidth of a patch antenna by

using the parameters, ,ur and er , of the substrate. Figure 5.1.1 and Figure 5.1.2 illustrate

this theory for a material with varying permeability and permittivity at a center frequency

of 1 GHz.

Bandwidth with Constant Permittivity ep=5 at 1 GHz

2 4
1.8-
1.6 4
1.4 1
1.2 --

 

[— Bandwiduj

 

0.8 J
0.6 ~
0.4 J
0.2 ~

Percent Bandwidth

 

 

1 2 3 4 5 6 7 8 9 1 0
Relative Permeability

Figure 5.1.1: Bandwidth for constant permittivity with permeability varying from 1 to 10.

66

 

 

Bandwidth with Constant Permeability mu=2 at 1 GHz

1.8 ~
1.6 4
1.4 ~
1.2 ~
1 a .
0.8 - [— BandWIdthl
0.6 J 2_
0.4 ~
0.2 1

0 I f I T I l I T T 1

1 2 3 4 5 6 7 8 9 10
Relative Permittivity

 

 

 

Percent Bandwidth

 

 

Figure 5.1.2: Bandwidth for constant permeability with permittivity varying from 1 to 10. l

 

Using Sonnet Lite, a patch antenna with sides of length 4340 mils, approximately 110
mm, is simulated in order to determine the VSWR. The center frequency according to
equation (5.1.1) is about 650 MHz, and the calculated bandwidth according to equation
(5.1.4) is going to be 0.448%. The relative permeability and permittivity of the substrate
are going to be almost equal in order to get the best matching. The values for permittivity
and permeability are 2.2 and 2 respectively. Figure 5.1.3 shows the resulting VSWR

graphed versus fiequency.

In order to obtain the bandwidth from this graph, the lower frequency where VSWR=2 is
subtracted from the higher frequency where VSWR=2 and the difference is divided by
the center frequency. This gives a bandwidth of 0.46%, which is close the bandwidth

found using equation (5.1.4).

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 VSWR vs. Frequency
M
a
9
n
i 1
t
u
d
e
0
642 644 646 648 650 652 654 656
3211...... ‘0: Frequency (MHz)

 

Figure 5.1.3: Bandwidth plot for permittivity of 2.2 and permeability of 2.

Now the data from the hexaferrite sample is used in equation (5.1.4) as well as the Sonnet
Lite program for further comparison. The permittivity is 5.1 and the permeability is 1.5
which was obtained from the sample with volume fraction #025. The bandwidth found
by calculation is going to be 0.15% around a center fiequency of 495 MHz. Figure 5.1.4
shows the VSWR graph using Sonnet Lite. The bandwidth found through this graph is
1.5%, which differs from the calculated amount by a factor of 10. This could be due to
the fact that the equation predicts such a small bandwidth that it is not easily simulated

using Sonnet Lite.

68

 

 

 

VSWR vs. Frequency

 

 

 

 

 

 

 

 

 

 

2 Y1
1% 41‘ -'

M (($328 161‘
a «5:9
9
n
i 1
t
u
d
e

0

475 485 495 505 515 525

5.8....4 8. Frequency (MHz)

 

 

Figure 5.1.4: Bandwidth plot for permittivity of 5.1 and permeability of 1.5.

For aid in describing the benefit of using a magneto-dielectric material as the substrate as
opposed to just a dielectric material, a non-magnetic material was modeled with this
formulation and simulated with Sonnet lite. The patch antenna modeled is the same as

seen above but with 1:, =1 and a, = 4.4 , which results in a bandwidth percentage of

only 0.22%. This is half the bandwidth percentage of the magneto dielectric material
modeled in the first example. Sonnet-lite, however does not give the same bandwidth as
the calculated results. The simulation actually shows an increase in bandwidth as seen in
figure 5.1.5. This is an area that deserves attention in future work. The goal would be to
setup an actual test procedure for getting real measured bandwidth results in order to

determine if this formulation is an accurate prediction.

69

 

 

mac—v-Dtomg

" Sofie.

2

1.8
1.6
1.4
1.2

1

0.8
0.6
0.4
0.2

0

5."? :l‘C

0.644 0.645 0.646 0.647 0.648 0.649 0.65 0.651 0.652 0.653 0.654 0.655

 

1

l

I

 

 

l l l l l l T T 1

Frequency (GHz)

1

 

Figure 5.1.5: Bandwidth plot for permittivity of 4.4 and permeability of l.

