THREE-DIMENSIONAL FINITE-DIFFERENCE TIME-DOMAIN SIMULATION

OF COAXIAL TRANSMISSION LINE FOR BROADBAND DIELECTRIC

CHARACTERIZATION

SARIMAH BINTI NOOR AHMAD

UNIVERSITI TEKNOLOGI MALAYSIA

THREE-DIMENSIONAL FINITE-DIFFERENCE TIME-DOMAIN SIMULATION

ON COAXIAL TRANSMISSION LINE FOR BROADBAND DIELECTRIC

CHARACTERIZATION

SARIMAH BINTI NOOR AHMAD

A thesis submitted in fulfillment of the

requirements for the award of the degree of

Master of Science (Physics)

Faculty of Science

Universiti Teknologi Malaysia

AUGUST 2014

iii

To my beloved parents,

Father, Noor Ahmad Bin Sakiban

Mother, Mestijah Binti Md Said

Thanks for all the sacrifies and love…

Dedicated to all my friends,

Izzati, Diana, Farhah, Asmahani, lab mates of computer instrumentation, and my

husband Muhd Dinie

For the support given will not forgotten

Thank you for supporting me

iv

ACKNOWLEDGEMENT

Bismillahirrahmanirrahim, Alhamdulillah.

Thanks to God the almighty for grace and mercy for His guidance and giving

me the strength to complete this project.

Special appreciation goes to my supervisor, Dr. Amiruddin Bin Shaari for his

supervision, advice and guidance from the early stage of this project as well as

giving me extraordinary experiences throughout the work. Sincere thanks to all my

friends and everyone that helped me throughout the dissertation, for their kindness,

and moral support during my study.

My deepest and special appreciation to my beloved family, my parents, Noor

Ahmad Bin Sakiban and Mestijah Binti Md Said for their support, endless love,

prayers, and encouragement from the beginning until the end of this project.

Finally, my further appreciation dedicated to the Ministry of High Education

for providing part of the financial support through GUP funding vot 03J69. To those

who indirectly contribute in this project, your kindness means a lot to me. May God

reward their services and sacrifies. God willings.

Sarimah, UTM Skudai

v

ABSTRACT

In this work, a part of a coaxial transmission line is used as a sample holder

where the propagation of electromagnetic waves in the range of 500 MHz is studied

using three-dimensional (3D) finite-difference time domain (FDTD) method. This

study presents the results from the numerical simulations of electromagnetic waves in a

mixture of dielectric materials. The effective relative permittivity of the mixture is

calculated by recording one of the electric field components (Ey) of the transmitted and

reflected electromagnetic pulses in the transmission line. The complex frequency

spectra of these time-domain signals are then obtained by taking the Fourier transforms

of the respective signals. These spectra are then used to calculate the complex

transmission and reflection coefficients for the sample. The analysis of raw data is

performed using open source package, GNU Octave. Finally a numerical procedure is

developed to convert the raw data into an effective dielectric property of the mixture of

materials. The influence of water contents on dielectric properties is studied using

samples made from different mixtures of soil, water, and air. The results show that the

effective dielectric permittivities of the mixtures are highly dependent on the soil’s

moisture content. Strong frequency dependence in the dielectric properties is observed

especially at the low end of frequency range which can be attributed to the presence of

the DC conductivity of water (5 𝜇S/m) in the mixture. In general the results are

consistent with those calculated using Maxwell-Garnett mixing formula especially at

the high end of the frequency range.

vi

ABSTRAK

Dalam kajian ini, sebahagian daripada talian penghantaran sepaksi digunakan

sebagai pemegang sampel di mana perambatan gelombang elektromagnet dalam julat

500 MHz dikaji dengan menggunakan kaedah perbezaan terhingga domain masa

(FDTD) tiga dimensi (3D). Kajian ini membentangkan hasil simulasi berangka daripada

gelombang elektromagnet dalam campuran bahan dielektrik. Ketelusan relatif berkesan

campuran dihitung dengan merekodkan salah satu komponen medan elektrik (Ey) dalam

denyut elektromagnet yang dipancar dan dipantulkan dalam talian penghantaran.

