MUHAMMAD OMER FAROOQ, MSc Dissertation, Communication-Engineering, 2009, Student ID 7367100 Complete

8/19/2019 MUHAMMAD OMER FAROOQ, MSc Dissertation, Communication-Engineering, 2009, Student ID 7367100 Complete

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Complex Permittivity of Dielectric Materials with

Periodic Discontinuities

A dissertation submitted to The University of

Manchester for the degree of Master of Science in the

Faculty of Engineering and Physical Sciences

2009

Muhammad Omer FAROOQ.

Student ID: 7367100


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Table of Contents

Table of Contents

Section Page Number

List of Figures……………………………………………………………....4

List of Tables………………………………………………………………..7

Abstract………………………………………………………………........11

Declaration………………………………………………………..……….12

Copyright………………………………………………………………….12

Acknowledgements………………………………………………………..13

Abbreviations……………………………………………………………...14

Symbols……………………………………………………………………16

1. Introduction............................................................................................18

1.1. Aims and Objectives of the project…...…….……………......…….20

2. Literature Review……………………………………………………...22

3.

Back-Ground Theory………………………………………..………...26

3.1. Dielectrics, Polarization and Permittivity…………..………...........26

3.1.1. Basic Concept of Polarization………………………………..27

3.1.2. Dipole in Time Harmonic Field………………………………29


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Table of Contents

3.4.2. TM Mode…………………………………………...………...40

3.5.

Scattering by Conducting Wedge……………...……….………….42

3.6. Maxwell-Garnett Mixing Theory…………………....……...……..43

3.7. S-Parameters...…………………………………………...………..44

4. Extraction of Complex Permittivity from S-Parameters…………...47

4.1. Two Independent Ways of Obtaining S-Parameters…………........47

4.2. Conversion of Scattering Matrix to ABCD Matrix………………..48

4.3. Example of 4-Layer Problem……………………………………...49

4.4. Deembedding of ABCD Matrix of Profiled Layer………...………50

4.5. Calculation of Complex Permittivity of Profiled Layer from its

ABCD Matrix………………………………….................................51

4.6. Software Used for the Calculation of Complex Permittivity……...53

5.

HFSS Simulation and Its Results..........................................................54

5.1. Introduction to the Software…………...…………………………..54

5.2.

Technique Used………………………………………………........54

5.2.1. The Finite Element Method…………………………………..54

5.2.2. Size of Mesh vs. Accuracy……………………...………........55

5.3. Meshing and its Effect on Simulation Results…...………………...55

5.3.1. General Observation during the simulation…………………..55

5.4. Developing the Model in HFSS and Results….……………….......56

5.4.1. Slots…………………………………………………………..57

5.4.2. 60 Degree Grooves…………………………………………...61


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Table of Contents

6.4. Different Sizes and Shapes of the Interface……………..…..…….73

6.4.1.

Rectangular slots…………………………………………......74

6.4.2. 60 Degrees Grooves………………………………………….76

6.4.3. 90 Degree Grooves……………………………………….......77

6.4.4. 120 Degree Grooves………………………………………….78

6.5. Previously Obtained Experimental Results and Problems with

Them……………………………………………………………….80

6.6. Technical Difficulties ……………...……………………………...82

6.7. Steps Involve in this Method ..........................................................83

6.8. Experimental Results……………………………………...………84

7.

Conclusions and Empirical Formulae Derived……………………..92

8. Reference..............................................................................................108

Appendix A Calibration of VNA…................................................................................111

Appendix B Already Known Experimental Results …………………………………..113

Appendix C Example for the Calculation of Complex Permittivity…………………...114

Appendix D C++ Programs Used for calculation of Complex Permittivity ……..........116

Appendix E Simulation Results………………………………………………..............126

Appendix F Experimental Results….………………………………………………….160

Appendix G Datasheet of Perspex….………………………………………………….180

Appendix H Why ( )*Im ε of experimental results are positive in number of

cases………………..................................................................................182


