CHARACTERIZATION OF ANISOTROPIC MATERIALS USING A
PARTIALLY-FILLED RECTANGULAR WAVEGUIDE
By
Junyan Tang

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering – Doctor of Philosophy
2015

ABSTRACT
CHARACTERIZATION OF ANISOTROPIC MATERIALS USING A
PARTIALLY-FILLED RECTANGULAR WAVEGUIDE
By
Junyan Tang
Engineered materials with special electromagnetic properties are gaining an interest for
applications where tailored material parameters are needed to meet stringent requirements.
As the number of applications and the complexity of the engineered materials increase, there
is a growing need for characterization methods that can accurately predict the constitutive
parameters of these materials. Rectangular waveguide methods are popular in characterizing
isotropic and anisotropic materials for many reasons, such as ease of sample preparation, high
signal strength due to field confinement, the ability to control the polarization of the applied
electric field, and high transmission efficiency over a broad frequency band. However, most
characterization techniques based on rectangular waveguides are implemented by placing
the sample against various metallic objects such as irises or posts, or by confining samples
in sample holders, which increases the measurement complexity and the extraction uncertainties. In this work, a partially-filled waveguide technique that overcomes the drawbacks
of the traditional methods is investigated for the characterization two types of anisotropic
materials: biaxial and gyromagnetic.
The first group of materials considered for this characterization method are biaxial materials, which have three nonzero entries in each material tensor, corresponding to six complex
material parameters that must be extracted. Typically three different samples are manufactured from the same material and placed into the rectangular waveguide in three orientations
in order to interrogate the material along three orthogonal axes. Instead of using three samples, this technique uses a single cubical sample of biaxial material which is placed at the
center of a guide and measured under three different rotations, providing the required number of reflection and transmission measurements to determine the six unique constitutive

parameters. The theoretical reflection and transmission coefficients are determined using a
modal analysis. The desired complex constitutive parameters can be obtained by minimizing
the difference between theoretical and measured data.
The second type of materials considered in this work are gyromagnetic. Gyromagnetic
materials have scalar permittivity and anisotropic permeability which can be described by
a tensor with three diagonal entries and two off-diagonal entries. Since the cross-sectional
dimensions of waveguides become large at low frequencies where the gyromagnetic properties are most pronounced, and sufficiently large samples that can fill the cross-section of the
waveguide are typically unavailable, a technique overcomes the limitation of sample size and
only requires the sample to fill part of the guide is beneficial. The measured refection and
transmission coefficients can be obtained from a gyromagnetic sample under various experimental configurations. The theoretical refection and transmission coefficients are determined
using a mode matching technique. A nonlinear least squares method is employed to extract
the gyromagnetic material parameters using optimization algorithms in Matlab.

ACKNOWLEDGMENTS

I would first like to thank the professors who are not on my committee. Thank you to Dr.
Qi for encouraging and helping me to transfer from Geography to ECE department. Thank
you to Dr. Kemple for supporting me at my conference talks and giving me lots of advice
during my Ph.D. study. Thank you to Dr. Havrilla for generously letting us use your lab
equipments for a long period of time, without which I would not have the measured data for
the completion of this work.
Thank you to Dr. Diaz for giving me the opportunity to work with the metamaterial
research group and also for his time and effort being my committee member. Thank you
to Dr. Chahal for his assistance and help during my transfer to the ECE department, and
many useful advice during my job search. Thank you to Dr. Shanker for his support all over
the years. I will miss all the classes I had with him.
Many thanks to Dr. Rothwell for accepting me as his student and transforming me
in many ways. I really appreciate all his effort over the years for educating, helping and
encouraging me. I will always cherish the fun times we had together and remember the
interesting stories he told us. He is and always will be my friend and mentor.
Special thanks to Dr. Raoul Ouedraogo and Dr. Benjamin Crowgey who guided me at
different stages of my study at MSU. Thank you for helping me through many difficult times
in my life. You two are like my big brothers and I know our friendship will never die. Thank
you to Korede Akinlabi-Oladimeji for his numerous amount of hours spent on reading my
writings and for his unconditional support when I am in trouble. Thank you to Andrew
Temme and Joshua Myers for the friendship we established over the years and I will miss
your interesting comments on my “Tang-ism”. Thank you to Dr. Kazuko Fuchi for being a
nice coworker during my study and also being a kind friend in my life. Thank you to Dr.
Jose Hejase for believing in me and supporting me many times in my life.
Thank you to my parents for the love and support over the years. No words can express

iv

my gratitude to them for letting me be myself.
Finally I want to thank my girlfriend Ranran Yang who is also going to be my wife in
the near future for her love and support. I know that we shall create an extraordinary life
together!

v

TABLE OF CONTENTS

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

viii

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

ix

CHAPTER 1 INTRODUCTION AND BACKGROUND . . . . . . . . . . . . . . . .

1

CHAPTER 2 Characterization of Biaxial Material Using a Partially-Filled Waveguide
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Theoretical Transmission and Reflection Coefficients Using Mode-Matching
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Field Structure in a Waveguide Partially Filled with Biaxial Material
2.2.2 Solution for S-Parameters Using Modal Expansions . . . . . . . . . .
2.2.3 Validation of Theoretical Analysis . . . . . . . . . . . . . . . . . . . .
2.2.4 Extraction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Error and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
9

CHAPTER 3 Characterization of Gyromagnetic Materials Using a Partially-Filled
Waveguide Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Theoretical Transmission and Reflection Coefficients Using Mode-Matching
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Characteristics of Gyromagnetic Material . . . . . . . . . . . . . . . .
3.2.2 Modal analysis of the gyromagnetic material in a partially-filled waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2.1 Modal behavior inside a rectangular waveguide partially filled
with gyromagnetic material . . . . . . . . . . . . . . . . . .
3.2.2.2 Modal spectrum dependence on the transverse position of the
sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Fields in the Waveguide . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Mode Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Validation of Theoretical Analysis . . . . . . . . . . . . . . . . . . . .
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10
13
16
20
26
30
39

44
44
45
45
47
50
59
62
73
83
86

CHAPTER 4 Extraction Process for Gyromagnetic Material Properties Using a PartiallyFilled Waveguide Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Least Square Method for Extraction . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.1 Extraction Process Using Matlab Optimization Algorithms . . . . . . 89
4.2.2 Validation of the Extraction Using Matlab Optimization Functions . 95
4.3 Error and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 97
vi

4.4
4.5

Extraction of Measured S-parameters . . . . . . . . . . . . . . . . . . . . . . 102
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

CHAPTER 5 EXPERIMENTAL SETUP AND RESULTS . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Characterization of Biaxial/Uniaxial Material . . . . . . . . . . . . . . . . .
5.2.1 Sample Construction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3.1 Repeatability Analysis . . . . . . . . . . . . . . . . . . . . .
5.2.3.2 Extracted Parameters of The Uniaxial Sample . . . . . . . .
5.2.3.3 Extracted Parameters of Materials with both Uniaxial Electric and Uniaxial Magnetic Properties . . . . . . . . . . . .
5.3 Characterization of Gyromagnetic Material . . . . . . . . . . . . . . . . . . .
5.3.1 Experimental System Setup . . . . . . . . . . . . . . . . . . . . . . .
5.3.1.1 Gyromagnetic Material Sample Specifications . . . . . . . .
5.3.1.2 Measured Magnetic Biasing Fields . . . . . . . . . . . . . .
5.3.1.3 Experiment Set-up . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2.1 Repeatability Test . . . . . . . . . . . . . . . . . . . . . . .
5.3.2.2 Extracted Parameters . . . . . . . . . . . . . . . . . . . . .
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120
120
121
121
124
126
128
129
135
144
144
149
152
153
156
157
159
162

CHAPTER 6 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . 166
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

vii

LIST OF TABLES

Table 2.1 Material parameters for a fictitious biaxial material. . . . . . . . . . .

26

Table 2.2 Material parameters for a fictitious biaxial material. . . . . . . . . . .

30

Table 2.3 Material parameters for a fictitious Uniaxial material. . . . . . . . . .

39

Table 4.1 Number of iterations for different initial guesses. . . . . . . . . . . . .

97

Table 4.2 VNA measurement uncertainties. . . . . . . . . . . . . . . . . . . . . . 100
Table 4.3 Extracted material parameters from measurements with different external magnetic biasing fields. . . . . . . . . . . . . . . . . . . . . . . . . 115
Table 4.4 Extracted parameters values from one optimization using measurements
of different external magnetic biasing fields. . . . . . . . . . . . . . . . 115
Table 5.1 Extracted H0 and material parameters for measurements of the highest
external magnetic biasing fields. . . . . . . . . . . . . . . . . . . . . . 163
Table 5.2 Extracted H0 and material parameters for measurements of the second
highest external magnetic biasing fields. . . . . . . . . . . . . . . . . . 163
Table 5.3 Extracted H0 and material parameters for measurements of the lowest
external magnetic biasing fields. . . . . . . . . . . . . . . . . . . . . . 164
Table 5.4 Extracted H0 and material parameters using three biasing fields

viii

. . . 164

LIST OF FIGURES

Figure 2.1

Waveguide contains cubical sample. . . . . . . . . . . . . . . . . . .

11

Figure 2.2

Cross-sectional view of the waveguide partially filled with biaxial material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Figure 2.3

Side view of a partially filled waveguide. . . . . . . . . . . . . . . . .

12

Figure 2.4

S-parameters computed for a Teflon test material. . . . . . . . . . .

22

Figure 2.5

S-parameters computed for a Teflon test material. . . . . . . . . . .

23

Figure 2.6

S-parameters computed for a biaxial test material with material parameters shown in Table 2.1. . . . . . . . . . . . . . . . . . . . . . .

24

S-parameters computed for a biaxial test material with material parameters shown in Table 2.1. . . . . . . . . . . . . . . . . . . . . . .

25

Figure 2.8

Four cube orientations used in the three-step process. . . . . . . . .

28

Figure 2.9

Extracted permittivity of the fictitious test material with parameters
shown in Table 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Extracted permeability of the fictitious test material with parameters
shown in Table 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Real part of the relative permittivities for a fictitious biaxial material
extracted using 200 random trials. Center line is the average of the
trials. Upper and lower lines show the 95% confidence interval. . . .

35

Figure 2.7

Figure 2.10

Figure 2.11

Figure 2.12

Imaginary part of the relative permittivities for a fictitious biaxial
material extracted using 200 random trials. Center line is the average
of the trials. Upper and lower lines show the 95% confidence interval. 36

Figure 2.13

Real part of the relative permeabilities for a fictitious biaxial material
extracted using 200 random trials. Center line is the average of the
trials. Upper and lower lines show the 95% confidence interval. . . .

37

Figure 2.14

Imaginary part of the relative permeabilities for a fictitious biaxial
material extracted using 200 random trials. Center line is the average
of the trials. Upper and lower lines show the 95% confidence interval. 38

Figure 2.15

Real part of the relative permittivities for a fictitious uniaxial material
extracted using 200 random trials. Center line is the average of the
trials. Upper and lower lines show the 95% confidence interval. . . .

ix

40

Figure 2.16

Imaginary part of the relative permittivities for a fictitious uniaxial
material extracted using 200 random trials. Center line is the average
of the trials. Upper and lower lines show the 95% confidence interval. 41

Figure 2.17

Real part of the relative permeabilities for a fictitious material uniaxial extracted using 200 random trials. Center line is the average of
the trials. Upper and lower lines show the 95% confidence interval. .

42

Figure 2.18

Imaginary part of the relative permeabilities for a fictitious uniaxial
material extracted using 200 random trials. Center line is the average
of the trials. Upper and lower lines show the 95% confidence interval. 43

Figure 3.1

Permeability of the gyromagnetic material for H0 = 1000 Oe. . . . .

48

Figure 3.2

Permeability of the gyromagnetic material for H0 = 1500 Oe. . . . .

48

Figure 3.3

Permeability of the gyromagnetic material for H0 = 1800 Oe. . . . .

49

Figure 3.4

Off-diagonal term δ of the permeability dyadic for H0 = 1800 Oe. .

49

Figure 3.5

Waveguide partially filled with a gyromagnetic sample. . . . . . . .

51

Figure 3.6

Cross-sectional view of a partially filled waveguide. . . . . . . . . . .

51

Figure 3.7

Side view of a partially filled waveguide. . . . . . . . . . . . . . . . .

52

Figure 3.8

Modal spectrum of the propagating modes for a centered sample. . .

61

Figure 3.9

Modal spectrum of the evanescent modes for a centered sample.

62

Figure 3.10

Modal spectrum of the propagating modes for sample shift s=0.1 inches. 62

Figure 3.11

Modal spectrum of the complex modes for sample shift s=0.1 inches.

Figure 3.12

Modal spectrum of the propagating modes for sample shift s=0.3 inches. 63

Figure 3.13

Modal spectrum of the complex modes for sample shift s=0.3 inches.

64

Figure 3.14

Modal spectrum of the propagating modes for sample shift s=0.75
inches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

. .

63

Figure 3.15

Modal spectrum of the complex modes for sample shift s=0.75 inches. 65

Figure 3.16

Propagation constants (propagating modes) for various sample shift
(s) at 2.6 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Propagation constants (complex modes) for various sample shift (s)
at 2.6 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Figure 3.17

x

Figure 3.18

Figure 3.19

Figure 3.20

Figure 3.21

Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25

Propagation constants (propagating modes) for various sample shift
(s) at 3.2 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Propagation constants (complex modes) for various sample shift (s)
at 3.2 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Propagation constants (propagating modes) for various sample shift
(s) at 3.8 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Propagation constants (complex modes) for various sample shift (s)
at 3.8 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Field distribution of the first propagating mode for a sample shift of
s=0 at 2.8 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Field distribution of the first propagating mode for a sample shift of
s=0.2 inches at 2.8 GHz. . . . . . . . . . . . . . . . . . . . . . . . .

74

Field distribution of the first propagating mode for a sample shift of
s=0 at 3.6 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Field distribution of the first propagating mode for a sample shift of
s=0.2 inches at 3.6 GHz. . . . . . . . . . . . . . . . . . . . . . . . .

76

Figure 3.26

Magnitude of the S-parameters simulated using proposed mode-matching
technique and FEM method under L-band (sample centered). . . . . 84

Figure 3.27

Phase of the S-parameters simulated using proposed mode-matching
technique and FEM method under L-band (sample centered). . . . .

84

Figure 3.28

Magnitude of the S-parameters simulated using proposed mode-matching
technique and FEM method under S-band (1 inch shift). . . . . . . 85

Figure 3.29

Phase of the S-parameters simulated using proposed mode-matching
technique and FEM method under S-band (1 inch shift). . . . . . .

85

3-D surface plot of the root squared error for different 4πMs and r
with H0 set to 3000 gauss. . . . . . . . . . . . . . . . . . . . . . . .

91

3-D surface plot of the root squared error for different 4πMs and r
with H0 set to 3000 gauss. . . . . . . . . . . . . . . . . . . . . . . .

92

3-D surface plot of the root squared error for different H0 and r with
4πMs set to 1000 gauss. . . . . . . . . . . . . . . . . . . . . . . . .

93

3-D surface plot of the root squared error for different H0 and r with
4πMs set to 1000 gauss. . . . . . . . . . . . . . . . . . . . . . . . .

93

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

xi

Figure 4.5

Problem setup and results panel of the optimization toolbox GUI. .

98

Figure 4.6

Option panel of the optimization toolbox GUI. . . . . . . . . . . . .

99

Figure 4.7

Iterative function values during one optimization. . . . . . . . . . . 100

Figure 4.8

Extracted H0 for a test sample of width d=34 mm under different
number of trials. The center dots are the average of the trials. Upper
and lower dots show one standard deviation. . . . . . . . . . . . . . 103

Figure 4.9

Extracted r for a test sample of width d=34 mm under different
number of trials. The center dots are the average of the trials. Upper
and lower dots show one standard deviation. . . . . . . . . . . . . . 103

Figure 4.10

Extracted 4πMs for a test sample of width d=34 mm under different
number of trials. The center dots are the average of the trials. Upper
and lower dots show one standard deviation. . . . . . . . . . . . . . 104

Figure 4.11

Extracted H0 for a test sample of width d=54 mm under different
number of trials. The center dots are the average of the trials. Upper
and lower dots show one standard deviation. . . . . . . . . . . . . . 104

Figure 4.12

Extracted r for a test sample of width d=54 mm under different
number of trials. The center dots are the average of the trials. Upper
and lower dots show one standard deviation. . . . . . . . . . . . . . 105

Figure 4.13

Extracted 4πMs for a test sample of width d=54 mm under different
number of trials. The center dots are the average of the trials. Upper
and lower dots show one standard deviation. . . . . . . . . . . . . . 105

Figure 4.14

Histogram of the H0 extracted from 200 random trials of simulated
S-parameters for a partially filled G1010 sample with various widths. 106

Figure 4.15

Histogram of the relative permittivity extracted from 200 random
trials of simulated S-parameters for a partially filled G1010 sample
with various widths. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Figure 4.16

Histogram of the 4πMs extracted from 200 random trials of simulated
S-parameters for a partially filled G1010 sample with various widths. 107

Figure 4.17

Histogram of the H0 extracted from 200 random trials of simulated Sparameters for a partially filled G1010 sample with various thicknesses.107

Figure 4.18

Histogram of the relative permittivity extracted from 200 random
trials of simulated S-parameters for a partially filled G1010 sample
with various thicknesses. . . . . . . . . . . . . . . . . . . . . . . . . 108

xii

Figure 4.19

Histogram of the 4πMs extracted from 200 random trials of simulated S-parameters for a partially filled G1010 sample with various
thicknesses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 4.20

Relative permittivity extracted from 200 random trials of simulated
S-parameters for a partially filled G1010 sample with various widths.
Center black line is the average of the trials. Upper and lower red
dashed lines show the 95% confidence interval. . . . . . . . . . . . . 109

Figure 4.21

Relative permeability values extracted from 200 random trials of simulated S-parameters for a partially filled G1010 sample with various
widths. Center black line is the average of the trials. Upper and lower
red dashed lines show the 95% confidence interval. . . . . . . . . . 109

Figure 4.22

Relative permeability values extracted from 200 random trials of simulated S-parameters for a partially filled G1010 sample with various
widths. Center black line is the average of the trials. Upper and lower
red dashed lines show the 95% confidence interval. . . . . . . . . . . 110

Figure 4.23

Relative permittivity extracted from 200 random trials of simulated
S-parameters for a partially filled G1010 sample with various thicknesses. Center black line is the average of the trials. Upper and lower
red dashed lines show the 95% confidence interval. . . . . . . . . . . 110

Figure 4.24

Relative permeability values extracted from 200 random trials of simulated S-parameters for a partially filled G1010 sample with various
thicknesses. Center black line is the average of the trials. Upper and
lower red dashed lines show the 95% confidence interval. . . . . . . 111

Figure 4.25

Relative permeability values extracted from 200 random trials of simulated S-parameters for a partially filled G1010 sample with various
thicknesses. Center black line is the average of the trials. Upper and
lower red dashed lines show the 95% confidence interval. . . . . . . . 111

Figure 4.26

Measured S-parameters of the highest biasing field and theoretical
S-parameters generated using extracted material parameters. . . . . 116

Figure 4.27

Measured S-parameters of the second biasing field and theoretical
S-parameters generated using extracted material parameters. . . . . 117

Figure 4.28

Measured S-parameters of the lowest biasing field and theoretical Sparameters generated using extracted material parameters. . . . . . 118

Figure 5.1

Uniaxial electric and isotropic magnetic material sample. . . . . . . 123

Figure 5.2

Material sample with both uniaxial electric and magnetic properties. 123

xiii

Figure 5.3

Waveguide measurement system. . . . . . . . . . . . . . . . . . . . . 125

Figure 5.4

Uniaxial material sample placed in the center of the waveguide at
orientation 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Figure 5.5

Sample inserted into the waveguide extension attached to port 1. . . 127

Figure 5.6

Magnitude of S11 from the uniaxial sample meausured 10 times at
orientation 1. The center blue line is the average value of 10 measurements. The upper and lower lines in red show the 95% confidence
interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Figure 5.7

Phase of S11 from the uniaxial sample meausured 10 times at orientation 1. The center blue line is the average value of 10 measurements.
The upper and lower lines in red show the 95% confidence interval. . 130

Figure 5.8

Magnitude of S21 from the uniaxial sample meausured 10 times at
orientation 1. The center blue line is the average value of 10 measurements. The upper and lower lines in red show the 95% confidence
interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Figure 5.9

Phase of S21 from the uniaxial sample meausured 10 times at orientation 1. The center blue line is the average value of 10 measurements.
The upper and lower lines in red show the 95% confidence interval. . 131

Figure 5.10

Magnitude of S11 from the uniaxial sample meausured 10 times at
orientation 2. The center blue line is the average value of 10 measurements. The upper and lower lines in red show the 95% confidence
interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 5.11

Phase of S11 from the uniaxial sample meausured 10 times at orientation 2. The center blue line is the average value of 10 measurements.
The upper and lower lines in red show the 95% confidence interval. . 132

Figure 5.12

Magnitude of S21 from the uniaxial sample meausured 10 times at
orientation 2. The center blue line is the average value of 10 measurements. The upper and lower lines in red show the 95% confidence
interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 5.13

Phase of S21 from the uniaxial sample meausured 10 times at orientation 2. The center blue line is the average value of the 10 measurements. The upper and lower lines in red show the 95% confidence
interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 5.14

Relative permittivities (mean values) of the biaxial sample extracted
using 10 measurement sets. Inset shows 2 − σ confidence interval. . 136

xiv

Figure 5.15

Relative permeabilities (mean values) of the biaxial sample extracted
using 10 measurement sets. Inset shows 2 − σ confidence interval. . 136

Figure 5.16

Extracted relative permittivities of the biaxial sample fitted to a fifthorder polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 5.17

Extracted relative permeabilities of the biaxial sample fitted to a
fifth-order polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 5.18

Comparison of A for the biaxial sample extracted using the partiallyfilled waveguide technique and the reduced-aperture waveguide technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 5.19

Comparison of B for the biaxial sample extracted using the partiallyfilled waveguide technique and the reduced-aperture waveguide technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 5.20

Comparison of C for the biaxial sample extracted using the partiallyfilled waveguide technique and the reduced-aperture waveguide technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 5.21

Comparison of µA for the biaxial sample extracted using the partiallyfilled waveguide technique and the reduced-aperture waveguide technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Figure 5.22

Comparison of µB for the biaxial sample extracted using the partiallyfilled waveguide technique and the reduced-aperture waveguide technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 5.23

Comparison of µC for the biaxial sample extracted using the partiallyfilled waveguide technique and the reduced-aperture waveguide technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 5.24

FGM125 completely filled in the waveguide. . . . . . . . . . . . . . . 142

Figure 5.25

Real and imaginary parts of relative permittivity extracted from 5
sets of measurements of FGM-125. Center line is the average of the
extracted parameter. Upper and lower lines show the 95% confidence
intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure 5.26

Real and imaginary parts of relative permeability extracted from 5
sets of measurements of FGM-125. Center line is the average of the
extracted parameter. Upper and lower lines show the 95% confidence
intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

xv

Figure 5.27

Relative permittivities (real part) of the uniaxial sample extracted
using 10 sets of measurements. Center line is the average of the
extracted parameter. Upper and lower lines show the 95% confidence
intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Figure 5.28

Relative permittivities (imaginary part) of the uniaxial sample extracted using 10 sets of measurements. Center line is the average
of the extracted parameter. Upper and lower lines show the 95%
confidence intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Figure 5.29

Relative permeabilityies (real part) of the uniaxial sample extracted
using 10 sets of measurements. Center line is the average of the
extracted parameter. Upper and lower lines show the 95% confidence
intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Figure 5.30

Relative permeabilityies (imaginary part) of the uniaxial sample extracted using 10 sets of measurements. Center line is the average
of the extracted parameter. Upper and lower lines show the 95%
confidence intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure 5.31

Machined G1010 sample. . . . . . . . . . . . . . . . . . . . . . . . . 151

Figure 5.32

Wooden fixture containing two magnets. . . . . . . . . . . . . . . . . 154

Figure 5.33

Magnetic field strength in the cross-section of the waveguide when no
spacer is inserted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure 5.34

Magnetic field strength in the cross-section of the waveguide when
the 5 mm spacers is inserted. . . . . . . . . . . . . . . . . . . . . . . 155

Figure 5.35

Magnetic field strength in the cross-section of the waveguide when
the 9.7 mm spacers is inserted. . . . . . . . . . . . . . . . . . . . . . 156

Figure 5.36

Polystyrene foam sample holder inside the waveguide with the magnets fixture attached. . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Figure 5.37

