Numerical S-Parameter Extraction and Characterization of Inhomogeneously Filled Waveguides

NUMERICAL S-PARAMETER EXTRACTION ANDCHARACTERIZATION OF INHOMOGENEOUSLY

FILLED WAVEGUIDES

By

Pedro Barba

A DISSERTATION

Submitted toMichigan State University

in partial fulfillment of the requirementsfor the degree of

DOCTOR OF PHILOSOPHY

Electrical and Computer Engineering

2006

ABSTRACT

NUMERICAL S-PARAMETER EXTRACTION AND

CHARACTERIZATION OF INHOMOGENEOUSLY FILLED

WAVEGUIDES

By

Pedro Barba

A numerical tool based on the finite element method (FEM) is developed in order

to assess the parameter uncertainty vulnerability in a novel inversion algorithm to

extract the electromagnetic constitutive parameters from a material sample. This

inversion algorithm relies heavily on the assumption, when having the cross-section

of the testing waveguide partially filled, that the material sample has to be perfectly

centered. In the present work, the effect of having the material sample displaced from

the center is measured by comparing its extracted constitutive parameters (ε, µ) with

the values corresponding to the perfectly centered case. The finite element method

formulation presented here, can also be used to provide the theoretical data (that

otherwise would have to be obtained via traditional mode matching techniques or

the hybrid mode decomposition), required for the inversion algorithms corresponding

to non-regular samples. The results of this work identify some of the cases in which

errors are originated from the sample preparation or from the measurement technique

utilized. This information is used to identify the band of frequencies in which the

error in the inversion algorithm can be minimized. The numerical method is further

extended to investigate the behavior of waveguides loaded with layered as well as

anisotropic materials.

To Edith, my lovely girlfriend

iii

ACKNOWLEDGMENTS

I would like to extend my appreciation and gratitude to my academic advisor, Dr.

Leo Kempel, for providing me with the opportunity to work under his guidance and

making me part of his research team. To Dr. Shanker Balasubramaniam, for helping

me write my computer codes faster and more efficiently. To Dr. Edward Rothwell,

for always having the time and willingness to answer all my questions. Also, Dr.

Gregory Kobidze, for his friendship and help during our time at the Computational

Electromagnetics Lab at MSU.

A very special Thank You to Dr. Barbara O’Kelly and Dr. Percy Pierre for

providing me with the opportunity to come to this wonderful institution where I have

spent the happiest years of my life.

My eternal gratitude for my parents Irma and Sergio, for always being there

unconditionally for me. My grandfather, Isauro Medina Hinojos, for always being an

inspiration in all my endeavors. To Edith, my lovely girlfriend, for spending all this

time with me and be willing to work with me no matter a what time, no matter for

how long. I love you!

This work was partially supported by the National Science Foundation under grant

ECS-0134236 and the Air Force Office of Scientific Research under grant FA9550-06-

1-0023. I would also like to gratefully acknowledge the Michigan State University

High Performance Computing Center (HPCC) for providing computational resources

for this project.

iv

“Wenn die Tugend geschlafen hat, wird sie frischer aufstehen.”

Menschliches, Allzumenschliches

Friedrich Nietzsche

v

TABLE OF CONTENTS

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

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

KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . xiii

CHAPTER 1Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 2Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 The Thru-Reflect-Line Calibration Technique . . . . . . . . . . . . . 32.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Scattering and Transmission Parameters . . . . . . . . . . . . 32.1.3 Derivation of Equations . . . . . . . . . . . . . . . . . . . . . 5

2.1.3.1 The Thru Measurement . . . . . . . . . . . . . . . . 72.1.3.2 The Line Measurement . . . . . . . . . . . . . . . . 82.1.3.3 The Reflect Measurement . . . . . . . . . . . . . . . 112.1.3.4 Postprocessing of Measured Standards . . . . . . . . 13

2.2 Derivation of the Reflection and Transmission Coefficients for a Fully-Filled Rectangular Waveguide. TE10 Mode . . . . . . . . . . . . . . 18

2.3 The Inversion Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Hybrid Modes and the Transverse Resonance Method . . . . . . . . . 322.4.1 Hybrid Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.2 The Transverse Resonance Method . . . . . . . . . . . . . . . 392.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

CHAPTER 3The Finite Element Method Formulation for Inhomogeneous Waveguides . . . 41

3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1.1 Domain Discretization . . . . . . . . . . . . . . . . . . . . . . 413.1.2 Interpolation Basis Functions . . . . . . . . . . . . . . . . . . 423.1.3 Formulation of the System of Equations Using The Ritz Method 50

3.1.3.1 Solution of Integrals . . . . . . . . . . . . . . . . . . 563.1.4 Solution of the System of Equations . . . . . . . . . . . . . . . 703.1.5 Numerical S-Parameter Computation . . . . . . . . . . . . . . 71

3.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

vi

CHAPTER 4Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 Error Generated by Cross-Sections Shifted from Center . . . . . . . . 794.1.1 Low-Contrast Material . . . . . . . . . . . . . . . . . . . . . . 814.1.2 High-Contrast Material . . . . . . . . . . . . . . . . . . . . . . 814.1.3 Magneto-Dielectric Materials . . . . . . . . . . . . . . . . . . 82

4.2 Layered Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2.1 Perpendicular in the Direction of Propagation . . . . . . . . . 834.2.2 Parallel to the Direction of Propagation. Horizontal and Verti-

cal Layering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 Anisotropic Formulation: A Ferrite . . . . . . . . . . . . . . . . . . . 85

CHAPTER 5Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

vii

LIST OF TABLES

Table 3.1 Definition for each volume-function ςei within each tetrahedron. . 46

Table 3.2 Definition for each edge on a tetrahedral element. . . . . . . . . . 47

Table 3.3 Definition for each edge on a triangular element with its constitu-tive nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Table 3.4 Definition for each area-function ψti within each triangular element. 63

Table 3.5 Parameters for the four-point triangular surface Gaussian integra-tion rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Table 4.1 Dimensions for a rectangular waveguide for the frequency bandsused to conduct the numerical simulations and inversion operations[27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

viii

LIST OF FIGURES

Figure 2.1 A two-port linear network with input and output signals. . . . . . 4

Figure 2.2 A two-port network with connectors. . . . . . . . . . . . . . . . . 6

Figure 2.3 The Thru standard connection. . . . . . . . . . . . . . . . . . . . 7

Figure 2.4 The Line standard connection. . . . . . . . . . . . . . . . . . . . 8

Figure 2.5 The Reflect standard connection. . . . . . . . . . . . . . . . . . . 12

Figure 2.6 Signal flow graph for the Reflect standard. . . . . . . . . . . . . . 13

Figure 2.7 Rectangular homogeneous source-free waveguide. . . . . . . . . . 18

Figure 2.8 Sideview of a fully-filled cross-section waveguide propagating theTE10 mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Figure 2.9 Extracted relative permittivity and permeability for an acrylic sam-ple using the algorithm in [10] . . . . . . . . . . . . . . . . . . . . 30

Figure 2.10 Extracted relative permittivity and permeability for an aluminasample using the algorithm in [10] . . . . . . . . . . . . . . . . . . 31

Figure 2.11 Vertically loaded waveguide to illustrate the LSE and LSM modedecomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 2.12 Vertically loaded waveguide. . . . . . . . . . . . . . . . . . . . . . 36

Figure 2.13 Horizontally loaded waveguide. . . . . . . . . . . . . . . . . . . . 37

Figure 2.14 Suspended sample rod loaded waveguide. . . . . . . . . . . . . . . 38

Figure 3.1 Tetrahedron element for waveguide mesh discretization. . . . . . . 41

Figure 3.2 Mesh for the waveguide cross-section with material sample inside. 42

Figure 3.3 Tetrahedron element showing its vertices and the interior point p. 45

Figure 3.4 Definition for a tetrahedron. Showing its nodes, edges and edgedirections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 3.5 Rectangular waveguide with obstacle . . . . . . . . . . . . . . . . 49

Figure 3.6 2-D Element for Ψt. . . . . . . . . . . . . . . . . . . . . . . . . . 60

Figure 3.7 Comparison between the FEM and theoretical closed-form solutionin a fully-filled d/a = 1 rectangular waveguide. Acrylic (ε = 2.5,µ = 1), sample length ` = 5 mm. . . . . . . . . . . . . . . . . . . 75

Figure 3.8 Comparison between the FEM and mode-matching solution in apartially-filled d/a = 0.5 rectangular waveguide. Acrylic (ε = 2.5,µ = 1), sample length ` = 7.5 mm. . . . . . . . . . . . . . . . . . 76

Figure 3.9 Comparison between the FEM and mode-matching solution in apartially-filled d/a = 0.25 rectangular waveguide. Acrylic (ε = 2.5,µ = 1), sample length ` = 7.5 mm. . . . . . . . . . . . . . . . . . 77

ix

Figure 3.10 Comparison between the FEM and theoretical closed-form solutionin a fully-filled d/a = 1 rectangular waveguide. Alumina (ε =9.0− j0.0027, µ = 1), sample length ` = 3 mm. . . . . . . . . . . 78

Figure 4.1 Vertically loaded waveguide with material sample shifted from thecenter by a distance δ. . . . . . . . . . . . . . . . . . . . . . . . . 80

Figure 4.2 Comparison of the transmission and reflection coefficients (magni-tude) for acrylic when the parameter δ is varied from 0 to 5 mmin ten steps. (ε = 2.5,µ = 1). . . . . . . . . . . . . . . . . . . . . 87

Figure 4.3 Comparison of the transmission and reflection coefficients (phase)for acrylic when the parameter δ is varied from 0 to 5 mm in tensteps. (ε = 2.5,µ = 1). . . . . . . . . . . . . . . . . . . . . . . . . 88

Figure 4.4 Percent error on the S-parameter magnitude resulting from shiftingthe center of the acrylic material sample from δ = 0 to δ = 5 mm. 89

Figure 4.5 Percent error on the S-parameter phase resulting from shifting thecenter of the acrylic material sample from δ = 0 to δ = 5 mm. . . 90

Figure 4.6 (a)Extracted relative permittivity ε for an acrylic sample when theparameter δ is increased from δ = 0 to δ = 5 mm, (b) error. . . . 91

Figure 4.7 Comparison of the transmission and reflection coefficients (mag-nitude) for alumina when the parameter δ is varied from 0 to 5mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Figure 4.8 Comparison of the transmission and reflection coefficients (phase)for alumina when the parameter δ is varied from 0 to 5 mm. . . . 93

Figure 4.9 Percent error on the S-parameter magnitude resulting from shiftingthe center of the alumina material sample from δ = 0 to δ = 5 mm. 94

Figure 4.10 Percent error on the S-parameter phase resulting from shifting thecenter of the alumina material sample from δ = 0 to δ = 5 mm. . 95

Figure 4.11 (a) Extracted relative permittivity ε for an alumina sample whenthe parameter δ is increased from δ = 0 to δ = 3.5 mm, (b) error. 96

Figure 4.12 S-parameters for a lossy-magneto-dielectric material (magRAM)sample (a)magnitude and (b) phase. . . . . . . . . . . . . . . . . 97

Figure 4.13 Waveguide with a layered material in the direction of propagationof the incident field. . . . . . . . . . . . . . . . . . . . . . . . . . 98

Figure 4.14 S-Parameters (magnitude) for a perpendicularly layered material.Material A: (ε = 9− j0.0027, µ = 1), Material B: (ε = 1, µ = 1). . 99

Figure 4.15 S-Parameters (phase) for a perpendicularly layered material. Ma-terial A: (ε = 9− j0.0027, µ = 1), Material B: (ε = 1, µ = 1). . . 100

x

Figure 4.16 Extracted relative permittivity for a layered material perpendicularin the direction of propagation. Material A: (ε = 9 − j0.0027,µ = 1), Material B: (ε = 1, µ = 1). . . . . . . . . . . . . . . . . . 101

Figure 4.17 S-Parameters (magnitude) for a perpendicularly layered material.Material A: (ε = 1, µ = 1), Material B: (ε = 9− j0.0027, µ = 1). . 102

Figure 4.18 S-Parameters (phase) for a perpendicularly layered material. Ma-terial A: (ε = 1, µ = 1), Material B: (ε = 9− j0.0027, µ = 1). . . 103

Figure 4.19 Extracted relative permittivity for a layered material perpendicu-lar in the direction of propagation. Material A: (ε = 1, µ = 1),Material B: (ε = 9− j0.0027, µ = 1). . . . . . . . . . . . . . . . . 104

Figure 4.20 Extracted relative permittivities for a layered material perpendicu-lar to the direction of propagation and the asymptotic permittivityfor a homogenized material. The plot on top shows the result whenMaterial A has a higher permittivity, the plot on the bottom Ma-terial A with a lower permittivity: (ε = 1, µ = 1), (ε = 9−j0.0027,µ = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 4.21 Waveguide with a layered material parallel to the direction of prop-agation. Horizontal layering. . . . . . . . . . . . . . . . . . . . . . 106

Figure 4.22 Waveguide with a layered material parallel to the direction of prop-agation. Vertical layering. . . . . . . . . . . . . . . . . . . . . . . 107

Figure 4.23 S-Parameters (magnitude) for a horizontally layered material. Ma-terial A: (ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1). . 108

Figure 4.24 S-Parameters (phase) for a horizontally layered material. MaterialA: (ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1). . . . . . 109

Figure 4.25 Extracted relative permittivity for a layered material parallel tothe direction of propagation. Horizontal layering. Material A:(ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1). . . . . . . 110

Figure 4.26 S-Parameters (magnitude) for a horizontally layered material. Ma-terial A: (ε = 2.5, µ = 1), Material B: (ε = 9− j0.0027, µ = 1). . 111

Figure 4.27 S-Parameters (phase) for a horizontally layered material. MaterialA: (ε = 2.5, µ = 1), Material B: (ε = 9− j0.0027, µ = 1). . . . . . 112

Figure 4.28 Extracted relative permittivity for a layered material parallel to thedirection of propagation. Horizontal layering. Material A:(ε = 2.5,µ = 1), Material B:(ε = 9− j0.0027, µ = 1). . . . . . . . . . . . . 113

Figure 4.29 S-Parameters (magnitude) for a vertically layered material. Mate-rial A: (ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1). . . . 114

Figure 4.30 S-Parameters (phase) for a vertically layered material. Material A:(ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1). . . . . . . 115

xi

Figure 4.31 Extracted relative permittivity for a layered material parallel tothe direction of propagation. Vertical layering. Material A: (ε =9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1). . . . . . . . . . 116

Figure 4.32 S-Parameters (magnitude) for a vertically layered material. Mate-rial A: (ε = 2.5, µ = 1), Material B: (ε = 9− j0.0027, µ = 1) . . . 117

Figure 4.33 S-Parameters (phase) for a vertically layered material. Material A:(ε = 2.5, µ = 1), Material B: (ε = 9− j0.0027, µ = 1). . . . . . . 118

Figure 4.34 Extracted relative permittivity for a layered material parallel to thedirection of propagation. Vertical layering. Material A: (ε = 2.5,µ = 1), Material B: (ε = 9− j0.0027, µ = 1). . . . . . . . . . . . . 119

Figure 4.35 Transmission and Reflection Coefficients (magnitude) for a mag-netized ferrite. 4πMs = 5000 Gauss, ∆H = 500 Oe, Ha = 200 Oe,Ho = 100 Oe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120




xii

KEY TO SYMBOLS AND ABBREVIATIONS

FEM: Finite Element Method

LSE: Longitudinal Section Electric

LSM: Longitudinal Section Magnetic

MagRAM: Magnetic Radar Absorbing Material

MUT: Material Under Test

NRW: Nicolson-Ross-Weir

TE: Transverse Electric

TM: Transverse Magnetic

TRL: Thru-Reflect-Line

TRM: Transverse Resonance Method

VNA: Vector Network Analizer

xiii

CHAPTER 1

INTRODUCTION AND BACKGROUND

Typically, a material may be described by its bulk electromagnetic constitutive param-

eters: electric permittivity ε and magnetic permeability µ. In general these quantities

are complex valued (in a time-harmonic scheme) and can either be scalar or tensor

functions. The extraction of these parameters by an indirect measurement such as

the amount of electromagnetic energy that they reflect and transmit, is known as

electromagnetic material characterization.