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List of Figures

List of Figures

Figure__ Page Number

Chapter-3

Figure-3.2.1 Polarization of non polar molecule in electric field……………………………27

Figure-3.2.2 Atomic dipole model.........................................................................…...............31

Figure-3.2.3 Oscillation of the electron about the nucleus……………………………………31

Figure-3.2.3 Plot of the real and imaginary parts of the resonant susceptibilityres

χ …………34

Figure-3.5.1 Standard rectangular wave guides……………………………………………….39

Figure-3.5.2 Rectangular Waveguide………………………………………………………….40

Figure-3.6.1 Electric line source near a two dimensional conducting wedge, reference at

bisector………………………………………………………………………...…42

Figure-3.7.1 An arbitrary N-port microwave network..........................................................…..46

Chapter-4

Figure-4.2.1. A two port network……………………………………………………………...48

Figure-4.3.1. 2 and 4-Layer Problem with the plane of calibration specified in each case……49

Chapter-5

Figure-5.4.1. 4-Layers of the problem with specified dimension used in simulating each


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List of Figures

Figure-6.1.3 5.8mm thick Cell which hold the sample inside the waveguide…………………71

Figure-6.1.4 Actual practical setup for the measurement of S-parameters…………………….72

Figure-6.2.1. Perspex prepared sample placed on the cell……………………………………..73

Figure-6.4.1. Picture of rectangular slots used………………………………………………...75

Figure-6.4.2. Profile layer having regular, periodic and rectangular grooves…………………75

Figure-6.4.3. Picture of all the samples of 60 Degrees Grooves………………………………76

Figure-6.4.4. Profile layer having regular and periodic grooves and discontinuities………….77

Figure-6.4.5. Picture of all the samples of 90 Degrees Grooves………………………………78

Figure-6.4.6. Profile layer having regular and periodic grooves and discontinuities………….78

Figure-6.4.7. Picture of all the samples of 120 Degrees Grooves……………………………..79

Figure-6.4.8. Profile layer having regular and periodic grooves and discontinuities with

dimensions……………………………………………………………………….80

Figure-6.5.1. Plot of real part of complex permittivity for slots and grooves of different angles

and material fill factor at 6GHz. In addition to this the Maxwell-Garnett mixing

curve is also plotted which is independent of frequency………………………...82

Figure-6.5.2. Plot of real part of complex permittivity for slots and grooves of different angles

and material fill factor at 8GHz In addition to this the Maxwell-Garnett mixing


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List of Figures

Figure-7.2. Plot of real part of complex permittivity using experimental data, Maxwell-Garnett

mixing rule and amended Maxwell-Garnett rule at 8GHz for the case of

slots………………………………………………………………………………95


mixing rule and amended Maxwell-Garnett rule at 6GHz for the case of 60 degree

grooves…………………………………………………………………………...98



grooves…………………………………………………………….......................98



grooves……………………………………………………..…………………...101



grooves………………………………………………………………………….101


mixing rule and amended Maxwell-Garnett rule at 6GHz for the case of 120

degree grooves………………………………………………………..………...103



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List of Tables

List of Tables

Chapter-3

Table 3.5.1: Characteristics of the rectangular wave guide…..…………………………...…...41

Chapter-5

Table-5.4.1. Complex permittivity of slots with 1.4mm pitch…..……………………………..58

Table-5.4.2. Complex permittivity of slots with 1.7mm pitch…..………………………..........58


Table-5.4.4. Complex permittivity of slots with 2.0mm pitch, with depth of 0.577mm………59

Table-5.4.5. Complex permittivity of slots with 2.5mm pitch…..……………………..............59

Table-5.4.6. Complex permittivity of slots with 3.0mm pitch…..………………..……………59



Table-5.4.9. Complex permittivity of slots with 5.0mm pitch…………………………............60

Table-5.4.10. Complex permittivity of 60 degree grooves having1.2mm pitch……………….61



Table-5.4.13. Complex permittivity of 60 degree grooves having1.7mm pitch………….........62

Table-5.4.14 Complex permittivity of 60 degree grooves having 2.0mm pitch……………….62