G1010 sample inserted into the waveguide extension attached to port
1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Figure 5.38

Magnitude of S11 from the G1010 sample meausured 10 times with
the highest magnetic biasing field strength. Center blue line is the
average value of 10 measurements. Upper and lower lines in red show
the 95% confidence interval (two standard deviations). . . . . . . . . 159

xvi

Figure 5.39

Phase of S11 from the G1010 sample meausured 10 times with the
highest magnetic biasing field strength. Center blue line is the average
value of 10 measurements. Upper and lower lines in red show the 95%
confidence interval (two standard deviations). . . . . . . . . . . . . . 160

Figure 5.40

Magnitude of S21 from the G1010 sample meausured 10 times with
the highest magnetic biasing field strength. Center blue line is the
average value of 10 measurements. Upper and lower lines in red show
the 95% confidence interval (two standard deviations). . . . . . . . . 160

Figure 5.41

Phase of S21 from the G1010 sample meausured 10 times with the
highest magnetic biasing field strength. Center blue line is the average
value of 10 measurements. Upper and lower lines in red show the 95%
confidence interval (two standard deviations). . . . . . . . . . . . . . 161

xvii

CHAPTER 1
INTRODUCTION AND
BACKGROUND
There has been a recent emergence in the development of materials with electromagnetic
properties tailored to the demands of different applications. Most materials that exist naturally are found to either exhibit high loss in the microwave region or to have minimal magnetic properties, which make them undesirable for many applications. Similarly, no known
natural materials exhibit negative permittivity and permeability simultaneously. Because
of the usefulness of these types of properties, there is great emphasis on the development
of engineered materials that offer a variety of possibilities to achieve applications that are
ordinarily out of reach. One well known example is the development of negative index materials by Dr. Pendry [1], later known as metamaterials. The properties of those types of
artificial materials have been studied, and the ability of manipulating material properties
through structural design enables applications such as cloaking [2]-[3]. The development of
novel antennas is another application that employs such materials [4]-[6]. Composite materials with inclusions of graphene nanoribbons or nano metallic particles are used in many
RF applications to obtain desired material properties [7], to miniaturize radio frequency
(RF) electronic components [8]-[9], or to create flexible substrates for wearable RF devices

1

[10]. Engineered materials with 3D structures such as honeycombs are used for broadband
microwave absorption [11]-[12], and 3-D printing technology is used to achieve complex 3D
objects with artificially controlled effective material properties [13]. As the number of applications and the complexity of the engineered materials increase, there is a growing need for
characterization methods that can accurately predict the constitutive parameters of these
materials.
Many characterization techniques suitable for measuring EM materials have been developed. The typical procedures for determining the permittivity and permeability involve
comparisons of the theoretical data and the measured data. The measured data are usually
acquired in a laboratory where the test samples are prepared to accommodate the experimental environment, and a vector network analyzer (VNA) is used to obtain the data.
Appropriate algorithms are then used to extract the complex constitutive parameters by
minimizing the difference between predicted and measured data. The majority of the existing techniques can be categorized into free space methods, resonant cavity methods, coaxial
line methods, and rectangular waveguide methods.
Free space methods have many advantages compared to other methods, such as accurate
results over a wide frequency range, ease of sample preparation, and no physical contact
between the sample and the test equipment [14]. They also provide characterization using
multiple measurements of different incident angles or polarizations [15]-[16] which can be
hard to achieve with other methods. However, these types of methods usually require a
large and flat test sample, and multiple reflections occur between the interrogating antennas
and the surface of the sample. As such, diffraction at the edge of the sample may lead to
significant increases in error. In addition to this, in some applications only transmission
data S21 is used since S11 is more sensitive to measurement uncertainties [17]-[18]. These
disadvantages make free space methods unsuitable for many materials.
Resonant cavity measurements are very accurate in obtaining permittivity and permeability. There are two types of resonant measurements that are commonly adopted. Perturba-

2

tion methods are suitable for all permittivity measurements, and measurements of magnetic
materials with medium to high loss [19]. A low loss measurement method is suitable for
measurements on low loss materials with larger sizes [20]. The resonant cavity method offers the ability to measure very small materials samples by using approximate expressions
for fields in the sample and cavity. However, this method usually requires high frequency
resolution VNAs, accurately machined samples to fit into the cavity, and is generally narrow
band. Therefore, characterization of materials over a wide frequency range is difficult using
this method.
Coaxial line methods are very popular for characterization of complex permittivity and
permeability since they provide broadband measurements using a single test. They are
usually easy to implement compared to the resonant cavity methods where special cavities
are required [21]-[22], and the required sample size is typically small. However, there are
many disadvantages in using these methods. First, samples must be precisely machined
so that there is no air gap between the sample and the coaxial line fixture. It is found
that even a very small air gap can cause huge errors for samples with high permittivity and
corrections have to be made to compensate for the errors caused by air gaps [23]-[24]. Second,
coaxial lines lack ruggedness and the thermal expansion or mechanical vibration between
the sample fixture and the test samples makes it very difficult to maintain consistent precise
measurements [25].
Rectangular waveguides are the most common waveguiding structure and they have several advantages over other characterization methods. Physically, they are more rugged than
coaxial lines, making them less sensitive to thermal expansion and vibration. Unlike the free
space method, resonant cavity method, or the coaxial lime method, sample preparation for
rectangular waveguides is much simpler, since the samples are usually rectangular sheets or
blocks and they do not require high precision. The large surface area of waveguides significantly reduces conductor losses that are relatively high in two wire transmission lines and
coaxial lines due to small conducting surface and roughness of the copper [26]. Also, at high

3

frequencies, energy transmission by coaxial cables becomes less efficient due to skin effect
while waveguides can still maintain a high transmitted signal strength. Also, the analytical
analysis of sample interaction is available [27].
The primary limitations of rectangular waveguides are their cost and physical size. The
inside surfaces of waveguides are often plated with silver or gold to reduce losses which make
them very expensive in many applications. The width of a waveguide is usually a half wavelength at the frequency of the wave to be transported. For instance, the WR650 L-band
waveguide (1.12 GHz-1.7 GHz) has dimensions 6.5 inches by 3.25 inches which makes it very
difficult to carry or set up and this makes the use of rectangular waveguide below 1 GHz
almost impractical. Limited operating bandwidth is also another drawback of rectangular
waveguides compared to other method such as coaxial lines methods. To allow the propagation of a single dominant mode, the lower end of the operating band is usually about 30%
higher than the cutoff frequency. The higher end of the band is also chosen be lower than
the cutoff frequency of the next higher order mode.
Many characterization techniques based on rectangular waveguide applicators have been
developed in which the samples can be placed inside the guide, against an open-ended flange,
or placed against various metallic objects such as irises, posts, slots or stubs. One of the
most widely used technique is the Nicolson-Ross-Weir (NRW) method [28]-[29] in which a
sample is machined to fill the cross-section of the waveguide and the transmission and reflection coefficients are measured. The benefit gained from using this method is its ability
to directly solve for the complex µ and

in closed form. However, a closed-form expression

is not always available for most anisotropic materials and in some cases it is even difficult to
obtain flat test samples which are large enough to fill the entire waveguide cross section. To
obtain the parameters of anisotropic materials without a closed form expression, iterative
solvers such as Newton’s method [30] or least squares approaches [31] are usually used. Several techniques have also been developed to reduce the size requirement of the test sample.
In [32], the authors utilize a stepped waveguide aperture to characterize isotropic materials

4

with small electrical size. This method reduces the cost or producing large, precise samples.
However, the extracted results using this method may be sensitive to the propagated error
from measurement uncertainties. In [33], the author presents characterization of PEC backed
lossy simple media using a waveguide slot aperture, but the extraction process require two
measurements of different sample thickness. Techniques that do not require metallic objects
such as reduced waveguide apertures or waveguide slots have also been developed to simplify
the measurement process. In [34], a dielectric sample is partially loaded in a rectangular
waveguide and measured. Newton’s method is used to search for a fit between simulated
and measured scattering parameters. In [35], the authors present a partially-filled waveguide
method that improves the accuracy of the material characterization. A mode-matching technique is developed to accommodate the high order modes excited at the discontinuity of the
waveguide and Newton’s method is employed to extract the electromagnetic properties of
samples with low-loss and high-loss parameters. To the best knowledge of the author, little
work has been done to achieve accurate characterization of complex anisotropic materials using only rectangular waveguides. The motivation for this dissertation is to develop
characterization methods for different complex anisotropic materials using a partially-filled
waveguide technique, which can be a stepping stone toward full tensor characterization.
Two types of anisotropic materials are considered in this work. The first type are biaxial
materials which have six nonzero entries in the material tensors, corresponding to six complex material parameters. Several techniques have been previously developed to determine
the complex material parameters such as a cavity perturbation technique [36] and a modified
free-space method which extracts the dielectric properties of the test samples. Characterization approaches based on rectangular waveguides have also been investigated to determine
the complex material parameters. Two issues are usually taken into consideration when
characterizing biaxial materials which contain multiple complex parameters. The first is the
determination of the total number of measured data which are sufficient for extracting all
the constitutive parameters, whether using reflection measurement data, transmission mea-

5

surement data or both. The second problem is finding the appropriate experimental setup
to collect the measured data that introduces few uncertainties.
In [37], the authors demonstrate a technique for the extraction of the six material parameters in a biaxial material. Closed form solutions are derived to solve three material
parameters using two different excitations of TE10 and TE20 mode. By rotating the sample, the other three parameters can be obtained. In order to obtain the measured data under
two excitations, a special waveguide fixture is built. However, only the extraction for a nondispersive material was conducted using two sets of measurements and no measurements of
an actual biaxial sample were carried out. Recently a method for characterizing the properties of biaxially anisotropic materials was developed using a reduced-aperture waveguide
system [38]. By using a sample holder of cubical shape, a single sample of biaxial material
may be measured in three different orientations, providing the required number of reflection and transmission measurements to determine the six unique constitutive parameters.
The fields in the sample region are computed analytically, and the mode-matching approach
is used to determine the theoretical S-parameters of the cascaded system consisting of the
sample holder and the empty waveguide transitions. This technique has the drawback that
the sample must fit tightly within the conducting sample holder (to eliminate air gaps), the
restricted aperture of the sample holder reduces the energy transmitted through the sample,
and a special sample holder must be constructed. To achieve the full characterization of
baixial media without requiring an additional sample holder, one work [39] was developed in
which the measured data is obtained from multiple independent reflection measurements of
samples partially filled in a shorted rectangular waveguide at different orientations. Iterative
solvers that compare the difference between FEM simulated results of the S11 characteristics
and the measured reflection data are used to determine the parameters. However, this work
only shows the characterization of permittivity properties of the biaxial sample and the use
of FEM simulations greatly slow down the extraction procedure. To overcome the drawbacks
seen in these works, this dissertation introduces a partially filled waveguide technique for full

6

biaxial material characterization. This approach eliminates the presence of gaps along the
sidewalls, reduces reflections from the conducting restriction, and does not require a special
sample holder. More importantly, an analytic solution is used instead of FEM simulations,
which greatly shortens the extraction time.
The second type of materials considered in this dissertation are gyromagnetic materials.
Gyromagnetic materials have scalar permittivity and an anisotropic permeability tensor that
contains two identical and one different diagonal entry, and two off-diagonal entries. Gyromagnetic materials are widely used in many applications such as filters, phase shifters, and
radar absorbing materials (RAM) and techniques that accurately extract the complex scalar
permittivity and permeability tensor components are desired. In [40] and [41] the authors
describe a method for measuring the complex scalar permittivity and permeability tensor
components of a gyromagnetic material using a partially filled waveguide technique. A sample that partially fills the cross-section of the guide is placed next to a dielectric slab. A
mode-matching technique is used to derive the theoretical reflection and transmission coefficients. Two sets of measurements are required to extract the material parameters. The first
set is taken when the gyromagnetic material is completely demagnetized to extract the scalar
permittivity properties. A second measurement with certain magnetic biasing configuration
is used to complete the extract for the permeability tensor. However, the authors failed
to demonstrate the accuracy of the extracted parameters and no information was given to
validate the extracted results compared to the theoretical values. In addition, this method
may suffer from the measurement uncertainties caused by air gaps between the test sample
and the dielectric slab, and the use of electromagnets with the assumption of known static
magnetic field strength may amplify the uncertainties in the extracted parameters. Recently,
a method for characterizing gyromagnetic materials using a reduced-aperture waveguide was
developed [42] in which the sample completely fills the cross-section of a narrow waveguide
section. This method has several drawbacks, such as the need of a special sample holder,
potential for air gaps along four walls, reduced energy transmission due to the restriction of

7

the narrow guide and no experimental validation is performed. These drawbacks leads to
part of the motivation for this work, which is to develop a accurate method for characterizing
gyromagnetic materials by placing a narrow sample into a full-aperture waveguide.

8

CHAPTER 2
Characterization of Biaxial Material
Using a Partially-Filled Waveguide
2.1

Introduction

This chapter presents a novel waveguide method for extracting full biaxial tensors using a
single sample that is partially filled in a waveguide. This method eliminates the presence
of gaps along the sidewalls, reduces reflections from the conducting restriction, and does
not require a special sample holder [42]. Since the extraction of material parameters is
accomplished through an inverse problem that minimizes the difference between the theoretical reflection and transmission coefficients and the measured S-parameters, a carefully
computed forward problem which computes theoretical S-parameters is needed for accurate
characterization. The theoretical reflection and transmission coefficients are determined using a mode-matching technique. This method allows the computational error in the forward
problem to be controlled by using an appropriate number of modes. Two samples made
of stacked layers of substrates were manufactured in the shape of a cube so that multiple
measurements can be obtained only from one sample which is inserted with different orientations in the guide. The sample was placed at the center of the cross-section of a waveguide,

9

and the transmission and reflection coefficients were measured for various orientations. The
material parameters were extracted to demonstrate the feasibility of the proposed approach.

2.2

Theoretical Transmission and Reflection Coefficients
Using Mode-Matching Analysis

Biaxial materials are anisotropic and have six non-zero complex permittivity and permeability tensor entries along the orthogonal axes A, B, and C. The material properties can be
described by the tensors




 A

= 0
 0

0
and

0
B
0

0 

0 


C



(2.1)



0
 µA 0

µ = µ0 
 0 µB 0

0
0 µC



,



(2.2)

where the values A , µA , etc., are relative parameters, and are complex quantities: A =
rA + j iA , µA = µrA + jµiA , etc.
The proposed measurement system is shown in Figure 2.1. A cubical sample of the
biaxial material is centered within the cross-section of a rectangular waveguide such that the
cross-sectional view is shown in Figure 2.2. The sample is machined into a cube of which the
side length d is same as the height of the waveguide aperture b. Moreover, no extra sample
holder is required to contain the cubical sample, and the sample can be placed directly into
the waveguide extension of the sending end (z < 0). The extension serves as a sample holder
and the length is properly chosen such that a single TE10 mode is incident on the sample
from the sending (z < 0) extension, and that a single mode appears at the receiving end
(z > d).

10

[Ɛ], [µ]

Ɛ0,µ 0

Ɛ0,µ 0

y

b
x

z

a

d
Figure 2.1 Waveguide contains cubical sample.

y

b

0𝑎

−

2

III

I

II

Ɛ0,µ 0

[Ɛ], [µ]

Ɛ0,µ 0
x

−

𝑑
+𝑠
2

𝑑
+𝑠
2

𝑎
2

Figure 2.2 Cross-sectional view of the waveguide partially filled with biaxial material.

11

To solve the inverse problem which requires minimizing the difference between the measured and the theoretical reflection and transmission coefficients, an accurate computation
of the S-parameters of the partially-filled waveguide system is desired. This is accomplished
by a mode-matching technique, which takes account of the higher order modes excited at
the interface between the empty waveguide and the sample region (0 < z < d), and allows
the computational accuracy to be easily controlled.
A single TE10 mode is assumed to be incident upon the sample, as shown in Figure 2.3.
Due to the discontinuity at the sample interface, an infinite number of waveguide modes
are reflected into the transmitting extension (z < 0), while an infinite number of waveguide
modes are transmitted into the sample region. The transmitted modal fields are incident on
the second sample interface z = d, and thus a spectrum of higher order modes is also reflected
back into the sample region, and transmitted into the receiving end (z > d). However, since
the electric field of the incident TE10 mode is even about x = 0, and the sample placement
is also symmetric about x = 0, the incident field will only couple to modes with a similar
symmetry. Also, the field structure of the empty waveguide is well known [43].Therefore,
only TEn0 modes with odd values of n are excited in the empty guides, with electric fields
that are even about x = 0.

y

[Ɛ], [µ]
TEn0
TEn0

Ɛ0,µ 0
TE10
TEn0
0

Ɛ0,µ 0

TEn0
z
Δ

Figure 2.3 Side view of a partially filled waveguide.

12

2.2.1

Field Structure in a Waveguide Partially Filled with Biaxial
Material

The material sample can be placed into the waveguide in various orientations. Assuming that
the orientation is chosen such that µA = µx , µB = µy and µC = µz , etc. As mentioned,
only TEn0 modes with odd values of n are excited in the empty guides. The field structure
of the empty waveguide modes is given by,
a
jωµo
sin k¯cn x −
E¯n (x, z) = − ¯
2
kcn

¯ n (x, z) = En (x, z) .
e±jβn z , H
Z¯n

(2.3)

s = nπ/a is the cutoff wavenumber for the nth TE
Here a is the width of the guide, k¯cn
n0
ωµ
mode (n = 1, 2, . . .), and Z¯n = ¯ o is the TE wave impedance. The real phase constant β¯n
βn
of the TEn0 mode is given by
β¯n =

k02 − (k¯cn )2

(2.4)

√
with k0 = ω µ0 0 being the free-space wavenumber.
The field structure of the waveguide modes in the sample region is more complicated and
can be found solving the wave equation for a biaxial medium (Section 2.3.3 [42]):
∂2
+ (kcn )2 Hz (x, z) = 0
2
∂x

(2.5)

µz 2
(kcn )2 =
ω µx y − (βn )2
µx

(2.6)

where

is the cutoff wavenumber. The fields are given as
HzI (x, z) = AI sin kcI e±jβn z
jωµz I
EyI (x, z) =
A cos kcI e±jβn z ,
I
kc

13

(2.7)
(2.8)

I is
where AI is the modal amplitude of the transverse field for the biaxial medium, and kcn
the cutoff wavenumber shown in (2.6). Note that there is also empty space in the sample
region. The field structure for this part is the same as that of the empty waveguide, and
thus the fields can be expressed as:
II e±jβn z
HzII (x, z) = AII sin kcn
jωµ0 II
a ±jβn z
EyII (x, z) = −
A cos kcII x −
e
,
II
2
kcn

(2.9)
(2.10)

where AII is the modal amplitude of the transverse field for the empty part in the sample
II is the cutoff wavenumber in the empty part of the sample region.
region, and kcn
With the expressions of the fields shown above, the boundary conditions are applied at
d
the interfaces x = a
2 and x = 2 . The complex propagation constants β are found by solving
the transcendental equation,
I d
µo kcI sin kcn
2

II a − d
sin kcn
2

II cos k I d
= µz kcn
c2

II
cos kcn

a−d
2

(2.11)

where the cutoff wavenumbers are related to the complex propagation constants by
I
kcn

2

=

µz 2
ω µx y − (βn )2
µx
2.
kcII 2 = k02 − βn

(2.12)
(2.13)

d
Note that the boundary conditions at x = − a
2 and x = − 2 are also satisfied due to symmetry.
Different approaches can be used to solve the transcendental equation for βn . One approach is to start with the solution for βn under the condition d = 0, which is found by
equations of the TEn0 modes in an empty guide, and then slowly increase the width of the
sample, solving for βn at each step using the previous solution as an initial guess. This approach has been used in [35] for thin samples. However, as the sample width increases, there
is an uncertainty of the modal types between propagating and evanescent. Thus, solving
14

the transcendental equation using previous results as initial guesses becomes problematic.
Instead, the following approach is used when the material loss is not too high. First the
imaginary parts of the material parameters are set to zero which implies a lossless material.
Then the real values of β are found by looking for zero crossings of the transcendental equa√
tion (2.11) between β = 0 and β = ω µx y , and as beta exceeds this range, no real roots
can be found. These zero crossings represent the propagating modes, and the modal number
n is assigned based on the order of the zero crossing. Next the phase constant β in the
transcendental equation is replaced by jδ, and real values of δ are found by finding zero
crossings starting from δ = 0 and continuing until a desired number of modes have been
found. These are the evanescent modes, and they are numbered continuing after the highest
order propagating mode. Finally, the imaginary parts of the material parameters and the
zero crossing values found earlier for β and δ are used as initial guesses in a root-search
method to find the complex propagation constants βn for both the propagating and the
evanescent modes.
With the complex propagation constants βn found, the modal fields are given for x > 0
by



I (x, z),
E n
En (x, z) =


II (x, z),
E n



I (x, z),
Hn
Hn (x, z) =


II (x, z),
Hn

15

0 ≤ x ≤ d2

(2.14)

d <x≤ a
2
2
0 ≤ x ≤ d2
d < x ≤ a,
2
2

(2.15)

where
jωµz
I x e±jβn z ,
cos kcn
I
kcn
a
II
d sin kc (x − 2 ) ±jβn z
I
sin kcn
e
,
2 cos k II d−a
c
2
I
II
I (x) = En (x) , H II (x) = En (x) ,
Hn
n
ZI n
Z II n
I (x, z) =
En

II (x, z) = − jωµz
En
II
kcn

(2.16)

(2.17)

(2.18)

with
ZI n =

2.2.2

ωµx
,
βn

Z II n =

ωµo
.
βn

(2.19)

Solution for S-Parameters Using Modal Expansions

The transverse fields can be expanded as an infinite sum of TEn0 modes, with modal amplitudes determined by applying appropriate boundary conditions. For the empty waveguide
extension region z < 0,
¯
Ey (x, z) = ai1 E¯1 (x)e−j β1 z +

∞

¯
arn E¯n (x)e+j βn z

(2.20)

¯ n (x)e+j β¯n z .
arn H

(2.21)

n=1
odd

¯ (x)e−j β¯1 z +
Hz (x, z) = −ai1 H
1

∞
n=1
odd

The modal amplitude ai1 represents the amplitude of the incident TE10 wave, which is
assumed to be known. Only odd mode indices are used such that no even mode is taken into

16

account. In the sample region, 0 < z < d, the transverse fields are
∞

−jβn z + a− e+jβn z E (x)
a+
n
ne
n

(2.22)

−jβn z + a− e+jβn z H (x).
−a+
n
ne
n

(2.23)

Ey (x, z) =
n=1
odd

∞
Hx (x, z) =
n=1
odd

Finally, since only transmited fields exist in the waveguide extension for z > d, the transverse
fields are
∞

¯
atn E¯n (x)e−j βn (z−d)

(2.24)

¯ n (x)e−j β¯n (z−d) .
atn H

(2.25)

Ey (x, z) =
n=1
odd

∞
Hx (x, z) = −
n=1
odd

The unknown modal amplitudes can be determined by applying the boundary conditions
on Ey and Hx at the interfaces z = 0 and z = d,
N

ai1 E¯1 (x) +

n=1
¯ (x) +
−ai1 H
1

N

−
a+
n + an En (x),

(2.26)

−
−a+
n + an Hn (x),

(2.27)

n=1

¯ n (x) =
arn H

N
n=1

n=1
N

N

arn E¯n (x) =

−jβn d + a− e+jβn d E (x) =
a+
n
ne
n

atn E¯n (x),

(2.28)

¯ n (x).
atn H

(2.29)

n=1

n=1
N

N

−jβn d + a− e+jβn d H (x) = −
−a+
n
ne
n

N
n=1

n=1

Note the number of modes N used for the empty waveguide is the same as that used for

17

sample regions. These equations above are a system of functional equations and can be
transformed into a system of linear equations by applying the testing operations of empty
waveguide modes to the four equations. The linear equations with testing operators are
expressed as
a
a
N
2 ¯
2 ¯
r
E1 (x)E¯m (x)dx +
an
En (x)E¯m (x)dx =
0
0
n=1

ai1
N

a+
n

n=1
ai1

n=1
−jβn d
a+
ne

n=1

a
2
0

0

a+
n

a
2
0

a
2

N

−jβn d
a−
ne

En (x)E¯m (x)dx +
n=1

0
a
2

n=1

a
2
0

En (x)E¯m (x)dx,

(2.30)

atn

N
¯ m (x)dx −
Hn (x)H

−jβn d
a−
ne

n=1
N
n=1

18

0
a
2

0

atn

¯ m (x)dx,
Hn (x)H

(2.31)

En (x)E¯m (x)dx =

0

n=1
−jβn d
a+
ne

0

¯ m (x)dx − a−
Hn (x)H
n

N

N

a
2

En (x)E¯m (x)dx + a−
n

a
a
N
2 ¯
2 ¯
r
¯
¯ m (x)dx =
H1 (x)Hm (x)dx −
an
Hn (x)H
0
0
n=1

N

N

a
2

a
2 ¯
En (x)E¯m (x)dx,

(2.32)

¯ m (x)dx =
Hn (x)H

0

a
2 ¯
¯ m (x)dx. (2.33)
Hn (x)H

The integrals are defined as,

Cmn =

a
2

E 1 (x)E m (x)dx
0
a
2
H 1 (x)H m (x)dx
Emn =
0
a
2
E n (x)E m (x)dx
Dmn =
0
a
2
Fmn =
H n (x)H m (x)dx.
0

(2.34)
(2.35)
(2.36)
(2.37)

The integrals can be computed in closed form and this results in a 4N × 4N system of
equations as follows,

+
ai1 Cm1

N

N

arn Cmn =

a+
n Dmn +

N

a−
n Dmn

n=1
n=1
n=1
N
N
N
+
i
r
a1 Em1 −
an Emn =
an Fmn −
a−
n Fmn
n=1
n=1
n=1
N
N
N
− D e+jβn d =
−jβn d +
a+
D
e
a
atn Cmn
n mn
n mn
n=1
n=1
n=1
N
N
N
−
+
−jβ
d
+jβ
d
n
n
an Fmn e
atn Emn ,
an Fmn e
−
=
n=1
n=1
n=1

(2.38)

(2.39)

(2.40)

(2.41)

which can be rewritten as a matrix equation,


−Cmn



 Emn



0


0

Dmn

Dmn

0






Fmn −Fmn
0



Dmn −Dmn −Cmn  

−Fmn Fmn
Emn

19

arn
a+
n
a−
n
atn





Cm1










 E

m1



 = ai1 





 0 



0

(2.42)

where
Dmn = Dmn e−jβn d ,

Dmn = Dmn e+jβn d

Fmn = Fmn e−jβn d ,

Fmn = Fmn e+jβn d .