Traditional methods for material characterization often use rectangular metallic

waveguides because of the simplicity of the geometry to produce suitable material

samples and the nearly universality of these components available in microwave labo-

ratories. The mathematical models that describe the behavior of the electromagnetic

fields in a rectangular waveguide are also simpler that those for many other geome-

tries.

One of the most popular methods to characterize materials is the Nicholson-Ross-

Wier (NRW) technique [2, 3]. The biggest advantage of this technique is that once

the scattering parameters of the material sample are known (from experimentation),

the permittivity and permeability for the test sample are then provided in closed

form. Certain conditions, however, must be met by the material sample to be char-

acterized properly. One of these conditions is that the sample has to fill the entire

cross-section of the waveguide. If the material is lossy or highly reflective, a poor

transmission coefficient will be obtained, yielding to poor results in the extracted

constitutive parameters. Other conditions for the material sample is that it must

be linear, homogeneous and isotropic. Also, the geometry of the sample must have

parallel front and rear faces, perpendicular to the waveguide walls [7].

1

An alternative method uses a two-dimensional root-search algorithm in lieu of the

NRW approach. This alternative method, for example, was used in [10] for solving

the case of a partially filled waveguide. When implementing this inversion method,

a word of caution is in order. The material sample being tested has to be perfectly

centered in the waveguide cross-section. The objective of this practice is to simplify

the mathematical analysis by exploiting the symmetry of the problem.

In this thesis a simulation tool is developed to assess the robustness of the inversion

algorithm mentioned above when perturbations are present on the experimental setup,

specially when the test sample is misplaced inside the waveguide (e. g. laterally

shifted from the cross-section center.) since this violates a major assumption during

the inversion algorithm. Other perturbations to the experimental setup, that need to

be numerically simulated, include the frontal and rear faces of the sample not being

parallel to each other or not being perpendicular to the waveguide walls. Chipping

of the material sample during its manufacturing process or during handling in the

laboratory could also be a contribution for error when extracting its constitutive

parameters.

It will be shown in the present work how different methods, mainly the

longitudinal-section electric (LSE) and the longitudinal-section magnetic (LSM) [11]

decompositions or the transverse resonance method (TRM) [22] fail to be practical in

modeling this inhomogeneous waveguide problem, especially when the material under

test consists of an increasingly number of layers or the material exhibits anisotropic

properties.

Since the finite element method (FEM) treats each element as a homogeneous

entity and the discretization of the computational domain can, for all intents and

purposes, conform to any shape, the finite element method is a powerful tool for

uncertainty analysis.

2

CHAPTER 2

PRELIMINARY WORK

2.1 The Thru-Reflect-Line Calibration Technique

2.1.1 Introduction

Errors resulting from imperfections of a measurement system can be classified as

either random or systematic. Systematic errors, like the ones resulting from the use

of equipment not being properly calibrated, are the repeatable errors that can be

measured and then mathematically removed from the measurement via calibration.

The Thru-Reflect-Line (TRL) calibration technique was first introduced in 1979

by Engen and Hoer [23]. Consider Figure 2.2 in which a two-port network is formed

with connectors A and B and a waveguide segment labeled here as “MUT” (mate-

rial under test). Then, the TRL calibration technique effectively removes the error

introduced into the measurement by connectors A and B when measuring the S-

parameters of the network. Also, at the end of the calibration, the reference planes

are at the boundaries of the MUT as shown on Figure 2.2, rather than the VNA

ports. The technique requires the measurement of three standards in addition to the

“total” measurement, which comprises of the S-parameters of the connectors A and

B altogether with the MUT. These standards are: 1) the Thru, corresponding to the

S-parameter measurement of a zero-length (or thru) connection between connector

A and connector B; 2) the Reflect, consisting of the S11 and S22 measurement of a

highly reflective one-port device, Γ; and 3) the Line which is the measurement of the

S-parameters of an empty transmission line of known length.

2.1.2 Scattering and Transmission Parameters

Figure 2.2 shows a two-port linear network. The network can be completely charac-

terized by means of the scattering parameters, which relate inward to outward waves

3

from each port, as shown in Equation (2.1). If the [S] matrix is symmetric it means

that the network is reciprocal. Also, for a lossless network [S] is unitary [27].

Figure 2.1. A two-port linear network with input and output signals.

vout1

vout2

=

S11 S12

S21 S22

vin1

vin2

(2.1)

A physical interpretation to the scattering parameters is to think of them as the

reflection and transmission coefficients of the network. These coefficients are complex

quantities consisting of a magnitude and a phase and are computed as follows:

S11 =vout1

vin1

∣∣∣∣∣vin2 =0

= R = |R| ejθr (2.2)

S21 =vout2

vin1

∣∣∣∣∣vin2 =0

= T = |T | ejθt (2.3)

with similar expressions for S22 and S12.

Another way to characterize this network is by relating the waves, ingoing and

outgoing, at port 1 to those at port 2. This relationship is known as the transmission

4

parameters (T-matrix).

vout1

vin1

=

t11 t12

t21 t22

vin2

vout2

(2.4)

If multiple networks are connected in series, it is possible to obtain one equivalent

transmission matrix for the whole array by multiplying all matrices in the same order

as their network position in the array [25]. By using the theory described in [34,

pp.539-541], two useful relationships to convert T-parameters to S-parameters and

vice versa are obtained:

[T] =

t11 t12

t21 t22

=

1

S21

S12S21 − S11S22 S11

−S22 1

(2.5)

[S] =

S11 S12

S21 S22

=

1

t22

t12 t11t22 − t12t22

1 −t21

(2.6)

2.1.3 Derivation of Equations

The derivation presented here follows that of Matthews and Song [24]. Let a MUT

be connected to a vector network analyzer (VNA) by using the connectors A and B

as shown in Figure 2.2. By first measuring the total S-parameters of the network, as

seen by the VNA, the effects of each network component can be de-embedded from

the total measurement by means of the transmission (T) parameters as follows. The

conversion from S- to T-parameters is done with the help of the formula (2.6).

[Sm] −→ [Tm]

[Tm] = [Ta] · [Tmut] ·[Tb

](2.7)

5

BMUTATo VNATo VNA

ReferencePlane“A” ReferencePlane“B”Figure 2.2. A two-port network with connectors.

where

[Tm] =

tm11 tm12

tm21 tm22

(2.8)

[Ta] =

ta11 ta12

ta21 ta22

(2.9)

[Tmut] =

tmut11 tmut12

tmut21 tmut22

(2.10)

[Tb

]=

tb11 tb12

tb21 tb22

(2.11)

[Tm] is the “measured” transmission matrix as seen by the VNA, [Ta] is the con-

nector “A” transmission matrix, [Tmut] is the “Material Under Test” transmission

matrix and[Tb

]is the connector “B” transmission matrix. The strategy is now to

accurately characterize the connectors A and B so that their contribution can be re-

moved from the measured values of the VNA. Then, the MUT transmission matrix

6

can be expressed as

[Tmut] = [Ta]−1 · [Tm] · [Tb]−1 (2.12)

2.1.3.1 The Thru Measurement

The Thru standard consists on joining both of the connectors A and B as shown in

Figure 2.3 and then measure its S-parameters, [St]. With the aid of Equation (2.6)To VNAA B

To VNAFigure 2.3. The Thru standard connection.

the resulting measured T-matrix for the Thru standard, [Tt], is written as

[St] −→ [Tt]

[Tt] = [Ta] · [Tthru] · [Tb

]= [Ta] · [Tb

](2.13)

where

[Tt] =

tt11 tt12

tt21 tt22

(2.14)

and the theoretical T-matrix is given by

[Tthru

]=

1 0

0 1

(2.15)

7

2.1.3.2 The Line Measurement

The Line standard measurement, shown on Figure 2.4, consists of placing an empty

waveguide section of length ` between connectors A and B. The S-parameters for this

configuration are then measured, obtaining[Sl

]Using Equation (2.6) the conversion

BLineATo VNATo VNA l

ReferencePlane“A” ReferencePlane“B”Figure 2.4. The Line standard connection.

[Sl

] −→ [Tl

]

is performed. The transmission matrix for the Line standard is given by

[Tl

]= [Ta] · [Tline

] · [Tb]

(2.16)

where

[Tl

]=

tl11 tl12

tl21 tl22

(2.17)

8

and its theoretical T-matrix is given by

[Tline

]=

e−γl 0

0 eγl

(2.18)

Using equations (2.13) and (2.16) the following is obtained:

[Tl

]= [Ta] · [Tline

] · [Ta]−1 · [Tt] (2.19)

Now, defining

[Tlt

]=

[Tl

] · [Tt]−1 =

tlt11 tlt12

tlt21 tlt22

(2.20)

with

tlt11 =1

|Tt|(tl11tt22 − tl12tt21

)(2.21)

tlt12 =1


)(2.22)

tlt21 =1


)(2.23)

tlt22 =1


)(2.24)

where |Tt| is the determinant of matrix [Tt]. Then equation (2.19) can be written as

[Tlt

] · [Ta] = [Ta] · [Tline]

(2.25)

or

tlt11 tlt12

tlt21 tlt22

·

ta11 ta12

ta21 ta22

=

ta11 ta12

ta21 ta22

·

e−γl 0

0 eγl

(2.26)

9

Carrying out the products of each side

tlt11ta11 + tlt12ta21 = ta11e−γl (2.27)

tlt11ta12 + tlt12ta22 = ta12eγl (2.28)

tlt21ta11 + tlt22ta21 = ta21e−γl (2.29)

tlt21ta12 + tlt22ta22 = ta22eγl (2.30)

Taking the ratios of the equations above (the ones that share the same sign on the

exponential function) the following set of quadratic equations are obtained:

tlt11

(ta11ta21

)2+

(tlt22 − tlt11

) (ta11ta21

)− tlt12 = 0 (2.31)

tlt21

(ta12ta22

)2+

(tlt22 − tlt11

) (ta12ta22

)− tlt12 = 0 (2.32)

These quadratic equations share the same set of solutions, namely

(ta11ta21

),

(ta12ta22

)=

1

2tlt21

(tlt11 − tlt22 ±

√(tlt22 − tlt11

)2 + 4(tlt21tlt12

) )

(2.33)

The choice of roots will be made later in §2.1.3.4. Now, from Equation (2.13), solving

for[Tb

][Tb

]= [Ta]−1 · [Tt] (2.34)

or alternatively written as

tb11 tb12

tb21 tb22

=

1

|Ta|

ta22 −ta12

−ta21 ta11

·

tt11 tt12

tt21 tt22

(2.35)

10

Carrying out the product, each element of the connector B transmission matrix is

obtained as a function of the thru measurement given in Equation (2.13) and the

transmission parameters of connector A.

tb11 =1

|Ta| (tt11ta22 − tt21ta12)

tb12 =1

|Ta| (tt12ta22 − tt22ta12)

tb21 =1

|Ta| (tt21ta11 − tt11ta21)

tb22 =1

|Ta| (tt22ta11 − tt12ta21) (2.36)

Taking the ratios

tb21tb22

=tt21 − tt11

(ta21ta11

)

tt22 − tt12

(ta21ta11

) (2.37)

tb12tb11

=tt12 − tt22

(ta12ta22

)

tt11 − tt21

(ta12ta22

) (2.38)

the following relationship is found

(ta11ta22

)·(

tb11tb21

)=

tt11 − tt21

(ta12ta22

)

tt22 − tt12

(ta21ta11

) (2.39)

2.1.3.3 The Reflect Measurement

The measurement of the Reflect standard, Figure 2.5, consists of placing a reflectome-

ter of value Γ at the end of either connector A or B and then measure its reflection

coefficient as seen from the VNA. Using the signal flow graph theory presented in [27]

and with the help of Figure 2.6, the reflection coefficient, as seen by the VNA, can be

expressed as a function of the reflection coefficient Γ and the intrinsic S-parameters

11

To VNAA or B ReflectStandardΓ

Figure 2.5. The Reflect standard connection.

of the connector as follows:

Sr11 = Sa11 +Sa12Sa21Γ

1− Sa22Γ

=

ta12ta22

+(ta11

ta22

)Γ

1 +(

ta21ta22

)Γ

(2.40)

The following ratio is obtained

ta11ta22

=1

Γ·

Sr11 −ta12ta22

1− Sr11ta21ta11

(2.41)

Following the same procedure on connector B using the same reflection coefficient Γ,

these expressions are found

Sr22 = Sb22 +Sb12Sb21Γ

1− Sb11Γ(2.42)

=−tb21

tb22+

tb11tb22

Γ

1− tb12tb22

Γ(2.43)

tb11tb22

=1

Γ·

Sr22 +tb21tb22

1 + Sr22tb12tb11

(2.44)

12

taking their product, the necessary expression is obtained

S11

S21

S12

S22 г

Figure 2.6. Signal flow graph for the Reflect standard.

(ta11ta22

)·(

tb22tb11

)=

(Sr11 −

ta12ta22

) (1 + Sr22

tb12tb11

)

(Sr22 +

tb21tb22

) (1− Sr11

ta21ta11

) (2.45)

2.1.3.4 Postprocessing of Measured Standards

The next step is to assign the correct root to Equation (2.33). This will be accom-

plished by using the value of the reflection coefficient Γ from the Reflect standard.

By placing a metallic plate during this measurement, and by recognizing that the

ratio ta12/ta22 is the reflection coefficient as seen by the VNA, then it is clear that

the root with the larger magnitude is assigned to the reflection coefficient, the ratio

ta12/ta22.

All of the three standards have been measured at this point. All what is left now

is to combine the results of these measurements to express the S-parameters of the

MUT from Equation (2.12). Combining Equation(2.39) with Equation (2.45) yields

to

ta11ta22

= ±

(tt11 − tt21

(ta12ta22

))(Sr11 −

ta12ta22

) (1 + Sr22

(tb12tb11

))

(tt22 − tt12

(ta21ta11

)) (Sr22 +

tb21tb22

) (1− Sr11

(ta21ta11

))

12

(2.46)

13

The sign of the ratio (2.46) is chosen in such a way that when using the same value

of Γ during the measurement of the Reflect standard, (2.41) and (2.46) yield to the

same numerical value. Also from Equation (2.39)

tb11tb22

=

tt11 − tt21

(ta12ta22

)

tt22 − tt12

(ta21ta11

)

·

(ta11ta22

)−1(2.47)

and by using the relation (2.5), the parameters Sa11 and Sa22 are constructed.

Sa11 =ta12ta22

(2.48)

Sa22 = −(

ta21ta22

)

= −(

ta21ta11

)·(

ta11ta22

)(2.49)

From Equation (2.6) the product Sa12 · Sa21 is found to be

Sa12 · Sa21 =ta1ata22

·[1−

(ta12ta22

)(ta21ta11

)](2.50)

For connector B, equations (2.5) and (2.36) are used to construct its S11 and S22

parameters as

Sb11 =tb12tb22

=tt12 − tt22 ·

(ta12ta11

)·(

ta11ta22

)

tt22 ·(ta11

ta22

)− tt12 ·

(ta21ta22

) (2.51)

Sb22 = −tt21 ·

(ta11ta22

)− tt11 ·

(ta21ta22

)

tt22 ·(

ta11ta22

)− tt12 ·

(ta21ta22

) (2.52)

14

Taking the determinant of [Sb] and using the previous two results

∣∣Sb∣∣ = Sb11 · Sb22 − Sb12 · Sb21 = −tb21tb12

tb22tb22− tb11

tb22+

tb21tb12tb22tb22

(2.53)

Sb12 · Sb21 =tb11tb22

+ Sb11 · Sb22 (2.54)

It is now necessary to separate the off-diagonal elements of [Sa] and [Sb]. With this

objective in mind, two assumptions will be made [24]: 1) The determinants of the

measured transmission matrices for the Thru and the Line standards are the same

|Tt| =∣∣Tl

∣∣ =

(Sa12Sa21

)·(

Sb12Sb21

)(2.55)

and 2) the determinants of the connectors A and B transmission matrices are also

the same.

|Ta| =∣∣Tb

∣∣ (2.56)

With these assumptions the ratio

Sa12Sa21

=Sb12Sb21

=

√St12St21

(2.57)

is found, decoupling both off-diagonal S-parameters on [Sa].