70

 

 

CHAPTER 6 — Conclusions and Future Work

6.1 - Conclusion

Upon comparing the results of the classical mixing formulas in this thesis compared to
actual measured data, it can be seen that most of the permittivity models are relatively
reliable. The Clausius Mosotti formula was shown that when used with a two phase
mixture of spherical particles dispersed in an environment will give the same exact
answer as the classical Maxwell-Gamett approach. These two mixing principles focus on
the inclusions more than the background and how they affect the mixture; Maxwell-
Gamett uses volume averaged field quantities, and Clausius Mosotti replaces the
inclusions with polarizations. The other two formulations; Coherent Potential and
Bruggeman’s philosophy, attempt to homogenize the material and find the effective
permittivity equally from the inclusions as from the environment based on their volume
fractions. Of all four mixing equations, Coherent potential formulation is the most

consistently accurate for all three volume fractions.

However, when applying the concept of duality and swapping the values of permeability
for permittivity in these same equations they do not yield as good of results as when
applied to permittivity. The Maxwell-Garnett formula gives much better results than
other approaches, but is not accurate enough to be considered a reliable method for
predicting permeability. As mentioned above, the Maxwell-Garnett and for this
geometry Clausius Mosotti, focus more on the inclusions than on the environment. This
was not as accurate for permittivity, but for permeability this does result in better data.
When a magnetic inclusion is placed in a non-magnetic background the inclusion will

have more control over the effective permeability and shows why these two formulations

71

will give better results than the homogenization approach used by Coherent Potential and
Bruggeman. This shows that the magnetic behavior of heterogeneous materials is not
easily predicted and it is necessary to analyze this problem on the quantum level for more

precise results.

Another problem is the use of spherical particles as the inclusions. In order to maintain
any of the magnetic properties of hexaferrite the volume fraction needed to be 0.4 or
above. The problem with high volume fractions like this is that the weight of the mixture
becomes not much less than if you had the hexaferrite in bulk. Figure 6.1.1 illustrates the
correlation between volume fraction and weight fraction for density ratios of 10:1, 5:1,

andlzl.

Volume % vs Welght %

100

 

, . 1 .
. . 1 . , .
1 1 1 I
. . ‘ ’. I
901 _. :11 » ; iiiiiiiii » . :
1 ’ -'
1 . ,
. I
801
1 . 1 .
1 3 1’” I
701.... . . . . . .. .. . ........ x ,, , , '
,
' d
I 1
1 Id ‘ I
60.1.. ‘ _._.. ....L_... _ ____.... ’a' ....._ i j ..
. 1 d . 1 1
. ' 1 1

. , . _1
I .
.
‘ /
50 4) .. .. . .-. . . 1.2. .. . . . .......... . . - ....-.-.._.. .. .......... ,I .. _
1 ’1 l
.
I l 1

Volume 1%)

40‘

30‘

201 ,

10*

 

 

 

 

0 10 20 30 40 50 60 70 80 90 1 00
WOlght Pl.)

[—---10:1 -' 5:1 ----- 1:11

 

 

Figure 6.1.1: Comparison of volume percent to weight percent.

72

This ratio is the density of the inclusion to density of the environment. For a density ratio
of 5:1, it can be seen from figure 6.1.1 that a volume fraction of 0.4, 40 % by volume,
would become greater than 75% of the weight for that mixture. Also, as the volume
fractions increases adjacent particles begin to interact with each other and can no longer
be considered mono-domain which will increase the difficulty of predicting permeability
and even permittivity. These results help to show that using spherical inclusions do not
yield appreciable magnetic results and do not allow for a reduction in weight. As
discussed in future work, other geometries will need to be considered in order to achieve

better results.

6.2 - Future Work

Up to this point the only type of sample that was tested and was attempted to model have
been COD composites. This means that the mixture is conducting along 0 dimensions [1].
This consists of spherical particles dispersed in a polymer background and can be seen in

Figure 6.2.1 along with the illustrations for C1D and C2D composites.

C2D C1D COD

W? [800508 """ 80

Figure 6.2.1: Types of composites.

 

 

 

 

qo

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

73

In the future it might be beneficial to try the C1D or the C2D composites for two reasons.
The first is for different results in the effective parameters based on their applications,
and second reason is for accuracy of predicting the effective parameters. Another
consideration is to use ferromagnetic materials; however these do not perform at as high

of frequencies.

Also, if the COD composites are to still be used the modeling will have to be done on the
quantum scale. Clearly the only way to fully predict the behavior of the magnetic
properties of these mixtures will be to model the interaction and movement of electrons
within a sample. Also, trying to compare the results of the mixing principles for other
samples than can be measured will help to better understand the ability of these equations

to predict permittivity and permeability and help realize their limitations.

Another aspect for future work is to setup a test procedure for the square patch antenna
example. The test will be able to see the bandwidth results and allow us to compare them

with the calculated results.

74

[21

[3]

[4]

[5]

[10]

[11]

[12]

[13]

[141

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76

   

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}