(2.43)

Each of the quantities Cmn and Dmn are N xN submatrices and arn and atn are the unknown
modal coefficients. By solving the matrix equation, the modal amplitudes can be found.
Since ai1 is the known amplitude of the incident TE10 mode, the theoretical S-parameters
(S thy ) can be computed from the equations
r
thy a
S11 = 1 ,
ai1

2.2.3

t
thy a
S21 = 1 .
ai1

(2.44)

Validation of Theoretical Analysis

It is important to validate the mode-matching technique by comparison to an alternative
method before using it for parameter extraction. Ideally, for the mode-matching technique,
an infinite number of modes should be taken into consideration in order to compute the
theoretical S-parameters with high accuracy. However, it is not necessary to compute the
forward problem with a large number of modes all the time. For example, it would be timeconsuming and needless to compute the S-parameters with an accuracy of ±0.01% if the
percentage error of the measured S-parameters is 1%. Thus, an appropriate number of modes
should be chosen so that reasonable accuracy can be achieved without drastically increasing
the computational time. One straightforward approach to achieve this is to increase the size
of N until the expected accuracy is achieved. This approach might be time consuming since
it requires repeatedly solving the matrix equation. One way to accelerate this process as N
is increased is to use an iterative matrix solver, with a previous solution as an initial guess
[?]. For some cases where a large number of iterative computation is required, extrapolation
methods such as linear or least squares extrapolation can be used as an alternative to improve
20

estimation of the S-parameters.
The validation starts with a lossless, isotropic sample of Teflon with a dielectric constant
of 2.1. The mode-matching technique was used to compute the S-parameters for the cubical
sample placed into an L-band waveguide system with waveguide dimensions of a = 16.51 cm
and b = 8.255 cm. The convergence criteria for computing the S-parameters are 0.01 dB in
magnitude, and 0.1◦ in phase. For this Teflon sample, convergence can be reached within
ten modes, and the number of modes is much smaller than the number of modes used in
[42] where more than two hundred modes may be required to reach the same convergences.
The S-parameters were also computed using the commercial full-wave solver HFSS. The
waveguide extensions with length of 12 inches were included in HFSS, and that length
was chosen to ensure that only the TE10 mode propagates at the waveguide ports. The
convergence tolerance (delta S) in HFSS was specified as 0.001, which is the absolute Sparameter difference between two iterations. It can be seen from Figures 2.4 and 2.5 that
excellent agreement is obtained between the two methods, validating the proposed modematching technique.
A second example, of a fictitious biaxial material with the material parameters shown in
Table 2.1, was considered to further validate this technique. The material is placed into the
same L-band waveguide as in the first example with an orientation such that the material
axes (A, B, C) are aligned with (x, y, z) respectively. The same S-parameter accuracy as the
first example was specified and the computed S-parameters are shown in Figure 2.6 and 2.7,
along with the results generated from HFSS. The excellent agreement further validates the
mode-matching technique with biaxial materials.
Note that the computational time of the mode-matching technique is significantly shorter
than that in the HFSS. Therefore, the extraction process using closed form expressions take
less time than those approaches using FEM simulated results.

21

Figure 2.4 S-parameters computed for a Teflon test material.

22

Figure 2.5 S-parameters computed for a Teflon test material.

23

Figure 2.6 S-parameters computed for a biaxial test material with material parameters shown
in Table 2.1.

24

Figure 2.7 S-parameters computed for a biaxial test material with material parameters shown
in Table 2.1.

25

Table 2.1 Material parameters for a fictitious biaxial material.
Parameter Value
1.5
A
1
B
2.0
C
µA
1.0
3.0
µB
µC
1.0

2.2.4

Extraction Process

Since there are six complex biaxial material parameters ( A , B , C , µA , µB , µC ) to be determined, a minimum of six independent measurements are required. These measurements
can be achieved by measuring S11 and S21 of a biaxial sample with the material axes A, B,
and C aligned along three properly chosen directions. With the following orientations,

(A, B, C) → (x, y, z)

(2.45)

(A, B, C) → (z, x, y)

(2.46)

(A, B, C) → (y, z, x)

(2.47)

meas , S meas , S meas , S meas , and S meas , S meas can be obtained,
S-parameters of S11,1
21,1
11,2
21,2
11,3
21,3
respectively. The material parameters can be determined by solving six nonlinear equations
with six complex unknowns,
thy
meas = 0
S11,n ( A , B , C , µA , µB , µC ) − S11,n
thy
meas = 0
S21,n ( A , B , C , µA , µB , µC ) − S21,n

n = 1, 2, 3

(2.48)

n = 1, 2, 3.

(2.49)

However, since standard root-solving methods such as Newton’s method typically require
accurate initial guesses, it is very difficult to find six proper guesses for this problem. In
addition, due to experimental error, one might not be able to find solutions to the system of
26

equations even with good initial guesses. Therefore, traditional methods are not advisable
for solving this problem, and an alternative method needs to be developed. In this work,
the six parameters are obtained by separating the system equations into several subsets of
system equations. The new subsets of equations contain fewer unknowns and equations. A
subset of the material parameters may be found by solving the first set of system equations
and then these parameters can be used as known quantities to solve the second set until all
parameters are extracted. This approach is feasible because only parameters B , µA , and
µC are involved in any orientation.
For a cubical sample, there are 24 different orientations that it can be inserted into the
waveguide, and this creates various measurement combinations. A three-step process was
developed to obtain the material parameters. First, measurements are implemented with
the orientations

giving the S-parameters

(A, B, C) → (x, y, z)

(2.50)

(A, B, C) → (−z, y, x)

(2.51)

meas , S meas
S11,1
21,1

and

meas , S meas
S11,2
21,2

respectively. The initial

orientation is shown in Figure 2.8 and labeled 1. Then the sample is rotated by 90◦ and
measured S-parameters are obtained in the orientation labeled 2. These measurements only
involve the parameters B , µA , and µC . Using three out of the four complex measurements
and Newton’s method, the first set of equations with three unknowns,

thy
meas = 0, n = 1,
S11,n ( B , µA , µC ) − S11,n
thy
meas = 0, n = 1, 2,
S21,n ( B , µA , µC ) − S21,n

(2.52)
(2.53)

can be solved. The reason that measurement of S21,2 is chosen over S11,2 is that it might
improve the characterization efficiency and less difficulties are encountered during the root
27

y

B

y

B
Z

C

Z

C

X

X
A

A
y

y

C

A
C Z
X

Z
B

X

B

A

Figure 2.8 Four cube orientations used in the three-step process.
search since S21 phase is generally more accurate and it is less sensitive to position error.
Next, another measurement is made using orientation 3 in Figure 2.8. This orientation

(A, B, C) → (y, −x, z)

(2.54)

meas , S meas and this measurement involves , µ , and µ . However, since µ
gives S11,3
A B
C
C
21,3
is known, A and µB can be found by solving the system of equations,
thy
meas = 0,
S11,3 ( A , µB ) − S11,3
thy
meas = 0,
S21,3 ( A , µB ) − S21,3

(2.55)
(2.56)

using Newton’s method. Finally, a last measurement is made under orientation 4 in Figure
2.8,
(A, B, C) → (z, x, y).
28

(2.57)

This measurement implicates C , µA , and µB . Since only C is unknown, it can be found
by solving the single complex equation
thy
meas = 0.
S21,4 ( C ) − S21,4

(2.58)

This process described above is the three-step approach that can be used to extract the six
complex material parameters. The drawback of this method is that many measurements are
required and since the three-step approach requires using previously solved parameters for
the latter two steps, measurement error from the first two steps can influence the extracted
results, sometimes leading to amplified errors in the extracted results.
For materials that are unixially anisotropic, where A = C and µA = µC , the threestep approach can be modified to reduce the impact of accumulated errors, thus improving
the extraction accuracy. The extraction process for uniaxial materials, which have only four
complex material parameters, can be reduced to two steps. First, a measurement is made
with orientation 1 shown in Figure 2.8. This measurement implicates the parameters B ,
µA , and µC . Since the value of µA is the same as µC for unixial material, only two complex
measurements are required to solve the following equations using Newton’s method:
thy
meas = 0,
S11,1 ( B , µC ) − S11,1
thy
meas = 0,
S21,1 ( B , µC ) − S21,1

(2.59)
(2.60)

Next, another measurement is made using orientation 3 in Figure 2.8 and this measurement
implicates A , µB , and µC . However, since µC is determined from the previous equations,
A and µB can be found by solving the system of equations
thy
meas = 0,
S11,3 ( A , µB ) − S11,3
thy
meas = 0.
S21,3 ( A , µB ) − S21,3

29

(2.61)
(2.62)

This approach reduces the number of measurements for characterizing a uniaxial material.
Thus the propagation of measurement errors can be reduced, which improves the accuracy
of the extracted material parameters.
To validate the proposed technique for characterizing bixial materials, S-parameters were
generated using the fictitious material parameters from Table 2.2 with the cubical sample in
four orientations. The three-step extraction approach was performed with the convergence
of the S-parameters assigned to be 0.01 dB in magnitude, and 0.01◦ in phase. The results of
the characterization are shown in Figure 2.9 and 2.10. Good agreement is achieved between
the characterized material parameters and the parameters shown in Table 2.2. Notice that
data gaps are seen in the exrtacted parameters over specific ranges of frequency. This
is a typical problem inherent to all guided-wave techniques where the sample thickness
approaches one half of a guided wavelength, includingthe Nicolson-Ross-Wier closed-form
method for isotropic materials [28]-[29]. Experience has shown that a frequency range within
approximately ±5% of the half-wavelength frequency should be avoided, and data within that
range is not displayed in the figures.
Table 2.2 Material parameters for a fictitious biaxial material.
Parameter
A
B
C
µA
µB
µC

2.2.5

Value
2.0-0.01j
4.0-0.02j
3.0-0.03j
1.5-0.02j
2.0-0.01j
2.5-0.03j

Error and Sensitivity Analysis

There are two types of error that should be taken into consideration during the extraction
procedure. The first type of error arises from inaccuracies in the forward problem, and usually
these errors in the theoretical reflection and transmission coefficients are introduced from
30

4.5
4

εrB

3.5

Relative Permittivity

3

εrC

2.5
2

εrA
0

1.5

εiA

−0.01

1

εiB

−0.02
−0.03
2.78

0.5

εiC
2.8

2.82 2.84 2.86

0
−0.5
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 2.9 Extracted permittivity of the fictitious test material with parameters shown in
Table 2.2.

31

2.5

µrC
µ

Relative Permeability

2

1.5

rB

µrA
0

1

µiB
µ
iA
µiC

−0.01
−0.02

0.5

−0.03
2.78

2.8

2.82 2.84 2.86

0

−0.5
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 2.10 Extracted permeability of the fictitious test material with parameters shown in
Table 2.2.

32

errors in the numerical solution or from difficulties in accurately modeling the experiment.
The analysis in this section uses closed-form expressions for the prediction of reflection and
transmission coefficients and therefore this first type of error is not implicated.
The second type of error is due to measurement inaccuracies and it can be categorized
into systematic error and random error. Systematic errors influence the experimental results to cause consistent inaccuracies. They mostly arise from the imperfect construction
of the experimental setup or fabricated sample or poorly performing test equipment. An
inaccurately machined sample, for example, can produce air gaps between the sample and
waveguide walls and cause error between the theoretical S-parameters and the measured
S-parameters. Random errors are caused by misalignment of waveguide sections, inaccurate sample positioning in the waveguide which varies from experiment to experiment, and
measurement uncertainty inherent in the VNA. It is difficult to model the S-parameter errors caused by inaccurate sample position, due to the analysis assumption that the sample
is located at the center of the cross-section of the guide. To reduce random errors in the
experiments, accurately machined samples are required along with alignment pins for waveguide sections. In comparison, uncertainties inherent to the VNA can be easily studied using
Monte Carlo techniques.
Different S-parameters uncertainties were determined for the extraction of the material
parameters of biaxial and uniaxial materials based on the HP 8510C Specifications & Performance Verification Program provided by Hewlett Packard. For the HP8510C, a statistical
variance of S11 is specified linearly in amplitude and phase, with values of σA = 0.004
11
◦
and σφ = 0.8 , respectivly. Variance of S21 is specified logarithmically in amplitude and
11
linearly in phase with values of σA = 0.04 dB and σφ = 2.0◦ , respectivly. Note that
21
21
this VNA is an old model and the error level is significantly larger than newer VNA models.
Also, these are worst case VNA uncertainties.
A Monte Carlo analysis of the propagation of VNA uncertainty for extraction of a bixial
material at S-band (2.6-3.95 GHz) was performed using the fictitious test material with

33

properties are given in Table 2.2. The cubical sample has a side length of 34.036 mm.
The forward problem was computed at 101 frequency points under the four orientations
described in Section 2.2.4. White Gaussian noise was introduced at each frequency point of
the generated S-parameters and the values of the standard deviations of the Gaussian noise
was chosen to be half of the values indicated by the HP 8510 Specifications & Performance
Verification Program. The data with noise mixed in was then used to extract the material
parameters according to the three-step approach. This process was repeated 200 times,
and the average values of the material parameters were calculated, along with the standard
deviations. Results are shown in Figures 2.11 to 2.14. In these figures, the center black
line is the average value of the 200 trials, while the two surrounding lines indicate the
95% confidence interval (2σ). It can be seen that data gaps similar to those displayed in
Figures 2.9 and 2.10 appear in the error analysis. In addition, new data gaps at 2.7 GHz
and 3.65 GHz are observed. It can be also noticed that the propagation errors for the real
part of the material parameters vary significantly with frequency. For example, in Figure
2.11, rB shows a low error at the lower frequency range of S-band and higher error at the
higher frequency ranges, while the opposite occurs for the propagation error of rA . For the
imaginary part of the material parameters, the propagation error is relatively low compared
to that of the real part except at 2.75, 3.4 and 3.75 GHz. Considering the overall propagation
error level and the corresponding uncertainties introduced, this extraction method for biaxial
material is reasonally robust.
A similar Monte Carlo error analysis was undertaken for a fictitious uniaxial material
to evaluate the impact of measurement uncertainties for the extraction method for uniaxial
material. The material parameters used are shown in Table 2.3. The material parameters are
chosen to have larger dielectric loss and magnetic loss compared to the test biaxial sample.
The standard deviations used to generate the gaussian noise are one quarter of the values
indicated by the HP 8510 Specifications & Performance Verification Program. The values are
σA = 0.001 and σφ = 0.2◦ , σA = 0.01 dB and σφ = 0.5◦ . The results are shown
11
21
11
21
34

4.5

εrB
r

Relative Permittivity ( ε )

4
3.5

εrC
3
2.5

εrA

2
1.5
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 2.11 Real part of the relative permittivities for a fictitious biaxial material extracted
using 200 random trials. Center line is the average of the trials. Upper and lower lines show
the 95% confidence interval.

35

Relative Permittivity ( εi )
εiB

εiA

0.2
0
−0.2
2.6

2.8

3

3.2

3.4

3.6

3.8

2.8

3

3.2

3.4

3.6

3.8

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

0.5
0
−0.5
2.6

εiC

0.2
0
−0.2
2.6

Figure 2.12 Imaginary part of the relative permittivities for a fictitious biaxial material
extracted using 200 random trials. Center line is the average of the trials. Upper and lower
lines show the 95% confidence interval.

36

2.8

µrB

r

Relative Permeability ( µ )

2.6
2.4
2.2

µrC

2
1.8

µrA

1.6
1.4
1.2
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 2.13 Real part of the relative permeabilities for a fictitious biaxial material extracted
using 200 random trials. Center line is the average of the trials. Upper and lower lines show
the 95% confidence interval.

37

µiA

0.2
0

Relative Permeability ( µi )
µiB

−0.2
2.6

2.8

3

3.2

3.4

3.6

3.8

2.8

3

3.2

3.4

3.6

3.8

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

0.2
0
−0.2
2.6

µiC

0.2
0
−0.2
2.6

Figure 2.14 Imaginary part of the relative permeabilities for a fictitious biaxial material
extracted using 200 random trials. Center line is the average of the trials. Upper and lower
lines show the 95% confidence interval.

38

in Figures 2.11 to 2.14. Since the number of orientations to generate the S-parameters are
reduced to two for uniaxial materials, and the number of unknowns under each orientation
are also reduced to two, this extraction approach for uniaxial material shows a much lower
error due to measurement uncertainties. It can be seen that almost no data gaps exist and
the overall error is consistent and small (less than 5% of the actual value). Therefore, this
extraction technique is very robust to random measurement errors.
Table 2.3 Material parameters for a fictitious Uniaxial material.
Parameter
A
B
µA
µB

2.2.6

Value
4.0-0.2j
5.0-0.3j
1.2-1.3j
1.4-1.1j

Summary

A partially-filled waveguide technique is introduced for measuring the permittivity and permeability of biaxially anisotropic materials. A single cubical sample is required and measured under different orientations in a waveguide to complete the extraction process. The
performance of the technique is evaluated using error analysis based on network analyzer
uncertainty. It is found that data gaps may occur during the error analysis when the electrical size of the sample approaches a half wavelength. However, the overall performance is
still considered to be acceptable. In addition, a simplified method for extracting unixially
anisotropic materials is established and this method has shown excellent performance under
moderate noise levels.

39

6
5.02

5

Relative Permittivity ( εr )

5.5

εrB

4.98
2.9

2.92

2.94

2.96

2.9

2.92

2.94

2.96

5
4.02

4

4.5

3.98

εrA
4

2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 2.15 Real part of the relative permittivities for a fictitious uniaxial material extracted
using 200 random trials. Center line is the average of the trials. Upper and lower lines show
the 95% confidence interval.

40

Relative Permittivity
εiA

−0.18
−0.19
−0.2
−0.21
−0.22

Relative Permittivity
εiB

2.6

2.8

3

3.2

3.4

3.6

3.8

2.8

3

3.2

3.4

3.6

3.8

−0.26
−0.28
−0.3
−0.32
−0.34
2.6

Figure 2.16 Imaginary part of the relative permittivities for a fictitious uniaxial material
extracted using 200 random trials. Center line is the average of the trials. Upper and lower
lines show the 95% confidence interval.

41

1.45

µrB
Relative Permeability( µr )

1.4
1.35
1.3
1.25

µrC
1.2
1.15
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 2.17 Real part of the relative permeabilities for a fictitious material uniaxial extracted
using 200 random trials. Center line is the average of the trials. Upper and lower lines show
the 95% confidence interval.

42

Relative Permeability
µiC

−1.29
−1.295
−1.3
−1.305
−1.31

Relative Permeability
µiB

2.6

2.8

3

3.2

3.4

3.6

3.8

2.8

3

3.2

3.4

3.6

3.8

−1.09
−1.1
−1.11
2.6

Figure 2.18 Imaginary part of the relative permeabilities for a fictitious uniaxial material
extracted using 200 random trials. Center line is the average of the trials. Upper and lower
lines show the 95% confidence interval.

43

CHAPTER 3
Characterization of Gyromagnetic
Materials Using a Partially-Filled
Waveguide Technique
3.1

Introduction

This chapter presents a technique for characterizing gyromagnetic materials using rectangular waveguides where the sample is magnetized perpendicular to the broad dimension of the
guide, and only part of the cross-section of a waveguide is filled. Since the extraction of material parameters is implemented through minimizing the difference between the measured
and the theoretically reflection and transmission coefficients, an accurate modeling method
is required for material characterization. This is achieved by computing the S-parameters of
the waveguide system with the material sample placed at a proper location using a modematching technique, which analyzes the behavior of the higher order modes excited at the
interface between the sample region and the empty waveguide region. A special case where
the material sample is placed at the center of the waveguide is also discussed and analyzed.
By using an appropriate number of modes, the theoretical S-parameters for samples under

44

various measurement conditions can be predicted accurately.

3.2

Theoretical Transmission and Reflection Coefficients
Using Mode-Matching Analysis

3.2.1

Characteristics of Gyromagnetic Material

The increased demand for developing new artificial materials has driven more researchers
to study the behavior and application of gyromagnetic materials at microwave frequencies.
The material parameters of gyromagnetic material can be expressed by the tensors




 r

= 0
 0

0

0
r
0

0 

0 


r

(3.1)

and




 µg 0 −jδ 


,
µ = µ0 
0
1
0




jδ 0 µg

(3.2)

where the dielectric constant is an isotropic scalar either real or complex, and the permeability tensor above is formulated for the case when a material is biased along the height of the
waveguide or in the y-direction. The variables δ and µ are complex numbers: δ = δ + jδ
and µg = µg + jµg . The expressions for δ and µ are given by

δ=

f fm
f 2 − f02

45

(3.3)

and

µ=

1−

f0 fm
f 2 − f02

.

(3.4)

In (3.3) and (3.4), f is the operating frequency, the saturation magnetic resonance frequency fm = 2.8 × 106 × (4πMs ), and the magnetic resonance frequency fo = 2.8 × 106 ×
H0 + j ∆H
2 . Here H0 is the strength of the internal static biasing magnetic field in Oersted (Oe), which can be determined from the expression H0 = Ha + N Ms , where N is the
demagnetization factor, Ha is the external magnetic field, and 4πMs is the saturation magnetization in gauss (G) that defines the external field magnitude above which all the spins of
unpaired electrons in the ferrite are aligned in the same direction. When the external static
magnetic field is parallel to a thin layer of ferrite, the demagnetization factor N becomes 0,
which results in H0 = Ha . The magnetic losses inherent in the magnetic materials is also
included as the linewidth ∆H. With the equations above it is possible to determine the
effect of the magnetic resonance on the value of permeability at different frequencies.
The permeability dyadic has a dependence on biasing magnetic field and frequency. The
diagonal term µ was plotted under different magnetic fields over frequency to help visualize
the dependance. Consider a gyromagnetic material with 4πMs = 1000 G and linewidth
∆H = 25. It is seen from Figure 3.1 that the magnetic resonance frequency occurs around
2.8 GHz with the internal magnetic biasing field of 1000 Oe. As H0 increases to 1500 Oe, it
is seen from Figure 3.2 that the magnetic resonance rises to 4.2 GHz. The occurrence of the
magnetic resonance in the frequency range of interest may cause the extraction procedure
to be more difficult due to the drastic variation of the relative material parameters. The
ideal biasing field strength should bring a magnetic resonance slightly above the frequency
band of interest, which keeps the permeability above zero while also maintaining a smooth
variation across the frequency band. In Figure 3.3, for a magnetic field of 1800 Oe, where the
magnetic resonance is at 5 GHz, the permeability value varies from 2 to 2.5 for the frequency

46

range of 2.6-4 GHz which is the frequency band interested in this work. By manipulating the
magnetic biasing field, one could obtain the desired permeability properties at the frequency
of interest. This unique property is appealing for developing new engineered material and
methodologies of characterizing this type of material are investigated. The behavior of the
off-diagonal term δ under the same magnetic field strength is also shown in Figure 3.4. It
can be seen both real and imaginary parts of δ are close to 0 within the frequency band of
interest.