Spektrum frekuensi kompleks bagi isyarat domain masa ini diperoleh dengan

menggunakan transformasi Fourier signal berkaitan. Spektrum ini kemudiannya

digunakan untuk menghitung pekali pantulan dan penghantaran kompleks bagi sampel.

Semua analisis data asal ini dilakukan dengan menggunakan pakej sumber terbuka,

GNU Octave. Kemudian, suatu prosedur berangka dibangunkan untuk menukar data

asal kepada sifat dielektrik berkesan campuran bahan. Pengaruh kandungan air pada

sifat dielektrik dikaji menggunakan sampel yang diperbuat daripada campuran tanah,

air, dan udara yang berbeza. Keputusan menunjukkan bahawa ketelusan dielektrik

berkeasan bagi campuran adalah amat bergantung kepada kandungan kelembapan

tanah. Pergantungan kuat terhadap frekuensi dalam sifat dielektrik dapat diperhatikan

terutamanya di hujung julat frekuensi rendah yang boleh dikaitkan dengan kehadiran

kekonduksian DC air (5 𝜇S/m) dalam campuran. Secara umumnya, keputusan adalah

konsisten dengan hasil yang dihitung menggunakan formula percampuran Maxwell-

Garnett terutamanya di hujung julat frekuensi tinggi.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES xi