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List of Tables



Table-5.4.24 Complex permittivity of 120 degree grooves having4.0mm pitch………………66

Table-5.4.25 Complex permittivity of 120 degree grooves having 4.5mm pitch……………...67





Chapter-6

Table-6.4.1. Slots of different sizes pitches and air fill factor…………………………………74

Table-6.4.2. Grooves with 60 degree angle with different pitches and air fill factor. ………...76

Table-6.4.3. Grooves with 90 degree angle with different pitches and air fill factor………….77

Table-6.4.4. Grooves with 120 degree angle with different pitches and air fill factor………...79

Table-6.8.1. Measured values of Complex permittivity for the case of slots with 4.0mm

pitch………………………………………………………………………………84


pitch………………………………………………………………………………84


pitch………………………………………………………………………………85


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List of Tables

Table-6.8.7. Measured values of Complex permittivity for the case of 60 degrees grooves with

1.3mm pitch……………………………………………………………………87


2.5mm pitch……………………………………………………………………87


3.0mm pitch……………………………………………………………………87

Table-6.8.10. Measured values of Complex permittivity for the case of 90 degrees grooves

with 2.0mm pitch………………………………………………………………88


with 4.0mm pitch………………………………………………………………88


with 5.0mm pitch………………………………………………………………89


with 6.0mm pitch………………………………………………………………89


with 4.0mm pitch………………………………………………………………89


with 5.0mm pitch………………………………………………………………90


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List of Tables

Chapter-7

Table:-7.1. Values of the real pert of complex permittivities for the case of slots obtained from

Experiments, Maxwell-Garnett mixing rule and from Amended Maxwell-Garnett

formula at 6GHz and at 8GHz…………………………………………………...94

Table:-7.2. Values of the real pert of complex permittivities for the case of 60 degrees grooves

obtained from Experiments, Maxwell-Garnett mixing rule and from Amended

Maxwell-Garnett formula at 6GHz and at 8GHz………………………………..97

Table:-7.3. Values of the real pert of complex permittivities for the case of 90 degrees grooves

obtained from Experiments, Maxwell-Garnett mixing rule and from Amended

Maxwell-Garnett formula at 6GHz and at 8GHz……………………….………100

Table:-7.4. Values of the real pert of complex permittivities for the case of 120 degrees

grooves obtained from Experiments, Maxwell-Garnett mixing rule and from

Amended Maxwell-Garnett formula at 6GHz and at 8GHz……………………102

Table-7.5 Amended Maxwell-Garnett formulae in the different cases…………………..…..104

Table-7.6 Comparison between simulation results and experimental one…………………...105


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Abstract

Abstract

When a waveguide cell is used to measure the complex permittivity of granular

material, such as wheat grains then errors are introduced because or the irregular interface

caused by the grain. This non-planer interface causes difficulties in measuring the complex

permittivity and introduces errors. In order to understand the phenomenon this project

includes designing, fabricating, testing and modelling a range of periodic discontinuous

surfaces around a wave guide cell with the intention of measuring S-parameters of such

interfaces using a Vector Network Analyser (VNA). From these measurements the complex

permittivity is calculated using a technique called deembedding of the characteristic (ABCD)

matrix of the profiled layer. The approach taken is to fabricate the samples and use the

available test equipment in MACS group material measurement laboratory. Samples were

loaded into the rectangular wave guide (WG-14) cell and the S-parameters were measured

using a VNA and complex permittivity was extracted using a C++ program, which uses the

mathematical technique of deembedding the ABCD matrix for the profiled layer and the

theoretical background of the waveguide theory. In addition the 3-D problem was also

modelled using a commercial software HFSS for different shapes and depth of the profiled

layer. The simulated and experimental results compared well. The outcomes of this project are

a modified Maxwell-Garnett model and methods for estimating the complex permittivity of

periodic interface surfaces as a function of the geometry of the profiled layer.


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Deceleration

Declaration

No portion of the work referred to in the dissertation has been submitted in support of

an application for another degree or qualification of this or any other university or

other institute of learning.