3.2.2

Modal analysis of the gyromagnetic material in a partiallyfilled waveguide

To compute the S-parameters of a rectangular waveguide partially filled by a gyromagnetic
material using a mode-matching technique, all the modal fields supported by the guide must
be formulated. The waveguide system considered is shown in Figure 3.5, consisting of a
rectangular waveguide partially filled with a gyromagnetic material sample biased along
the y-axis. Waveguide extensions are attached with lengths chosen to be such that only a
single TE10 rectangular waveguide mode exists at the measurement port of the transmitting
extension (z < 0) and the receiving extension (z > d). A cross-section view of the waveguide
system with sample is shown in Figure 3.6.
A single TE10 rectangular waveguide mode is incident on the sample, as shown in Figure
3.7. An infinite number of waveguide modes are reflected into the transmitting extensions,
while an infinite number of waveguide modes are transmitted into the sample region, 0 <
z < d. On the next interface at z = d, these waveguide modes become incident, and thus
a spectrum of waveguide modes is reflected into the sample region and transmitted into the
receiving end of the empty waveguide extensions. Since the TE10 mode is invariant in the
y-direction and even about x=a/2 and the gyromagnetic material is magnetized along the
y-axis in this case, it is assumed that all excited higher order modes are also y-invariant.
Therefore, only TEn0 modes are taken into account to describe the field distributions in
47

30
20

Re[μ]

10

µ

0
−10
−20
−30

Im[μ]

−40
−50
2

2.5

3

3.5

4
Frequency (GHz)

4.5

5

5.5

6

Figure 3.1 Permeability of the gyromagnetic material for H0 = 1000 Oe.

30
20

Re[μ]

10

µ

0
−10
−20
−30

Im[μ]
−40
−50
2

2.5

3

3.5

4
Frequency (GHz)

4.5

5

5.5

Figure 3.2 Permeability of the gyromagnetic material for H0 = 1500 Oe.

48

6

30
20

Re[μ]

10
0
µ

4
−10
−20

3
2
1

−30
−40

Im[μ]

0
−1
2.8

−50
2

3
2.5

3.2

3.4

3

3.6
3.5

3.8

4
Frequency (GHz)

4.5

5

5.5

6

Figure 3.3 Permeability of the gyromagnetic material for H0 = 1800 Oe.

40

30

0.5
0

20

−0.5
−1

δ

−1.5
10

Im[δ]

−2
2.8

3

3.2

3.4

3.6

3.8

0

Re[δ]

−10

−20
2

2.5

3

3.5

4
Frequency (GHz)

4.5

5

5.5

Figure 3.4 Off-diagonal term δ of the permeability dyadic for H0 = 1800 Oe.

49

6

each of the empty waveguide regions.

3.2.2.1

Modal behavior inside a rectangular waveguide partially filled with gyromagnetic material

For y-invariant TEn0 modes in the empty region of a rectangular waveguide, the wave
equation [42] is expressed as

∂2
+ kc2 Ey = 0,
2
∂x

(3.5)

where
kc2 = k02 − β 2

(3.6)

is the cutoff wavenumber and k0 = ω 2 µ0 0 is the free-space wavenumber. The solution to
the wave equation is expressed as
Ey (x, z) = (Asin kc x + Bcos kc x)e±jβz .

(3.7)

Applying the boundary condition Ey (x = a
2 ) = 0, the fields in the empty waveguide where
only TEn0 modes are excited are given by
a
Ey (x, z) = A sin kc (x + ) e±jβz
2
jkc
a
Hz (x, z) = A
cos kc (x + ) e±jβz .
ωµo
2

50

(3.8)
(3.9)

[Ɛ], [µ]

Ɛ0,µ 0
H0

Ɛ0,µ 0

y

b
x

z

a

d
Figure 3.5 Waveguide partially filled with a gyromagnetic sample.

y

b

0𝑎

−

2

III

I

II

Ɛ0,µ 0

[Ɛ], [µ]

Ɛ0,µ 0
x

𝑑
− +𝑠
2

𝑑
+𝑠
2

Figure 3.6 Cross-sectional view of a partially filled waveguide.

51

𝑎
2

y

[Ɛ], [µ]
TEn0
TEn0

Ɛ0,µ 0
TE10
TEn0
0

Ɛ0,µ 0

TEn0
z
Δ

Figure 3.7 Side view of a partially filled waveguide.
In the empty part of the sample region ( d2 + s < x < a
2 ), the field in region II of Figure 3.6
can be written as



II
II
sin (kc x)
sin (kc x)
 e±jβz , (3.10)
EyII (x, z) = AII
+ B II cos(kcII x)
a−d−2s
a−d−2s
II
II
cos(kc
)
cos(kc
)
2
2
where
kcII = ko2 − β 2 .

(3.11)

The denominator in (3.10) was chosen for computational purpose to eliminate large values
calculated from sinusoidal functions when kc is complex. Applying the boundary condition
EyII = 0 at x = a
2 yields
a
a
AII sin (kcII ) + B II cos (kcII ) = 0,
2
2

(3.12)

and the field in region II is given by the following equation,
EyII (x, z) =

a
AII
sin (kcII x) − tan(kcII )cos (kcII x) e±jβz .
2
cos(kcII a−d−2s
)
2

52

(3.13)

Using 3.13, the fields in region II can be rewritten as
sin kcII (x − a
2)
e±jβz
a−d−2s
II
cos kc (
)
2
cos kcII (x − a
2)
e±jβz .
a−d−2s
II
)
cos kc (
2

EyII (x, z) = AII
II
j ∂Ey
= AII
HzII (x, z) =
ωµo ∂x

jkcII
ωµo

(3.14)

(3.15)

Similarly, the transverse fields in the other empty part (region III) of the sample region can
be found by using



III
III
sin (kc x)
sin (kc x)
 e±jβz ,
EyIII (x, z) = AIII
+ B III cos kcIII x
a−d+2s
a−d+2s
III
III
)
)
cos kc (
cos kc (
2
2
(3.16)
where
kcIII = kcII =

ko2 − β 2 ,

(3.17)

and
a
a
−AIII sin kcIII
+ B III cos kcIII
=0
2
2

(3.18)

given by the boundary condition EyIII = 0 at x = − a
2 . Thus, the electric field in region III
is given by
EyIII (x, z) =

AIII
cos kcII ( a−d+2s
)
2

a
sin kcII x + tan kcII
cos kcII x
2

53

e±jβz . (3.19)

Therefore, the fields in region III are expressed by
sin kcII (x + a
2)
e±jβz
a−d+2s
II
cos kc
2
cos kcII (x + a
2)
e±jβz .
a−d+2s
II
cos kc
2

EyIII (x, z) = AIII
II
jk II
j ∂Ey
= AIII c
HzIII (x, z) =
ωµo ∂x
ωµo

(3.20)

(3.21)

To find the fields in the region I occupied by the sample, first start with the solutions for
the wave equation in the sample region (Section 2.2.4 [42])




cos (kc x)  ±jβz
sin (kc x)
EyI (x, z) = AI
+ BI
e
,
cos kc d2
cos kc d2

(3.22)

and
u/ω
HzI (x, z) = −j
δ 2 − µ2

∂EyI
∂x

∓

βδ I
E
µ y

,

(3.23)

so that
cos(kc x)
sin(kc x)
u/ω
A I kc
− B I kc
∓
HzI (x, z) = −j
2
2
δ −µ
cos kc d2
cos kc d2
AI

βδ sin(kc x)
βδ cos(kc x)
∓ BI
µ cos k d
µ cos k d
c2
c2

e±jβz .

(3.24)

where
δ2
kc2 = ω 2 µε − ω 2 µε − β 2
µ2

(3.25)

is the cutoff wavenumber for region I.
Since Ey and Hz are continuous at the interface between the sample and the empty
waveguide, four equations can be established by applying the boundary conditions for both

54

Ey and Hz . 1) Ey ( d2 + s) is continuous
−AII sin kcII ( a−d−2s
)
sin kc ( d2 + s)
sin kc ( d2 + s)
2
I
I
=A
+B
cos(kc d2 )
cos(kc d2 )
cos kcII ( a−d−2s
)
2

(3.26)

2) Hz ( d2 + s) is continuous

AII

jkcII
ωµo

= −j

u/ω
δ 2 − µ2
AI

βδ
µ

sin kc (s + d2 )
sin kc (s + d2 )
I
− B kc
∓
cos(kc d2 )
cos(kc d2 )
d
sin kc (s + d2 )
βδ cos kc (s + 2 )
I
∓B
(3.27)
µ
cos(kc d2 )
cos(kc d2 )
AI kc

3) Ey (− d2 + s) is continuous
−AIII sin kcII ( a−d+2s
)
sin kc (s − d2 )
sin kc (s − d2 )
2
I
I
=A
+B
cos(kc d2 )
cos(kc d2 )
cos kcII ( a−d+2s
)
2

(3.28)

4) Hz (− d2 + s) is continuous

AIII

jkcII
ωµo

= −j

u/ω
2
δ − µ2
AI

βδ
µ

sin kc (s − d2 )
sin kc (s − d2 )
I
− B kc
∓
cos(kc d2 )
cos(kc d2 )
d
sin kc (s − d2 )
βδ cos kc (s − 2 )
I
∓B
(3.29)
µ
cos(kc d2 )
cos(kc d2 )
AI kc

55

Substituting AII in terms of AI and B I from (3.26), the expression (3.27) becomes
d
II a−d−2s )
kcII (δ 2 − µ2 ) sin kc (s + 2 ) cot kc (
2
I
A
+
µµo
cos(kc d2 )
d ) cot k II ( a−d−2s )
c
2
2
=
cos(kc d2 )
cos kc (s + d2 )
sin kc (s + d2 )
AI kc
− B I kc
∓
d
cos(kc 2 )
cos(kc d2 )
d
d
βδ sin kc (s + 2 )
βδ cos kc (s + 2 )
I
I
A
∓B
.
µ
µ
cos(kc d2 )
cos(kc d2 )

k II (δ 2 − µ2 ) cos kc (s +
BI c
µµo

(3.30)

The equation above can be written as
± AI + F ± B I = 0,
FAA
AB

(3.31)

where
d
II a−d−2s )
kcII (δ 2 − µ2 ) sin kc (s + 2 ) cot kc (
2
±
+
FAA = −
d
µµo
cos(kc 2 )
d
cos kc (s + d2 )
βδ sin kc (s + 2 )
∓
kc
µ
cos(kc d2 )
cos(kc d2 )
d
II a−d−2s )
kcII (δ 2 − µ2 ) cos kc (s + 2 ) cot kc (
2
±
FAB = −
−
d
µµo
cos(kc 2 )
d
sin kc (s + d2 )
βδ cos kc (s + 2 )
kc
∓
.
µ
sin(kc d2 )
cos(kc d2 )

56

(3.32)

(3.33)

Similarly, using (3.28), the expression (3.29) becomes
d
II a−d+2s )
kcII (δ 2 − µ2 ) sin kc (s − 2 ) cot kc (
2
I
A
+
µµo
cos(kc d2 )
d
II a−d+2s )
kcII (δ 2 − µ2 ) cos kc (s − 2 ) cot kc (
2
I
B
µµo
cos(kc d2 )
cos kc (s − d2 )
sin kc (s − d2 )
I
I
+A kc
− B kc
cos(kc d2 )
cos(kc d2 )
d
d
βδ sin kc (s − 2 )
βδ cos kc (s − 2 )
I
I
∓A
∓B
= 0.
µ
µ
cos(kc d2 )
cos(kc d2 )

(3.34)

The equation above can also be written as
± AI + F ± B I = 0,
FBA
BB

(3.35)

where
d
II a−d+2s )
kcII (δ 2 − µ2 ) sin kc (s − 2 ) cot kc (
2
±
+
FBA =
d
µµo
cos(kc 2 )
d
cos kc (s − d2 )
βδ sin kc (s − 2 )
kc
∓
µ
cos(kc d2 )
cos(kc d2 )
d
II a−d+2s )
kcII (δ 2 − µ2 ) cos kc (s − 2 ) cot kc (
2
±
FBB =
−
µµo
cos(kc d2 )
d
sin kc (s − d2 )
βδ cos kc (s − 2 )
kc
∓
.
µ
sin(kc d2 )
cos(kc d2 )
Equation (3.31) and (3.35) may be written in matrix form as


   
±
±
I
 FAA FAB   A  0

  =  .
±
±
FBA FBB
BI
0
57

(3.36)

(3.37)

The determinant of the 2 x 2 matrix has to be zero for a nontrivial solution for AI and B I
to exist , which yields
± F ± − F ± F ± = 0.
FAA
BA AB
BB

(3.38)

± , F ± , F ± , and F ± can be written as
The expressions FAA
BB
AB BA
± = (Γ ± ∆)SC − CC
FAA
1
1
1
± = (Γ ∓ ∆)SC + CC
FBA
2
2
2

± = (Γ ± ∆)CC + SC
FAB
1
1
1
± = (Γ ∓ ∆)CC + SC ,
FBB
2
2
2

(3.39)

where
k II (δ 2 − µ2 )
Γ1 = − c
cot
kc µµo
k II (δ 2 − µ2 )
Γ2 = − c
cot
kc µµo
sin kc (s + d2 )
SC1 =
CC1
cos (kc d2 )
sin kc ( d2 − s)
CC2
SC2 =
cos (kc d2 )

kcII (

a − d − 2s
)
2

a − d + 2s
)
2
cos kc (s + d2 )
=
cos (kc d2 )
cos kc ( d2 − s)
=
cos (kc d2 )
βδ
.
∆=
µkc
kcII (

(3.40)

Then (3.38) can be written as

[(Γ1 ±∆)SC1 −CC1 ][(Γ2 ∓∆)CC2 +SC2 ]−[(Γ1 ±∆)CC1 +SC1 ][(Γ2 ∓∆)SC2 +CC2 ] = 0.
(3.41)
Finally, this equation above can be further simplified into

(Γ1 ± ∆)(Γ2 ∓ ∆) sin(kc d) − (Γ2 ∓ ∆) cos(kc d) + (Γ1 ± ∆) cos(kc d) − sin(kc d) = 0, (3.42)
and thus the characteristic equation for the modal spectrum of the gyromagnetic material
58

sample partially placed in the waveguide is given by

(Γ1 ± ∆)(Γ2 ∓ ∆) sin(kc d) − (Γ1 + Γ2 ) cos(kc d) − sin(kc d) = 0,

(3.43)

A special case of the characteristic equation can be obtained when s = 0, which means the
sample is placed at the center of the cross-section of the waveguide extensions. This situation
results in a much simplified characteristic equation
(Γ + T )(ΓT − 1) = ∆2 T,

(3.44)

where
k II (δ 2 − µ2 )
a−d
Γ=− c
)
cot kcII (
kc µµo
2
βδ
∆=
µkc
d
T = tan kc
.
2

3.2.2.2

(3.45)

Modal spectrum dependence on the transverse position of the sample

Based on the characteristic equation derived above, the modal spectrum can be plotted
for different transverse positions of the sample in the waveguide to understand the modes
in the waveguide. Since the measurements for extraction were performed using an S-band
waveguide, the investigation of the modal behavior was carried out in S-band (2.6-3.95 GHz).
Consider a 1.34 inches wide lossless gyromagnetic sample with 4πMs =1000 G and r =14.7
and H0 of 2800 Oe which is placed at the center of the waveguide cross-section. The propagation constant β can be found by solving (3.44). Of many possible root solving algorithms,
the Newton’s method is chosen for its simplicity and capability of finding complex roots.
Figure 3.8 shows the ω − β diagram for the propagation modes with real values of β. When
the sample is centered, more than one propagating mode exists in the sample region, and
59

the number of propagating modes varies with frequency. Roots with imaginary values are
plotted in Figure 3.9. Since an infinite number of evanescent modes can be found, only the
first five evanescent modes are shown in the plot. Note that for any root found from (3.44),
either real or imaginary, one can always find a second root of the equation by changing the
sign of the first root. Therefore, only the roots with positive sign are shown in those two
figures.
Figure 3.10 shows all the propagating modes when the sample is shifted by s =0.1 inches.
It can be seen that the propagating modes existing in the sample region are close to those in
the case where the sample is centered. However, since (3.43) has different signs depending
on the direction of wave propagation, two different roots of the same mode at any given
frequency can be found, one for each propagating direction. It may appear in some cases
that the two roots are very close, but as long as s = 0, two roots should be different. The
upper half of Figure 3.10 shows the propagating modes for the wave propagating along +z
direction, while the lower half shows the modes for the −z direction. The signs of the roots
found in the −z direction are chosen to be minus for clear graphical demonstration since any
real root with both signs still satisfies the characteristic equation.
When s = 0, purely imaginary roots cannot satisfy (3.44). Instead of evanescent modes,
complex roots are found. These roots are found in complex conjugate pairs. For instance,
for any complex root of a+jb (a and b are real numbers), one is able to find that a-jb
is also a valid root. Complex modes have been studied by many researchers and they
are observed in circular waveguides [44]-[45], dielectric waveguides [46]-[48] or structures
with high-permittivity inhomogeneity [49]. The physical existence of complex modes in
waveguides does not violate the principle of conservation of energy as long as both roots
of each pair are included, as is discussed in [50]. The first 5 complex modes for s = 0.1
inches are shown in Figure 3.11. The red curve shows the imaginary part of the complex
root and the black curve shows the real part. It can be noticed that the imaginary parts
of the roots are very close to the purely imaginary roots found for the center case (Figure

60

3.9), which indicates a similar rate of attenuation. Also, it can be seen that the real parts
of the complex roots are much smaller than the imaginary parts. Similar modal behavior
can be observed in Figures 3.12-3.15 when the sample is shifted by 0.3 and 0.75 inches. The
number of propagating modes may vary as the shift increases; for example, the number of
propagating modes changes from four to three as the shift goes from 0.3 inches to 0.75 inches.
To further investigate the impact of sample shift on the modes, the computed roots β are
plotted as a function of the shift. Figures 3.16-3.21 show the relationship between β and s at
three different frequencies. The variation of the roots for the propagating modes is less than
10% when the shift is between 0 and 0.4 inches. However, once the shift is larger than 0.4
inches, a noticeable change can be observed in the roots, especially for the low order modes.
In any cases, the variation of s does not cause the complex modes to change dramatically.
It can be seen that both the real and imaginary parts of the complex roots stay within a
certain range, and the real part remains a relatively small number as the sample location is
changed.
400
350
300

β

250
200
150
100
50
0

2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

Figure 3.8 Modal spectrum of the propagating modes for a centered sample.

61

4

350
300
250

β

200
150
100
50
0

2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 3.9 Modal spectrum of the evanescent modes for a centered sample.
400
300
200
100
β

0
−100
−200
−300
−400

2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 3.10 Modal spectrum of the propagating modes for sample shift s=0.1 inches.

3.2.3

Fields in the Waveguide

Different modes existing in the waveguide with gyromagnetic sample can be found through
solving the characteristic equation (3.43). As a result, the fields in the sample region (0 <
62

300
200

β

100
0
−100
−200
−300
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 3.11 Modal spectrum of the complex modes for sample shift s=0.1 inches.
400
300
200

β

100
0
−100
−200
−300
−400

2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 3.12 Modal spectrum of the propagating modes for sample shift s=0.3 inches.
x < d) can be plotted and the modes found can be validated. To obtain the expression for

63

300
200

β

100
0
−100
−200
−300
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 3.13 Modal spectrum of the complex modes for sample shift s=0.3 inches.
400
300
200

β

100
0
−100
−200
−300
−400

2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 3.14 Modal spectrum of the propagating modes for sample shift s=0.75 inches.
the fields in the sample region, use (3.35) to obtain
±
FBA
I
I
B = −A ± .
FBB
64

(3.46)

300
200
100
β

0
−100
−200
−300
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 3.15 Modal spectrum of the complex modes for sample shift s=0.75 inches.
260
240
220
200

β

180
160
140
120
100
80
60
0

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

Figure 3.16 Propagation constants (propagating modes) for various sample shift (s) at
2.6 GHz.
This equation can be written as
B I = −AI G± ,
65

(3.47)

1.5

Re(β)

1
0.5
0
−0.5
−1
0

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

0

Im(β)

−100
−200
−300
0

Figure 3.17 Propagation constants (complex modes) for various sample shift (s) at 2.6 GHz.
where

±
(Γ2 ∓ ∆) sin kc (s − d2 ) + cos kc (s −
F
G± = BA
± =
FBB
(Γ2 ∓ ∆) cos kc (s − d2 ) − sin kc (s −

d)
2
.
d)
2

(3.48)

For (3.26) and (3.28), substitute B I with (3.47) to give
sin kc ( d2 + s)
cos kc ( d2 + s)
I
±
−A G
cos(kc d2 )
cos(kc d2 )
sin kc (s − d2 )
cos kc (s − d2 )
a − d + 2s
III
III
I
I
±
A
tan kc (
) =A
−A G
.
2
cos(kc d2 )
cos(kc d2 )
a − d − 2s
−AII tan kcII (
) = AI
2

66

(3.49)

(3.50)

300
250

β

200
150
100
50
0
0

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

Figure 3.18 Propagation constants (propagating modes) for various sample shift (s) at
3.2 GHz.
Based on this, the relationship between AI , AII and AIII can be obtained as
AII = AI W ±

(3.51)

AIII = AI U ± ,

(3.52)

where
cos kc ( d2 + s)
sin kc ( d2 + s)
−
cos(kc d2 )
cos(kc d2 )
sin kc ( d2 − s)
cos kc (s − d2 )
±
−
−G
cos(kc d2 )
cos(kc d2 )

W± =

U± =

G±

67

cot kcII (

a − d − 2s
)
2

a − d + 2s
cot kcII (
) .
2

(3.53)

(3.54)

2

Re(β)

1
0
−1
−2
0

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

−50

Im(β)

−100
−150
−200
−250
−300
0

Figure 3.19 Propagation constants (complex modes) for various sample shift (s) at 3.2 GHz.
Now Ey and Hx within the empty part of the material region (region II and III) can be
written as
sin kcII (x − a
2)
cos kcII ( a−d−2s
)
2
sin kcII (x − a
β
2)
II
±
Hx (x, z) = ±W
ωµo cos k II ( a−d−2s )
c
2
sin kcII (x + a
2)
III
±
Ey (x, z) = U
cos kcII ( a−d+2s
)
2
sin kcII (x + a
β
2)
III
I
±
Hx (x, z) = ±A U
ωµo cos k II ( a−d+2s )
c
2
EyII (x, z) = W ±

68

e±jβz

(3.55)

e±jβz

(3.56)

e±jβz

(3.57)

e±jβz ,

(3.58)

400

350

β

300

250

200

150
0

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

Figure 3.20 Propagation constants (propagating modes) for various sample shift (s) at
3.8 GHz.
and the fields inside the material (region I) can be expressed as



HxI (x, z) =

δ/ω
δ 2 − µ2


sin
(k
x)
cos
(k
x)
c
c  ±jβz
EyI (x, z) = AI 
e
(3.59)
− G±
d
cos(kc 2 )
cos(kc d2 )


sin(kc x)
βµ
cos(kc x)
βµ
AI 
(kc G± ∓
)+
(kc ± G± ) e±jβz(3.60)
.
δ
δ
cos(kc d2 )
cos(kc d2 )

To summarize, the fields in the partially-filled waveguide are described as

69

2

Re(β)

1
0
−1
−2
0

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

0.1

0.2

0.3

0.4
Shift (inch)

0.5

0.6

0.7

0.8

0

Im(β)

−100

−200

−300
0

Figure 3.21 Propagation constants (complex modes) for various sample shift (s) at 3.8 GHz.



sin kcII (x− a

2)

±

W
e±jβz


a−d−2s
II

cos kc (
)


2



cos (kc x) ±jβz
sin (kc x)
±
− G±
e
E(x, z) =
d

cos(kc 2 )
cos(kc d2 )






sin kcII (x+ a

2)
±


U
e±jβz


a−d+2s
II
cos kc (
)
2

70

d +s
2

x

a
2

− d2 + s

x

d +s
2

−a
2

x

− d2 + s,

(3.61)

and



sin kcII (x− a

2)
β

±

±W
e±jβz

ωµ
o

a−d−2s
II

cos kc (
)


2




δ/ω
sin(kc x)

± βµ


 δ 2 −µ2 cos(k d ) (kc G ∓ δ )+
c2
H(x, z)± =

cos(kc x)

βµ

(kc ± G± ) e±jβz


d
δ

cos(kc 2 )






sin kcIII (x+ a

2)
β

±jβz
±
 ±U ωµ


o cos k II ( a−d+2s ) e

c
2

d +s
2

x

a
2

x

d +s
2

(3.62)
− d2 + s
−a
2

− d2 + s.

x

The fields for the special case where the sample is placed at the center of the waveguide can
be found by setting s to 0, and thus G± , W ± and U ± can be further simplified into
−(Γ ∓ ∆) tan(kc d2 ) + 1
±
G =
(Γ ∓ ∆) + tan(kc d2 )

(3.63)

d
a−d
W ± = G± − tan (kc ) cot kcII (
)
2
2
d
a−d
U ± = −G± − tan (kc ) cot kcII (
) ,
2
2

(3.64)
(3.65)

where Γ is shown in (3.45). Finally, the fields for the special case are found to be


sin kcII (x− a

2)

±

W
e±jβz


a−d−2s
II

cos kc (
)


2



cos (kc x) ±jβz
sin (kc x)
e
− G±
E(x, z)± =
d
d)

cos(k
)
cos(k

c
c

2
2


II (x+ a )


sin
k
c

2


U±
e±jβz


a−d+2s
II
cos kc (
)
2

71

d +s
2

x

a
2

− d2 + s

x

d +s
2

−a
2

x

− d2 + s,

(3.66)

and



sin kcII (x− a

2)
β

±

±W
e±jβz

ωµ
o

a−d−2s
II

cos kc (
)


2




δ/ω
sin(kc x)

± βµ


 δ 2 −µ2 cos(k d ) (kc G ∓ δ )+
c2
H(x, z)± =

cos(kc x)

βµ

(kc ± G± ) e±jβz


d
δ

cos(kc 2 )






sin kcIII (x+ a

2)
β

±jβz
±
 ±U ωµ


o cos k II ( a−d+2s ) e

c
2

d +s
2

x

a
2

x

d +s
2

(3.67)
− d2 + s
−a
2

x

− d2 + s.