Sa12 =

√√√√Sa12 · Sa21 ·√

St12St21

(2.58)

Sa21 =Sa12 · Sa21

Sa12(2.59)

The same process applies for the off-diagonal S-parameters for the B connector.

All the information necessary to construct the elements of Tmut as expressed in

equations (2.10) and (2.12) are now available. Using equations (2.6) and (2.5), and

15

using the elements of [Sa] and [Sb] as shown above, the T-parameters for the MUT

are obtained as:

tmut11 =ta22

(tm11tb22 − tm12tb21

)− ta12(tm21tb22 − tm22tb21

)

(ta11ta22 − ta12ta21)(tb11tb22 − tb12tb21

) (2.60)

tmut12 =ta22


)− ta12(tm22tb11 − tm21tb12

)


) (2.61)

tmut21 =ta11


)− ta21(tm11tb22 − tm12tb21

)


) (2.62)

tmut22 =ta11


)− ta21(tm12tb11 − tm11tb12

)


) (2.63)

Finally, the S-parameters for the MUT are given by:

[Smut] =

Smut11 Smut12

Smut21 Smut22

(2.64)

With each individual element given by:

Smut11 =tmut12tmut22

(2.65)

Smut12 =1

tmut22(tmut11tmut22 − tmut12tmut21) (2.66)

Smut21 =1

tmut22(2.67)

Smut22 = −tmut21tmut22

(2.68)

Or alternatively, expressing the results in terms of only: 1) the scattering param-

eters of connector A, [Sa ], 2) the scattering parameters of connector B, [Sb ], and

3) the total measured scattering parameters, as seen from the VNA, [Sm ]; the final

expressions, as presented in [24], are:

16

Smut11 =Sb11 · (Sa11 · Sm22 − |Sm|) + (Sm11 − Sa11) ·

∣∣Sb∣∣

Sb11 · (Sm22 · |Sa| − Sa22 · |Sm|) + (Sm11Sa22 − |Sa|) ·∣∣Sb

∣∣ (2.69)

Smut12 =−Sm12 · Sa21 · Sb21


∣∣ (2.70)

Smut12 =−Sm21 · Sa12 · Sb12


∣∣ (2.71)

Smut22 =Sb22 · (Sa22 · Sm11 − |Sa|) + Sm22 · |Sa| − Sa22 · |Sm|


∣∣ (2.72)

Where the operation | · | represents the determinant of the argument.

17

2.2 Derivation of the Reflection and Transmission Coefficients for a Fully-

Filled Rectangular Waveguide. TE10 Mode

a

b

x

y

o

^

^

Figure 2.7. Rectangular homogeneous source-free waveguide.

Consider the metallic waveguide cross-section shown in Figure 2.7. The structure

has infinite length in the ±z directions. Because of its solenoid nature in a source-free

region, the electric flux density vector can be written as

∇ ·D = 0 −→ D = −∇× F

or

E = −1ε ∇× F (2.73)

and obeys the homogeneous vector Helmholtz wave equation

∇2 F + k2 F = 0 (2.74)

18

The objective is to reproduce TEz modes, which means

Ez = 0 −→ Fx = Fy = 0 and Fz 6= 0 (2.75)

Hence, Equation (2.74) is written as the homogeneous scalar Helmholtz wave equation

∇2 Fz + k2 Fz = 0 (2.76)

The vector potential Fz can be expressed as the product of three independent func-

tions

Fz(x, y, z) = f(x) g(y) h(z) (2.77)

Substituting (2.77) into (2.76) subject to the boundary conditions

Ey(x = 0) = Ey(x = a) = 0 (2.78)

Ex(y = 0) = Ex(y = b) = 0 (2.79)

and assuming a wave propagating in the +z direction, the following is obtained.

Fz(x, y, z) = Ao cos(kx x) cos(ky y) exp(−jkz z) (2.80)

where

k2 = ω2εµ = k2x + k2

y + k2z (2.81)

kz =

√k2 −

(π

am

)2 −(π

bn)2

, m, n = 0, 1, 2, 3, . . . (2.82)

19

In addition, Ao is a constant. The fields E and H are then constructed by using

Ex = −1

ε

∂ Fz∂y

Hx = − j

ωεµ

∂2 Fz∂x∂z

Ey =1

ε

∂ Fz∂x

Hy = − j

ωεµ

∂2 Fz∂y∂z

Ez = 0 Hz = − j

ωεµ

(∂2

∂z2+ k2

)Fz (2.83)

From the previous relations and the solution provided in Equation (2.80), each com-

ponent for the electric and magnetic fields are expressed as:

Ex = Aoky

εcos(kx x) sin(ky y) exp(−jkz z) (2.84)

Ey = −Aokxε

sin(kx x) cos(ky y) exp(−jkz z) (2.85)

Ez = 0 (2.86)

Hx = Aokxkzωεµ

sin(kx x) cos(ky y) exp(−jkz z) (2.87)

Hy = Aokykz

ωεµcos(kx x) sin(ky y) exp(−jkz z) (2.88)

Hz = −jAok2x + k2

y

ωεµcos(kx x) cos(ky y) exp(−jkz z) (2.89)

Defining the constant Eo = −Aokx/ε, the TEz dominant mode is obtained by letting

m = 1 and n = 0 on Equations (2.82) and (2.84) to (2.89), yielding

k2 = ω2εµ = k2x + k2

z (2.90)

kz10 =

√ω2εµ−

(π

a

)2(2.91)

as the dominant wave number. The modal fields for the mode of interest are given

20

by:

Ex = 0 (2.92)

Ey = Eo sin(kx x) exp(−jkz z) (2.93)

Ez = 0 (2.94)

Hx = −Eokzωµ

sin(kx x) exp(−jkz z) (2.95)

Hy = 0 (2.96)

Hz = jEokxωµ

cos(kx x) exp(−jkz z) (2.97)

A side view of a loaded waveguide is shown in Figure 2.8. A dielectric and magnetic

discontinuity exists in Region 2, defined in z1 < z < z2, filling the entire cross-

section in the xy plane. Regions 1 and 3 share the same electromagnetic properties.

On Figure 2.8, the coefficients A, B, C and D are the constant amplitudes of the

incident and reflected electric fields in each of the waveguide regions; and the functions

Ψ1,2(x,±z) are given by:

Ψ1(x,−z) = sin ( kx x ) · exp (−jk1z ) (2.98)

Ψ1(x, z) = sin ( kx x ) · exp ( jk1z ) (2.99)

Ψ2(x,−z) = sin ( kx x ) · exp (−jk2z ) (2.100)

Ψ2(x, z) = sin ( kx x ) · exp ( jk2z ) (2.101)

Suppose that an incident field coming from z < z1 hits the material discontinuity at

z = z1, partially transmitting into Region 2 and reflecting back into Region 1. Then,

the total electric and magnetic fields in Region 1 can be expressed as the superposition

21

of their incident and reflected quantities in this way:

Etot1 = sin(kxx) [ A exp(−jk1z) + B exp(jk1z) ] y (2.102)

Htot1 =

(k1

ω µ1

)sin(kxx) [−A exp(−jk1z) + B exp(jk1z) ] x

+ j

(kx

ω µ1

)cos(kxx) [ A exp(−jk1z) + B exp(jk1z) ] z

(2.103)

Following the same procedure as in Region 1, the total electric and magnetic fields in

Region 2 are given by

ℓ0

Region 1A·Ψ (x, -z)1

B·Ψ (x, z)1

E·Ψ (x, -z)1

F·Ψ (x, z)1

Region 2(ε , µ )2 2(ε , µ )1 1

2D·Ψ (x, z)2

z1 z2

yz Region 3

(ε , µ )1 1C·Ψ (x, -z)

Figure 2.8. Sideview of a fully-filled cross-section waveguide propagating the TE10mode.

22

Etot2 = sin(kxx) [ C exp(−jk2z) + D exp(jk2z) ] y (2.104)

Htot2 =

(k2

ω µ2

)sin(kxx) [−C exp(−jk2z) + D exp(jk2z) ] x

+ j

(kx

ω µ2

)cos(kxx) [ C exp(−jk2z) + D exp(jk2z) ] z

(2.105)

Assuming that no reflection is produced from z > z2 (` →∞), then the constant F

shown in Figure 2.8 is set equal to zero, producing in this way, only a transmitted

wave. The total electric and magnetic fields in Region 3 are then given by

Etot3 = E sin(kxx) exp(−jk1z) y (2.106)

Htot3 = E

(exp(−jk1z)

ω µ1

)[−k1 sin(kxx) x + j kx cos(kxx) z ] (2.107)

Tangential electric and magnetic field continuity must be maintained across each

section interface. This means that at the first waveguide discontinuity, z = z1, the

two conditions that must be satisfied are:

Etan1 (z = z1) = Etan

2 (z = z1)

A exp(−jk1z1) + B exp(jk1z1) = C exp(−jk2z1) + D exp(jk2z1)

(2.108)

and

Htan1 (z = z1) = Htan

2 (z = z1)

A

(k1µ1

)exp(−jk1z1) − B

(k1µ1

)exp(jk1z1) =

C

(k2µ2

)exp(−jk2z1) −D

(k2µ2

)exp(k2z1)

(2.109)

23

Similarly, at the second waveguide discontinuity, z = z2

Etan2 (z = z2) = Etan

3 (z = z2)

C exp(−jk2z2) + D exp(jk2z2) = E exp(−jk1z2)

(2.110)

and

Htan2 (z = z2) = Htan

3 (z = z2)

C

(k2µ2

)exp(−jk2z2) − D

(k2µ2

)exp(jk2z2) = E

(k1µ1

)exp(−jk1z2)

(2.111)

From (2.110) and (2.111) two of the unknowns are expressed as a function of the

transmitted wave amplitude E.

C =1

2· µ2k2

(k1µ1

+k2µ2

)· exp(z2(k2 − k1)) · E (2.112)

D =1

2·(

1 − k1µ1

· µ2k2

)· exp(−z2(k1 + k2)) · E (2.113)

from (2.108) and (2.109) A and B are expressed as a function of the previous constants

24

found.

A =1

2· µ1k1·(

k1µ1

+k2µ2

)· exp(z1(k1 − k2) · C

+1

2· µ1k1

·(

k1µ1

− k2µ2

)· exp(z1(k1 − k2) ·D (2.114)

B =

[1 − 1

2· µ1k1·(

k1µ1

+k2µ2

)]· exp(−z1(k1 + k2)) · C

+

[1 − 1

2· µ1k1

·(

k1µ1

− k2µ2

)]· exp(z1(k2 − k1)) ·D (2.115)

Using Equations (2.112) and (2.113) in (2.114) the following ratio is found

E

A=

exp (jk1(z2 − z1))

cos (k2 (z2 − z1)) + j

(k1µ1

)2+

(k2µ2

)2

2

(k1µ1

)(k2µ2

)

sin (k2 (z2 − z1))

(2.116)

similarly using Equations (2.112) and (2.113) in (2.115), the second ratio is expressed

as

B

E= j

(k1µ1

)2−

(k2µ2

)2

2

(k1µ1

)(k2µ2

)

· sin (k2 (z2 − z1)) · exp (−jk1(z2 + z1)) (2.117)

From the previous two ratios found, (2.117) and (2.116), taking their product provides

25

the third and last ratio

B

A=

[B

E

]·[E

A

]

=

j

(k1µ1

)2−

(k2µ2

)2

2

(k1µ1

)(k2µ2

)

exp (−j2k1z1)

cos (k2 (z2 − z1)) + j

(k1µ1

)2+

(k2µ2

)2

2

(k1µ1

)(k2µ2

)

sin (k2 (z2 − z1))

(2.118)

Recognizing that the reflection coefficient is related to B/A and that the transmission

coefficient is related to E/A, the S11 parameter evaluated at the z = z1 plane is then

expressed as

S11|z = z1=

B sin(kxx) exp(jk1z1)

A sin(kxx) exp(−jk1z1)

=

[B

A

]exp(j2k1z1)

=

j

(k1µ1

)2−

(k2µ2

)2

2

(k1µ1

)(k2µ2

)

cos (k2 (z2 − z1)) + j

(k1µ1

)2+

(k2µ2

)2

2

(k1µ1

)(k2µ2

)

sin (k2 (z2 − z1))

(2.119)

26

while the S21 parameter evaluated at z = z2 is expressed as

S21|z = z2=

E sin(kxx) exp(−jk1z2)

A sin(kxx) exp(−jk1z1)

=

[E

A

]exp(−jk1(z2 − z1))

=

1

cos (k2 (z2 − z1)) + j

(k1µ1

)2+

(k2µ2

)2

2

(k1µ1

)(k2µ2

)

sin (k2 (z2 − z1))

(2.120)

27

2.3 The Inversion Algorithm

The objective is to solve the system of simultaneous equations given by

Sthy11 (ω, ε, µ)− S

exp11 (ω) = 0

Sthy21 (ω, ε, µ)− S

exp21 (ω) = 0 (2.121)

where Sthyij and S

expij for ij = 11, 21 are the theoretical and experimental S-

parameters of the material sample, assuming a symmetric S-matrix. The theoret-

ical S-parameters are obtained by means of the mode matching technique. Equation

(2.121) is solved using the iterative complex two-dimensional Newton’s root search

algorithm. A successful solution to (2.121) will provide the permittivity and perme-

ability of the material sample in question.

In short, the method developed in [4, 5, 6, 10] consists in:

1. Expansion of the electric and magnetic fields into orthogonal modes in each

region of the waveguide.

2. Application of boundary conditions to the tangential components of the fields

at each material/waveguide interface.

3. Application so symmetry properties of the incident TE10 mode and geometry

of the waveguide (e.g. centering the material in the cross-section.)

4. Test the resulting equations with orthogonal modes to obtain a linear system

of equations.

5. Solve the linear system to obtain Sthy11 (ω, ε, µ) and S

thy21 (ω, ε, µ).

6. Solve the system of equations (2.121) iteratively and extract the permittivity ε

and permeability µ.

28

The algorithm is validated in this work by feeding the theoretical values for the S-

parameters developed in (2.119) and (2.120) for two different non-magnetic materials.

The first case is acrylic (ε = 2.5+ j0), Figure 2.9, with a sample length of 7.5mm and

the second is alumina (ε = 9.0, tan δ = 0.003), Figure 2.10, with a sample length of

3mm. Both cases fill the entire waveguide cross-section (a = 22.86mm, b = 10.16mm)

and are tested in the X-Band (8-12 GHz). The agreement throughout the whole band

is consistent with the values of permittivity and permeability used to generate the

theoretical S-parameters for both materials.

29

8 9 10 11 12−0.5

0

0.5

1

1.5

2

2.5

3Extracted Relative Permittivity for Acrylic

Frequency (GHz)

epsilon, realepsilon, imaginary

(a) Relative permittivity.

8 9 10 11 12−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Frequency (GHz)

Extracted Relative Permeability for Acrylic

mu, realmu, imaginary

(b) Relative permeability.

Figure 2.9. Extracted relative permittivity and permeability for an acrylic sampleusing the algorithm in [10]

30

8 9 10 11 12

0

2

4

6

8

10

Extracted Relative Permittivity for Alumina

Frequency (GHz)

epsilon, realepsilon, imaginary

(a) Relative permittivity.

8 9 10 11 12−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Frequency (GHz)

Extracted Relative Permeability for Alumina

mu, realmu, imaginary

(b) Relative permeability.

Figure 2.10. Extracted relative permittivity and permeability for an alumina sampleusing the algorithm in [10]

31

2.4 Hybrid Modes and the Transverse Resonance Method

2.4.1 Hybrid Modes

In solving for the eigenfunctions and eigenvalues for any of the waveguide struc-

tures shown in Figure 2.12, Figure 2.13 and Figure 2.14 by trying to decompose the

electric and magnetic fields in TEz or TMz , as it was done for the empty waveg-

uide in §2.2, will not lead to the correct imposition of boundary conditions in the

air/material interface [21]. Instead, a decomposition of fields known as hybrid modes

will be used [11, 21, 22]. The term hybrid arises from the fact that each decompo-

sition that is a solution for this problem is a combination of TEz and TMz modes.