Copyright

1. Copyright in text of this dissertation rests with the author. Copies (by any

process) either in full, or of extracts, may be made only in accordance with

instructions given by the author. Details may be obtained from the appropriate

Graduate Office. This page must form part of any such copies made. Further

copies (by any process) of copies made in accordance with such instructions

may not be made without the permission (in writing) of the author.

2. The ownership of any intellectual property rights which may be described in

this dissertation is vested in the University of Manchester, subject to any prior

agreement to the contrary, and may not be made available for use by third

parties without the written permission of the University, which will prescribe

the terms and conditions of any such agreement.

3. Further information on the conditions under which disclosures and

exploitation may take place is available from the Head of the School of


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Acknowledgements

Acknowledgements

I would like to express my sincerest gratitude to my supervisor Professor Andrew Gibson, to

whom I will always be indebted, for his inestimable help and guidance through out the MSc

course and during the dissertation stage.

I would also like to extend my deepest appreciation to Dr. Arthur D Haigh for taking interest

in this project and guiding me at all the time when I needed.

I am thankful to the people working in the workshop on the D-floor of Sackville Street

Building (SSB) for making the samples in time and the precision of the samples.

In the end I would like to thank my parents for their unconditional support particularly my

mother who called me daily from Pakistan, to pray for my success and keeping my moral up

without telling about her health which is not good, so that I can concentrate on my studies.


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Abbreviations

Abbreviations

2D Two dimensional

3D Three Dimensional

AFF Air Fill Factor

CP Complex Permittivity

CPF Coherent Potential Formula

DC Direct Current

DUT Device under Test

EM Electromagnetic

EMA Effective Medium Approximation

EWS Electromagnetic Wave Scattering

FDTD Finite Difference time domain

FEM Finite Element Method

GTD Geometrical Theory of Diffraction

HM Hybrid Mode

HFSS High Frequency Structure Simulator

MG Maxwell-Garnett

MMA Methacrylate monomer

MMIC Monolithic Microwave Integrated Circuit


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Abbreviations

TEM Transverse electromagnetic mode

TM Transverse magnetic mode

VDU Visual Display Unit

VNA Vector network analyser

VSWR Voltage standing wave ratio.

WG Waveguide


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Symbols

Symbols

eff ε = effective permittivity.

0ε = Absolute permittivity of free space.

*ε = Relative complex permittivity.

r ε = Relative permittivity.

ε ′ = Real part of complex permittivity.

ε ′′ = Imaginary part of complex permittivity

f M f = = Material fill factor.

f

A = Air fill factor.

P

= Macroscopic polarization vector.

p

= Dipole moment.

e χ = Electric susceptibility.

res χ = Susceptibility at resonance.

e= Electronic charge.

m = Mass of electron.

d γ = Damping factor.

ω = Angular frequency.

I = Electric current.

V = Voltage


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Symbols

σ = Electric conductivity.

t

E = Tangential component of electric field.

t H = Tangential component of magnetic field.

β = Phase constant.

ck = Cut-off wave number.

k = Wave number.

λ = Free space wavelength.

cλ = cut-off wavelength.

gλ = Guide wavelength.

pv = Phase velocity.

d α = Attenuation constant.

( )δ tan = Loss tangent.

+nV = Amplitude of the voltage wave incident on port “n”

−

nV = Amplitude of the voltage wave reflected from port “n”


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Introduction

1. Introduction: -

Microwave processing is a field of increasing importance and is receiving more attention

particularly in processing materials with a broad range of compositions, sizes and shapes. In

recent years, microwave processing research and development have been expanded into many

new areas such as calculation and measurement of the complex permittivity of materials like

ceramics, polymers, composites, and chemicals as a function of frequency and temperature.

For example it was shown that the complex permittivity of material must be known to control

the microwave processing of ceramics ]23[ . The real and imaginary parts of the complex

permittivity, ε ′ andε ′′ respectively are parameters that describe the behaviour of a dielectric

material under the influence of a microwave field. Both affect the power absorbed and the

half-power depth. They also describe how microwaves penetrate and propagate through an

absorbing material, reflect and scatter from the dielectric material, and influence the

volumetric heating of a given material.