LIST OF FIGURES xii

LIST OF SYMBOLS xvi

LIST OF ABBREVIATIONS xix

LIST OF APPENDICES xx

1 INTRODUCTION

1.1 Background of Study 1

1.1.1 Effective Permittivity of Dielectric Mixtures 2

1.1.2 Transmission Line Method 3

1.2 Problem Statement 4

1.3 Objective of Research 4

1.4 Scope of Study 5

viii

1.5 Significance of Study 5

2 THEORY

2.1 Electromagnetic Theory 6

2.2 Coaxial and Shielding of Transmission Lines 8

2.3 Principle of The Transmission Line Method 11

2.4 Scattering Matrices S-Parameters 14

2.5 Plane Wave Reflection and Transmission at an

Interface

16

2.6 Material Properties 18

2.6.1 Material Classifications 19

2.6.2 Intrinsic Impedance and Propagation

Coefficient

20

2.7 Dielectrics, Permittivity, and The Electric Field in a

Dielectric

20

2.7.1 Dielectric Properties of Inhomogeneous

Media

21

2.7.2 Dielectric Properties Measuring Technique 22

2.8 Transmission and Reflection Line Measurements

Method

24

2.9 Dielectric Mixtures 25

2.9.1 Three- Dimensional Mixing Models 27

2.9.2 Theoretical Bounds for The Effective

Permittivity

28

2.9.3 Principle of Numerical Mixture Calculations 28

2.9.4 Maxwell-Garnett Formula for Multiphase

Composites

30

2.10 FDTD Simulation 32

2.10.1 Time Domain Measurements 33

2.10.2 Stability Requirements and Boundary

Conditions

34

ix

3 METHODOLOGY

3.1 EastFDTD 36

3.2 Modelling Steps 37

3.2.1 Modelling In Windows 37

3.3 Simulation Steps 38

3.3.1 Step 1: Establish The Project and Set Project

Properties

38

3.3.2 General Parameters 39

3.3.3 Boundary Parameters 40

3.3.4 Calculation Domain and Mesh Grid Setting 42

3.3.5 Calculation Model 44

3.4 Step 2: Establish Simulation Model 45

3.5 Step 3: Create Source 47

3.6 Step 4: Create Records and Data PostProcessing 48

3.6.1 Point 48

3.7 Source (Natural Parameter in The Domain) 49

3.7.1 Gaussian Pulse 49

3.7.2 Spatial Character Parameter 50

3.7.2.1 3D Gaussian Beam 50

3.8 Utilities 52

3.8.1 Calculation In LINUX 52

3.8.2 Visualization Tools 53

3.8.3 Calculating The Transmission and

Reflectance Spectra

53

3.9 Transmission Line Measurements 55

3.9.1 Transmission Line Modeling 55

3.10 Measuring S-parameters From FDTD Simulation 60

x

3.10.1 Effective Material Properties 61

3.11 Three - Dimensional Mixing Models 62

3.11.1 Simulation Setup 62

3.12 Random Mixtures 63

3.12.1 Local Permittivity Near Material Interfaces 64

4 RESULTS

4.1 Properties of TEM Waves in Coaxial Transmission

Line

66

4.1.1 TEM Waves Along z-axis on The

Transmission Line

67

4.1.2 TEM Waves around of The Transmission

Line

69

4.1.2.1 Three Points Check TEM Waves at

Four Difference Axis (X and Y)

69

4.2 Air–Filled Transmission Line 70

4.3 Material-Filled Transmission Line 71

4.4 Calculations of Transmission and Reflection

Coefficient

72

4.5 Random Mixtures 77

4.5.1 Mixture without Clustering (Dilute Mixture) 77

4.5.2 Three Phase Mixture 78

4.6 Soil and Sand Mixture Models 79

4.7 Wet Granular Materials 83

5 CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions 90

5.2 Recommendations 92

REFERENCES 94

Appendices A - F 99 - 112

xix

LIST OF ABBREVIATIONS

ABC - Absorbing Boundary Conditions

3D - Three dimensional

FDTD - Finite Difference Time Domain

PML - Perfectly match layer

UPML - Unixial anisotropic perfectly match layer

PEC - (Ideal electric conductor)

PMC - (Ideal magnetic conductor)

ta - Polarization in the next direction of the axis of the light incident

axis

tb - Polarization in the more next direction of the axis of the light

incident axis

ra - Reflection in the next direction of the axis of the light incident

axis

rb - Reflection in the more next direction of the axis of the light

incident axis

TEM - Transverse electromagnetic

xx

LIST OF APPENDICES

APPENDIX TITLE PAGE

A INSTALLATIONS 99

B INPUT SOURCE 101

C RAW DATA OUTPUT 102

D GNU OCTAVE SCRIPT 103

E CALCULATIONS (DIELECTRIC PERMITTIVITY

AND CONDUCTIVITY)

108

F MATERIAL-FILLED RESULTS 112

xii

LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 Cross section of coaxial cable 8

2.2 Coaxial line (shielded) 9

2.3 E and H field in a coaxial transmission line 13

2.4 Signal Flows in a Two-Port Network 15

2.5 Plane wave reflection and transmission in a medium

interface

18

3.1 Linux server IP setting dialog box 37

3.2 EastFDTD modeling interface 37

3.3 Create project document 38

3.4 Global preferences setting 40

3.5 Specific parameter setting 41

3.6 Computational domain and time setting 43

3.7 Calculation modes 44

3.8 Symbol setting 45

3.9 Material parameter setting 46

3.10 Structure parameter setting 46

3.11 Gauss source parameter setting 47

3.12 Point record setting 48

3.13 The time domain parameter meaning of Gaussian pulse. 50

3.14 Parameter setting of Gaussian pulse 51

3.15 The field distribution schematic diagram of the field of each

component of the electric field at the cross section of beam-

waist of 3D Gaussian beam.

52

3.16 The styles of data files 54

3.17 Dimensions of coaxial transmission line sample holder 56

xiii

3.18 Three different readings at the X-axis (620 mm, 500 mm,

and 580 mm) to calibrate the simulation process on the

coaxial transmission line holder. Five difference points to

record each fields along the transmission lines at z-axis

where Point 1 (290 mm), Point 2 (445 mm), Point 3 (600

mm), Point 4 (755mm), and Point 5 (910 mm)

57

3.19(a) Four different points along the circular Z-axis to calibrate

the simulation process on the coaxial transmission line

holder. Four difference points (Point A, B, C and D) to

record each fields around the center of cylinder model (600

mm along z-axis)

57

3.19(b) Three different points along the Z-axis to calibrate the

simulation process on the coaxial transmission line holder

(Point A at 350 mm along z-axis, Point B at 600 mm along

z-axis and Point C at 850 mm along z-axis.