Hybrid is used interchangeably with the terms LSE or LSM (longitudinal section

electric/magnetic) decomposition. Modes LSEx or y mean TEx or y and similarly

LSMx or y mean TMx or y. Structures exhibiting a material discontinuity in the x

direction are solved in such a way that either no component of the electric field exists

in this direction, meaning it is LSEx; or that the magnetic field lacks the x compo-

nent, meaning it is LSMx. The same principle is applied to material discontinuities

in y.

Take for instance the geometry of the waveguide shown in Figure 2.11, the same

geometry used by the inversion algorithm [10] when the material sample is centered.

Assuming the interior of the structure is free of sources

∇ ·D = 0 −→ D = −jωµε∇×Πh (2.122)

the electric and magnetic fields are written as a function of the magnetic Hertzian

potential as

E = −jωµ∇×Πh (2.123)

H = ∇×∇×Πh (2.124)

32

a

b

x

y

o

^

^x x1 2

Figure 2.11. Vertically loaded waveguide to illustrate the LSE and LSM mode de-composition.

The Hertzian potential obeys the wave equation

∇2 Πh + k2 Πh = 0 (2.125)

The discontinuity of the material exists in the x direction. In order to obtain LSEx

modes, it is necessary to define the magnetic hertzian potential as traveling waves in

the +z direction with only one component, the x -component, as follows:

Πh = ϕ(x, y) exp(−jkz z) x (2.126)

The wave equation (2.125) then becomes

∇2T ϕ(x, y) +

(k2 − k2

z

)ϕ(x, y) = 0 (2.127)

where

∇2T =

∂2

∂x2+

∂2

∂y2(2.128)

33

The tangential field components Ez and Hy are constructed using

Ez = jωµ exp (−jkzz)∂

∂yϕ(x, y) (2.129)

Hy = exp (−jkzz)∂2

∂x∂yϕ(x, y) (2.130)

Applying the separation of variables technique as it was done in §2.2, solutions that

satisfy the boundary conditions at the waveguide walls are obtained for ϕ(x, y).

ϕ1(x, y) = A1 sin(kx1 x) cos(ky y), 0 ≤ x ≤ x1

ϕ2(x, y) =[A2 sin(kx2 x) + B2 cos(kx2 x)

]cos(ky y), x1 ≤ x ≤ x2

ϕ3(x, y) = A3 sin(

kx1 (a − x))

cos(ky y), x2 ≤ x ≤ a

(2.131)

where the wavenumber ky is given by

ky =mπ

b, m = 1, 2, 3, . . . (2.132)

and from the separation of variables technique, the characteristic equations that relate

all the wave numbers are:

k21 = k2

x1+ k2

y + k2z (2.133)

k22 = k2

x2+ k2

y + k2z (2.134)

since the phase in the direction of propagation has to be preserved

k2z = k2

1 − k2x1

− k2y (2.135)

= k22 − k2

x2− k2

y (2.136)

34

Matching the tangential fields Ez and Hy at x = x1 and x = x2 and gives four

equations for the constants A1, A2, B2 and A3. Solving for these constants gives the

transcendental equation:

k2x1

tan(

kx2 ( x2 − x1 ))

+ kx1kx2 tan(

kx1 ( a − x2 ))

+ kx1kx2 tan(

kx1x1

)

− k2x2

tan(

kx2 ( x2 − x1 ))· tan

(kx1x1

)· tan

(kx1 ( a − x2 )

)= 0

(2.137)

For the particular case of having the test material perfectly centered in the cross-

section (e. g. a = x1 + x2), Equation (2.137) reduces to

2 kx1kx2 tan(

kx1x1

)− k2

x2tan

(kx2 ( x2 − x1 )

)· tan2

(kx1x1

)

+ k2x1

tan(

kx2 ( x2 − x1 ))

= 0

(2.138)

which can be decoupled into the set

kx2 tan(

kx1x1

)= − kx1 tan

(kx22

( x2 − x1 )

)(2.139)

kx1 cot(

kx1x1

)= kx2 tan

(kx22

( x2 − x1 )

)(2.140)

Equations (2.139), (2.140) and (2.135) are solved numerically for kx1, kx2 and kz .

These values are then substituted in equations (2.123) and (2.124) to construct the

fields in all three regions for all modes.

35

a

b

x

y

o

^

^(a) Along the side wall.

a

b

x

y

o

^

^(b) With off-set.

Figure 2.12. Vertically loaded waveguide.

36

a

b

x

y

o

^

^(a) Along the bottom wall.

a

b

x

y

o

^

^(b) With off-set.

Figure 2.13. Horizontally loaded waveguide.

37

a

b

x

y

o

^

^(a) With rectangular cross-section.

a

b

x

y

o

^

^(b) With arbitrary cross-section.

Figure 2.14. Suspended sample rod loaded waveguide.

38

2.4.2 The Transverse Resonance Method

The cross-sections depicted on Figure 2.12, Figure 2.13 and Figure 2.14 can also be

modeled as an equivalent transmission line circuit, formed with different sections,

each having different impedance values. In doing so, a transcendental equation in the

form of (2.137) is obtained. This approach is referred to as the transverse resonance

method (TRM) [22].

For obtaining the propagation constants for the electric and magnetic fields, the

TRM approach is a much more straightforward and simple solution when compared

to the hybrid mode decomposition method from §2.4.1. But when it comes to recon-

struct the distribution of the fields, the TRM lacks in the ability to provide with the

information to do so. For this reason this approach will not be further discussed in

this work.

2.4.3 Conclusion

If any of the geometries shown in Figure 2.12, Figure 2.13 or Figure 2.14 is changed by

increasing the number of material discontinuities they possess (e.g. become layered),

then the complexity of a solution also increases, since more boundary conditions need

to be satisfied at each interface. A hybrid decomposition solution for this problem

becomes impractical, specially if the materials in question are anisotropic in nature.

Furthermore, if a material sample, like the one depicted in Figure 2.14 is considered,

a solution in closed form is not possible, even if the material geometry is rectangular.

For this case a variational/perturbational approach is required [11].

With this in mind, it is convenient to develop an alternate tool to assess the

robustness of the inversion algorithm described in §1 and §2.3. As a requirement the

simulation tool has to be able to handle eigenproblems with material inhomogeneities

and dispersive materials having complex geometries.

The finite element method (FEM) is particularly suitable for modeling three-

39

dimensional bodies with complex geometry features. It can also incorporate materials

of any composition without the need to reformulate the problem [1, 13, 16].

The FEM formulation for three-dimensional inhomogeneous waveguides is the

subject of the next chapter.

40

CHAPTER 3

THE FINITE ELEMENT METHOD FORMULATION FOR

INHOMOGENEOUS WAVEGUIDES

3.1 Formulation

3.1.1 Domain Discretization

Because of the versatility to conform to many shapes, the element chosen to discretize

the waveguide space is a tetrahedron, shown in Figure 3.1. As a first step, the

Figure 3.1. Tetrahedron element for waveguide mesh discretization.

cross-section of the waveguide is drawn in two dimensions, with the material sample

aligned. Then, a mesh consisting of triangles is generated. The number of triangles

increases with the electrical density of the material sample. The two-dimensional

mesh is extruded into depth with a finite number of layers, producing triangular-

prism elements which in turn are partitioned into tetrahedra. In this way the three-

dimensional waveguide is generated.

41

Figure 3.2. Mesh for the waveguide cross-section with material sample inside.

3.1.2 Interpolation Basis Functions

Within each tetrahedron, the unknown field can be interpolated from each node value

by using the the first order polynomial

ρe(x, y, z) = ae + bex + cey + dez (3.1)

The value of the field at each vertex (node) of the tetrahedron is therefore

ρe1(x1, y1, z1) = ae + bex1 + cey1 + dez1 (3.2)




42

Each coefficient in Equations (3.2)-(3.5) can be expressed as a function of the coor-

dinate values for each vertex. This coefficients are then given by

ae =1

6V e

∣∣∣∣∣∣∣∣∣∣∣∣∣

ρe1 ρe

2 ρe3 ρe

4

xe1 xe

2 xe3 xe

4

ye1 ye

2 ye3 ye

4

ze1 ze

2 ze3 ze4

∣∣∣∣∣∣∣∣∣∣∣∣∣

=1

6V e(ae1ρe

1 + ae2ρe

2 + ae3ρe

3 + ae4ρe

4)

(3.6)

be =1

6V e

∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 1 1

ρe1 ρe

2 ρe3 ρe

4

ye1 ye

2 ye3 ye

4

ze1 ze2 ze

3 ze4

∣∣∣∣∣∣∣∣∣∣∣∣∣

=1

6V e(be1ρe

1 + be2ρe2 + be3ρe

3 + be4ρe4)

(3.7)

ce =1

6V e

∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 1 1

xe1 xe

2 xe3 xe

4

ρe1 ρe

2 ρe3 ρe

4

ze1 ze

2 ze3 ze4

∣∣∣∣∣∣∣∣∣∣∣∣∣

=1

6V e(ce1ρe

1 + ce2ρe2 + ce3ρe

3 + ce4ρe4)

(3.8)

de =1

6V e

∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 1 1

xe1 xe

2 xe3 xe

4

ye1 ye

2 ye3 ye

4

ρe1 ρe

2 ρe3 ρe

4

∣∣∣∣∣∣∣∣∣∣∣∣∣

=1

6V e(de1ρe

1 + de2ρe

2 + de3ρe

3 + de4ρe

4)

(3.9)

where | · | is the determinant operator. The volume for each tetrahedron is given by

V e =1

6· abs(

∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 1 1

xe1 xe

2 xe3 xe

4

ye1 ye

2 ye3 ye

4

ze1 ze

2 ze3 ze

4

∣∣∣∣∣∣∣∣∣∣∣∣∣

) =1

6· abs(Ae + Be + Ce + De) (3.10)

43

Expanding each determinant on Equations (3.6) through (3.10) and grouping them,

each element constant is then obtained in terms of its nodal coordinates

ae1 = (ye

3ze4 − ye

4ze3)xe

2 + (ye4ze

2 − ye2ze

4)xe3 + (ye

2ze3 − ye

3ze2)xe

4 (3.11)

ae2 = (ye

4ze3 − ye

3ze4)xe

1 + (ye1ze

4 − ye4ze

1)xe3 + (ye

3ze1 − ye

1ze3)xe

4 (3.12)

ae3 = (ye

2ze4 − ye

4ze2)xe

1 + (ye4ze

1 − ye1ze

4)xe2 + (ye

1ze2 − ye

2ze1)xe

4 (3.13)

ae4 = (ye

3ze2 − ye

2ze3)xe

1 + (ye1ze

3 − ye3ze

1)xe2 + (ye

2ze1 − ye

1ze2)xe

4 (3.14)

be1 = (ye3 − ye

4)ze2 + (ye4 − ye

2)ze3 + (ye

2 − ye3)ze

4 (3.15)

be2 = (ye4 − ye

3)ze1 + (ye1 − ye

4)ze3 + (ye

3 − ye1)ze

4 (3.16)

be3 = (ye2 − ye

4)ze1 + (ye4 − ye

1)ze2 + (ye

1 − ye2)ze

4 (3.17)

be4 = (ye3 − ye

2)ze1 + (ye1 − ye

3)ze2 + (ye

2 − ye1)ze

3 (3.18)

ce1 = (xe4 − xe

3)ze2 + (xe

2 − xe4)ze

3 + (xe3 − xe

2)ze4 (3.19)

ce2 = (xe3 − xe

4)ze1 + (xe

4 − xe1)ze

3 + (xe1 − xe

3)ze4 (3.20)

ce3 = (xe4 − xe

2)ze1 + (xe

1 − xe4)ze

2 + (xe2 − xe

1)ze4 (3.21)

ce4 = (xe2 − xe

3)ze1 + (xe

3 − xe1)ze

2 + (xe1 − xe

2)ze3 (3.22)

de1 = (xe

3 − xe4)ye

2 + (xe4 − xe

2)ye3 + (xe

2 − xe3)ye

4 (3.23)

de2 = (xe

4 − xe3)ye

1 + (xe1 − xe

4)ye3 + (xe

3 − xe1)ye

4 (3.24)

de3 = (xe

2 − xe4)ye

1 + (xe4 − xe

4)ye3 + (xe

3 − xe1)ye

4 (3.25)

de4 = (xe

3 − xe2)ye

1 + (xe1 − xe

3)ye2 + (xe

2 − xe1)ye

3 (3.26)

Ae = [(xe3 − xe

4)ye2 + (xe

4 − xe2)ye

3 + (xe2 − xe

3)ye4]ze

1 (3.27)

Be = [(xe4 − xe

3)ye1 + (xe

1 − xe4)ye

3 + (xe3 − xe

1)ye4]ze

2 (3.28)

Ce = [(xe2 − xe

4)ye1 + (xe

4 − xe1)ye

2 + (xe1 − xe

2)ye4]ze

3 (3.29)

De = [(xe3 − xe

2)ye1 + (xe

1 − xe3)ye

2 + (xe2 − xe

1)ye3]ze

4 (3.30)

44

Equation (3.1) can now be rewritten as

ρe(x, y, z) =4∑

i=1

ςei (x, y, z)ρei (3.31)

The function ςei is a first-order polynomial given by

ςei (x, y, z) =1

6V e (aei + bei x + cei y + de

i z), i = 1, 2, 3, 4 (3.32)

where i is one of the four nodes of the tetrahedral element. The meaning of ςei is

better explained with the aid of Figure 3.3 shown below. ςe1 is the normalized partial

volume of element e defined by point p and nodes 2, 3 and 4. When point p is at

node 1, the partial volume is equal to six times that of the element volume, hence

ςe1 = 1. When point p is anywhere on the face opposite to node 1 (face 234), then

ςe1 = 0. The same principle applies to ςe2, ςe3 and ςe4. Table 3.1 describes the nodes

related to each ςei . The vector basis function, Nei , is now defined as

1

2

3

4

p

Figure 3.3. Tetrahedron element showing its vertices and the interior point p.

45

Function vertex 1 vertex 2 vertex 3 vertex 4

ςe1 p 2 3 4

ςe2 p 1 3 4

ςe3 p 1 2 4

ςe4 p 1 2 3

Table 3.1. Definition for each volume-function ςei within each tetrahedron.

Nei = èi

(ςei1

(x, y, x)∇ςei2(x, y, z)− ςei2

(x, y, z)∇ςei1(x, y, z)

)

= èi

(ςei1∇ςei2

− ςei2∇ςei1

)

=

(èi

(6V e)2

)[(ai1bi2 − ai2bi1) + (ci1bi2 − ci2bi1)y + (di1bi2 − di2bi1)z

]x

+

(èi

(6V e)2

)[(ai1ci2 − ai2ci1) + (bi1ci2 − bi2ci1)x + (di1ci2 − di2ci2)z

]y

+

(èi

(6V e)2

)[(ai1di2 − ai2di1) + (bi1di2 − bi2di1)x + (ci1di2 − ci2di1)y

]z

(3.33)

where the notation i1 stands for “node 1” of edge “i” and i2 stands for “node 2” of

edge “i”, all on the element “e.” Figure 3.4 and Table 3.2 define how a tetrahedron

element is constituted with nodes, edges and their relationship.

Equation (3.33) is a tangential vector finite element (TVFE) basis function, also

called CT-LN (Continuous Tangential - Linear Normal) basis function [17]. This type

of basis function was first introduced in 1980 by Nedelec [18]; however, they were first

described by Whitney [19] about 45 years ago. The basis function Ni posses some

46

1

2

3

4

1 2 3

4

5

6

Figure 3.4. Definition for a tetrahedron. Showing its nodes, edges and edge directions.

Edge i Node i1 Node i2

1 1 2

2 1 3

3 1 4

4 2 3

5 4 2

6 3 4

Table 3.2. Definition for each edge on a tetrahedral element.