So complex permittivity of the material determines not only electrical but also affects the

thermal performance of the material and in general it is a function of frequency and

temperature, hence it is very important to measure this electrical property of materials to

characterize them. Knowledge of complex permittivity of the materials at microwave

frequency is a very important in the description of their physical and chemical properties. It is


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Introduction

permittivity of water is very high (according to Cole-Cole equation it is approximately “-10”

at 30 degree centigrade) hence there is lot of electrical energy is being converted in to heat

energy in water at this frequency. As almost all the food contains water hence all microwave

ovens are designed at the above mentioned frequency in order to heat the food stuff. Also to

dry wheat grain and medicines in pharmaceutical industry using microwave techniques as

they may contain water contents hence the frequency of applied electromagnetic wave is so

adjusted to have the maximum heating effect (or loss) that is at that frequency the imaginary

part of the complex permittivity is maximum in magnitude. This needs the complete

knowledge of the complex permittivity of water as a function of frequency. Some more

includes electrosurgery in which the internal bleeding from the tiny blood vessels can be

stopped and diathermy which is the way of making the muscles of human body relaxed using

electrically induced heat shows the importance of the measurement of complex permittivity

and its trends with the frequency.

Often in practice, when particular materials have to be characterized using microwave

technique it is necessary to go through some sample preparation and in order to extract the

complex permittivity from the calibrated measurement setup. Real materials used in industrial

and food processing, frequently came in particulate (powder) form. For example in agro

chemicals, pharmaceutical and food processing industries pallets and grains occur frequently.

These type of granular materials are the most difficult to prepare the experimental


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Introduction

In this project we have designed the systematic approach to understand the irregular interface

problem. These irregularities are implemented in the form of grooves and slots of different

dimensions angles and fill-factor (which depends upon the pitch of grooves or slots) on a

Perspex layer (material used in this work) which is then loaded in a rectangular wave guide

cell (WG-14) to measure S-parameters by using Vector Network Analyzer (VNA). For the

deeper understanding of the problem simulations of the profiled layers in the rectangular

waveguide are performed on commercial software HFSS to get S-parameters. Then these S-

parameters are used to calculate complex permittivity of the profiled layer using a technique

called “Deembedding a characteristic matrix” using the complex permittivity of the Perspex

layer which is calculated to be 02.062.2* j−=ε used the S-parameters which are taken from

the VNA and the length of the layers. It is demonstrated that the approximate formulae like

Maxwell-Garnett equations produces a close fit with very small error to measured and

simulated effective permittivity for the case of slots. Different angles like 60, 90 and 120

degrees of grooves produces different scattering patterns results in different complex

permittivity does not seems to obey original Maxwell-Garnett mixing rule. Hence 8 new

different formulae are calculated here empirically using the numerical techniques which uses

matrix algebra to solve homogeneous simultaneous linear equations and experimental data for

different shapes of discontinuities (slots and grooves). Out of the 8 different formulas 4 are

valid at 6GHz and the others 4 formulae are valid for 8GHz. There are some interesting


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Introduction

2) Exploration of the microwave measurement techniques.

3)

Understanding the concept of ray tracing.

4) Familiarization with software like “High frequency Structure simulator” (HFSS)

which is very sophisticated three dimensional electromagnetic simulation tool.

5) Familiarization with the operation and working of vector network analyzer (VNA).

6) Finally the most important objective of this project is the development of model to

estimate the complex permittivity of 2-phase dielectric medium with regular and

periodic discontinuities by loading them in a rectangular waveguide cell (WG-14).

The project was very challenging and informative. It provide lot of inside and knowledge

about electromagnetic wave theory and electromagnetic wave scattering and because of this

the change in the complex permittivity of the profiled layer.