58

3.20 Empty specimen coaxial transmission line holder 58

3.21 Coaxial transmission line holders with presence of material 59

3.22 Dielectric conformal approximations for a small gap 65

4.1 The signal at Point 3 at 600 mm along z-axis for reading 1 67

4.2 The signal from five different points on all three readings

(P=Point record)

68

4.3 The signal at Point B along the z-axis (-X, +X, -Y, and +Y

axis) to record each fields around the center of cylinder

model (600 mm along z-axis)

69

4.4 The signal at Point A (350 mm), Point B (600 mm) and

Point C (850 mm) along z-axis

70

4.5 Transmission and reflection signals on the air-filled

transmission line

71

4.6 Transmission and reflection signals on the material-filled

transmission line

72

4.7 Spectra of reflected signals for the air filled and material 74

xiv

filled transmission lines

4.8 Spectra of transmitted signals for the air filled and material

filled transmission lines

74

4.9 Magnitude of the transmission and reflection coefficients

versus frequency

75

4.10 Phases of the transmission and reflection coefficients versus

frequency

75

4.11 Real (red line) and imaginary (green line) part of the

relative dielectric permittivity calculated from the FDTD

simulations

76

4.12 DC conductivity calculated from the FDTD simulations 76

4.13 The effective relative permittivity of dilute mixtures from

FDTD simulation compared to those mixture formulas at

500 MHz

77

4.14 The effective relative permittivity of three mixtures (air-

solid-water) from FDTD simulation compared to those

mixture formulas at 500 MHz

78

4.15 Frequency dependency of effective relative permittivity of

sandy loam at soil volumetric wetness of 4.3%

79

4.16 Frequency dependency of effective relative permittivity of

sandy loam at soil volumetric wetness of 24.3%

80

4.17 Frequency dependency of the effective relative permittivity

of soils calculated from FDTD simulations

81

4.18 Effective relative complex permittivity on volumetric

wetness at frequency 100 MHz and 400 MHz

82

4.19 Effective relative complex permittivity of sand

dependencies on volumetric wetness at a frequency of 100

MHz and 400 MHz

82

4.20(a) Measured relative permittivity vs. volumetric moisture

content of pure sand from FDTD simulation compared to

Maxwell-Garnett formula at 400 MHz frequency

84

xv

4.20(b) Measured relative permittivity vs. volumetric moisture

content of pure sand from FDTD simulation compared to

Maxwell-Garnett formula at 100 MHz frequency

84

4.21(a) Measured relative permittivity vs. volumetric moisture

content of sand + 10% clay from FDTD simulation

compared to Maxwell-Garnett formula at 400 MHz

frequency

85

4.21(b) Measured relative permittivity vs. volumetric moisture

content of sand + 10% clay from FDTD simulation

compared to Maxwell-Garnett formula at 100 MHz

frequency

85

4.22(a) Measured relative permittivity vs. volumetric moisture

content of sand + 20% clay FDTD simulation compared to

Maxwell-Garnett formula at 400 MHz frequency

86

4.22(b) Measured relative permittivity vs. volumetric moisture

content of sand + 20% clay FDTD simulation compared to

Maxwell-Garnett formula at 100 MHz frequency

86

4.23(a) Conductivity vs. volumetric moisture content of pure sand

from FDTD simulation compared to Maxwell-Garnett

formula at 400 MHz frequency

87

4.23(b) Conductivity vs. volumetric moisture content of sand + 10%

clay from FDTD simulation compared to Maxwell-Garnett

formula at 400 MHz frequency

88

4.23(c) Conductivity vs. volumetric moisture content of sand + 20%

clay from FDTD simulation compared to Maxwell-Garnett

formula at 400 MHz frequency

88

xvi

LIST OF SYMBOLS

α - Polarizability

β - Fitting parameter for constant phase element

ɛ - Dielectric permittivity or dielectric constant

ɛr - Complex dielectric permittivity

ɛeff - Effective dielectric permittivity

σ - Conductivity

μ - Relative permeability

μr - Complex relative permeability

μeff - Effective relative permeability

n - Refractive index

neff - Effective refractive index

λ - Wavelength

λn - Wavelength in the medium

δ - Loss angle

c - Speed of light in vacuum or air (300mm/ns)