Also, the fields inside an empty waveguide are given by

Em (x) = −

a
jωµo
sin kcm (x − )
kcm
2
Em
Hx (x) =
,
Zm

(3.68)

where

kcm =

mπ
, m = 1, 2, 3 · · ·
a
ωµo
Zm =
.
βm

The fields in the waveguide can be plotted to validate the boundary condition. Considering the same material parameters used in Section 3.2.2.2, the propagation constant β of the
first order propagating mode is computed under different conditions. For modes propagating
in both the +z and -z directions, Figures 3.22-3.25 show the normalized fields of Ey ,Hx and
Hz at two frequencies with s=0 and s=0.2 inches. It is obvious that Ey and Hz are always
continuous at the boundary of the sample, which satisfies the imposed boundary conditions.
As the sample is shifted by 0.2 inches, one can see that the field confined in the material is
also shifted as expected. At 3.6 GHz, a similar trend can be seen while the field distribu-

72

tion is different from that at 2.6 GHz. With the validation of the modes, a mode matching
technique can be developed to compute the reflection and transmission coefficients.

E+y

E−y

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

H+

0

0.02

H−

x

x

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

H+z

0

0.02

H−z

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

0

0.02

Figure 3.22 Field distribution of the first propagating mode for a sample shift of s=0 at
2.8 GHz.

3.2.4

Mode Matching

Now the transverse fields in the material region and empty region of the waveguide are known
and can be expressed as an infinite summation of modal fields. The next step is to determine
the modal amplitudes by applying different boundary conditions. From Figure 3.7, it can be

73

+
y

−
y

E

E

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

H+

0

0.02

H−

x

x

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

H+z

0

0.02

H−z

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

0

0.02

Figure 3.23 Field distribution of the first propagating mode for a sample shift of s=0.2 inches
at 2.8 GHz.
seen that for the empty waveguide region z < 0,

−jβ 1 z +
Ey (x, z) = ai1 E −
1 (x)e
−jβ 1 z +
Hx (x, z) = ai1 H −
1 (x)e

N
n=1
N
n=1

74

jβ z
arn E +
n (x)e n

(3.69)

jβ z
arn H +
n (x)e n

(3.70)

+
y

−
y

E

E

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

H+

0

0.02

H−

x

x

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

H+z

0

0.02

H−z

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

0

0.02

Figure 3.24 Field distribution of the first propagating mode for a sample shift of s=0 at
3.6 GHz.
The coefficient ai1 is the amplitude of the incident TE10 wave, which is supposed to be
known. In the material region 0 < z < ∆, the transverse fields are expressed as
N
Ey (x, z) =

−
−jβn z +
a−
n En (x)e

N

+
jβn (z−∆)
a+
n En (x)e

n=1
n=1
N
N
−
−
+
−jβ
z
jβn (z−∆) .
n
+
a+
Hx (x, z) =
an Hn (x)e
n Hn (x)e
n=1
n=1

75

(3.71)

(3.72)

+
y

−
y

E

E

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

H+

0

0.02

H−

x

x

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

H+z

0

0.02

H−z

1
1
0.5
0

0.5

−0.02

0

0

0.02

−0.02

0

0.02

Figure 3.25 Field distribution of the first propagating mode for a sample shift of s=0.2 inches
at 3.6 GHz.
The fields in the empty waveguide region z > ∆ are
N
Ey (x, z) =

− (x)e−jβn (z−∆)
atn En

n=1
N
− (x)e−jβn (z−∆) .
Hx (x, z) =
atn Hn
n=1

(3.73)

(3.74)

The modal amplitude can be determined by applying the boundary conditions on Ey
and Hx at the interfaces at z=0 and z = ∆. The continuity of Ey at z=0 gives

ai1 E −
1 (x) +

N
n=1

arn E +
n (x) =

N

−
a−
n En (x) +

n=1

N
n=1

76

+
−jβn ∆ .
a+
n En (x)e

(3.75)

The continuity of Hx at z=0 gives
N

ai1 H −
1 (x) +

arn H +
n (x) =

N

−
a−
n Hn (x) +

n=1

n=1

N

+
−jβn ∆ .
a+
n Hn (x)e

(3.76)

n=1

The continuity of Ey at z = ∆ gives
N

−
−jβn ∆ +
a−
n En (x)e

n=1

N

+
a+
n En (x) =

n=1

N

− (x).
atn En

(3.77)

n=1

The continuity of Hx at z = ∆ gives
N

−
−jβn ∆ +
a−
n Hn (x)e

n=1

N

+
jβn (z−∆) =
a+
n Hn (x)e

N

− (x).
atn Hn

(3.78)

n=1

n=1

To convert the set of equations (3.75) - (3.78) into a set of linear equations, the testing
operations of empty waveguide modes using (3.69) are applied to the four equations and
yield,

−
Cm1

+
Cmn

a
a
2 −
2 +
+
N
i
r
a1
E 1 (x)E m (x)dx + n=1 an
E n (x)E +
m (x)dx
a
a
−2
−2
−
+
Dmn
Dmn
a
a
2 −
2 +
+
−
+
N
N
−jβn ∆ (3.79)
= n=1 an
En (x)E m (x)dx + n=1 an
En (x)E +
m (x)dxe
a
a
−2
−2
−
+
Em1
Emn
a
2
+
i
a1
H−
1 (x)H m (x)dx +
a
−2

a
N ar 2 H + (x)H + (x)dx
m
n=1 n − a n
2

77

−
Fmn

+
Fmn

a
a
2
2
+
−
−
+
+ (x)H + (x)dx(3.80)
N
N
−jβ
∆
n
= n=1 an
Hn (x)H m (x)dx + n=1 an e
Hn
m
a
a
−2
−2
−
+
Dmn
Dmn
a
a
N a− e−jβn ∆ 2 E − (x)E + (x)dx + N a+ 2 E + (x)E + (x)dx
n
m
m
n=1 n
n=1 n − a n
−a
2
2
−
Cmn
a
2 −
N
t
= n=1 an
En (x)E +
m (x)dx(3.81)
a
−2
−
+
Fmn
Fmn
a
a
N a− e−jβn ∆ 2 H − (x)H + (x)dx + N a+ ejβn (z−∆) 2 H + (x)H + (x)dx
n
n
m
m
n=1 n
n=1 n
−a
−a
2
2
−
Emn
a
2
− (x)H + (x)dx.(3.82)
N
t
= n=1 an
Hn
m
a
−2
This gives the set of linear equations

− +
ai1 Cm1

N

+ =
arn Cmn

N

−
a−
n Dmn +

N

+ −jβn ∆
a+
n Dmn e

n=1
n=1
n=1
N
N
N
−
−
−
+
+ −jβn ∆
i
r
a1 Em1 +
an Emn =
an Fmn +
a+
n Fmn e
n=1
n=1
n=1
N
N
N
−
−
−
+
+
−jβ
∆
n
an Dmn e
+
an Dmn =
atn Cmn
n=1
n=1
n=1
N
N
N
− e−jβn ∆ +
− .
+F + =
F
a
a−
atn Emn
n mn
n mn
n=1
n=1
n=1

(3.83)

(3.84)

(3.85)

(3.86)

Let eN = e−jβn ∆ , The linear equation can be expressed as a 4 × 4 block matrix equation
where each block is N × N

78


  
+
i
−
r
eN D
0   a  a1 f 



  
−
+
−E +
  a−   ai g − 
F
e
F
0

   1 
N


  = 

 0
  
−
+
−
eN D
D
−C 

  a+   0 


  
t
−
+
−
0
a
0
eN F
F
−E


−C +

D−

(3.87)

− = C − and g − = E − . Finally, the S parameters can be obtained from
where fm
m
m1
m1
ar1
S11 =
ai1
at
S21 = 1
ai1

(3.88)

(3.89)

The matrix entries can be determined by calculating the corresponding integrals. Begin
with
± =
Cmn

a
2 ±
E 1 (x)E +
m (x)dx
a
−2

a
nπ
a
mπ
a
2
=
sin
(x − ) sin
(x − )dx
a
2
a
2
−a
2
± = a . if m = n, C ± = 0.
if m = n, Cmn
mn
2

(3.90)

(3.91)
(3.92)

or
± = aδ
Cmn
mn
2

(3.93)

Similarly
± =
Emn

a
2
+
H±
1 (x)H m (x)dx
a
−2

a
1
1 a
2 ±
=±
E 1 (x)E +
(x)dx = ±
δmn ,
m
a
zmzn −
(Z m )2 2
2
79

(3.94)

(3.95)

where

Zm =

ωµo
.
βn

(3.96)

± can be expressed as
Next, Dmn
± =
Dmn

a
2 ±
En (x)E +
m (x)dx
a
−2

a
II
− d2 +s
a sin kc (x + 2 )
U ± sin k cm (x − )
dx
=
2 cos k II a−d+2s
−a
c
2
2 

d +s
cos kc x 
a
sin kc x
2
+
− G±
sin k cm (x − ) 
dx
d
d
2
− 2 +s
cos kc 2
cos kc d2
a
a
II
a sin kc (x − 2 )
2
±
+
W sin k cm (x − )
dx.
d +s
2 cos k II a−d−2s
c
2
2

(3.97)

This can be rewritten as
± =
Dmn

±
1
Wn±
Un±
A +
B + Gn I C +
D ,
Imn
Imn
Imn
mn
a−d+2s
d
d
a−d−2s
III
II
cos kc
cos kcn 2
cos kcn 2
cos kc
2
2
(3.98)

and each of the coefficients Imn can be obtained by calculating the corresponding integrals.

80

Here
− d2 +s
a
a
sin k cm (x − )sin kcIII (x + )dx
a
2
2
−2
a−d +s
2
=
sin k cm (u − a)sin kcIII udu
0
a−d+2s
III
sin(k cm − kc )
sin(k cm + kcIII ) a−d+2s
1
m
2
2
= (−1)
−
III
III
2
k cm − kc
k cm + kc
d +s
a
2
B
sin k cm (x − )sin kcn xdx
Imn =
2
− d2 +s
akcm + (k − k )(s + d )
cn
cm
sin[ ak2cm − (k cm + kcn )(s + d2 )] sin
2
2
=
−
2(k cm + kcn )
2(k cm − kcn )
akcm + (k − k )(s − d )
cm
cn
sin[ ak2cm − (k cm + kcn )(s − d2 )] sin
2
2
−
+
2(k cm + kcn )
2(k cm − kcn )
d +s
a
2
C
sin k cm (x − )cos kcn xdx
Imn =
2
− d2 +s
akcm − (k + k )(s + d )
cn
cm
cos[ ak2cm + (kcn − k cm )(s + d2 )] cos
2
2
=−
−
2(k cm − kcn )
2(k cm + kcn )
akcm − (k + k )(s − d )
cn
cm
cos[ ak2cm + (kcn − k cm )(s − d2 )] cos
2
2
+
+
2(k cm − kcn )
2(k cm + kcn )
a
D = 2 sin k (x − a )sin k II (x − + a )dx
Imn
cm
c
d +s
2
2
2
a−d −s
2
=
sin k cm (u − a)sin kcII udu
0
a−d−2s
II
sin(k cm + kcII ) a−d−2s
1 sin(k cm − kc )
2
2
=
−
.
II
II
2
k cm − kc
k cm + kc
A =
Imn

81

(3.99)

(3.100)

(3.101)

(3.102)

± can be expressed as
Similarly, Fmn
± =
Fmn

β
±Un± ωµno

± βn µ B
A
Imn
δ/ω kc G ∓ δ Imn
+
+
d
zm
zm
δ 2 − µ2
cos kcIII a−d+2s
cos
k
c
2
2
β
β
µ
±
± n C
D
±Wn ωµno
δ/ω kc ± G
Imn
δ Imn +
,
zm
zm
δ 2 − µ2
cos kc d2
cos kcIII a−d−2s
2

(3.103)

A , I B , I C and I D are the same as those in the D
where all the coefficients Imn
mn case.
mn mn
mn
When the material is placed at the center of the waveguide, where s=0, the expressions
of the integrals can be further simplified into,
− d2
a
a
sin k cm (x − )sin kcIII (x + )dx
2
2
−a
2
a−d
2
=
sin k cm (u − a)sin kcIII udu
0
III
sin(k cm + kcIII ) a−d
sin(k cm − kc ) a−d
1
m
2
2
−
= (−1)
2
k cm − kcIII
k cm + kcIII
d
a
2
B
Imn =
sin k cm (x − )sin kcn xdx
d
2
−2
d
d
mπ sin[(k cm + kcn ) 2 ] sin(kcn + k cm ) 2
−
= cos
2
(k cm − kcn )
(k cm + kcn )
d
a
2
C
Imn =
sin k cm (x − )cos kcn xdx
2
− d2
d
d
mπ sin[(k cm − kcn ) 2 ] sin(kcn + k cm ) 2
= − sin
+
2
(k cm − kcn )
(k cm + kcn )
a
D = 2 sin k (x − a )sin k II (x − + a )dx
Imn
cm
c
d
2
2
2
a−d
2
=
sin k cm (u − a)sin kcII udu
0
a−d
II
sin(k cm + kcII ) a−d
1 sin(k cm − kc ) 2
2 .
=
−
II
II
2
k cm − kc
k cm + kc
A =
Imn

82

(3.104)

(3.105)

(3.106)

(3.107)

It is very important to mention that since the propagation constant (β) is different along
the two propagating directions when s = 0, appropriate β values should be used to calculate
the matrix entries. For instance, to compute the entries of D− and eN D+ , the β determined
for +z direction should be used to compute D− , while the β value for the −z direction should
be used to compute both eN and D+ . However, for the special case where the sample is
centered, it is not necessary to follow this rule since both values of β for the two propagating
directions for any mode are the same.

3.2.5

Validation of Theoretical Analysis

Since HFSS does not support assigning permeability tensors of a material with off-diagonal
terms, an in-house FEM code wrote by Dr. Tuncer was used as an alternative method to
validate the mode matching technique. For a fictitious gyromagnetic sample of d=1.4 inches
wide and ∆=0.4 inches thick, with r =14.7, 4πMs =1000 G, Ho =900 Oe and linewidth
∆H=0, this sample is simulated within a L-band (1.12-1.7 GHz) waveguide with the sample
positioned at the center of the guide. To obtain a convergence to the third decimal place,
the number of unknowns or meshing elements used in the FEM code is 60000. The number
of modes used in the mode matching technique is chosen 20 and to achieve an accuracy
of 0.1◦ in phase and 0.01 dB in magnitude. The calculated S-parameters are compared in
Figure 3.28 and 3.29. Similarly, the same fictitious sample is simulated within a S-band (2.63.95 GHz) waveguide with a shift s=1 inch and Ho =2800 Oe. The calculated S-parameters
are shown in Figure ?? and ??. Good agreement is achieved between the two methods,
which validate the proposed mode matching technique.

83

0

−5

Magnitude (dB)

−10

−15

−20

−25
FEM S11
FEM S21
Modal Analysis S11
Modal Analysis S21

−30

−35

1.15

1.2

1.25

1.3

1.35

1.4
1.45
Frequency(GHz)

1.5

1.55

1.6

1.65

Figure 3.26 Magnitude of the S-parameters simulated using proposed mode-matching technique and FEM method under L-band (sample centered).
200
150

FEM S11
FEM S21
Modal Analysis S11
Modal Analysis S21

100

Phase (°)

50
0
−50
−100
−150
−200

1.15

1.2

1.25

1.3

1.35

1.4
1.45
Frequency(GHz)

1.5

1.55

1.6

1.65

Figure 3.27 Phase of the S-parameters simulated using proposed mode-matching technique
and FEM method under L-band (sample centered).

84

0

−5

Magnitude (dB)

−10

−15

−20

−25
FEM S11
FEM S21
Modal Analysis S11
Modal Analysis S21

−30

−35
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 3.28 Magnitude of the S-parameters simulated using proposed mode-matching technique and FEM method under S-band (1 inch shift).
200
150

FEM S11
FEM S21
Modal Analysis S11
Modal Analysis S21

100

Phase (°)

50
0
−50
−100
−150
−200
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 3.29 Phase of the S-parameters simulated using proposed mode-matching technique
and FEM method under S-band (1 inch shift).

85

3.3

Summary

A mode matching technique is developed for computing the S-parameters of gyromagnetic
materials partially filling a rectangular waveguide. This method can be adopted to develop
an extraction process which eliminates the need for a large sample required by standard
waveguide techniques in which a sample completely fills the rectangular waveguide cross
section. The mode matching technique is validated using an in-house FEM code and a special
case where the material is placed at the center is also discussed and analyzed. An inverse
problem which extracts the material parameters from measured or theoretical S-parameters
is introduced in the following chapter.

86

CHAPTER 4
Extraction Process for Gyromagnetic
Material Properties Using a
Partially-Filled Waveguide Technique
4.1

Introduction

This chapter presents a method for characterizing gyromagnetic materials using two complex
measurements (S11 and S21 ). The material parameters are obtained through a nonlinear
least square method that minimizes the difference between the measured and the theoretical
reflection and transmission coefficients. This procedure is implemented by utilizing optimization algorithms in Matlab. The adopted optimization algorithms in Matlab are described,
and an error analysis using these optimization algorithms with different sample configurations is considered to show that the proposed characterization technique is very robust
to measurement error. In addition, an extraction process which employs a series of measured S-parameters under different magnetic biasing fields to extract a single set of material
parameters is introduced.

87

4.2

Least Square Method for Extraction

The three complex constitutive parameters (µg , κ, and r ) describing a gyromagnetic material can be extracted using a frequency by frequency characterization. However, since each
measurement contains only two complex measured quantities (S11 and S21 ), this results in
an underdetermined system. Alternatively, based on the fact that the value of r of the
gyromagnetic samples of interest is typically very small, a frequency by frequency characterization of four real physical parameters (4πMs , H0 , ∆H, and r ) can be achieved by
using four real measurements (S11,r , S11,i , S21,r and S21,i ). In this work, the extraction
process is considered to be frequency-independent since three of the physical parameters
4πMs , H0 , and ∆H are frequency-independent, and the permittivity varies little over the
measurement band. Therefore, the frequency-independent assumption is sufficient for accurate characterization. A nonlinear least-square inversion method is adopted to extract the
frequency-independent parameters using measured and theoretical values of S11 and S21 at
a number of frequencies.
The root squared error between theoretical and measured S-parameters is given as

Nf
Error =

2
2
meas (f ) − S thy (f ) + S meas (f ) − S thy (f )
S11,r
j
j
j
j
11,r
11,i
11,i

j=1
Nf
+

2
2
meas (f ) − S thy (f ) + S meas (f ) − S thy (f ) ,
S21,r
j
j
21,r j
21,i
21,i j

(4.1)

j=1
where S thy are the theoretical S-parameters, and S meas are the measured (or simulated)
data, fj is the j th frequency point, and r and i denote the real and imaginary parts of
the S-parameters, respectively. Due to the complexity of the inverse problem that requires
extraction of multiple parameters, traditional methods such as Newton’s method often have

88

difficulty finding the global minimum. Matlab provides a variety of multi-variables optimization algorithms which are easy to implement. Therefore, in this work, optimization
algorithms included in the Matlab optimization tool box are utilized to find accurate approximations of the target material parameters.

4.2.1

Extraction Process Using Matlab Optimization Algorithms

To test the feasibility of the extraction method, theoretical S-parameters are used as a substitute for measured S-parameters. By reducing the root squared error to zero, the extracted
parameters should match the parameters used to generate the theoretical S-parameters exactly. At the present, the inversion procedure does not consider characterization of ∆H,
since the theoretical implementation assumes lossless permeability. Therefore, a method
that can accurately find the three physical parameters (4πMs , H0 ,and r ) is sought.
In this work, the extraction procedure is based on the integration of the compiled forward problem with the optimization solvers included in the Matlab optimization toolbox.
The optimization toolbox provides solvers for finding parameters that minimize or maximize objective functions which can either contain a single variable or multiple variables.
The toolbox includes various solvers such as linear programming, quadratic programming,
nonlinear optimization, and multi-objective optimization, that can offer different domain
searching algorithms for a variety of problems which can be either continuous or discrete,
and constrained or unconstrained. As a result, the solution of an inverse problem can be
obtained by choosing an appropriate solver.
To solve the inverse problem which requires minimizing the root squared error shown in
(4.4), the behavior of the problem must be investigated so that proper searching algorithms
can be chosen for extracting the three parameters. One way to achieve this is to plot the
root squared error against different material parameters. Due to the number of the variables
involved, which makes it difficult to graphically demonstrate the relationship between the
error and all of the parameters simultaneously, the errors are compared with two variables at
89

a time so that a 3-D surface plot can be used. The theoretical S-parameters of a gyromagnetic
sample with 4πMs = 1000 G, H0 = 3000 Oe, and r = 14.7 were generated and substituted
for the measured S-parameters. Figure 4.1 shows the error as a function of 4πMs and r
when H0 is set to 3000 G. It can be seen that the surface plot forms a valley where the
bottom dark regions represent small error between the measured data and theoretical data.
Figure 4.2 shows the same surface plot of the errors but rotated at a different angle. It is
noticed that a minimum can be achieved for a 4πMs value of around 1000 and an r of 14.7
where the total error is expected to be zero. Similar behavior can be noticed for the errors
corresponding to different H0 and r in Figures 4.3 and 4.4. A minimum can also be be
seen at the bottom of the valley where the variables reach the values used to generate the
“measured” S-parameters. Several local minima can be seen around the global minimum
which emphasizes the need for an solver to find the global minimum in a short period of
time.
The Matlab version used for solving the inverse problem was version 7.14/R2012a, which
operates under a Windows 64-bit environment. The optimization toolbox included in this
release is version 6.2. A new release of the toolbox can be installed to obtain an enhancement of the optimization solvers or to obtain more solvers. However, this version of the
toolbox is sufficient for users to take advantage of multiple solvers for different optimization
problems. Though the use of this toolbox will be discussed in this section, it is still highly
recommended for users to read the description of the optimization toolbox in the help menu
before proceeding to actual problems.
There are two ways to access to the optimization toolbox. One is to use the commands
in the Matlab prompt directly where the options can be set through different commands,
while the other is to use the graphical user interface (GUI) with which one can have access
to all the information and resources in a single page. To activate the toolbox, one can either
type optimtool in the command window or open it from the directory “Start - Toolboxes
- Optimization - Optimization Tools” at the main user interface. In this section, the

90

40

Error

30

20

10
15.2
0
700

15
800

14.8
900

14.6
1000

1100

14.4
1200

1300

14.2
εr

4πMs

Figure 4.1 3-D surface plot of the root squared error for different 4πMs and r with H0 set
to 3000 gauss.