47

desirable properties. First, it has a zero divergence value, as shown in Equation

(3.34).

∇ ·Nei =

èi

(6V e)2

[(bei1

bei2− bei2

bei1) + (cei1

cei2− cei2

cei1) + (de

i1dei2− de

i2dei1

)]

= 0

(3.34)

Second, it exhibits a constant, non-zero curl, as it is shown in Equation(3.35). This

property ensures continuity of the field across elements sharing faces containing edge

i.

∇×Nei = èi∇×

[ςei1∇ςei2

− ςei2∇ςei1

]

= 2èi

[∇ςei1

−∇ςei2

]

=

(2èi

(6V e)2

) (( cei1

dei2− de

i1cei2

) x + ( dei1

bei2− bei1

dei2

) y

+ ( bei1cei2

− cei1bei2

) z)

6= 0 (3.35)

With this definition of the basis (interpolation) function Ni, the electric field inside

a tetrahedron can be expressed as

Eei =

6∑

i=1

Nei (x, y, z) Ee

i

= [Ne] · [Ee]T

= [Ne]T · [Ee]

(3.36)

48

Figure 3.5. Rectangular waveguide with obstacle

49

3.1.3 Formulation of the System of Equations Using The Ritz Method

Consider Figure 3.5, in which a rectagular waveguide is loaded with any irregular

shaped obstacle (although, the obstacle shown here has a rectangular prism shape for

convenience), having constitutive parameters (ε, µ). The waveguide is excited with

the dominant TE10 mode. On surface S1, the surface at which the incident field

is impressed, the total electric field E(x, y, z) can be written as the superposition of

both, the incident and the reflected fields in the same manner as Equation(2.102)

E(x, y, z1) = Einc(x, y, z1) + Eref (x, y, z1)

=[Eo sin

(π

ax)

exp(−jkz10 z1) + R Eo sin(π

ax)

exp(jkz10 z1)]y

(3.37)

Similarly, the total field at surface S2 is expressed as

E(x, y, z2) = Etrans(x, y, z2)

= T Eo sin(π

ax)

exp(jkz10 z2) y

(3.38)

where R and T are the reflection and transmission coefficients, respectively. The

TE10 mode guided wavenumber is given by Equation(2.91).

Uniqueness require that all tangential field components have to be specified on

all of the six faces surrounding the waveguide domain. All tangential electric field

components have to vanish at the perfectly conducting walls of the waveguide. But

on surfaces S1 and S2, however, impedance boundary conditions are required [20].

Beginning on surface S1, with the total electric field expression written on Equation

(3.37), an expression relating the tangential magnetic and electric fields on this surface

50

is written following these steps

n×∇× E = −z×∇×(Einc + Eref

)

= −jkz10Einc + jkz10E

ref

= jkz10E − 2jkz10Einc (3.39)

so that

n×∇× E − jkz10E = −2jkz10Einc (3.40)

Noting that

n× n× E = −E (3.41)

then Equation (3.40) is finally written as

n×∇× E + (jkz10) n× n× E = (−2jkz10) Einc (3.42)

which is the boundary condition at surface S1. In a similar manner, the tangential

magnetic and electric fields at surface S2 are related by the boundary condition

n×∇× E + (jkz10) n× n× E = 0 (3.43)

which implies that no energy is being reflected from surface S2, that is to say, surface

S2 is a perfectly matched layer. It is important to mention that when generating the

discretization mesh for Figure 3.5, the surfaces S1 and S2 are placed sufficiently far

away from the material sample, so that higher order modes excited by the material

discontinuity decay and vanish when the fields are recovered at these surfaces.

51

The boundary value problem for Figure 3.5 can be summarized as

[∇×(

1

µ∇×

)− ω2ε ] E = 0 P.D.E. (3.44)

n× E = 0 on PEC walls (3.45)

n×∇× E + (jkz10) n× n× E = (−2jkz10) Einc on S1

n×∇× E + (jkz10) n× n× E = 0 on S2

The finite element method calls for finding the values of the electric field that will make

the first variation of the functional for Equation (3.44) stationary. The functional,

from the generalized variational principle involving lossy media and inhomogeneous

boundary conditions, is given by

F(ρ) =1

2〈Lρ, ρ〉 − 1

2〈Lρ, u〉+

1

2〈ρ, Lu〉 − 〈ρ, f〉 (3.46)

where 〈a, b〉 is the inner product defined as

〈a, b〉 =

∫

Ωa b dΩ (3.47)

Ω is the domain of validity for a and b, u is a function that satisfies the boundary

conditions (3.42) and (3.43); f is the forcing function (which for this case is zero,

since all field sources are nonexistent) for the differential equation (3.44); and L is the

wave equation operator ∇×(

1µ ∇×

)− ω2ε . After performing these substitutions,

applying the boundary conditions above with the aid of the first and second vector

52

Green’s theorems [15]

∫∫∫

V[ ϕ ( ∇×U ) · ( ∇×V ) − U · ( ∇× ϕ∇×V ) dV

=

∮

Sϕ ( U×∇×V ) · n dS (3.48)

∫∫∫

V[ V · ( ∇× ϕ∇×U ) − U · ( ∇× ϕ∇×V ) ] dV

=

∮

Sϕ ( U×∇×V −V ×∇×U ) · n dS

(3.49)

the functionals for the electric field E and isotropic media

Fiso (E) = 12

∫∫∫

V

[1

µr(∇× E) · (∇× E)−

(k2oεr

)E · E

]dV

+

∫∫

S1

[(jkz10

2

)(n× E) · (n× E)− (2jkz10) E · Einc

]dS

+

∫∫

S2

[(jkz10

2

)(n× E) · (n× E)

]dS

(3.50)

and anisotropic media are then obtained.

Fanis (E) = 12

∫∫∫

V

[(∇× E) · [ νr ] · (∇× E)− k2

o E · [ εr] · E]dV

+

∫∫

S1

[(jkz10

2

)(n× E) · (n× E)− (2jkz10) E · Einc

]dS

+

∫∫

S2

[(jkz10

2

)(n× E) · (n× E)

]dS

(3.51)

53

The relative-dielectric permittivity tensor [ εr ] is of the form

[ εr ] =

εxx εxy εxz

εyx εyy εyz

εzx εzy εzz

(3.52)

and the relative-magnetic permeability tensor [ µr ] is related to its inverse [ νr ] in

the following fashion

[ νr ] =

νxx νxy νxz

νyx νyy νyz

νzx νzy νzz

=

µxx µxy µxz

µyx µyy µyz

µzx µzy µzz

−1

= [ µr ]−1 (3.53)

Substitution of the electric field expansion (3.36) into the isotropic functional (3.50)

and the anisotropic functional (3.51) leads to their discretized versions:

F (E) =1

2

Ntot∑

e=1

EeT[ Ie1 + Ie2 ]Ee +1

2

NS1+ NS2∑

e=1

EeT[ Ie3 ]Ee

− 1

2

NS1∑

e=1

Ie4 EeT (3.54)

F (E) =1

2

Ntot∑

e=1

EeT[ Ie5 + Ie6 ]Ee +1

2

NS1+ NS2∑

e=1

EeT[ Ie3 ]Ee

− 1

2

NS1∑

e=1

Ie4 EeT (3.55)

where Ntot is the total number of tetrahedra in the waveguide mesh, NS1is the

number of tetrahedra on surface S1 and NS2is the number of tetrahedra on surface

54

S2. The matrices [ Ie1 ], [ Ie2 ], [ Ie3 ], [ Ie5 ]and [ Ie6 ] and the vector Ie4 are given by

[ Ie1 ] =

∫∫∫

Ve

(1

µer

)∇ ×Ne T · ∇ ×Ne dV (3.56)

[Ie2

]=

∫∫∫

Ve

(−k2

oεr

)Ne T · Ne dV (3.57)

[Ie3

]=

∫∫

S1∪S2(jkz10) n×Ψt T · n×Ψt dS (3.58)

Ie4 =

∫∫

S1(−2jkz10) n×Ψt T · Einc × n dS (3.59)

[Ie5

]=

∫∫∫

Ve∇ ×Ne T · [ νr ] · ∇ ×Ne dV (3.60)

[Ie6

]=

∫∫∫

Ve−k2

o Ne T · [ εr ] · Ne dV (3.61)

Noting that the formal derivative of the functionals (3.54) and (3.55) is given by [1]

δF(E) =∂

∂ EiF(E) (3.62)

and using the partial differentiation rules for matrices and vectors as described in [16]

∂

∂ x ( C · x ) = C (3.63)

∂

∂ x(xT · [A ] · x

)= 2 [A ] · x (3.64)

then the first variation of the functionals (3.54) and (3.55) after setting them equal

to zero, δ F(E) = 0, as the Ritz procedure requires; and enforcing the boundary con-

ditions (3.42), (3.43) and (3.45), the following linear system of equations is obtained

[Ie1 + Ie2 + Ie3] · E = Ie4 (3.65)

[Ie5 + Ie6 + Ie3] · E = Ie4 (3.66)

where the system (3.65) solves the isotropic problem depicted in Figure 3.5 while

55

(3.66) solves the anisotropic version of the same geometry.

3.1.3.1 Solution of Integrals

The solution of the integrals (3.56 - 3.61) is performed within each element, assuming

the homogeneity of the tetrahedra. Using the result from (3.35) and taking the dot

product in the integrand of (3.56)

τij = (∇×Nei ) · (∇×Ne

j)

=

(4 `i `j

(6V e)4

)·(( cei1

dei2− de

i1cei2

)( cej1dej2− de

j1cej2

)

+ ( dei1

bei2− bei1

dei2

)( dej1

bej2− bej1

dej2

)

+ ( bei1cei2

− cei1bei2

)( bej1cej2

− cej1bej2

))

= constant, i, j = 1, 2, 3, . . . , 6

(3.67)

Since the integrand τij is always a constant, the value of (3.56) reduces to the volume

of the tetrahedron multiplied by a set of constants as follows:

56

[ Ie1 ]ij =

∫∫∫

Ve

(1

µer

)∇ ×Ne

i T · ∇ ×Nej dV

=

∫∫∫

Ve

(τij

µer

)dV

=

(τij

µer

) ∫∫∫

VedV

=

(Ve

µer

) (τij

)

=

(4 Veì `j

µe (6V e)4

)·(( cei1

dei2− de

i1cei2

)( cej1dej2− de

j1cej2

)

+ ( dei1

bei2− bei1

dei2

)( dej1

bej2− bej1

dej2

)

+ ( bei1cei2

− cei1bei2

)( bej1cej2

− cej1bej2

))

,

i, j = 1, 2, 3, . . . , 6

(3.68)

As for the second integrand, it involves the dot product of the vector basis functions,

χij(x, y, z).

χij(x, y, z) =(−k2

oεr

)· (Ne

i ) · (Nej)

=(−k2

oεr

)·(èi `

ej

)(ςei1∇ςei2

− ςei2∇ςei1

)·(ςej1∇ςej2

− ςej2∇ςej1

)

=(−k2

oεr

)·(èi `

ej

)·(ςei1

ςej1

(bei2

bej2+ cei2

cej2+ de

i2dej2

)

− ςei1ςej2

(bei2

bej1+ cei2

cej1+ de

i2dej1

)

− ςei2ςej1

(bei1

bej2+ cei1

cej2+ de

i1dej2

)

+ ςei2ςej2

(bei1

bej1+ cei1

cej1+ de

i1dej1

))(3.69)

Using the formula from [16]

∫∫∫

Ve

(ςe1

)r (ςe2

)s (ςe3

)t (ςe4

)u dV =3! r! s! t! u!

(3 + r + s + t + u)!· Ve (3.70)

57

and applying it to the integrand (3.69), all the elements of the integral [I2]ij are then

obtained as follows

[Ie2

]11 =

(−k2

oεr

)·(

è1 è1360 Ve

)· ( φe

22 − φe12 + φe

11)

(3.71)

[Ie2

]12 =

(−k2

oεr

)·(

è1 è2720 Ve

)· ( 2 φe

23 − φe21 − φe

13 + φe11

)(3.72)

[Ie2

]13 =

(−k2

oεr

)·(

è1 è3720 Ve

)· ( 2 φe

24 − φe21 − φe

14 + φe11

)(3.73)

[Ie2

]14 =

(−k2

oεr

)·(

è1 è4720 Ve

)· ( φe

23 − φe22 − 2 φe

13 + φe12

)(3.74)

[Ie2

]15 =

(−k2

oεr

)·(

è1 è5720 Ve

)· ( φe

22 − φe24 − φe

12 + 2 φe14

)(3.75)

[Ie2

]16 =

(−k2

oεr

)·(

è1 è6720 Ve

)· ( φe

24 − φe23 − φe

14 + φe13

)(3.76)

[Ie2

]22 =

(−k2

oεr

)·(

è2 è2360 Ve

)· ( φe

33 − φe13 + φe

11)

(3.77)

[Ie2

]23 =

(−k2

oεr

)·(

è2 è3720 Ve

)· ( 2 φe

34 − 2 φe13 − 2 φe

14 + φe11

)(3.78)

[Ie2

]24 =

(−k2

oεr

)·(

è2 è4720 Ve

)· ( φe

33 − φe23 − φe

13 + 2 φe12

)(3.79)

[Ie2

]25 =

(−k2

oεr

)·(

è2 è5720 Ve

)· ( φe

23 − φe34 − φe

12 + φe14

)(3.80)

[Ie2

]26 =

(−k2

oεr

)·(

è2 è6720 Ve

)· ( φe

13 − φe33 − 2 φe

14 + φe34

)(3.81)

[Ie2

]33 =

(−k2

oεr

)·(

è3 è3360 Ve

)· ( φe

44 − φe14 + φe

11)

(3.82)

[Ie2

]34 =

(−k2

oεr

)·(

è3 è4720 Ve

)· ( φe

34 − φe24 − φe

13 + φe12

)(3.83)

58

[Ie2

]35 =

(−k2

oεr

)·(

è3 è5720 Ve

)· ( φe

24 − φe44 − 2 φe

12 + φe14

)(3.84)

[Ie2

]36 =

(−k2

oεr

)·(

è3 è6720 Ve

)· ( φe

44 − φe34 − φe

14 + 2 φe13

)(3.85)

[Ie2

]44 =

(−k2

oεr

)·(

è4 è4360 Ve

)· ( φe

33 − φe23 + φe

22)

(3.86)

[Ie2

]45 =

(−k2

oεr

)·(

è4 è5720 Ve

)· ( φe

23 − 2 φe34 − φe

22 + φe24

)(3.87)

[Ie2

]46 =

(−k2

oεr

)·(

è4 è6720 Ve

)· ( φe

34 − φe33 − 2 φe

24 + φe23

)(3.88)

[Ie2

]55 =

(−k2

oεr

)·(

è5 è5360 Ve

)· ( φe

22 − φe24 + φe

44)

(3.89)

[Ie2

]56 =

(−k2

oεr

)·(

è5 è6720 Ve

)· ( φe

24 − 2 φe23 − φe

44 + φe34

)(3.90)

[Ie2

]66 =

(−k2

oεr

)·(

è6 è6360 Ve

)· ( φe

44 − φe34 + φe

33)

(3.91)

The function φeij is defined as

φeij = bei b

ej + cei c

ej + de

i dej (3.92)

And, of course,[

Ie2

]ij

=[

Ie2

]ji

. The integrand of[

Ie3

]and

[Ie4

]require the tan-

gential component of the surface basis function on surfaces S1 and S2. This tangential

component translates directly into the face of each tetrahedron that constitutes either

surface. For simplicity, the author chooses to treat each tetrahedra face lying on S1

and S2 as two-dimensional triangular basis functions, whose outward pointing normal

n is always directed to the outside of the mesh (e.g. −z for S1 and +z for S2).

Following an analogous procedures as the one for the three-dimensional tetrahe-

dron, the two-dimensional triangular element is constructed as shown on Figure 3.6

and with the edge definition as given on Table 3.3.

59

y

x

1

2

3

32

1(x , y )1 1

(x , y )2 2

(x , y )3 3

p•

Figure 3.6. 2-D Element for Ψt.