v - Phase velocity

xvii

π - Radian

θ - Phase angle

t - Time

k - Phase constant or wave number

f - Frequency

fc - Centre frequency

fco - Cut – off frequency

fmax, fm - Maximum frequency

fmin - Minimum frequency

Γ - Reflection coefficient

Δ - Delta or interval

Δt - Time interval

ω - Angular frequency

rin - Inner radius

rout - Outer radius

Nt - Fitting parameter for constant phase element

B - Magnetic induction vector

D - Electric displacement vector

E - Electric field

Ei - Incident electric field

Er - Reflected electric field

xviii

Et - Transmitted electric field

H - Magnetic field

J - Current density vector

L - Length of transmission line

S - Pointing vector

S11 - Reflection coefficient

S21 - Transmission coefficient

│S11│ , │S21│- Magnitude of the S-parameters (Complex S-parameters)

T - Transmission coefficient

Z - Impedance

z - Effective impedance

xi

LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 Comparison Between The Measurement Methods 23

2.2 Polarization Factors For Ellipsoids 31

4.1 Composition For Soil Model 84

F.1 Attenuation and Relative Permittivity of Subsurface

Material Measured at 500 MHz

113

F.2 Dielectric Constant of Soil at Various Porosity Values 114

1

CHAPTER 1

INTRODUCTION

1.1 Background of Study

Accurate extraction of reflection coefficient, 𝛤 from finite-different time-domain

(FDTD) simulations is an important practical problem. The value of 𝛤 is used to obtain

the S-parameters of the analysed circuit. A common approach consists of matching the

outputs by means of high quality absorbing boundary conditions (ABC’s) which

permits to assume 𝑆11 to be equal to 𝛤 at the input. Recently, a more elaborate method

has been proposed by Sharkov in 1995, which multiple calculations of reflection

coefficient at different ports allow to obtain the S parameters with good accuracy, even

for relatively poor ABC’s. In either case, the main problem of Γ extraction resides in

separating the incident and reflected waves at the considered port. To solve this

problem, two basic approaches have been reported so far (Sharkov, 1995).

The first approach consists of running the simulation twice or concurrently on

two models, one run is for the analyzed circuit and the other for a nonreflecting

structure with the identical input. The second simulation provides a pure incident wave.

It can be reduced to the two-dimensional (2D) simulation of the cross section of the line

(V.J.Brankovic, 1995), but at the expense of repcating it independently for each relevant

frequency. Although good results in applying this approach have been reported, it is

2

clear advantage is the necessity of double simulation, which significantly increases the

employed computer resources and complicates the use of program.

The second approach (W.K. Gwarek, 1988) assumes that for reference, either

wave impedance or characteristic impedance of the line is a priori known (analytically

or heuristically). After application of correction factors due to the shifts between the E

and H fields in time and space, this method gives very good results for homogeneously

filled transverse electromagnetic (TEM) lines (J. Fang, 1995) which are when the wave

impedance does not change with frequency or waveguides when it changes according to

an analytically known rule. However, in application to inhomogeneous quasi-TEM

lines, even if the quasi-static wave impedance is known, significant errors appear due to

dispersive properties that are difficult to predict.

1.1.1 Effective Permittivity of Dielectric Mixtures

The study of the electromagnetic behavior and characterization of dielectric

mixtures is a classical problem which was already been addressed by Maxwell more

than a century ago. Nowadays, this initial interest has not decayed at all. On the

contrary, new artificial materials have renewed and drastically increased this attention

due to the design of new and even unknown applications non-existing in natural media.

The quasi-static initial works have been substituted or completed with high frequency

applications, such as perfect lens manufacturing, sub-wavelength microwave devices,

enhanced radiation by small antennas and others.

In addition, the availability of powerful computers has also modified the way in

which these and other related problems are addressed. In this sense, the first attempts to

model dielectric mixtures were concentrated on obtaining theoretical formulas to predict

the effective permittivity of composite materials consisting of mixtures with several

homogeneous components.