91

40
35
30

Error

25
20
15
10
5
0
700

800

900

1000

1100
4πMs

1200

1300

14

14.5

15

εr

Figure 4.2 3-D surface plot of the root squared error for different 4πMs and r with H0 set
to 3000 gauss.

92

35
30

Error

25
20
15
10
5
0
3400

15

3200

14.8

3000

14.6

2800

14.4

2600

ε

H

o

r

Figure 4.3 3-D surface plot of the root squared error for different H0 and r with 4πMs set
to 1000 gauss.

35
30
25

Error

20
15
10
5
0
15

14.8

14.6

14.4

2600

2800

3000

3200

3400

H

εr

o

Figure 4.4 3-D surface plot of the root squared error for different H0 and r with 4πMs set
to 1000 gauss.
93

access to the toolbox is implemented through the GUI where the settings can be seen and
modified easily. The use of built-in commands will be discussed in the following sections.
The use of the appropriate solver must be ensured to eliminate the possibility of finding
local minimum. Additionally, the computational cost should be considered for those solvers
that satisfy the goal. For this problem, as was previously shown in the 3D surface plot,
the goal is to minimize the root squared error which demonstrates a nonlinear relationship
with three variables. Many algorithms are able to optimize for a nonlinear problem, such
as genetic algorithm (GA) which is very useful for problems that are highly nonlinear, pattern search (PS) algorithms that find local minima using pattern searches, and simulated
annealing (SA) which periodically reset the searching radius to find the global minimum.
There are other algorithms for nonlinear problems in which constraints can be defined or
excluded. Common unconstrained algorithms in Matlab are fminunc and fminsearch, while
the constrained options are fmincon, fminbnd and fseminf. Note that most optimization
solvers are only local solvers, and different solvers take different time durations to find the
solutions. An accurate solver that can reduce the process time significantly is preferred.
Studies [51] have been done to compare the time cost of each algorithm to find the solution
for Rastrigin’s function with default settings. It has been shown that the nonlinear algorithms fminsearch and fmincon tend to cost less time to find the solution than PS, SA and
GA. Among all the solvers mentioned above, GA has been found to be the least efficient
costing the most amount of time and iterations. To test the performance of different solvers,
the measured S-parameters were substituted in for the theoretical ones, it was found out
that either fminsearch or fmincon was able to find the solution in a short period of time.
In this problem which uses theoretical S-parameters as the measured data, the fminsearch
was used as an unconstrained solver to demonstrate that the solution can be found quickly
for an unbounded region even with poor initial guesses. The fminsearch function finds the
minimum of a unconstrained scalar function with multiple variables based on the NelderMead method. The Nelder-Mead method is a technique for minimizing a multi-dimensional

94

objective function for which derivatives may not be known. The method uses the idea of
forming N + 1 vertices in N dimensions to approximate a local optimum with an assumption
that the function varies smoothly. In this problem, since there are three variables, a tetrahedron in three-dimensional space will be used to find the local optimum. Results found using
this solver are described in the following section.

4.2.2

Validation of the Extraction Using Matlab Optimization
Functions

To implement the inverse problem, the GUI of the optimization tool is initialized by following
the initializing steps mentioned previously. There are three panels displayed in the toolbox
GUI. The first panel is the problem setup and results panel which is shown in Figure 4.5. One
can call any of the solvers in the toolbox and assign constraints for the objective function.
The second panel shown in Figure 4.6 is the options panel where the control parameters such
as tolerances can be set for the algorithm. The third one is the quick reference panel that
allows users access to help documents and examples using the chosen algorithm. This panel
is not displayed in this section.
To initialize the optimization for the inverse problem which minimize the difference between theoretical S-paramters and actual measured S-paramters, the fminsearch solver was
selected. Two entries in the the problem setup and results panel were required to be specified.
As can be seen from Figure 4.5 the first entry to be specified is the name of the objective
function. In this case, the name of the function that computes the root squared error in
(4.4) was entered with the “@” sign as a prefix. The second entry is the initial guess for the
objective function. As was described previously, the selected solver is a local optimization
algorithm with no constraints. Therefore, it is important to specify a reasonable initial guess
so that a minimum can be found successfully. One of the initial guesses that was used to
initialize the optimization was [2800, 14.5, 1060] with the format of a vector. In the options
panel, the stopping criteria were chosen to be the default, where the termination tolerance
95

for both the function value and variables is 10−4 . The function value was clicked in the
plot function sub-panel to display the iterative function values. Also, the level of display is
selected to be iterative in the display sub-panel. Once all the parameters were set up, the
optimization could be launched by clicking the start button.
During the optimization procedure, the computed root squared error was plotted for each
iteration. Once the the optimization is completed by reaching the stopping criteria, one is
able to see that plot with the final results. The function values over different iterations with
the initial guess mentioned earlier is shown in Figure 4.7. Notice that the error decreases as
the number of iterations increases, and the computed error ends up as 2.8 × 10−8 which indicates that the difference between the theoretical S-parameters and measured S-parameters
is insignificant. The final results for the three variables were displayed in the problem setup
and results panel once the optimization was over. The returned results match the theoretical
values for the three parameters to the fifth decimal place. Therefore, this solver has shown
its ability to optimize this inverse problem.
As was stated earlier, a reasonable initial guess is required for local solvers, and different
random initial guesses alter the optimization time and results, thus a test of the employed
solver with different initial guesses was conducted to understand their impact on the results.
Table 4.1 shows the number of iterations required to find the solution. The first column
shows five different initial guesses used to start the optimization. It can be seen that different
guesses lead to different numbers of iterations to complete the optimization, and generally
a closer guess to the correct solution may reduce the number of iterations significantly. For
instance, the cost in terms of iterations for a guess of [3700, 15.2, 1230] is more than double
that of the one with [2800, 14.5, 1060], which implies the optimization time consumption can
be dependent on the starting guess. Furthermore, varying the tolerances in the optimization
option panel can also alter the number of iterations. It is obvious that the number of
iterations reduced as the tolerances were changed from the default setting (10−4 ) to 10−2 .
An average of 22% in iteration reduction is achieved for these five trials. Therefore, increasing

96

the tolerances will speed up the optimization.
Table 4.1 Number of iterations for different initial guesses.
Initial Guess
Number of Iteration (Tols=10−4 )
[2500, 14, 1260]
261
154
[2800, 14.5, 1060]
[2900, 14.9, 1030]
227
245
[3500, 13.6, 860]
[3700, 15.2, 1230]
324

4.3

Number of Iteration (Tols=10−2 )
148
133
165
206
297

Error and Sensitivity Analysis

As is described in Section 2.4, a Monte Carlo technique was used to evaluate effects on the
propagation of the random measurement error inherent to the VNA. Random measurement
errors of the VNA can be attributed to many factors such as temperature or humidity of
the operating environment. The uncertainties of the measured S-parameters from the VNA
can be obtained from the manufacturer. Table 4.2 lists the S-parameter uncertainties for
an HP8510C network analyzer system using the software package HP 8510 Specifications &
Performance Verification Program and for an Agilent E5071C ENA using Agilent E5071C
ENA Network Analyzer Data Sheet [52], which are the two VNA models used in the experiments. The value of the uncertainties for S21 are quantified by the manufacturer by
performing a “through” measurement 32 times over the full frequency range of the VNA
with a cable connected between two ports. Similarly, the uncertainties for S11 are obtained
by performing a “short” measurement 32 times over the frequency span of the analyzer with
the port terminated by a short. Standard deviation of the measured data at each frequency is
then calculated and then converted into an uncertainty (noise) level in magnitude and phase.
For the HP8510C, statistical variance of S11 is indicated linearly in amplitude and logarithmically in phase. Thus the uncertainty for S11 (σA ) carries no units while all of the rest
11
of the uncertainty levels contain units. It can be noticed that the E5071C demonstrates a
97

Figure 4.5 Problem setup and results panel of the optimization toolbox GUI.
98

Figure 4.6 Option panel of the optimization toolbox GUI.

99

Current Function Value: 2.78923e−08
3.5

Function value (Error)

3

2.5

2

1.5

1

0.5

0

0

20

40

60

80

100

120

140

160

Iteration
Figure 4.7 Iterative function values during one optimization.
much smaller uncertainty level than the HP8510C. However, to evaluate the impact of the
possible maximum measurement error from VNA for extracted parameters, the uncertainty
levels from HP8510C were used.
Table 4.2 VNA measurement uncertainties.

σA
11
σφ
11
σA
21
σφ
21

HP8510C network analyzer Agilent E5071C network analyzer
0.004
0.004 dB
◦
0.8
0.035◦
0.04 dB
0.003 dB
◦
2.0
0.035◦

The fictitious material described previously with H0 = 3000 Oe, r = 14.7, and 4πMs =
1000 G was used in the the Monte Carlo analysis of the propagation of VNA uncertainty.
The test sample is assumed to have a width of d = 34 mm that is the same as the height
of the S-band waveguide, and a thickness of 10 mm. The forward problem was computed
100

with 101 frequency points over the S-band from 2.6 to 3.95 GHz and 20 modes were used
to achieve an accuracy of the magnitude less than 0.01 dB and an accuracy of the phase
less than 0.1◦ . White gaussian noise was then added to each of the S-parameter sets, and
the noisy data was used to extract the material parameters. The noise levels indicated
by the HP 8510 Specifications & Performance Verification Program [53] were used. This
procedure is described in Section 2.2.5. One hundred and sixty trials were implemented for
this Monte Carlo analysis, and the average values of the material parameters using different
number of trials were calculated, along with the standard deviations. Figures 4.8 - 4.10
show the extracted parameter values corresponding to different numbers of random trials
that were used. The black dots are the average values of the trials and red dots show the
average value plus or minus one standard deviation. In these figures, it can be seen that
the difference between the average values and the fictitious material parameters are within
1% after a trial number of 110. The standard deviation for H0 using 160 trials is 34 Oe
which is equivalent to 1.1% relative error. Also, the standard deviations for r and 4πMs
are 0.01 and 15.2G, corresponding to relative errors of 0.7% and 1.5%, respectively. A
similar analysis was performed for another test sample with a width of 54 mm and a same
thickness of 10 mm. Figures 4.11 - 4.13 show the results of the analysis for a wider sample.
It can be seen that the extracted parameters share the same trend of average values as the
narrow sample. However, narrower error bars are observed which indicate a reduction in
the sensitivity of the extracted parameters to measurement errors. Note that the noise level
used in this error analysis is the maximum error that could be introduced into experiments,
and the error in the extracted material parameters due to S-parameter noise is still relatively
small. Therefore, this extraction technique has shown a robustness to VNA uncertainties.
Monte Carlo analysis of the uncertainties in sample widths were employed to evaluate the impact of inaccurately measured sample width on the extracted parameters. The
uncertainty of the sample width was obtained from measuring the actual G1010 sample (Section 5.3.1.1) 20 times and the standard deviation of these measured results were calculated

101

(0.034 mm). The biasing magnetic field is H0 = 3000 Oe, while the saturation magnetization is 4πMs = 1000 G. This H0 results in a FMR occurring at 8.40 GHz. The forward
problem was solved at 101 frequency points over the portion of S-band from 2.6 GHz to
3.95 GHz. With the S-parameters computed using altered sample widths, these data was
used to extract the material parameters. Two hundred trials were used in the Monte Carlo
analysis, and the average values of the material parameters and the standard deviations were
calculated. Histograms of different extracted parameters are shown in Figure 4.14-4.16 to
provide graphical representations of the distribution of the parameters. Figures 4.20-4.22
show the effects on the permittivity and the permeability characterization. Note that the
differences between the theoretical values of G1010 used to generate the S-parameters employed in the Monte Carlo analysis and the average values are within 0.5%. Therefore, the
theoretical values of G1010 are not plotted in these figures. Similar procedure was taken to
evaluate the impact of inaccurately measured sample thickness on the extracted parameters.
The meausured uncertainties of the sample thickness is 0.005 mm. Histograms of different
extracted parameters are shown in Figure 4.17-4.19. Figures 4.23-4.25 show the effects on
the permittivity and the permeability characterization.

4.4

Extraction of Measured S-parameters

As was shown in the previous section, the fminsearch solver was tested to show its capability
of finding the correct parameters by using a fictitious material. However, a constrained
guess region may be required for processing the measured data not only because accurate
initial guesses are usually unavailable before measuring an actual gyromagnetic sample, but
also to reduce the optimization time. In this section, three sets of measured data for the
same material biased under different external magnetic fields are used to demonstrate the
extraction process using measured S-parameters using Matlab optimization function. The
details of the measurement procedures are included in the next chapter, and this section only

102

3080
3060
3040

H (Gauss)

3020
3000
2980
2960
2940
2920
2900
0

50

100

150

Number of Trials

Figure 4.8 Extracted H0 for a test sample of width d=34 mm under different number of
trials. The center dots are the average of the trials. Upper and lower dots show one standard
deviation.
14.75
14.74
14.73

Permitivitty (ε )
Relative Permittivityr

14.72
14.71
14.7
14.69
14.68
14.67
14.66
14.65
0

20

40

60

80
Number of Trials

100

120

140

160

Figure 4.9 Extracted r for a test sample of width d=34 mm under different number of
trials. The center dots are the average of the trials. Upper and lower dots show one standard
deviation.

103

1040
1030

4πMs (Gauss)

1020
1010
1000
990
980
970
960
0

50

100

150

Number of Trials

Figure 4.10 Extracted 4πMs for a test sample of width d=34 mm under different number of
trials. The center dots are the average of the trials. Upper and lower dots show one standard
deviation.
3150

3100

H (Gauss)

3050

3000

2950

2900

2850
0

50

100

150

Number of Trials

Figure 4.11 Extracted H0 for a test sample of width d=54 mm under different number of
trials. The center dots are the average of the trials. Upper and lower dots show one standard
deviation.
104

14.73

14.72

Permitivitty
(εr)
Relative
Permittivity

14.71

14.7

14.69

14.68

14.67
0

20

40

60

80
Number of Trials

100

120

140

160

Figure 4.12 Extracted r for a test sample of width d=54 mm under different number of
trials. The center dots are the average of the trials. Upper and lower dots show one standard
deviation.

1050
1040
1030

4πMs (Gauss)

1020
1010
1000
990
980
970
960
950
0

50

100

150

Number of Trials

Figure 4.13 Extracted 4πMs for a test sample of width d=54 mm under different number of
trials. The center dots are the average of the trials. Upper and lower dots show one standard
deviation.

105

60

Number of trials

50

40

30

20

10

2700

2800

2900
3000
3100
Internal Magnetic Biasing Field H0(Oe)

3200

3300

Figure 4.14 Histogram of the H0 extracted from 200 random trials of simulated S-parameters
for a partially filled G1010 sample with various widths.

120

Number of trials

100

80

60

40

20

14.68

14.685

14.69

14.695

14.7
14.705
Relative Permittivity εr

14.71

14.715

14.72

14.725

Figure 4.15 Histogram of the relative permittivity extracted from 200 random trials of simulated S-parameters for a partially filled G1010 sample with various widths.

106

70

60

Number of trials

50

40

30

20

10

850

900

950
1000
1050
Saturation Magnetization 4πMs (G)

1100

1150

Figure 4.16 Histogram of the 4πMs extracted from 200 random trials of simulated Sparameters for a partially filled G1010 sample with various widths.

80
70

Number of trials

60
50
40
30
20
10

2700

2800

2900
3000
3100
Internal Magnetic Biasing Field H0(Oe)

3200

3300

Figure 4.17 Histogram of the H0 extracted from 200 random trials of simulated S-parameters
for a partially filled G1010 sample with various thicknesses.

107

70

60

Number of trials

50

40

30

20

10

14.62

14.64

14.66

14.68

14.7
14.72
Relative Permittivity εr

14.74

14.76

14.78

14.8

Figure 4.18 Histogram of the relative permittivity extracted from 200 random trials of simulated S-parameters for a partially filled G1010 sample with various thicknesses.

70

60

Number of trials

50

40

30

20

10

800

850

900

950
1000
1050
Saturation Magnetization 4πMs (G)

1100

1150

1200

Figure 4.19 Histogram of the 4πMs extracted from 200 random trials of simulated Sparameters for a partially filled G1010 sample with various thicknesses.

108

14.71
14.708

Relative Permittivity

14.706
14.704
14.702
14.7
14.698
14.696
14.694
14.692
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

Figure 4.20 Relative permittivity extracted from 200 random trials of simulated S-parameters
for a partially filled G1010 sample with various widths. Center black line is the average of
the trials. Upper and lower red dashed lines show the 95% confidence interval.

1.8

Relative Material Parameters

1.6

µ’g

1.4
1.2
1
0.8
0.6
0.4
0.2

µ’’g

0
−0.2
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 4.21 Relative permeability values extracted from 200 random trials of simulated Sparameters for a partially filled G1010 sample with various widths. Center black line is the
average of the trials. Upper and lower red dashed lines show the 95% confidence interval.
109

Relative Material Parameters

0.05

δ’’
0
−0.05
−0.1

δ’

−0.15
−0.2
−0.25
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 4.22 Relative permeability values extracted from 200 random trials of simulated Sparameters for a partially filled G1010 sample with various widths. Center black line is the
average of the trials. Upper and lower red dashed lines show the 95% confidence interval.

14.76

Relative Permittivity

14.74

14.72

14.7

14.68

14.66

14.64
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

Figure 4.23 Relative permittivity extracted from 200 random trials of simulated S-parameters
for a partially filled G1010 sample with various thicknesses. Center black line is the average
of the trials. Upper and lower red dashed lines show the 95% confidence interval.
110

1.8

Relative Material Parameters

1.6

µ’g

1.4
1.2
1
0.8
0.6
0.4
0.2

µ’’

g

0
−0.2
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 4.24 Relative permeability values extracted from 200 random trials of simulated Sparameters for a partially filled G1010 sample with various thicknesses. Center black line is
the average of the trials. Upper and lower red dashed lines show the 95% confidence interval.

0.1

Relative Material Parameters

0.05

δ’’

0
−0.05

δ’

−0.1
−0.15
−0.2
−0.25
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 4.25 Relative permeability values extracted from 200 random trials of simulated Sparameters for a partially filled G1010 sample with various thicknesses. Center black line is
the average of the trials. Upper and lower red dashed lines show the 95% confidence interval.
111

focuses on developing a proper procedure for extraction of measured data using the Matlab
optimization functions.
With initial guesses of variables known, fmincon is able to determine solutions faster
for a bounded region using measured data compared to fminsearch which does not require
bounds to be assigne. This solver finds the minimum of a constrained nonlinear function with
multiple variables based on the gradient method. It requires both the objective function and
constraint function to be continuous. Function fmincon has four algorithms, with several
options for computing Hessians (matrices of second-order partial derivatives of a function):
the trust region reflective algorithm, the SPQ algorithm, the active set algorithm, and the
interior point algorithm. The trust region reflective algorithm requires a user supplied Hessian or uses a finite-difference approximation for the Hessian. The active set algorithm and
SPQ algorithm compute a quasi-Newton approximation of the Hessian matrix. The interior
point algorithm uses a combination of the different methods mentioned above to define the
Hessian matrix. Appropriate algorithms need to be chosen for specific problems.
Instead of using the graphical user interface (GUI) in the Matlab optimization toolbox
as shown in the previous section, the implementation of the optimization using measured
data is achieved through Matlab commands, since this method is more convenient for setting
up the optimization parameters. Below is an example using fmincon with commands for
measured data.
H0=[2990,14.7,1020];

options = optimset(‘Algorithm’,‘interior-point’,‘Display’,‘iter’,‘PlotFcns’,
@optimplotfval,‘TolFun’,1e-4,‘TolX’,1e-5);

A = []; b = []; Aeq=[]; beq=[]; lb=[2600,14.0,640];
ub=[3800,15.2,1360]; nonlcon = [];

112

[H_opt,fval] = fmincon(@S_param_function_meas, H0,A,b,Aeq,beq,
lb,ub,nonlcon,options);
First, the initial guess was defined as a vector. Second, the optimization options were
specified. As it can be seen that in the options, the optimization algorithm chosen for this
problem is the interior point algorithm. This hybrid algorithm tends to take less time in
optimization; this was demonstrated by comparing it with other algorithms using the same
measured data. Also, a plot of the iterative function value can be requested in the options
column, which allows the user to see the status of the function value during optimization.
Third, the input argument for the objective function was defined. In this problem, only
the lower and upper boundary of the initial guess were assigned and they were decided
upon based on the reasonable material properties of the sample and estimated magnetic
biasing field. In the last step, the optimization was launched by using function fmincon
with the name of the objective function, which calculates the root squared error between the
theoretical data and measured data for this problem. The parameters H opt and fval are
the returned optimized results and final function value respectively.
Measured data was used to evaluate the performance of this method. A typical measurement contains 801 frequency points, it would be time-consuming to compute the forward
problem using so many points. Moreover, the S-parameters are more sensitive to the variations of the material parameters at frequencies near the resonance frequency. Thus, the
critical frequency points which are close to the resonance frequencies of the measurement
were chosen. However, for some resonance frequencies in the measured data, few frequency
points are available to describe that resonance. Therefore, only the frequency points that
are near the main resonance frequency are considered for processing. For example, the blue
curves in Figure 4.26 show the magnitude and phase of the S-parameters measured under the
highest magnetic biasing field. It can be seen that there is a resonance near 2.7 GHz, but it
is also not sensitive to the variation of the material parameters. For this measured data, 120
frequency points were chosen from 801 points around the main resonance frequency which
113

covers a frequency range of 3.62 − 3.82 GHz. The lower and upper bounds of external magnetic biasing field H0 are set to be ±400 of the intial guess. The bounds of r are determined
based on the specification sheet offered by the manufacturer of the G1010 sample which are
14 and 15.2. Upper and lower bounds of 4πMs are assigned to be 640 and 1360 G. These
bounds offer the solver fairly large regions within which reasonable results can be obtained.
Note that since only H0 varies between different measurements, the lower and upper bounds
for r and 4πMs were kept the same.
The first row in Table 4.3 shows the returned results for the measured data with highest
biasing field. Since this chapter only focuses on the extraction procedure, a discussion of
the extracted results are included in the chapter 5. Theoretical S-parameters using the
extracted parameters were generated and compared with the measured data. It can be seen
from Figure 4.26 that good agreement is achieved between the measured S-parameters and
theoretical S-parameters calculated using extracted values. Similarly, two other measured
data taken under two other different magnetic biasing fields were processed with the same
setup to extract the results. However, for these two measured data, some blips around the
main resonance frequency can be noticed in Figures 4.27 and 4.28 which may be caused
by the air gaps between the sample and waveguide wall. It is found that the best way to
eliminate the negative impact of these data on extraction is to avoid these points rather than
smoothing them. Thus, 60 frequency points on either side of the main resonance frequency
were selected carefully without including the blips. As a result, the same number of the
frequency points (120) can still be obtained to process the optimization. The extracted
results from these two measurements are shown in the second and third rows of Table 4.3.
Good agreement is still achieved between the measured S-parameters and the theoretical
S-parameters as shown in Figures 4.27 and 4.28.
Since for the same test sample, r and 4πMs should be the same, an alternative extraction
process can be established where all the parameters are extracted using all the data measured
with various biasing fields. To achieve this, the least-square error, which was optimized for

114

Table 4.3 Extracted material parameters from measurements with different external magnetic
biasing fields.

Measurement 1
Measurement 2
Measurement 3

H0 (Oe)
3201
1889.7
1406.4

r
14.7371
14.9862
14.2254

4πMs (G)
1049.5
1000.3
1051.2

a single measurement using (), is minimized using three different measured data. The same
number of points were chosen for each set of measured data. Initial guesses for five variables
(three different magnetic biasing field strengths for each measurement, r and 4πMs ) were
then specified along with the lower and upper bounds. Table 4.4 shows the resulting extracted
values of the parameters. The values are reasonably close to the previous values that were
extracted from separate data sets. This method may reduce the impact of the experimental
errors that are contained in each of the measurements by using three different measured
data, thus giving more accurate results.
Table 4.4 Extracted parameters values from one optimization using measurements of different
external magnetic biasing fields.
Parameter
H0 - 1
H0 - 2
H0 - 3
r
4πMs

4.5

Value
3024.3
1763.8
1450.0
14.4754
1061.5

Summary

A method for characterizing gyromagnetic materials from two complex measurements using Matlab optimization algorithms is introduced. The adopted optimization algorithm in
Matlab was validated using a fictitious sample and the performance of the characteriza115

S11 Mag (dB)

20
0
−20
−40
−60
2.6

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

S21 Mag (dB)

0
−20
−40
−60
−80
2.6

S11 Phase (°)

200
100
0
−100
−200
2.6

S21 Phase (°)

100
0
−100
−200
2.6

Theoretical data
Measured data

Figure 4.26 Measured S-parameters of the highest biasing field and theoretical S-parameters
generated using extracted material parameters.