Edge i Node i1 Node i2

1 1 2

2 2 3

3 3 1

Table 3.3. Definition for each edge on a triangular element with its constitutive nodes.

60

Any quantity, ξt(x, y), inside the triangle shown in Figure 3.6 can be readily

approximated by means of the first order polynomial

ξt(x, y) = et + ftx + gty (3.93)

evaluating this polynomial at each node, three different values are obtained for the

interpolation function

ξt1(x1, y1) = et + ftx1 + gty1 (3.94)

ξt2(x2, y2) = et + ftx2 + gty2 (3.95)

ξt3(x3, y3) = et + ftx3 + gty3 (3.96)

then, the value of ξt at any point (x, y) inside the triangular element, can be expressed

as a linear superposition of its own value at each vertex as

ξti(x, y) =3∑

i=1

ψti(x, y)ξti (3.97)

where the function ψti is the two-dimensional triangular basis function in area coor-

dinates:

ψti(x, y) =

1

2At

(eti + ft

i x + gtiy

)i = 1, 2, 3 (3.98)

To illustrate the meaning of area coordinates, suppose that the point p inside the

triangular element depicted in Figure 3.6 has the coordinates (xp, yp). Then, each

61

basis function ψti is expressed as:

ψt1(xp, yp) =

1

2At

(et1 + ft

1xp + gt1yp

)=

Area of triangle p23

Area of triangle 123=

∆ p23

∆ 123

(3.99)

ψt2(xp, yp) =

1

2At

(et2 + ft

2xp + gt2yp

)=



∆ p13

∆ 123

(3.100)

ψt3(xp, yp) =

1

2At

(et3 + ft

3xp + gt3yp

)=



∆ p12

∆ 123

(3.101)

From which it follows naturally

ψt1(x, y) + ψt

2(x, y) + ψt3(x, y) = 1 (3.102)

It is also important to note that ψti = 1 at node i and ψt

i = 0 at the edge opposite

to node i. Each constant for ψti(x, y) in Equation (3.98) is found as a function of the

triangular element vertex coordinates, giving

et1 = xt2yt

3 − xt3yt

2 (3.103)

et2 = xt3yt

1 − xt1yt

3 (3.104)

et3 = xt1yt

2 − xt2yt

1 (3.105)

ft1 = yt

2 − yt3 (3.106)

ft2 = yt

3 − yt1 (3.107)

ft3 = yt

1 − yt2 (3.108)

gt1 = xt

3 − xt2 (3.109)

gt2 = xt

1 − xt3 (3.110)

gt3 = xt

2 − xt1 (3.111)

62

Function vertex 1 vertex 2 vertex 3

ψt1 p 2 3

ψt2 p 1 3

ψt3 p 1 2

Table 3.4. Definition for each area-function ψti within each triangular element.

The area of the triangular element is also found to be

At =1

2

∣∣∣ft1gt

2 − ft2gt

1

∣∣∣ (3.112)

Similarly as it was done in §3.1.2 for the three-dimensional basis function 3.33, the

Whitney vector finite element for a two-dimensional case is generated by operating

on the function ψti(x, y) with the wronskian operator as follows

Ψti = `ti

[ψti1

(x, y)∇ψti2

(x, y)− ψti2

(x, y)∇ψti1

(x, y)]

= `ti

[ψti1∇ψt

i2− ψt

i2∇ψt

i1

]

=`i

2At

[ (fti2

ψti1− ft

i1ψti2

)x +

(gti2

ψti1− gt

i1ψti2

)y

](3.113)

The basis function Ψti shares the same properties as its three-dimensional counter-

part, Nei , in the sense that it is divergence-free (see Equation(3.34)) and that it offers

a finite, constant and different-from-zero curl (see Equation(3.35)). With these prop-

erties, the electric field Ex, y, z1,2 on each triangle constituting the surfaces S1 or

S2, can also be written as the expansion of the two dimensional basis function Ψti as

63

follows

Et(x, y, z1,2) =3∑

i=1

Ψti(x, y)Ei(z1,2) (3.114)

The integrands of [Ie3] and Ie4 call for the tangential component of the basis function

Ψti on either surface. With this objective in mind, the new basis function, Λt

i is now

defined as

Λti = n×Ψt

i

=`i

2At

[ (gti2

ψti1− gt

i1ψti2

)x −

(fti2

ψti1− ft

i1ψti2

)y

](3.115)

The integral (3.58) can now be re-written as as

[Ie3

]=

∫∫

S1∪S2(jkz10) Λt T · Λt dS (3.116)

or in its discretized version

[Ie3

]ij =

∫∫

S1∪S2(jkz10) ·

(`i `j(2At

)2

)·[ (

gti2

gtj2

+ fti2

ftj2

)ψti1

ψtj1

−(gti2

gtj1

+ fti2

ftj1

)ψti1

ψtj2−

(gti1

gtj2

+ fti1

ftj2

)ψti2

ψtj1

+(gti1

gtj1

+ fti1

ftj1

)ψti2

ψtj2

]dS, i, j = 1, 2, 3

(3.117)

Using the formula [16]

∫∫

S

(ψt1

)r (ψt2

)s (ψt3

)tdS =

2! r! s! t!

( 3 + r + s + t ) !A (3.118)

64

all the components of [Ie3] are found:

[Ie3

]11 =

(è1 è124 At

)·

[ft2 ft

2 + gt2 gt

2 − ft1 ft

2 + gt1 gt

2

+ ft1 ft

1 + gt1 gt

1

](3.119)

[Ie3

]12 =

(è1 è248 At

)·

[ft2 ft

3 + gt2 gt

3 − ft2 ft

2 + gt2 gt

2

− 2(

ft1 ft

3 + gt1 gt

3

)+ ft

1 ft2 + gt

1 gt2

](3.120)

[Ie3

]13 =

(è1 è348 At

)·

[ft2 ft

1 + gt2 gt

1 − 2(ft2 ft

3 + gt2 gt

3

)

− ft1 ft

1 + gt1 gt

1 + ft1 ft

3 + gt1 gt

3

](3.121)

[Ie3

]22 =

(è1 è224 At

)·

[ft3 ft

3 + gt3 gt

3 − ft2 ft

3 + gt2 gt

3

+ ft2 ft

2 + gt2 gt

2

](3.122)

[Ie3

]23 =

(è2 è348 At

)·

[ft3 ft

1 + gt3 gt

1 − ft3 ft

3 + gt3 gt

3

− 2(

ft2 ft

1 + gt2 gt

1

)+ ft

2 ft3 + gt

2 gt3

](3.123)

[Ie3

]33 =

(è3 è324 At

)·

[ft1 ft

1 + gt1 gt

1 − ft1 ft

3 + gt1 gt

3

+ ft3 ft

3 + gt3 gt

3

](3.124)

65

The basis function Ψti is also used for the computation of Ie4

Ie4 i =

∫∫

S1(−2jkz10) n×Ψt T · Einc × n dS

=

∫∫

S1(−2jkz10) Λt T · z× Einc dS

=3∑

i=1

∫∫

S1α ì

[ (eti1

gti2− eti2

gti1

)

+(fti1

gti2− ft

i2gti1

)x

]sin

(π

ax)

dxdy

= α

3∑

i=1

[pi · I41 + qi · I42

](3.125)

and each of its parameters are defined as

α = j

(2kz10Eo(

2 At)2

)· exp (−jkz10z1) (3.126)

pi =(

eti1gti2− eti2

gti1

)· ì , i = 1, 2, 3 (3.127)

qi =(

fti1

gti2− ft

i2gti1

)· ì , i = 1, 2, 3 (3.128)

I41 =

∫∫

Ssin

( π

ax

)dxdy , S = S1 (3.129)

I42 =

∫∫

Sx sin

( π

ax

)dxdy , S = S1 (3.130)

The integrals I41 and I42 are evaluated numerically using the gaussian quadrature

rule for triangles with four sampling points. The values for each parameter in this

integration rule are given in Table 3.5.

∫∫

SF (x, y) dxdy ≈ At

4∑

i=1

Wi F(

ζi1, ζi

2, ζi3

)(3.131)

The approximation for I41 is written as

66

i ζi1 ζi2 ζi3 Wi

1 13

13

13 −27

48

2 35

15

15

2548

3 15

35

15

2548

4 15

15

35

2548

Table 3.5. Parameters for the four-point triangular surface Gaussian integration rule.

I41 = −27

48· At · sin

(π

aθ1

)+

25

48· At · sin

(π

aθ2

)

+25

48· At · sin

(π

aθ3

)

+25

48· At · sin

(π

aθ4

)(3.132)

and for I42 as

I42 = −27

48· At · θ1 · sin

(π

aθ1

)+

25

48· At · θ2 · sin

(π

aθ2

)

+25

48· At · θ3 · sin

(π

aθ3

)

+25

48· At · θ4 · sin

(π

aθ4

)(3.133)

67

both integrals having the arguments

θ1 =1

3x1 +

1

3x2 +

1

3x3 (3.134)

θ2 =3

5x1 +

1

5x2 +

1

5x3 (3.135)

θ3 =1

5x1 +

3

5x2 +

1

5x3 (3.136)

θ4 =1

5x1 +

1

5x2 +

3

5x3 (3.137)

For the anisotropic case, [Ie5] and [Ie6], the tensors for [ νr ] and [ εr ] are decomposed

and the product of the integrands is carried out. For the first anisotropic integral,

[Ie5], the integrand is a constant. Like it was performed with [Ie1], the integrand is just

multiplied by the volume of the tetrahedron element.

[Ie5

]ij =

∫∫∫

Ve∇ ×Ne

i T · [ νr ] · ∇ ×Nej dV

=

(4 Ve `i`j

(6Ve)4

) [( cei1

dei2− de

i1cei2

) νexx ( cej1

dej2− de

j1cej2

)

+ ( cei1dei2− de

i1cei2

) νexy ( de

j1bej2

− bej1dej2

)

+ ( cei1dei2− de

i1cei2

) νexz ( bej1

cej2− cej1

bej2)

+ ( dei1

bei2− bei1

dei2

) νeyx ( cej1

dej2− de

j1cej2

)

+ ( dei1

bei2− bei1

dei2

) νeyy ( de

j1bej2

− bej1dej2

)

+ ( dei1

bei2− bei1

dei2

) νeyz ( bej1

cej2− cej1

bej2)

+ ( bei1cei2

− cei1bei2

) νezx ( cej1

dej2− de

j1cej2

)

+ ( bei1cei2

− cei1bei2

) νezy ( de

j1bej2

− bej1dej2

)

+ ( bei1cei2

− cei1bei2

) νezz ( bej1

cej2− cej1

bej2)]

(3.138)

For the integral [Ie6], four terms involving the product of the basis functions are

68

obtained. Just like it was done with [Ie2], the integrals given by∫Ve

(ςeim

ςejn

)dV

for i, j = 1, 2, . . . , 6 and m,n = 1, 2 are computed using the formula (3.70).

[Ie6

]ij =

∫∫∫

Ve−k2

o Nei T · [ εr ] · Ne

j dV

= −(

ϑeij

)·

∫

Veςei1

ςej1dV

[bi2

(εxxbj2

+ εxycj2+ εxzdj2

)

+ ci2

(εyxbj2

+ εyycj2+ εyzdj2

)

+ di2

(εzxbj2

+ εzycj2+ εzzdj2

)]

−∫

Veςei1

ςej2dV

[bi2

(εxxbj1

+ εxycj1+ εxzdj1

)

+ ci2

(εyxbj1

+ εyycj1+ εyzdj1

)

+ di2

(εzxbj1

+ εzycj1+ εzzdj1

)]

−∫

Veςei2

ςej1dV

[bi1

(εxxbj2

+ εxycj2+ εxzdj2

)

+ ci1

(εyxbj2

+ εyycj2+ εyzdj2

)

+ di1

(εzxbj2

+ εzycj2+ εzzdj2

)]

+

∫

Veςei2

ςej2dV

[bi1

(εxxbj1

+ εxycj1+ εxzdj1

)

+ ci1

(εyxbj1

+ εyycj1+ εyzdj1

)

+ di1

(εzxbj1

+ εzycj1+ εzzdj1

)]

(3.139)

where the constant ϑeij is defined as:

ϑeij = −

(k2o `i`j

( 6 Ve )2

)(3.140)

69

3.1.4 Solution of the System of Equations

The linear system (3.65) above is solved by using the biconjugate gradient method

(BiCG) for antisymetric systems [33]. The BiCG method solves the system [A] x =

y. The notation Aa stands for the adjoint of matrix A, the superscript ∗ for the

conjugation of a complex quantity and the inner product 〈x, y〉 is defined as xy∗T .

x1 = 0

r1 = p1 = y

w1 = q1 = y∗

For n = 1, · · · , N DO

an =〈rn , qn〉〈A pn , wn〉

xn+1 = xn + an pn

rn+1 = rn − an A pn

qn+1 = qn − a∗n Aa wn

pn+1 = rn+1 + cn pn

wn+1 = qn+1 + c∗n wn

cn =〈rn+1 , qn+1〉

rn , qn

until

‖rn+1‖‖y‖ ≤ tolerance

70

3.1.5 Numerical S-Parameter Computation

The TRL calibration technique described in detail in §2.1 is applied for the compu-

tation of the S-parameters of the material under test (MUT). Refering to Figure 3.5,

the region of the waveguide defined between the surface S1 at z = z1 and the frontal

face of the MUT would be referred to as, what is called in §2.1, Connector A; While

for the region delimited by surface S2 at z = z2, and the rear face of the MUT as

Connector B.

Using the definition given in §2.1.2 Figure 2.2, Equation (2.2) and using the prop-

erty of orthogonality of the waveguide modes, the scattering parameter S11 is derived

as

S11 |z=z1=

vout1 (z = z1)

vin1 (z = z1)

=

∫∫S1

[ (E(x, y, z1)− Einc(x, y, z1)

)· (sin (π

a x)

y) ]

dxdy

∫∫S1

(Einc(x, y, z1)

)· (sin (π

a x)

y)

dxdy

=

(2 exp (jkz10 z1)

a b Eo

) ∫∫

S1E(x, y, z1) ·

(sin

(π

ax)

y)

dxdy − 1

(3.141)

and with Equation (2.3), the scattering parameter S21 is derived as

S21 |z=z2=

vout2 (z = z2)

vin1 (z = z1)

=

∫∫S2

( E(x, y, z2) ) · (sin (πa x

)y)

dxdy

∫∫S1

(Einc(x, y, z1)

)· (sin (π

a x)

y)

dxdy

=

(2 exp (jkz10 z1)

a bEo

) ∫∫

S2E(x, y, z2) ·

(sin

(π

ax)

y)

dxdy

(3.142)

71

Using the same principle as the two previous cases, S12 is expressed as

S12 |z=z2=

vout1 (z = z1)

vin2 (z = z2)

=

(2 exp (−jkz10 z2)

a bEo

) ∫∫

S1E(x, y, z1) ·

(sin

(π

ax)

y)

dxdy

(3.143)

and S22 as

S22 |z=z2=

vout2 (z = z2)

vin2 (z = z2)

=

(2 exp (−jkz10 z2)

a b Eo

) ∫∫

S2E(x, y, z2) ·

(sin

(π

ax)

y)

dxdy − 1

(3.144)

Using the expansion of the electric field on the surfaces S1 and S2 given by Equation

(3.114) and substituting it in all four expressions above, the discrete versions of the

S-parameters for the MUT are given by

S11 = α1

3∑

i=1

[pi · I41 + qi · I42

] − 1 (3.145)

S21 = α1

3∑

i=1

[pi · I41 + qi · I42

](3.146)

S12 = α2

3∑

i=1

[pi · I41 + qi · I42

](3.147)

S22 = α2

3∑

i=1

[pi · I41 + qi · I42

] − 1 (3.148)

with the integrals I41 and I42 as defined in §3.1.3.1 and evaluated at the surface

72

S = S1 for S11 and S12; and S = S2 for S21 and S22.