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Literature Review

CHAPTER No 2

2. LITERATURE REVIEW

The history of microwave measurement of the complex permittivity of the heterogeneous

mixture and discontinuous interface problem and their electromagnetic analysis is very old.

There are number of different mixing formulas some of them are empirical. Analytical

formulas are also available for the calculation of complex permittivity for example those

derived by Tischer ]24[ . In addition to this there are number of different experimental

measurement techniques available for determining the complex permittivity of a material. The

choice of measurement technique depends on number of factors like frequency range, sample

size restriction, expected value of complex permittivity, conducting and non-conducting

nature of the samples, required measurement accuracy, physical conditions of measurements

like temperature and pressure, material properties like homogeneous or isotropic, cost and

form of material like liquid or powder ]7&4,3,1[ .

One of the methods used for the measurement of complex permittivity is the transmission line

technique which is used in this project. This technique was first introduced by Tischer ]24[ .

Tischer describe the method of measurement of electromagnetic properties of plasma in

section of wave guide as a test section. This approach has the advantage that it has the

l ti l l ti b id i th bl b d l bl Ti h i hi


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Literature Review

and it requires high resolution from the instrumentation. Open ended co-axial probe

techniques require no sample preparation; it is broad band, simple and convenient, ideal for

lossy materials and used particularly for liquid and sami-liquid materials. In the free space

technique the complex permittivity of the material is computed for the measurement of

transmission co-efficient and reflection co-efficient. It is a convenient technique for non

conducting materials, best for hostile environments like high temperature and pressure, good

for on-line microwave measurements and useful for large and flat materials.

Calculating or predicting the effective response of an inhomogeneous medium or discontinues

interface problem to incident electric and magnetic excitation is a complex and interesting

problem. It requires the careful and rigorous electromagnetic and polarizability analysis of the

material. Many different mixing rules have been suggested ]29[ in addition to the Maxwell

Garnett Theory ]1[ . For different classes of materials formulas are developed which predict the

value of complex permittivity. Chiral]30[

and the other magneto-electric]31[

mixing rules have

appeared in literature in the recent past. These are the deduction from the Maxwell Garnett

paper which was published in 1904. Sihvola ]6[ gave the mixing formula to estimate the

macroscopic properties of the heterogeneous two phase mixture by treating one of the

components as a background medium and the other one as inclusions. These inclusions are

assumed to be spherical and randomly distributed through out the background medium. The

simplest dielectric mixing formula for the effective permittivity effε is named after Maxwell


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Literature Review

Maxwell-Garnett expression that accounts for the density fluctuations of the second medium

on the background medium was given. Sihvola[ ]6

states that “Maxwell-Garnett (MG) mixing

rule has been widely used in dielectrics studies but critics say that it fails to predict the

behaviour of mixtures with high volume fractions or large dielectric constant between the

components”, and agreement with this observation is one of the conclusions of this project

dissertation.

As the large community of scientists don’t believe in the complete validity of Maxwell-

Garnett mixing rule, thus other mixing formulas are also been suggested which can predict the

value of complex permittivity of the mixture of mediums. In ]32[ , a family of mixing rule has

been presented according to the following

( ) ( )oeff o

o

oeff oeff

oeff f

ε ε ν ε ε

ε ε

ε ε ν ε ε

ε ε

−++

−=

−++

−

22 (2.2)

where the additional dimensionless constant parameters “ν ” determines the nature of mixing

rule. For 0=ν in equation (2.2) reduces to Maxwell-Garnett rule equation (2.1). Sihvola in

his paper [ ]6 summarized some of mixing formulas in these words, “Other integer values for ν

gives other well known mixing rules. The value 3 gives the so called coherent potential

formula (CPF). In solid state physics, CPF is known as the GKM rule after Gyorffy, Korrings

and Mills. Correspondingly, 2=ν gives the Bottcher mixing rule [ ]33 . In remote sensing

[ ]34


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Literature Review

In this project complex permittivity is measured experimentally for different shapes and

discontinuities of Perspex layers which can be consider homogeneous mixtures of Perspex

and air. An empirical formula is developed for complex permittivity of the mixture of Perspex

and air at 6GHz and 8GHz as a function of material fill factor (MFF) denoted by f M or f

which is defined as the ratio of the volume of inclusion (material) to the total volume of the

mixture. Finally this formula is related to equation (2.2) mentioned in ]32[ and value of ν are

calculated for different cases.