116

S11 Mag (dB)

0
−10
−20
−30
−40
2.6

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

S21 Mag (dB)

0
−20
−40
−60
2.6

S11 Phase (°)

200
100
0
−100
−200
2.6

S21 Phase (°)

200
100
0
−100
−200
2.6

Theoretical data
Measured data

Figure 4.27 Measured S-parameters of the second biasing field and theoretical S-parameters
generated using extracted material parameters.

117

S11 Mag (dB)

0
−10
−20
−30
−40
2.6

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2

3.4

3.6

3.8

4

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

S21 Mag (dB)

0
−20
−40
−60
2.6

S11 Phase (°)

200
100
0
−100
−200
2.6

S21 Phase (°)

200
100
0
−100
−200
2.6

Theoretical data
Measured data

Figure 4.28 Measured S-parameters of the lowest biasing field and theoretical S-parameters
generated using extracted material parameters.

118

tion technique was evaluated using error analysis based on network analyzer uncertainty. It
has been found that the technique performs very well for different sample sizes with high
VNA uncertainties. Extraction methods for processing measured data are also established
to extract material parameters either from a single measurement, or multiple measurements.

119

CHAPTER 5
EXPERIMENTAL SETUP AND
RESULTS
5.1

Introduction

This chapter presents experimental procedures and results for characterizing different anisotropic
materials using a partially-filled rectangular waveguide technique. Two types of anisotropic
material were constructed and measured at S-band to validate the proposed techniques. The
first type is a biaxial material which has different material parameters depending on the
incident wave polarization. This type of material can be described using two three by three
tensor matrices that each consist of three nonzero entries. To extract all six parameters of
the biaxial material, usually three different samples are required to perform three sets of
reflection and transmission measurements. However, one can extract all six parameters with
the partially filled waveguide technique from one well constructed sample. This extraction
method is the three-step approach shown in Section 2.2.4 which requires three reflection
and transmission measurements of a cubical sample at different orientations. A simplified
extraction approach for a uniaxial material which is a special case of a biaxial material is also
developed and shown in Section 2.2.4. Since two parameters in the tensors are identical, with

120

A = C and µA = µC , only four unknown parameters exist. The number of measurements
required to complete the extraction for this material is reduced to two.
The second type of the material is the gyromagnetic material which can be described by
an isotropic complex permittivity and a three by three permeability tensor with off-diagonal
terms which vary with different magnetic biasing conditions. Due to manufacturing difficulties, these material often come with a limited size which makes using the filled aperture
extraction method unfeasible for them. This can be solved by using a partially filled waveguide technique where the properly machined sample partially fills the waveguide aperture.
The extraction process can be achieved by either using a single measurement or multiple
measurements with different experimental configurations, such as with different magnetic
biasing fields or sample locations. Matlab optimization algorithms were used in the inverse
problem which minimizes the difference between theoretical and measured S-parameters to
expedite the extraction process. This is described in Section 4.4.

5.2

Characterization of Biaxial/Uniaxial Material

To experimentally validate the technique of extracting biaxial material and uniaxial material
parameters, two samples were built and measured in an S-band waveguide system. The
construction of the material sample and the details for setting up the waveguide system are
discussed in the following section. The repeatability of the measurements and error analysis
of the measured results are also assessed.

5.2.1

Sample Construction

Two samples were built to implement the extraction process using experimental results.
The first sample is a cube which was constructed using layers of alternating dielectrics.
This produces a material with unixial electric properties, but isotropic magnetic properties.
Due to the unavailability of the commercial anisotropic dielectric and anisotropic magnetic

121

material, this unixial dielectric sample was measured as a biaxial sample to validate the
feasibility of the three-step approach. This sample was built by gluing layers of Rogers
RO3010 and Rogers RT/druroid 5870 circuit boards together using super glue from Loctite.
The manufactured cubic sample is shown in Figure 5.1 where the light color layers are Rogers
RO3010. The RO3010 board has a dielectric constant of r1 = 10.2 and a loss tangent of
tan δ1 = 0.0022 with a thickness of 1.27 mm. The RT/druroid 5870 board has a dielectric
constant of r2 = 2.33 and a loss tangent of tanδ2 = 0.0012 with a thickness of 3.4 mm.
The second sample was built by gluing layers of Rogers RT/druroid 5870 with a thickness
of 1.27 mm and Eccosorb FGM-125 together, where the designation 125 represents the
thickness of the substrate (125 mils). This produces a sample with both unixial electric
properties, and uniaxial magnetic properties. The manufactured sample is shown in Figure
5.2 where the thick dark substrate is FGM-125. The reason that the first and last layer
of the sample is chosen to be Rogers 5870 is to reduce the sample abrasion from repeated
measurements, since FGM-125 is soft and rubbery. This sample was used to evaluate the
simplified approach for extracting uniaxial materials.
The ideal side length of the cubes is supposed to be the height of S-band waveguide
which is 34.036 mm. However, due to machine error, the final dimensions of the biaxial and
uniaxial samples along the direction that the layers inside the sample are horizontal are:
34.05 mm × 34.11 mm × 34.09 mm and 34.01 mm × 34.14 mm × 33.94 mm, respectively.
Note that a perfect sample should be slightly larger than the dimension of the waveguide
wall so that the air gap between the sample and waveguide wall can be eliminated. However,
it was observed that an air gap of less than 0.2 mm occurred when the uniaxial sample was
placed into the waveguide. It is also important to mention that different faces of each sample
are marked to give consistent orientation in the measurements.

122

Figure 5.1 Uniaxial electric and isotropic magnetic material sample.

Figure 5.2 Material sample with both uniaxial electric and magnetic properties.

123

5.2.2

Measurement Techniques

To experimentally validate the proposed technique at S-band, the S-parameters from different samples were measured and the material parameters were extracted. The waveguide
system used for the measurements is a S-band calibration kit from Maury Microwave which
consists of two 152.4 mm long waveguide section, which acted as extensions, with coaxial
transitions attached at the ends. An additional brass plate and a 30.5 mm waveguide section
is also included in the calibration kit which is used as short and line respectively during the
calibration procedure. The two waveguide extensions were connected to an Aglient E5071C
VNA through two 7 mm to 3.5 mm adapters. The whole system is shown in Figure 5.3. The
VNA was then calibrated using the calibration kit described above based on a Thru-ReflectLine (TRL) method with the corresponding algorithm stored in the VNA. The calibration
kit consists of two offset shorts and a load. It was found that by performing the TRL calibration method in a certain order, the calibration error can be reduced. For the calibration
implemented for the measurements of the cubical samples, the 30.5 mm waveguide section
(line) was used first. Then the brass plate was used twice to calibrate each of the ports.
Finally, a through measurement was performed to complete the calibration. The typical
variation in the value of the magnitude of a through measurement taken right after the calibration is within ±600 µdB. The reason that the best calibration results can be achieved by
following this order might be due to the fact that the error introduced by the misalignment
of the waveguide sections can be reduced by performing the through measurement at the
end. In addition, a crucial step for achieving consistent calibration and measured results
is to have the same configuration for waveguide assembly. To achieve this, an adjustable
torque wrench with the torque set to 50 Lb-in was used to tighten the screws used to assemble the waveguide sections or the calibration components. Note that there are 8 holes
on the periphery of the waveguide flanges and the calibration components. Two screws on
the diagonal were inserted and were tightened using the torque wrench. The next two bolts

124

were placed so they were not adjacent to the previous bolts. The remaining four bolts were
installed in a similar manner. This procedure maintains an even pressure on the waveguide
flange and therefore reduces error caused by misalignment. Two wooden holder were built
and attached at the end of the waveguide ports to minimize the physical vibration during
the calibration process. The measurements were made with VNA settings of -5 dBm source
power, 64 averages, 801 frequency points within S-band (2.65 GHz-3.95 GHz), and an IF
bandwidth of 70 kHz. Two fixed torque wrenches with torques of 8 lb-in and 12 lb-in were
also used to tighten the 3.5 mm and 7 mm adapters respectively, during the calibrations and
measurements when each time the waveguide system was dissembled or assembled.

Figure 5.3 Waveguide measurement system.
One advantage of using a partially-filled waveguide technique is that no sample holder is
required. Instead, the sample was placed inside the waveguide extension connected to Port 1.
The sample was placed at the center of the waveguide using a foam spacer and one surface of
the sample must be parallel to the waveguide flange to ensure that the propagation direction
125

of the incident wave aligns with the sample axis. This latter condition was implemented
by making one surface of the sample flush with the waveguide opening. Figure 5.4 shows
the unixial sample placed at the center of a waveguide extension at orientation 1 which is
described in Section 2.2.4. For the measurements of cubical samples shown in Figure 5.5, the
sample is inserted into the waveguide extension attached to port 1. The S-parameters of the
sample were measured by the VNA at the calibration plane. However, the extraction method
requires using S-parameters referenced to the sample planes z=0 and z=d. Therefore, the
measured S-parameters have to be mathematically de-embedded to the sample planes. The
S-parameter can be shifted from the calibration plane to the sample planes by multiplying
by e−jβD in the direction of the wave propagation. Here, β is the propagation constant
of the empty waveguide extension which can be found using (2.4), and D is the appropriate
distance of the shift. For the measurement of biaxial or uniaxial samples, the shifting can
be achieved using the following expressions:
s = S c e−2jβd
S21
21
s = S c e−jβd ,
S11
11

(5.1)
(5.2)

s and S s are the S-parameters referenced to the sample planes while S c and S c
where S11
21
11
21
are the measured S-parameters referenced to the calibration plane. This approach is also
described in appendix B. [42].

5.2.3

Experiment Results

The VNA was calibrated ten times using the waveguide calibration kit. Ten sets of measurements for each material sample were conducted and the S-parameters were saved for extraction. For the uniaxial dielectric and isotropic magnetic material (biaxial sample), three
orientations for each set of measurement were required to extract all six parameters. For
the uniaxial dielectric and uniaxial isotropic magnetic material (uniaxial sample), only two

126

Figure 5.4 Uniaxial material sample placed in the center of the waveguide at orientation 1.

Calibration
Plane

Sample

Port 1
0

Port 2
𝑑

Waveguide Extension

Z
Waveguide Extension

Figure 5.5 Sample inserted into the waveguide extension attached to port 1.

127

orientations were required since a simplified process was applied to extract the four parameters. Error analysis of both measurement repeatability and extracted material parameters
was undertaken.

5.2.3.1

Repeatability Analysis

In Section 2.2.5, a Monte-Carlo technique was used to evaluate the propagation of random
error from the VNA. In this section, a statistical analysis of the variance of S11 and S21
of the measured data was processed to evaluate the repeatability of the experiment.The repeatability error describes the measurement uncertainties. In these experiments, the sources
of measurement uncertaities may be attributed to imperfect calibration of the VNA, changes
in the environment (temperature, humidity,etc...), misalignment of waveguide sections, and
inaccurate sample positioning in the waveguide. A statistical analysis of these errors can
help determine the sensitivities of S-parameters to the experimental setup.
Since the experimental setups for both samples are the same, the sources of errors for
measuring both samples are identical. The measured S-parameters of the uniaxial sample
were chosen for the repeatability analysis because only two measurements were required for
each set of measurements and thus it would be more convenient to analyze them. Figures 5.6
- 5.9 show the average S-parameters and the standard deviation generated from 10 repeated
measurements of the uniaxial material at the first orientation. In these figures, the two
surrounding red lines indicate the 95% confidence interval of ±2 standard deviations. Note
that the two standard deviations of the magnitude of both S11 and S21 are less than 0.1 dB,
which indicates highly consistent measured results. From Figures 5.7 and 5.9, one can see
that the confidence interval of S11 is approximately 1 degree and that of S21 is around 0.5
degree. These values are much larger than the phase uncertainties of the VNA (0.035 degree)
. The overall statistical results of the measured S-parameters indicate good consistency of
the experimental setup. The same procedure was undertaken for the measured S-parameters
of the second orientation which is also described in Section 2.2.4 and the results are shown

128

in Figures 5.10 - 5.13. The confidence intervals for both of the magnitude of S11 and S21
are less than 0.04 dB, and confidence interval of the phase is also less than 1 degree. Note
that highly consistent results are obtained for the magnitude of the S-parameters from the
10 measurements, while acceptable consistency is shown for the phase. The inconsistency in
the phase may due to an unstable sample position in the waveguide which causes a change
in the measured phase. A well constructed foam fixture could be used to achieve better
measurement consistency. This is left as future work.

5.2.3.2

Extracted Parameters of The Uniaxial Sample

As outlined in Section 5.2.2, a uniaxial dielectric and isotropic magnetic material sample
was placed in the cross-section of a waveguide under different orientations and the transmission and reflection coefficients were measured, providing sufficient data for the three-step
approach to find both the permittivity and permeability of the sample. An estimation of the
material parameters from the sample should be acquired to validate the proposed technique.
The following formulas [54] can be used to determine the approximate material parameters
− r2 t1
− r1
r2
r1 r2 t1 + t2
1

−1
,

(5.3)

t1
A = C = 2r + ( r1 − 2r ) t + t ,
1
2

(5.4)

B=

where r1 = r1 (1 − j tan δ1 ), r2 = r2 (1 − j tan δ2 ) and (t1,t2) are the thicknesses of
the substrates. Substituting the variables with the parameters from each substrate gives
B = 2.95 − j0.0038 and A = C = 4.47 − j0.0081. It is expected that the formula for
B will be less accurate than the formula for A and C , due to the significant internal
reflections within the layers when the cube is oriented such that the interfaces are normal to
the wave propagation. It is also expected that A would be slightly different from C since
the commercial boards are slightly anisotropic.
Figures 5.14 and 5.15 show the average values of the permittivity and permeability ex-

129

−2

−10

−4

−10.2

−6

−10.4
3.38

3.4

3.42

−8
−10
−12

S

11

Magnitude (dB)

0

−14
−16
−18
−20
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.6 Magnitude of S11 from the uniaxial sample meausured 10 times at orientation 1.
The center blue line is the average value of 10 measurements. The upper and lower lines in
red show the 95% confidence interval.

120
46
100

44

40

60

38
3.18

3.2

3.22 3.24 3.26

40

S

11

Phase (°)

42
80

20
0
−20
−40
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.7 Phase of S11 from the uniaxial sample meausured 10 times at orientation 1. The
center blue line is the average value of 10 measurements. The upper and lower lines in red
show the 95% confidence interval.

130

0
−20.6
−20.8
−5

−21
−21.2

Magnitude (dB)

−10

3.38

3.4

3.42

S

21

−15

−20

−25

−30
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.8 Magnitude of S21 from the uniaxial sample meausured 10 times at orientation 1.
The center blue line is the average value of 10 measurements. The upper and lower lines in
red show the 95% confidence interval.

−160
−169

−162

−169.5

−164

−170
3.34 3.36 3.38

S21 Phase (°)

−166

3.4

3.42

−168
−170
−172
−174
−176
−178
−180
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.9 Phase of S21 from the uniaxial sample meausured 10 times at orientation 1. The
center blue line is the average value of 10 measurements. The upper and lower lines in red
show the 95% confidence interval.

131

0
−9.5

−2

−9.6
−4

−9.7

S11 Magnitude (dB)

−6

−9.8
3.38

3.4

3.42

−8
−10
−12
−14
−16
−18
−20
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.10 Magnitude of S11 from the uniaxial sample meausured 10 times at orientation
2. The center blue line is the average value of 10 measurements. The upper and lower lines
in red show the 95% confidence interval.
120
100

30
28

80

26
24

S11 Phase (°)

60
40

22
3.22 3.24 3.26 3.28

3.3

20
0
−20
−40
−60
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.11 Phase of S11 from the uniaxial sample meausured 10 times at orientation 2. The
center blue line is the average value of 10 measurements. The upper and lower lines in red
show the 95% confidence interval.
132

0
−17.6
−17.8

−5

S21 Magnitude (dB)

−18
−10

3.38

3.4

3.42

−15

−20

−25

−30
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.12 Magnitude of S21 from the uniaxial sample meausured 10 times at orientation
2. The center blue line is the average value of 10 measurements. The upper and lower lines
in red show the 95% confidence interval.
140
86

130

84

120

82
110
S21 Phase (°)

3.34 3.36 3.38

3.4

3.42

100
90
80
70
60
50
40
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.13 Phase of S21 from the uniaxial sample meausured 10 times at orientation 2. The
center blue line is the average value of the 10 measurements. The upper and lower lines in
red show the 95% confidence interval.
133

tracted from 10 sets of measurements of the uniaxial sample using the extraction approach
for biaxial maeterial. The results for A and C are slightly higher than the estimated values,
while B is very close to the predicted value. The extracted values of µr are all close to
unity since the sample is non-magnetic. The smaller insets in the figures show 95% (2 − σ)
confidence intervals for a narrower frequency range. The average value of each parameter is
individually shown in Figures 5.18 - 5.23. The narrow confidence intervals over the entire
band indicate that noticeable variations in the extracted parameters are due to systematic
errors, such as imperfect machining and alignment of the sample layers, or the presence of
glue between sample layers, or air gaps between the sample and the waveguide walls. Also,
data gaps in the frequency range between 3 and 3.15 GHz caused by ill-conditioning of the
extraction process are observed. The data gaps are due to experimental uncertainties that
are amplified near frequencies where the sample is a half-wavelength long. This behavior
is a drawback of using a cubical sample and it is discussed in Section 2.2.5. For these two
figures, extraction within a frequency band approximately ±5% near the half-wavelength
frequency is avoided, and so data within that range is not displayed in the figures. Since the
material parameters do not vary dramatically with frequency, it is possible to approximate
the values within the gap region by interpolation. It is found that a fifth order polynomial
fits the data of extracted parameters well over the entire frequency band. Figures 5.16 and
5.17 show the extracted parameters obtained by fitting a fifth-order polynomial to the data
across the entire band. It can be seen that a good agreement with the estimated values is
achieved.
As is described in Section 2.2 and 2.2.4, the partially-filled waveguide technique has
many advantages, such as having no requirement for a sample holder, the reduced impact of
air gaps, and faster computational time when compared to the reduced aperture waveguide
technique [38]. The material parameters extracted for the same sample using these two
techniques are compared and discussed. The average values of each parameter using the
partially-filled waveguide are individually shown in Figures 5.18 - 5.23 and are compared

134

with the results obtained using a reduced-aperture waveguide. Similar results are seen for
the parameters A , C , µB , and µC . However, for the results of B and µA , the partiallyfiled waveguide technique shows more consistent and reasonable results. The values of B
in Figures 5.19 extracted from a reduced aperture waveguide show more variations, and the
imaginary part of B is positive over a wide frequency range, which contradicts the fact
that the sample is lossy material. Similar behavior is also observed for µA in Figure 5.22.
Note that the half wavelength difficulties also occur with the reduced-aperture waveguide
technique, but due to the different propagation constants of the modes, a different gap range
of 3.55-3.75 GHz is seen. Moreover, for A and µB extracted using a reduced aperture
waveguide, a secondary gap near 2.85 GHz is also observed while no extra data gap is
observed for the partially-filled waveguide technique. To overcome the problem of data gaps,
combining data from both techniques may reduce the impact of half-wavelength difficulties.
This is left as future work.

5.2.3.3

Extracted Parameters of Materials with both Uniaxial Electric and Uniaxial Magnetic Properties

As is described in Section 5.2.3.2, a reasonable initial estimation of the material parameters
for anisotropic sample can expedite the extraction process. Similarly, an estimation for the
uniaxial sample is desired. However, the sample has both uniaxial electric and uniaxial magnetic properties, and no formulas were found to approximate the permeability of the sample.
However, formulas (5.3) can still be used to estimate the permittivity of the sample. The
electric properties of Eccosorb FGM-125 can be obtained from the manufacturer specification sheet. However, the data sheet published by the manufacturer only shows the material
parameters across a wide frequency range of 1-18 GHz with few data points. Thus, more
permittivity and permeability data for the FGM-125 in the S-band is needed to estimate the
sample parameters.
The material parameters of the FGM-125 were extracted at S-band using the Nicolson-

135

Figure 5.14 Relative permittivities (mean values) of the biaxial sample extracted using 10
measurement sets. Inset shows 2 − σ confidence interval.

Figure 5.15 Relative permeabilities (mean values) of the biaxial sample extracted using 10
measurement sets. Inset shows 2 − σ confidence interval.

136

Figure 5.16 Extracted relative permittivities of the biaxial sample fitted to a fifth-order
polynomial.

Figure 5.17 Extracted relative permeabilities of the biaxial sample fitted to a fifth-order
polynomial.

137

Figure 5.18 Comparison of A for the biaxial sample extracted using the partially-filled
waveguide technique and the reduced-aperture waveguide technique.

Figure 5.19 Comparison of B for the biaxial sample extracted using the partially-filled
waveguide technique and the reduced-aperture waveguide technique.

138

Figure 5.20 Comparison of C for the biaxial sample extracted using the partially-filled waveguide technique and the reduced-aperture waveguide technique.

Figure 5.21 Comparison of µA for the biaxial sample extracted using the partially-filled
waveguide technique and the reduced-aperture waveguide technique.

139

Figure 5.22 Comparison of µB for the biaxial sample extracted using the partially-filled
waveguide technique and the reduced-aperture waveguide technique.

Figure 5.23 Comparison of µC for the biaxial sample extracted using the partially-filled
waveguide technique and the reduced-aperture waveguide technique.

140

Ross-Wier closed-form method [28] since this technique is well developed and it is easy to
implement both theoretically and experimentally. A piece of FGM-125 having the same size
as that used in the S-band waveguide aperture was cut and backed by a foam block to keep
it vertically straight in the waveguide extension. The measurement repeatability error was
assessed by measuring the FGM-125 sample 5 times, with the VNA calibrated using the
S-band waveguide at the start of each set of measurements. A cross-section view of the
waveguide completely filled with FGM-125 is shown in 5.24. The permittivity extracted
from the 5 measurements is shown in Figure 5.25, while the permeability is shown in Figure
5.26. The center lines in these figures show the average value of the extracted values while
the upper and lower solid lines show the 95% confidence levels, or ±2 standard deviations.
It can be seen that the confidence intervals for these parameters are wide and increase at
higher frequencies. This might be due to the inconsistent placement of the FGM-125 sheet
in the waveguide. It is also observed that the variation of permittivity across the S-band
is not significant and dielectric loss is small while the opposite is true for the permeability.
With the parameters of the FGM-125 at S-band and the parameters of Rogers 5870, the
approximate permittivity of the sample can be determined using (5.3). At 2.6 GHz, the
formulas give B = 4.55 − j0.0326 and A = C = 5.91 − j0.0779.
The measured data from 10 sets of measurement of the second sample were used to
extract the material parameters. Each set contains measurements of the sample given at
two orientations. The same de-embedding procedure described in (5.9) was used to shift the
reference plane from the calibration plane to the sample planes. Figures 5.27 - 5.30 show the
average value of the permittivity and permeability extracted from 10 sets of measurements
of the sample. The results for A and B are lower than the estimated value and this may be
due to the reduced thickness of the FGM-125 during the sample construction or excessive glue
being applied between the layers. The variations in the extracted permittivity are smaller
when compared with Figures 5.14 and 5.15 which are obtained using the three-step approach.
Also, the overall trend of permeability across the S-band is smoother. However, data gaps

141

Figure 5.24 FGM125 completely filled in the waveguide.
10

Real

Relative Permitivity

8

6

4

2

Imaginary
0

−2
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

Figure 5.25 Real and imaginary parts of relative permittivity extracted from 5 sets of measurements of FGM-125. Center line is the average of the extracted parameter. Upper and
lower lines show the 95% confidence intervals.
142

4

3

Real

Relative Permiability

2

1

0

Imaginary

−1

−2

2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

Figure 5.26 Real and imaginary parts of relative permeability extracted from 5 sets of measurements of FGM-125. Center line is the average of the extracted parameter. Upper and
lower lines show the 95% confidence intervals.

143

which are seen in Figures 5.14 and 5.15 still appear in A and µB . The half wavelength
difficulties are experienced when using the three-step approach, but such problems were
not seen in the test using a fictitious material sample (Figures 2.15-2.18). Since these gaps
only appear in A and µB which are obtained after the completed extraction of B and
µA (Section 2.2.4), this suggests the gaps are consequences of amplified propagation of
experimental uncertainties that lead to a failure for the root-solving algorithm to obtain the
parameters. Note that the gap widths vary between each set of measured data, but they all
fall into the frequency ranges of 2.96-3.01 GHz and 3.62-3.7 GHz. Therefore, data in those
two regions are avoided. The overall results suggest this method can tolerate measurement
uncertainties and is useful for extracting parameters from uniaxial samples.