α1 =

(2 exp (jkz10 z1)

a b Eo(2 At

)2

)(3.149)

α2 =

(2 exp (−jkz10 z2)

a b Eo(2 At

)2

)(3.150)

pi =(

eti1gti2− eti2

gti1

)· `i , i = 1, 2, 3

qi =(

fti1

gti2− ft

i2gti1

)· `i , i = 1, 2, 3

I41 =

∫∫

Ssin

( π

ax

)dxdy , S = S1,2

I42 =

∫∫

Sx sin

( π

ax

)dxdy , S = S1,2

73

3.2 Validation

The FEM code developed in section 3.1 is validated on X-band (8-10GHz) by com-

paring its results with the theoretical S-parameters formulated in section 2.2 and the

mode-matching technique developed in [10]. The two compared cases are a low- and

high-contrast materials.

The first example, Figure 3.7 , consists of acrylic (ε = 2.5, µ = 1) as a material

sample with length ` = 5 mm which fills the waveguide corss-section entirely. The

dimensions of the waveguide are a = 22.86 mm, b = 10.16 mm [27]. In the second

example, Figure 3.8, the acrylic sample partially fills the waveguide cross-section by

50% (d/a = 0.5) and has a length ` = 7.5 mm. Finally, for the low-contrast material

case, the acrylic-to-waveguide width is d/a = 0.25 with a sample length of ` = 7.5 mm;

the results are depicted in Figure 3.9.

For the high-contrast material validation, a sample of alumina (ε = 9 − j0.0027)

with length ` = 5 mm and d/a = 1 is simulated. Figure 3.10 shows the magnitude

and phase of the reflection and transmission coefficients.

74

8 9 10 11 12−7

−6

−5

−4

−3

−2

−1

0S−Parameters, Magnitude

Frequency (GHz)

dBS11 TheoryS21 TheoryS11 FEMS21 FEM

S21’s

S11’s

(a) Reflection and transmission coefficients, magnitude

8 9 10 11 12−150

−100

−50

0

50

100

150

200

Frequency (GHz)

Deg

rees

S−Parameters, Phase

S21 TheoryS11 TheoryS11 FEMS21 FEM

S21’s

S11’s

(b) Reflection and transmission coefficients, phase

Figure 3.7. Comparison between the FEM and theoretical closed-form solution in afully-filled d/a = 1 rectangular waveguide. Acrylic (ε = 2.5, µ = 1), sample length` = 5 mm.

75

8 9 10 11 120

0.2

0.4

0.6

0.8


Frequency (GHz)

S21, FEMS21, Mode MatchingS11, FEMS11, Mode Matching

S11’s

S21’s


8 9 10 11 12−200

−150

−100

−50

0

50

100

150

200S−Parameters, Phase

Frequency (GHz)

Deg

rees


S21’s

S11’s


Figure 3.8. Comparison between the FEM and mode-matching solution in a partial-ly-filled d/a = 0.5 rectangular waveguide. Acrylic (ε = 2.5, µ = 1), sample length` = 7.5 mm. 76

8 9 10 11 120

0.2

0.4

0.6

0.8

1

Frequency (GHz)

S−Parameters, Magnitude


S11’s

S21’s


8 9 10 11 12−150

−100

−50

0

50

100

150

200

Frequency (GHz)

Deg

rees

S−Parameters, Phase


S21’s

S11’s


Figure 3.9. Comparison between the FEM and mode-matching solution in a partial-ly-filled d/a = 0.25 rectangular waveguide. Acrylic (ε = 2.5, µ = 1), sample length` = 7.5 mm. 77

8 9 10 11 12−9

−8

−7

−6

−5

−4

−3

−2

−1


dB

Frequency (GHz)

S11 TheoryS21 TheoryS11 FEMS21 FEM

S11’s

S21’s


8 9 10 11 12−200

−150

−100

−50

0

50

100

150

200S−Parameters, Phase

Frequency (GHz)

Deg

rees S11 Theory

S21 TheoryS21 FEMS11 FEM

S11’s

S21’s


Figure 3.10. Comparison between the FEM and theoretical closed-form solution ina fully-filled d/a = 1 rectangular waveguide. Alumina (ε = 9.0 − j0.0027, µ = 1),sample length ` = 3 mm.

78

CHAPTER 4

RESULTS

4.1 Error Generated by Cross-Sections Shifted from Center

The error measurement is performed using the waveguide configuration shown in

Figure 4.1 and the dimensions provided on Table 4.1 [27]. The length ` of the sample

(its length in the z direction) as depicted in Figure 3.5, is set to a value between λz/4

and λz/2, where λz is the guiding wavenumber (2π/kz10) [4, 5, 6]. The shifting

parameter δ is varied from 0 mm to 5 mm, a distance that can realistically represent

variations common in a laboratory environment. For each step the S-parameters of

the sample are measured and then inverted using the algorithm in [10] to extract its

constitutive parameter(s). From this information, the error of the extracted ε and

µ is quantized using the value entered during the forward simulation as a reference.

The error is computed with the formula

error =

∣∣∣∣x − xo

xo

∣∣∣∣ · 100% (4.1)

where “x” is the extracted constitutive parameter and “xo” is the reference value.

Three cases will be simulated and then inverted: 1) a low-contrast lossless dielectric

material (acrylic), 2) a high-contrast dielectric material with low loss (alumina) and

3) a lossy magneto-dielectric material (magRAM). Each of the previous three cases

will be simulated using a constant ratio of d to a, where, as Figure 4.1 shows, d is the

width of the material sample and a is the width of the waveguide. The hight of the

sample is always b.

79

a

b

x

y

o

^

^d

a2 δ

Figure 4.1. Vertically loaded waveguide with material sample shifted from the centerby a distance δ.

Band finitial (GHz) ffinal (GHz) a (mm) b (mm)

X 8.0 12.0 22.86 10.11

S 2.6 3.95 72.14 34.04

Table 4.1. Dimensions for a rectangular waveguide for the frequency bands used toconduct the numerical simulations and inversion operations [27].

80

4.1.1 Low-Contrast Material

The numerical simulation is performed in X-band, with the waveguide dimensions

specified on Table 4.1. The lossless low-contrast material chosen is acrylic. During the

computation of the S-parameters the constitutive parameters for the acrylic material

sample where ε = 2.5 and µ = 1 with d/a = 0.5 and a sample length ` = 5 mm.

The acrylic sample was shifted in ten equal steps from the center of the waveguide,

δ = 0 mm, to δ = 5 mm. The resulting scattering parameters are depicted in Figure

4.2 and Figure 4.3. The error for each parameter is shown in Figure 4.4 and Figure 4.5.

The percent error plot for the extracted relative permittivity, Figure 4.6, clearly shows

that its error is maximum and minimum when also the error on the magnitudes of the

transmission and reflection coefficients are maximum and minimum. The error peaks

14.5 % at both ends of the frequency band at δ = 5 mm. It also shows that for each

displacement of δ there exists a frequency, fo, for which there is no error, no matter

what the displacement of the sample is. The frequency fo increases approximately

from 10 GHz to about 10.6 GHz for increasing values of δ. Each frequency fo possesses

a band for which the error is fairly small. Out of this band the error increases rapidly.

4.1.2 High-Contrast Material

The slightly lossy high-contrast material for this run of simulations is alumina, par-

tially filling the cross-section of the waveguide by 50%. The alumina sample has a

relative permittivity of ε = 9.0 and a loss tangent (tan δ) of 0.0003. The sample

length is ` = 3 mm. The simulation is also performed on X-band incrementing the

shifting parameter δ in ten steps, from δ = 0 mm to δ = 5 mm. The reflection and

transmissions coefficients appear in Figure 4.7 (magnitude) and Figure 4.8 (phase).

The percent change for the transmission and reflection coefficients for both, magni-

tude and phase, are shown in Figure 4.9 and Figure 4.10 respectively. The error on

the extracted permittivity (ε) is shown in Figure 4.11. It is important to point out

81

that there exists a limit at which the displacement parameter δ can be increased for

this material. It was found that at displacements equal to δ = 4 mm the inversion

algorithm does not converge for a large number of frequency points, and as δ keeps

increasing (δ ≥ 4.5 mm) the algorithm does not converge at all. This is due to the

fact that with such a big variation from the assumed pair (ε, µ) it is not expected

that a root for Equation (2.121) can be found. As it was shown in the previous case,

acrylic, the extracted relative permittivity error for alumina also exhibits maxima

and minima throughout the whole frequency band for all the δ tested. The error

characteristic also follows those of the percent change of the magnitude of the trans-

mission and reflection coefficients. But in contrast to acrylic, alumina exhibits two

frequencies, f1 ≈ 8.6 GHz and f2 ≈ 10.5 GHz, at which the error is zero. Each of

these frequencies, f1 and f2, have a narrow band about them for which the error is

fairly small. Out of these frequency bands, the error increases rapidly.

4.1.3 Magneto-Dielectric Materials

As a third and last example a dispersive lossy magneto-dielectric material was simu-

lated on S-band (see Table 4.1 for waveguide dimensions.) The material chosen was

magnetic radar absorbing material (MagRAM) with sample- to waveguide width of

d/a = 0.09 and its length ` = 3 mm. The sample constitutive parameters used for

the S-parameter extraction were obtained from [8]. As it can be seen on Figure 4.2

and Figure 4.3, the effect of shifting the material sample from the center δ = 0 to

δ = 3.5 mm is negligible. This effect can be attributed to the magnetic field distribu-

tion in the cross section of the waveguide. As Equation (2.97) shows, the z component

of the magnetic field exhibits a null at the center of the cross-section, the position at

which the sample is located. As the sample is shifted sideways, this component of

the magnetic field increases, but not as rapidly as required for it to have a significant

contribution on the measurement of the reflection and transmission coefficients. In

order to to appreciate the effect of shifting a material sample, it would have to be

82

placed as an initial position, at either sidewall of the waveguide, where the z compo-

nent of the magnetic field is maximum and starts to decay towards the center of the

waveguide.

4.2 Layered Materials

By creating composite mixtures of different dielectric and magnetic materials in layer-

ing structures, the overall effective dielectric permittivity and magnetic permeability

can be engineered to obtain a desired value for an specific frequency or throughout

a whole band. One advantage of obtaining the effective constitutive parameters of a

sample in this way, is to reduce its cost, since using homogenous samples might be a

more expensive way to proceed [26]. In this section three cases for layering structures

are studied: 1) structures layered in the direction of propagation, 2) structures layered

horizontally and parallel to the direction of propagation, and 3) layered vertically and

parallel to the direction of propagation.

4.2.1 Perpendicular in the Direction of Propagation

The structure is shown in Figure 4.13. The length of the overall length ` is kept

constant and equal to 3 mm while the number of layers is incremented progressively

in odd numbers, in this way the material of the same type (material “A” as shown

in Figure 4.13) is always on either side of the waveguide. This practice ensures the

reciprocity of the sample, and the algorithm in [10] can be applied. The first case

consists of material of type-A being alumina(ε = 9, tan δ = 0.0003 and µ = 1) and

material type-B free-space, as an approximation for foam. The number of layers is

incremented from three to nineteen, and then the S-parameters are computed at a

fixed frequency fo = 9 GHz. The reflection and transmission coefficients are depicted

in Figure 4.14 (magnitude) and Figure 4.15 (phase). The results are validated using

the wave-matrix technique [11, 25, 26]. The transmission parameters for each layer

are obtained by using the theoretical reflection and transmission coefficients for a

83

fully-filled rectangular waveguide by using the equations derived in §2.2, Equation

(2.119) and Equation (2.120). From these two Equations, assuming the reciprocity

of the materials involved, the relationship (2.6) is used to convert the previous coeffi-

cients to T-parameters. This process is repeated for each layer, then each T-matrix is

multiplied in the same order as the layer in the waveguide to produce a “total” trans-

mission matrix. From this result and using the relationship (2.5) the S-parameters

for the total structure are then obtained.

[Ttot] =

tA11 tA12

tA21 tA22

·

tB11 tB12

tB21 tB22

· · ·

tB11 tB12

tB21 tB22

·

tA11 tA12

tA21 tA22

(4.2)

Figure 4.16 shows the extracted relative permittivity as a function of the number of

layers. As the number of layers increases the material homogenizes to the approxi-

mation [26]

εeff = εA · VA + εB · VB (4.3)

where VA is the volume fraction of material A and VB is the volume fraction for

material B. When the thickness of layer A is the same as that of layer B, Equation

(4.3) is written as

εeff =

(1

2 N

)· [ ( N + 1 ) · εA + ( N − 1 ) · εB

](4.4)

where N is the total number of layers in the structure. Clearly, from Equation (4.4),

the effective relative permittivity εeff becomes the mean average of both dielectric

permittivities εA and εB when the number of layers is sufficiently large (n → ∞).

εeff =εA + εB

2(4.5)

84

Figure 4.17, Figure 4.18 and Figure 4.19 show the results when the values of εA and

εB are reversed. Figure 4.20 shows the extracted permittivities for both cases and

the asymptotic approximation in (4.5).

4.2.2 Parallel to the Direction of Propagation. Horizontal and Vertical

Layering

The structures are constructed by filling the entire cross-section of the waveguide with

horizonal and vertical layers, as shown in Figure 4.21 and Figure 4.22 respectively.

The first exercise consists on having a high-contrast material in the outer layers

(material A) and then alternating with a low-contrast material (material B). The

constitutive parameters for material A are those of alumina and for B acrylic, with

the same values for ε and µ as described in previous sections. The values are then

reversed. The number of layers are increased from three to nineteen. The simulations

are performed on X-band. For both cases it is seen from the scattering parameters

and from the extracted relative permittivity that the material homogenizes as the

number of layers increases. Figure 4.23 through Figure 4.34 show the results for the

reflection and transmission coefficients as well as the extracted values for the relative

permittivity.

4.3 Anisotropic Formulation: A Ferrite

Anisotropic materials, like ferrites, find multiple applications in microwave circuits.

These applications include directional devices such as isolators, circulators, gyrators

[27], phase shifters, microwave switches, and microstrip antenna applications [28]. The

simulation of a ferrite is performed using the Polder model described in [27] and with

the specifications in [29, 30]. Figure 4.35 through Figure 4.38 show the magnitude

of the reflection and transmission coefficients for the same ferrite at different applied

magnetic fields. The ferrite specifications are: saturation magnetization 4πMs = 5000

Gauss, anisotropy field Ha = 200 Oe, linewidth ∆H = 500 Oe. The applied magnetic

85

fields are Ho = 100, 300, 500 and 800 Oe, all in the y direction. The sample width

is d = 4 mm and its length ` = 25 mm. The ferrite is place along the waveguide

sidewall at x = 0.

86

8 9 10 11 120.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Frequency (GHz)

Transmission Coefficient, Magnitude

0 mm

0 mm

5 mm

5 mm

(a)

8 9 10 11 120.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

Frequency (GHz)

Reflection Coefficient, Magnitude

0 mm

5 mm

5 mm

(b)

Figure 4.2. Comparison of the transmission and reflection coefficients (magnitude) foracrylic when the parameter δ is varied from 0 to 5 mm in ten steps. (ε = 2.5,µ = 1).

87

8 9 10 11 12−105

−100

−95

−90

−85

−80

−75

−70

−65

−60

−55

Frequency (GHz)

Deg

rees

Transmission Coefficient, Phase

0 mm

0 mm 5 mm

5 mm

(a)

8 9 10 11 12140

150

160

170

180

190

200

210

220

Frequency (GHz)

Deg

rees

Reflection Coefficient, Phase

0 mm

0 mm

5 mm

5 mm

(b)

Figure 4.3. Comparison of the transmission and reflection coefficients (phase) foracrylic when the parameter δ is varied from 0 to 5 mm in ten steps. (ε = 2.5,µ = 1).

88

8 9 10 11 120

5

10

15

20

25

30

Frequency (GHz)

% E

rro

r

Percent Error on Transmission Coefficient, Magnitude

0% @ 0 mm

5 mm

5 mm

(a)

8 9 10 11 12

0

2

4

6

8

10

12

14

16

Percent Error on Reflection Coefficient, Magnitude

Frequency (GHz)

% E

rro

r

5 mm

0% @ 0mm

5 mm

(b)

Figure 4.4. Percent error on the S-parameter magnitude resulting from shifting thecenter of the acrylic material sample from δ = 0 to δ = 5 mm.