3

The Maxwell Garnett (MG) mixing rule yields;

𝜀𝑒𝑓𝑓 − 𝜀2

𝜀𝑒𝑓𝑓 + (𝑑 − 1)𝜀2= 𝑝1

𝜀1 − 𝜀2

𝜀1 + (𝑑 − 1)𝜀2 (1.1)

Another famous mixing rule is the Bruggeman formula (D.A.G Bruggeman, 1935);

𝑝1 𝜀1 − 𝜀𝑒𝑓𝑓

𝜀1 + 𝑑 − 𝑎 𝜀𝑒𝑓𝑓= 𝑝2

𝜀2 − 𝜀𝑒𝑓𝑓

𝜀2 + 𝑑 − 1 𝜀𝑒𝑓𝑓 (1.2)

In both equations, the dimensionally d equals 2 if the problem is two-dimensional or

equal to 3 in the case of considering a three-dimensional media. Besides the two simple

approximation described above, many other mixing models exists. Nevertheless, it can

be shown that in most case, it is impossible to completely determine the effective

permittivity by the volume fractions and effective permittivity of phases.

1.1.2 Transmission Line Method

The transmission line method (P.B. Johns, 1971) is used for the modeling of

composite mixtures. As it happens with the FDTD method, transmission line method is

a low frequency numerical method which has been extensively used for the modeling of

wave propagation problems, mainly electromagnetic nature but also for problems in

acoustic or particle diffusion. The method is not only a numerical model to solve certain

phenomenon, but also a conceptual approach which does not consider analytical

equations governing the phenomenon, but directly considers the original phenomenon

by means of equivalent transmission line. This conceptual nature of transmission line

method makes this method a powerful tool transmission which allows considering

challenging problems from a hybrid numerical-theoretical point of view in an elegant

and suitable way.

4

1.2 Problem Statement

Beginning with the development of finite difference equations and leading to the

complete FDTD algorithm, the FDTD modeling of problems can be undertaken with

computer resources readily available to individual user. FDTD methods are relatively

forward and intuitively follow from a physical understanding of Maxwell’s equation.

The full range of useful quantities can be calculated such as the transmission or

reflection of light. In this research, the suitable coaxial transmission line for broadband

materials for high efficiency will be finder. To find the suitable one, it needs to do a lot

of sample and consume a lot of time to make the sample and test it. So, by simulating

each material, it can compute and found the better sample holder for broadband

material. It also include under several theoretical approaches for modeling dielectric

properties of heterogeneous media. Strong frequency dependence in the dielectric

properties is observe especially at the low and high end of frequency range which can

be attribute to the presence of the DC conductivity of water in the mixture. Effective

dielectric properties of heterogeneous mixtures can be obtained from FDTD simulation

using wave propagation on coaxial transmission line as a sample holder.

1.3 Objective of the Research

The main objectives of this study are:

1. To simulate the effectiveness of the coaxial transmission line for characterization of

dielectric materials for over a range frequency using FDTD method.

2. To determine the broadband dielectric properties of heterogeneous materials using

part of the transmission line as a sample holder.

3. To compare the effectiveness of dielectric properties obtained to those calculated

using mixture formula.

5

1.3 Scope of Study

This research are studied using three-dimensional finite-difference time-domain

(3D FDTD) method comes with commercial package otherwise known as EastFDTD.

This research is using a transmission line model as a sample holder and uses the input

for range of frequencies 100-500 MHz. This research includes three mixtures using

sand, water, air, and soil models.

1.5 Significance of Study

FDTD methods are well suited for analysis of problems with complex

geometrical features as well as those containing arbitrarily inhomogeneous materials.

FDTD method actually can save more costs than the vector analyzer network (VNA)

and others method to get the propagation electromagnetic wave due to the different

frequencies. Also, the FDTD method does not require the derivation of a Green’s

function or the solution of matrix equation. FDTD methods have emerged as the

methods with arguably the broadest range of applicability. This is especially true for

electromagnetic problems involving complex and dispersive media, photonics

applications, and modeling of high speed circuits and devices.

94

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