5.3

Characterization of Gyromagnetic Material

This section focuses on the experimental validation of extracting material parameters of a
sample of gyromagnetic material using a partilly-filled waveguide technique. The advantages
of of this technique are that no special sample holder is required and since typically the size
of gyromagnetic sample available on the market is limited, thus this method eliminates the
need for the large sample required by standard waveguide techniques in which a sample
completely fills the rectangular waveguide cross section. Modal analysis has been carried
out to determine the reflection and transmission coefficients of the dominant mode. An
optimization algorithm described in Section 4.4 was employed which minimizes the difference
between the measured and the theoretical reflection and transmission coefficients to extract
the gyromagnetic material parameters and internal magnetic biasing strength.

5.3.1

Experimental System Setup

Since the extraction of all the constitutive parameters of the gyromagnetic material is based
on an optimization process which minimizes the difference between the measured and the

144

5.8
5.6

εrA

Relative Permittivity ( εr )

5.4
5.2
5
4.8
4.6
4.4

εrB

4.2
4
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 5.27 Relative permittivities (real part) of the uniaxial sample extracted using 10 sets
of measurements. Center line is the average of the extracted parameter. Upper and lower
lines show the 95% confidence intervals.

145

0.3

Relative Permittivity ( εi )

0.2
0.1

ε

iB

0
−0.1
−0.2

ε

iA

−0.3
−0.4
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 5.28 Relative permittivities (imaginary part) of the uniaxial sample extracted using
10 sets of measurements. Center line is the average of the extracted parameter. Upper and
lower lines show the 95% confidence intervals.

146

2

µ

rB

Relative Permeability ( µr )

1.8

1.6

µrA

1.4

1.2

1
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 5.29 Relative permeabilityies (real part) of the uniaxial sample extracted using 10
sets of measurements. Center line is the average of the extracted parameter. Upper and
lower lines show the 95% confidence intervals.

147

−0.5

µiB

Relative Permeability ( µi )

−0.6
−0.7
−0.8
−0.9
−1
−1.1

µiA

−1.2
−1.3
−1.4
−1.5
2.6

2.8

3

3.2
3.4
Frequency(GHz)

3.6

3.8

Figure 5.30 Relative permeabilityies (imaginary part) of the uniaxial sample extracted using
10 sets of measurements. Center line is the average of the extracted parameter. Upper and
lower lines show the 95% confidence intervals.

148

theoretical reflection and transmission coefficients, the measurement system should be constructed in a way that best matches the theoretical model under which the S-parameters are
predicted. The measurement system consists of waveguide ports and waveguide extensions,
and a wooden fixture containing two permanent magnets which are used to produce a consistent but not quiet uniform magnetic biasing field in the area where the sample is located.
Different magnetic biasing fields can be obtained by separating the two magnets at certain
distance. Multiple S-parameters were taken for the extraction under different magnetic biasing strengths. Details of the experiments including sample preparation and system set-up
are presented and discussed.

5.3.1.1

Gyromagnetic Material Sample Specifications

As is shown in Section 3.1.1, the material parameters of gyromagnetic material can be
described by tensors

 r

= 0
 0

0


0
r
0

0 

0 


r

(5.5)

and




 µg 0 −jδ 



µ = µ0 
 0 1 0 ,


jδ 0 µg

(5.6)

where the permeability tensor is formulated when a material is biased along the height of
the waveguide or in the y-direction. The variables r , δ, and µg are complex numbers:
r = r + j r , δ = δ + jδ and µg = µg + jµg . Here the off-diagonal parameter δ and µg
are given by

149

δ=

f fm
f 2 − f02

(5.7)

and

µg =

1−

f0 fm
f 2 − f02

,

(5.8)

In (5.7) and (5.8), f is the operating frequency, fm = 2.8 × 106 × 4πMs , and f0 = 2.8 ×
H
106 × H0 + j 2 . Here H0 is the strength of the internal static biasing magnetic field
in Oersted and Ms the saturation magnetization typically expressed as 4πMs in gauss. The
magnetic losses inherent in the magnetic materials are also taken account by the linewidth
H. The gyromagnetic material empolyed in the measurements is a sample of G1010, a
commercial material manufactured by Trans-Tech, Inc [55]. The typical range of the material
parameters of the G1010 sample are shown in the sample specification sheet. The dielectric
constant r is within the range of 13.97-15.44, the dielectric loss tangent tan(δe ) is within
the range of 0 − 0.0002, the linewidth is within the range of 0-30 Oe and 4πMs is within
the range of 950 − 1050 G. Note that the manufacture also provided the tested material
parameters of the corresponding purchased sample, in which, r is listed as 14.24, the loss
tangent tan(δe ) is 0.0005, the linewidth

H is 19 Oe, and 4πMs is 989 G.

Notice that both dielectric and magnetic losses of the G1010 material are small, and since
it is seen in Figure 4.14-4.16 that good agreement between the theoretical S-parameters generated from a lossless G1010 sample and measured data, it is assumed that a good approximation of the measured S-parameters can be obtained by only incorporating parameters
of r , 4πMs . The extraction of the full properties of the G1010 sample will be carried
out in the future and is discussed in the future work section. Furthermore, the extraction
of H0 as a third parameter is also conducted along with the two other parameters. The
extracted values of the physical parameters are compared with the values provided by the
150

manufacture to evaluate the performance of the proposed method, and the extracted internal
magnetic biasing field strength H0 is also compared with the average value obtained using
a gaussmeter.
The G1010 sample is a hard ceramic material which cannot be machined by regular CNC
milling machine. Therefore, the G1010 sample was purchased from Island Ceramic Grinding,
Inc [56] where it was ground into the desired dimensions to fit into the S-band waveguide.
The final dimension of the purchased sample is 34.04 mm by 34.04 mm with a thickness of
5 mm. It is very important to mention that this material is very brittle, and thus it should
be handled with care to protect the corners from chipping so that multiple measurements
can be taken from a sample with the same dimensions. Figure 5.31 shows the final machined
G1010 sample.

Figure 5.31 Machined G1010 sample.

151

5.3.1.2

Measured Magnetic Biasing Fields

As is mentioned in Section 3.1.1, when the external static magnetic field is parallel to a
thin layer of ferrite sample, the internal magnetic field H0 is nearly equal to the external
magnetic biasing field Ha . In this work, the applied magnetic field is parallel to the broad
face of the gyromagnetic sample, and that results in the magnetic field at the surface of
the sample being equal to the internal magnetic field. Typically, an external biasing source
offers an evenly distributed magnetic field which can be obtained by either a proper solenoid
or certain permanent magnets. Permanent magnets are employed here because they are
inexpensive and easy to implement. However, few magnets are available on market that
can simultaneously meet the need for desired field strength and a large size which have a
relatively more even field at the center. In the end, magnets with high field strength and
also having a proper size for biasing the sample were purchased from K&J Magnets [?]. Two
magnets were purchased which are 4 inches × 1 inch × 1 inch sized neodymium blocks with
magnetization direction through the thickness (1 inch side). A large square magnet would be
preferred since a more even magnetic field can be obtained near the center, but the cost will
increase tremendously and it could be very dangerous for the experimentalist to use larger
magnets since the pulling force may extend to 500 lb. To prevent possible accidents during
the experiments and also to create a consistent configuration of the magnetic biasing field,
a wooden fixture was built to set the two magnets apart at a desired distance and allow the
fixture to be attached to the outside of the waveguide extension. A photo of the fixture is
shown in Figure 5.32.
Since the sample is measured inside the waveguide with the magnet fixture attached, it is
crucial to understand what the actual external biasing magnetic field for the G1010 sample
with the presence of the waveuguide is. A Bell 5070 series guassmeter [58] with a 4 inches
probe was used to to measure the magnetic distribution in the cross-section of the waveguide
where the sample was placed at. The probe of the gaussmeter was placed at different location
in the waveguide and the magnetic field at that point along the y direction (perpendicular to
152

the broad size of the waveguide) was measured. A total of 84 points were measured for the
entire cross-section of the waveuide, and that procedure was repeated three times to obtain
the average field distribution inside the waveguide. Figure 5.33 shows the magnetic field
strengths inside the waveguide; the average of all measured value is 3346.2 G. It is noticed
that a stronger field can be obtained near the surface of the magnets and the magnetic field
is strongest near the center of the magnets. Also, as the probe moves away from the surface
of the magnet, the measured magnetic field decreases until the probe reaches the center point
between the two magnets.
Two pairs of spacers made of rigid polystyrene foam with thicknesses of 5 mm and
9.7 mm were inserted between the waveguide walls and the fixture to increase the separation
distance of the magnets in order to obtain a weaker field strength. Figures 5.34 and 5.35
show the magnetic field strengths under these two separations, with average measured fields
of 1978.4 and 1647.3 G, respectively. It is obvious that the average magnetic field decreases
dramatically with the insertions of thin spacers, and the variations of the field within the
guide follow the same trend as when no spacer was added.

5.3.1.3

Experiment Set-up

To prevent the sample from moving in the waveguide during the measurement procedure,
sample holders made of polystyrene foam was built to hold the G1010 sample at various
positions of the waveguide. The geometry of the holders were designed to facilitate the placement and the removal of the sample. Figure 5.36 shows one waveguide extension containing
one sample holder with the magnet fixture attached. Measurements of the S-parameters
of the G1010 sample with different biasing field strengths were taken using the Agilent
E5071C VNA. The calibration procedure using the S-band waveguide and VNA settings are
described in Section 5.2.2. The VNA was calibrated 10 times using the Thru-Reflect-Line
(TRL) method and three measurements with different magnetic biasing strengths were taken
under each calibration. In the end, a total of 30 measurements were obtained for the case

153

Figure 5.32 Wooden fixture containing two magnets.
where the sample was located at the center of the waveguide.
Like the measurements of cubical uniaxial samples shown in Section 5.2.2, the G1010
sample was inserted into the waveguide extension attached to port 1; the placement of the
sample is shown in Figure 5.37. The S-parameters of the sample were measured by the VNA
at the calibration plane. However, the extraction method requires using S-parameters referenced to the sample planes z=d+∆ and z=d. Therefore, the measured S-parameters have to
be mathematically de-embedded to the sample planes. The S-parameter can be shifted from
the calibration plane to the sample planes by multiplying by e−jβD in the direction of the
wave propagation. Here, β is the propagation constant of the empty waveguide extension,
and D is the distance of an appropriately chosen shift, depending on whether S11 or S21
is corrected. For the measurement of the sample shown in Figure 5.31, the reference plane

154

Y-Positions (mm)
5.6 11.3 16.9 22.6 28.3 34
0

6.5 13

19.5 26 32.5 39 45.5 52 58.5 65 72
X-Positions (mm)

Y-Positions (mm)
5.6 11.3 16.9 22.6 28.3 34

Figure 5.33 Magnetic field strength in the cross-section of the waveguide when no spacer is
inserted.

0

6.5 13

19.5 26 32.5 39 45.5 52 58.5 65 72
X-Positions (mm)

Figure 5.34 Magnetic field strength in the cross-section of the waveguide when the 5 mm
spacers is inserted.

155

Y-Positions (mm)
5.6 11.3 16.9 22.6 28.3 34
0

6.5 13

19.5 26 32.5 39 45.5 52 58.5 65 72
X-Positions (mm)

Figure 5.35 Magnetic field strength in the cross-section of the waveguide when the 9.7 mm
spacers is inserted.
shift can be achieved using the following expressions:
s = S c e−2jβ(d+∆)
S21
21
s = S c e−jβd ,
S11
11

(5.9)
(5.10)

s and S s are the S-parameters referenced to the sample planes while S c and S c
where S11
21
11
21
are the measured S-parameters referenced to the calibration plane, and d is the thickness
of the sample which is 5 mm and ∆ is the inserted distance of the sample which is 45 mm
through out the measurements.

5.3.2

Experiment Results

Ten sets of measurements for three different biasing field strengths were conducted and the
S-parameters were mathematically de-embedded to the desired sample planes. Measurement
repeatability was evaluated and the processed data was used to extract the three frequency
156

Figure 5.36 Polystyrene foam sample holder inside the waveguide with the magnets fixture
attached.
independent parameters.

5.3.2.1

Repeatability Test

As is shown in Section 5.2.2.3, a repeatability test can demonstrate if the designed experimental setup is able to produce consistent measurements. A statistical analysis of the variance
of S11 and S21 of the G1010 sample was processed to evaluate the repeatability of the experiments. Figures 5.38 - 5.41 show the average S-parameters and the standard deviation
generated from 10 repeated measurements of the G1010 sample when it was placed at the
center of the waveguide with no spacer added to the magnet fixture (highest magnetic field
strength). Note that the two standard deviations of the magnitude of both S11 and S21 are
less than 0.1 dB at frequencies away from the resonance frequencies, which indicates consistent measured data near the non-resonance frequencies. However, for the S-parameters near

157

Calibration
Plane: Z=0

G1010
Sample

Port 2

Port 1
Δ

𝑑

Waveguide Extension 1

Waveguide Extension 2

Figure 5.37 G1010 sample inserted into the waveguide extension attached to port 1.
the resonance frequencies, large confidence intervals are observed, which represents poor repeatability of the measured S-parameters and high sensitivity of the S-parameters near those
frequencies.
It can be seen that three a secondary dip near the main resonance frequency. This
phenomenon was not seen in the first two sets of measurements and it became more noticeable
as chips at the edge of the sample began to appear. Therefore, the cause of this unwanted
dip is most likely due to the air gap between the sample and the waveguide wall due to the
chipped edge of the sample. From Figure 5.39 and 5.41, one can see that the phases of S11
and S21 have a large confidence intervals, with two standard deviations of approximately 6
degrees for S11 and 0.5 degrees for S21 . For the phase near the resonance frequencies, larger
confidence intervals are seen which also show the high sensitivity of the measured data near
these frequencies. A dip caused by air gaps is also seen in these two figures. To avoid having
these dips influence the values of the extracted material parameters, the frequencies where
158

those dips occur are avoided during the extraction process. The overall statistical results
of the measured S-parameters indicates acceptable consistency of the experiment set-up.
A foam holder with better rigidness may be required to achieve higher consistency of the
measured data. It is also very important to keep the sample in good physical condition such
that the impact from air gaps can be minimized.
0

−5

S11 Magnitude (dB)

−10

−0.6
−0.7

−15

−0.8
−0.9
3.37

−20

3.38

3.39

3.4

3.41

−25

−30

−35
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.38 Magnitude of S11 from the G1010 sample meausured 10 times with the highest
magnetic biasing field strength. Center blue line is the average value of 10 measurements.
Upper and lower lines in red show the 95% confidence interval (two standard deviations).

5.3.2.2

Extracted Parameters

A Matlab optimization function was used to process the measured data to extract the desired
parameters. The details of the optimization function and its settings and the extraction
procedure are described in Section 4.4. For each measurement, 120 frequency points near
the main resonance frequency were chosen for the optimization. These points were manually
chosen for each measurement to avoid the additional resonance caused by the air gap. To be
consistent, 60 collocated points below the resonance frequency and 60 collocated points above
the resonance frequency were chosen. Tables 5.1-5.3 show the extracted H0 and material
159

400
300

100
0

S

11

Phase (°)

200

−100

130

−200

120
110

−300
−400
2.6

3.22
2.8

3.24

3

3.26
3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.39 Phase of S11 from the G1010 sample meausured 10 times with the highest
magnetic biasing field strength. Center blue line is the average value of 10 measurements.
Upper and lower lines in red show the 95% confidence interval (two standard deviations).
0

−5

S21 Magnitude (dB)

−10

−15

−7.8
−8

−20

−8.2
3.32

−25

3.34

3.36

−30

−35
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.40 Magnitude of S21 from the G1010 sample meausured 10 times with the highest
magnetic biasing field strength. Center blue line is the average value of 10 measurements.
Upper and lower lines in red show the 95% confidence interval (two standard deviations).

160

250
200
150

−78

100

−80

S

21

Phase (°)

−82
50

−84

0

3.34 3.35 3.36 3.37 3.38

−50
−100
−150
−200
−250
2.6

2.8

3

3.2
3.4
Frequency (GHz)

3.6

3.8

4

Figure 5.41 Phase of S21 from the G1010 sample meausured 10 times with the highest
magnetic biasing field strength. Center blue line is the average value of 10 measurements.
Upper and lower lines in red show the 95% confidence interval (two standard deviations).
parameters from 10 sets of measurements. The lower and upper bounds of the parameters
used in the optimization are also included, and these bounds were chosen to be much larger
than that of the range offered by the manufacturer specification sheet in order to evaluate
the performance of this extraction method. The same function tolerance (10−4 ) and variable
tolerance (10−5 ) were used for processing all the measured data. Each extraction process
for a single measurement takes approximately 20 minutes on an Intel Core i7 computer with
8 Gb of RAM.
It can be seen from Table 5.1 for the biasing configuration that offers the highest magnetic
field strength, the variation of extracted values of H0 , r and 4πMs are small. The average
value of H0 from 10 measurements is 3143 Oe which is close to the average measured magnetic
field strength 3346 G. The average value of r is 14.68 while the manufacturer specification is
14.24. The average extracted 4πMs is 968 G which is also very close to 989 G which is given
by the manufacturer. The results for the second biasing configuration are shown in Table
5.2. The average value of the extracted H0 is 1855.5 Oe which is also slightly lower than the
161

measured value of 1978 G. The average extracted r and 4πMs at this biasing configuration
are higher than those found with the highest biasing field. The standard deviation of the
extracted H0 is smaller than the previous case while the standard deviations for r and
4πMs are larger. For the third biasing configuration, which produces the lowest magnetic
field strength, the average extracted H0 is 1436.9 while the average measured field strength is
1647 G. Also, the average r and 4πMs are close to the previous two cases. It is noticed that
the standard deviation of 4πMs is significantly reduced. Considering that the Linewidth ∆
is not included in the mode matching, the extracted results using different measured data
indicate this method is able to extract material parameters that are reasonably close to the
expected value.
A second extraction approach using multiple biasing fields was carried out and the results
are compared with the extracted values found using a single biasing field. The same frequency
points used in the previous extraction were used for this approach, and thus each optimization
involves 360 points. In addition, the number of variables used in this approach is 5 since
three different H0 variables are used in the optimization. Table 5.4 shows the extracted
values using this approach along with the average values. The obtained average values are
compared to the average values using a single biasing field. It can be noticed that the average
values are closer to the average values using the highest biasing field than the other two.
In addition, the standard deviation of the extracted parameters using this approach is less
compared to the results using a single biasing field. This may indicate that this approach is
less sensitive to the change of one measurement since three different measured data are used
simultaneously.

5.4

Summary

This chapter presents experimental procedures and results for characterizing different anisotropic
materials using a partially-filled rectangular waveguide technique. Two types of anisotropic

162

Table 5.1 Extracted H0 and material parameters for measurements of the highest external
magnetic biasing fields.

Initial guess
Lower bound
Upper bound
Measurement 1
Measurement 2
Measurement 3
Measurement 4
Measurement 5
Measurement 6
Measurement 7
Measurement 8
Measurement 9
Measurement 10
Average
Standard Deviation

H0 (Oe)
3180
2600
3800
3201
3063.6
2990
3792.6
2990
2990
2990
3110.2
3122.6
3180
3143
242.2

r
14.7
14.0
15.2
14.7371
14.7921
14.4449
14.6992
14.4169
14.4683
14.4016
14.9589
15.0485
14.7909
14.6758
0.233

4πMs (G)
990
680
1320
1049.5
867.2
1020
1180
1019.9
1020
1019.9
768.4
744.95
990
968
134.4

Table 5.2 Extracted H0 and material parameters for measurements of the second highest
external magnetic biasing fields.

Initial guess
Lower bound
Upper bound
Measurement 1
Measurement 2
Measurement 3
Measurement 4
Measurement 5
Measurement 6
Measurement 7
Measurement 8
Measurement 9
Measurement 10
Average
Standard Deviation

H0 (Oe)
1900
1600
2400
1889.7
1903.6
1879.7
1932.7
1853.9
1831.1
1759.3
1904.3
1704.2
1896.4
1855.5
72.1

163

r
14.7
14.0
15.2
14.9862
15.1758
14.2979
15.1469
14.8553
14.6690
14.2584
15.0163
15.1883
15.1241
14.8718
0.352

4πMs (G)
1020
680
1320
1000.3
1016.6
1317.7
992.8
1062.6
1079.9
1157.7
1016.3
715.5
1023.3
1038.3
150.4

Table 5.3 Extracted H0 and material parameters for measurements of the lowest external
magnetic biasing fields.

Initial guess
Lower bound
Upper bound
Measurement 1
Measurement 2
Measurement 3
Measurement 4
Measurement 5
Measurement 6
Measurement 7
Measurement 8
Measurement 9
Measurement 10
Average
Standard Deviation

H0 (Oe)
1410
1300
1900
1406.4
1410.1
1407.7
1409.5
1544.2
1410
1405.9
1410
1430.1
1535.2
1436.9
54.6

r
14.59
14.0
15.2
14.2254
14.3933
14.1718
14.0098
14.9269
14.3005
14.2385
14.3528
14.5105
15.1612
14.429
0.355

4πMs (G)
1050
800
1200
1051.2
1020
1050.7
1020.2
998.7
1019.9
1051.3
1049.5
1019.9
991.9
1027.3
22.3

Table 5.4 Extracted H0 and material parameters using three biasing fields

Initial guess
Lower bound
Upper bound
Measurement 1
Measurement 2
Measurement 3
Measurement 4
Measurement 5
Measurement 6
Measurement 7
Measurement 8
Measurement 9
Measurement 10
Average
Standard Deviation

H0 (Oe) 1st
3200
2600
3400
3024.3
3040.9
3028.1
3038.3
3400
3033
3031.9
3359.6
3192.5
3031
3147.9
147.1

H0 (Oe) 2nd
1890
1600
2400
1763.8
1694.1
1744.6
1731.8
1778.1
1745
1740.6
1699.9
1680
1741.9
1732
31.3

164

H0 (Oe) 3rd
1490
1300
1900
1450
1425.7
1449.9
1449.3
1511.2
1446.6
1431.3
1402.4
1397.9
1462.1
1442.6
32.1

r
14.2
14.0
15.2
14.4754
15.0732
14.3955
14.4207
14.6775
14.4419
14.3478
15.1998
15.1935
14.577
14.6802
0.343

4πMs (G)
1020
680
1320
1061.5
796.7
1070.8
1054.5
1026.5
1058.9
1076.6
704.5
683.9
1057
959.1
162.2

materials were measured at S-band to validate the corresponding extraction techniques. The
first type of the characterized material is biaxial. Two cubical samples made of stacked layer
substrates were constructed and measured to evaluate the proposed technique which characterize biaxial and unixial materials. It is found that the technique performs very well
for characterizing uniaxial material and also for the biaxial material at the frequency band
where the electrical length of the sample does not approach a half wavelength. The second
type of the material is gyromagnetic. The extraction procedure of this material is based
on an optimization algorithm in Matlab and it was validated by laboratory measurements
of a gyromagnetic test sample. Extracted material parameters from different experimental configurations or different extraction approaches show good agreement with the values
measured or supplied by the manufacturer.

165

CHAPTER 6
Conclusions and Future Works
In this dissertation, A partially-filled waveguide characterization technique is introduced for
two types of materials: biaxial and gyromagnetic. The extraction of the full tensors of a biaxial sample can be achieved using a single sample that is partially filled in a waveguide. This
method eliminates many drawback of the previous works which were developed to characterize biaxial materials. The theoretical reflection and transmission coefficients are determined
using a closed form solution. The inverse problem is to minimize the difference between
the theoretical reflection and transmission coefficients and the measured S-parameters using
an iterative solver. A simplified characterization approach is described for the extraction
of uniaxial materials. The sensitivity analysis for VNA uncertainties is demonstrated. Two
cubical samples made of stacked layer substrates were constructed and measured to validate
the proposed technique. To overcome the half-wavelength difficulties mentioned in 5.2.3.2,
investigations of new extraction approaches will be carried out in the future, such as combining data produced from reduced aperture waveguide techniques and from the technique
proposed in this work.
A method for characterizing gyromagnetic materials partially filled in a guide is also
described. The extraction procedure of this material is based on optimization algorithms
in Matlab and sensitivity analysis for VNA uncertainties and uncertainties of the sample

166

dimensions demonstrates promising performance of this method. Multiple measured data
with various applied biasing fields was obtained from a gyromagnetic sample. Extracted
material parameters from different experimental configurations or different extraction approaches show good agreement with the values measured or supplied by the manufacturer.
The extraction of the full properties of the gyromagnetic sample will be carried out in the
future along with the extraction from measurements of various sample positions.

167

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