89

8 9 10 11 12

0

2

4

6

8

10

12Percent Error on Transmission Coefficient, Phase

Frequency (GHz)

% E

rro

r

5 mm

0% @ 0mm

(a)

8 9 10 11 120

2

4

6

8

10

12

14

Frequency (GHz)

% E

rro

r

Percent Error on Reflection Coefficient, Phase

0% @ 0 mm

5 mm

5 mm

(b)

Figure 4.5. Percent error on the S-parameter phase resulting from shifting the centerof the acrylic material sample from δ = 0 to δ = 5 mm.

90

8 9 10 11 122

2.2

2.4

2.6

2.8

3

Frequency (GHz)

Extracted Relative Permittivity for Acrylic

5 mm

5 mm

(a)

8 9 10 11 120

5

10

15

20

25

Frequency (GHz)

% E

rro

r

Error on Extracted Permittivity for Acrylic

5 mm

5 mm

0% error @ 0 mm

(b)

Figure 4.6. (a)Extracted relative permittivity ε for an acrylic sample when the pa-rameter δ is increased from δ = 0 to δ = 5 mm, (b) error.

91

8 9 10 11 120.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Transmission Coefficient, Magnitude

Frequency (GHz)

5 mm

5 mm 5 mm

0 mm 0 mm

(a)

8 9 10 11 120.3

0.4

0.5

0.6

0.7

0.8

0.9

1Reflection Coefficient, Magnitude

Frequency (GHz)

5 mm

5 mm

0 mm

(b)

Figure 4.7. Comparison of the transmission and reflection coefficients (magnitude)for alumina when the parameter δ is varied from 0 to 5 mm.

92

8 9 10 11 12

−120

−100

−80

−60

−40

−20

0Transmission Coefficient, Phase

Frequency (GHz)

Deg

rees

5 mm

0 mm

(a)

8 9 10 11 12140

150

160

170

180

190

200Reflection Coefficient, Phase

Frequency (GHz)

Deg

rees

5 mm

0 mm

0 mm

5 mm5 mm

0 mm

(b)

Figure 4.8. Comparison of the transmission and reflection coefficients (phase) foralumina when the parameter δ is varied from 0 to 5 mm.

93

8 9 10 11 120

10

20

30

40

50

60

70Percent Error on Transmission Coefficient, Magnitude

Frequency (GHz)

% E

rro

r

5 mm

5 mm

(a)

8 9 10 11 120

10

20

30

40

50

60

70

Frequency (GHz)

% E

rro

r

Percent Error on Reflection Coefficient, Magnitude

5 mm

5 mm

(b)

Figure 4.9. Percent error on the S-parameter magnitude resulting from shifting thecenter of the alumina material sample from δ = 0 to δ = 5 mm.

94

8 9 10 11 120

10

20

30

40

50

60

70

80

90Percent Error on Transmission Coefficient, Phase

Frequency (GHz)

% E

rro

r

5 mm

(a)

8 9 10 11 120

5

10

15

20Percent Error on Reflection Coefficient, Phase

Frequency (GHz)

% E

rro

r

5 mm

5 mm

(b)

Figure 4.10. Percent error on the S-parameter phase resulting from shifting the centerof the alumina material sample from δ = 0 to δ = 5 mm.

95

8 9 10 11 122

4

6

8

10

12

14

16

Frequency (GHz)

Extracted Relative Permitivity for Alumina

3.5 mm

3.5 mm

3.5 mm

(a)

8 9 10 11 120

10

20

30

40

50

60

70

80

Frequency (GHz)

% E

rro

r

Percent Error on Extracted Permitivity for Alumina

3.5 mm

0% Error @ 0 mm

(b)

Figure 4.11. (a) Extracted relative permittivity ε for an alumina sample when theparameter δ is increased from δ = 0 to δ = 3.5 mm, (b) error.

96

2.6 2.8 3 3.2 3.4 3.6 3.80.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Frequency (GHz)

Transmission and Reflection Coefficients, Magnitude

S11’s

S21’s

0 mm

3.5 mm

3.5 mm

(a)

2.6 2.8 3 3.2 3.4 3.6 3.8180

200

220

240

260

280

300

320

340

Frequency (GHz)

Deg

rees

Transmission and Reflection Coefficients, Phase

S11’s

S21’s

(b)

Figure 4.12. S-parameters for a lossy-magneto-dielectric material (magRAM) sample(a)magnitude and (b) phase.

97

∆Ζο ℓ

a

byz x

(a) AB • • •

A A AB B∆Ζο

ℓ

y z(b)

Figure 4.13. Waveguide with a layered material in the direction of propagation of theincident field.

98

4 6 8 10 12 14 16 18−5

−4.9

−4.8

−4.7

−4.6

−4.5

−4.4

−4.3

Number of Layers

dB


S21, FEMS21, Wave Matrix

(a)

4 6 8 10 12 14 16 18−2

−1.95

−1.9

−1.85

−1.8

−1.75

−1.7

−1.65Reflection Coefficient, Magnitude

Number of Layers

dB


(b)

Figure 4.14. S-Parameters (magnitude) for a perpendicularly layered material. Ma-terial A: (ε = 9− j0.0027, µ = 1), Material B: (ε = 1, µ = 1).

99

4 6 8 10 12 14 16 18276

277

278

279

280

281

282Transmission Coefficient, Phase

Number of Layers

Deg

rees


(a)

4 6 8 10 12 14 16 18186

187

188

189

190

191

192

Number of Layers

Deg

rees



(b)

Figure 4.15. S-Parameters (phase) for a perpendicularly layered material. MaterialA: (ε = 9− j0.0027, µ = 1), Material B: (ε = 1, µ = 1).

100

4 6 8 10 12 14 16 185

5.2

5.4

5.6

5.8

6Extracted Effective Relative Permittivity

Number of Layers

Figure 4.16. Extracted relative permittivity for a layered material perpendicular inthe direction of propagation. Material A: (ε = 9−j0.0027, µ = 1), Material B: (ε = 1,µ = 1).

101

4 6 8 10 12 14 16 18−4.2

−4

−3.8

−3.6

−3.4

−3.2

Number Of Layers

dB



(a)

4 6 8 10 12 14 16 18

−2.8

−2.7

−2.6

−2.5

−2.4

−2.3

−2.2

−2.1

Number of Layers

dB



(b)

Figure 4.17. S-Parameters (magnitude) for a perpendicularly layered material. Ma-terial A: (ε = 1, µ = 1), Material B: (ε = 9− j0.0027, µ = 1).

102

4 6 8 10 12 14 16 18284

285

286

287

288

289

290

291

292

Number of Layers

Deg

rees



(a)

4 6 8 10 12 14 16 18194

195

196

197

198

199

200

201


Number of Layers

Deg

rees


(b)

Figure 4.18. S-Parameters (phase) for a perpendicularly layered material. MaterialA: (ε = 1, µ = 1), Material B: (ε = 9− j0.0027, µ = 1).

103

4 6 8 10 12 14 16 184

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9Extracted Effective Relative Permittivity

Number of Layers

Figure 4.19. Extracted relative permittivity for a layered material perpendicular inthe direction of propagation. Material A: (ε = 1, µ = 1), Material B: (ε = 9−j0.0027,µ = 1).

104

4 6 8 10 12 14 16 184

4.5

5

5.5

6Extracted Relative Permittivity for the Two Cases

Number of Layers

A : High−EpsilonB : Low−Epsilon

A : Low−Epsilon B : High−Epsilon

Asymptotic Approx.for Eps.

Figure 4.20. Extracted relative permittivities for a layered material perpendicularto the direction of propagation and the asymptotic permittivity for a homogenizedmaterial. The plot on top shows the result when Material A has a higher permittivity,the plot on the bottom Material A with a lower permittivity: (ε = 1, µ = 1),(ε = 9− j0.0027, µ = 1).

105

∆Yο ℓ

a

byz x

(a)

a

b

x

y

•A

AB•••••

(b)

Figure 4.21. Waveguide with a layered material parallel to the direction of propaga-tion. Horizontal layering.

106

ℓ

a

byz x

∆Xο

(a)

a

b

x

y

A AB A B

(b)

Figure 4.22. Waveguide with a layered material parallel to the direction of propaga-tion. Vertical layering.

107

8 9 10 11 120.55

0.6

0.65

0.7

0.75

Frequency (GHz)


3 Layers

13 Layers

5 Layers


8 9 10 11 12

0.7

0.75

0.8

0.85

Frequency (GHz)


3 Layers 5 Layers

13 Layers


Figure 4.23. S-Parameters (magnitude) for a horizontally layered material. MaterialA: (ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1).

108

8 9 10 11 12−100

−95

−90

−85

−80

−75

−70

−65Transmission Coefficient, Phase

Deg

rees

Frequency (GHz)

13 Layers

3 Layers

5 Layers


8 9 10 11 12170

175

180

185

190

195

200

205

Frequency (GHz)

Deg

rees


3 Layers 5 Layers

13 Layers


Figure 4.24. S-Parameters (phase) for a horizontally layered material. Material A:(ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1).

109

8 9 10 11 12

4

4.5

5

5.5

Frequency (GHz)

Extracted Effective Relative Permittivity

3 layers

5 layers

13 layers

7 layers

Figure 4.25. Extracted relative permittivity for a layered material parallel to thedirection of propagation. Horizontal layering. Material A: (ε = 9 − j0.0027, µ = 1),Material B: (ε = 2.5, µ = 1).

110

8 9 10 11 12

0.68

0.7

0.72

0.74

0.76

0.78


Frequency (GHz)

3 layers

5 layers

13 layers


8 9 10 11 120.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

Frequency (GHz)


13 layers

3 layers

5 layers


Figure 4.26. S-Parameters (magnitude) for a horizontally layered material. MaterialA: (ε = 2.5, µ = 1), Material B: (ε = 9− j0.0027, µ = 1).

111

8 9 10 11 12−85

−80

−75

−70

−65

−60

Deg

rees


Frequency (GHz)

3 layers

5 layers

13 layers


8 9 10 11 12185

190

195

200

205


Deg

rees

Frequency (GHz)

3 layers

13 layers


Figure 4.27. S-Parameters (phase) for a horizontally layered material. Material A:(ε = 2.5, µ = 1), Material B: (ε = 9− j0.0027, µ = 1).

112

8 9 10 11 123.4

3.45

3.5

3.55

3.6

3.65

3.7

3.75

3.8

Extracted Relative Permittivity

Frequency (GHz)

3 layers

13 layers

5 layers

Figure 4.28. Extracted relative permittivity for a layered material parallel to thedirection of propagation. Horizontal layering. Material A:(ε = 2.5, µ = 1), MaterialB:(ε = 9− j0.0027, µ = 1).

113

8 9 10 11 120.45

0.5

0.55

0.6


Frequency (GHz)

13 layers

3 layers 3 layers

5 layers

7 layers 9 layers

(a)

8 9 10 11 120.75

0.8

0.85

0.9Reflection Coefficient, Magnitude

Frequency (GHz)

3 layers

13 layers

5 layers

7 layers 9 layers

(b)

Figure 4.29. S-Parameters (magnitude) for a vertically layered material. Material A:(ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1).

114

8 9 10 11 12

−100

−95

−90

−85

−80


Frequency (GHz)

Deg

rees

layers 3

layers 5

layers 13layers 7

(a)

8 9 10 11 12170

175

180

185

190

195


Frequency (GHz)

Deg

rees

layers 13

layers 3

layers 3

layers 7

layers 5

(b)

Figure 4.30. S-Parameters (phase) for a vertically layered material. Material A:(ε = 9− j0.0027, µ = 1), Material B: (ε = 2.5, µ = 1).

115

8 9 10 11 125.7

5.8

5.9

6

6.1

6.2

6.3

6.4

6.5Extracted Relative Permittivity

Frequency (GHz)

Layers: 3Layers: 5Layers: 7Layers: 9Layers: 11Layers: 13 Layers: 11

Layers: 13

Layers: 9

Figure 4.31. Extracted relative permittivity for a layered material parallel to thedirection of propagation. Vertical layering. Material A: (ε = 9 − j0.0027, µ = 1),Material B: (ε = 2.5, µ = 1).

116

8 9 10 11 120.4

0.45

0.5

0.55

0.6

0.65

Frequency (GHz)


layers 3

layers 5

layers 7

layers 13

(a)

8 9 10 11 120.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9


Frequency (GHz)

layers 3

layers 5

layers 7

layers 13

(b)

Figure 4.32. S-Parameters (magnitude) for a vertically layered material. Material A:(ε = 2.5, µ = 1), Material B: (ε = 9− j0.0027, µ = 1)

117

8 9 10 11 12−115

−110

−105

−100

−95

−90

−85

−80

−75Transmission Coefficient, Phase

Frequency (GHz)

Deg

rees

layers 3

layeres 5

layers 13

(a)

8 9 10 11 12150

155

160

165

170

175

180

185

190


Frequency (GHz)

Deg

rees

layers 3

layers 5

layers 13

(b)

Figure 4.33. S-Parameters (phase) for a vertically layered material. Material A:(ε = 2.5, µ = 1), Material B: (ε = 9− j0.0027, µ = 1).

118

8 9 10 11 12

5.5

6

6.5

7

7.5

8

8.5

9Extracted Relative Dielectric Permittivity

Frequency (GHz)

layers 3

layers 5

layers 7layers 9

layers 13

Figure 4.34. Extracted relative permittivity for a layered material parallel to thedirection of propagation. Vertical layering. Material A: (ε = 2.5, µ = 1), Material B:(ε = 9− j0.0027, µ = 1).

119

8 9 10 11 12−9

−8

−7

−6

−5

−4

−3

−2

Frequency (GHz)

dB


(a)

8 9 10 11 12

−22

−20

−18

−16

−14

−12

−10

−8

−6

Frequency (GHz)


dB

(b)

Figure 4.35. Transmission and Reflection Coefficients (magnitude) for a magnetizedferrite. 4πMs = 5000 Gauss, ∆H = 500 Oe, Ha = 200 Oe, Ho = 100 Oe.

120

8 9 10 11 12−9

−8

−7

−6

−5

−4

−3

−2


Frequency (GHz)

dB

(a)

8 9 10 11 12−25

−20

−15

−10

−5Reflection Coefficient, Magnitude

Frequency (GHz)

dB

(b)


121

8 9 10 11 12−9

−8

−7

−6

−5

−4

−3

−2

Frequency (GHz)

dB


(a)

8 9 10 11 12−30

−25

−20

−15

−10

−5

Frequency (GHz)

dB


(b)


122

8 9 10 11 12−9

−8

−7

−6

−5

−4

−3

−2


Frequency (GHz)

dB

(a)

8 9 10 11 12

−40

−35

−30

−25

−20

−15

−10

Frequency (GHz)

dB


(b)


123

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

In the present work, the FEM was used to assess the error originated from misplacing

a material sample within a waveguide and whose constitutive parameters are being

extracted by using the algorithm described in [10]. It was found that for low-contrast

materials the repercussions of shifting the sample are tolerable if displacements are

present. A maximum error of 22% was found at 5 mm. For a high-contrast material,

however, errors of nearly 80% were found for displacements of 3.5 mm. The inversion

algorithm did not converge when greater displacement were simulated. It was also

found that no matter how far the sample is placed from the center of the waveguide,

there is always at least a frequency at which the error in non-existent. The shape of

the frequency characteristic of the error distribution for the extracted relative permit-

tivity is heavily determined by the percent change of the magnitude of its reflection

and transmission coefficients. When dealing with a magneto-dielectric material (Ma-

gRAM), there is no significant change of its S-parameters when the sample is shifted

from the center of the waveguide. The numerical tool developed proved to be very

effective in measuring the reflection and transmission coefficients of ferrite samples as

well as layered materials.

Additional cases that represent important sources of error for [10] remain to be

simulated. This sources include the geometry of the material sample not being per-

fectly rectangular, or the material being dented or chipped.

124

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125

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