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CHAPTER No 3

3. BACKGROUND THEORY

This chapter includes the background theory needed to understand the work done in this

dissertation project which includes the revision of electrical properties of materials like

polarization, permittivity and concept of complex permittivity. Waveguides, different modes of

electromagnetic waves and Maxwell-Garnett theory is also given here briefly. Finally the

importance and the theory of S-parameters are discussed. For very basic electromagnetic

theory, the reader can go to the references at the end and the appendices.

3.1. Dielectrics and Polarization

All the materials are made up of atoms or molecules, to understand the behaviour of the

material in the electric filed, it is batter to understand the behaviour of atoms and molecules in

electric field.

There are two types of molecules

1) Non-Polar

2) Polar

Following is the description of the behaviour of these two types of molecules in the time

independent and time dependent electric field.

Electric dipole is induced in the non-polar molecule if it is placed in the static electric field


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2. The entire polarized molecules align themselves parallel to each other and to oppose

the applied electric field.

The following Figure-3.2.1 explains the phenomenon of polarization.

Figure-3.2.1 Polarization of non polar molecule in electric field

In case of the polar molecules in weak electric field only alignment takes place such that the

electric field of the molecule is in opposite direction to that of applied field. At high electric

field the separation between positively charged centre and negatively charges electronic cloud

tends to increase and as the result electric potential energy is stored in the dielectric material

against the applied electric field. This separation increases with the increase in the magnitude

of the applied electric field and when the magnitude of electric field reduces it reduces. But

this happen in some specified range of the magnitude of applied electric field which is

determined by the material. If the electric field of high magnitude is applied so that the


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positive and negative charges against the Coulombs force of attraction and hence produces an

array of microscopic dipoles. These charges are known as bound charges. The molecules can

be arranged in an ordered and predictable manner or may exhibit random positioning and

orientation, as would occur in an amorphous material or a liquid depending upon the nature of

the material and the applied field. The molecule may or may not exhibit permanent dipole

moments (existing before the field is applied), and if they do, they will usually have random

orientations throughout the material volume. The macroscopic polarization vector P

rises

because of the displacement of the charges is defined as the dipole moment per unit volume

and is given mathematically as

∑∆×

=→∆ ∆

=

v N

i

iv

pv

limP0

0

1 (3.1.1)

Where,

v∆ = Very small volume ( 3m ).

N = Total number of dipoles in the volume v∆ .

i p

= Dipole moment of thi molecule/atom (C.m) which is defined as iii d Q p

×= , Where “ iQ ”

is the positive charge out of the two bound charges in thi molecule/atom and “ id

” is a

vector quantity whose magnitude is the distance between the positive and the negative

charges within a molecule/atom and directed towards the positive charge from the


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E P e

χ ε 0= (3.1.2)

Where the electric susceptibility, e χ is the very interesting part of the dielectric constant, and

can be related to this as

er χ ε +=1 (3.1.3)

Therefore to understand the concept and nature of the dielectric constant “ r ε ” we have to

understand the concept and nature of the electric susceptibility “ e χ ” this finally helps us to

understand the behaviour of the polarization P

.

3.1.2. Dipole in the time harmonic field

To understand the concept of the behaviour of the electric dipole in the time harmonic field

that is propagating as a wave through the material one has to go deep into the concept of

polarization and the phase difference between E

and D

field. The result of applying time

dependent E

on dielectric material is oscillating dipole moments are setup, and these in turn

establish a polarization wave that propagates through the material. The effect is to produce

the polarization function ( )t zP ,

, having the same functional form as the field ( )t z E ,

which

produces this. The atoms/molecules don’t move physically throughout the material, but their

oscillating dipole moments collectively exhibit wave motion. This is very important and deep

understanding of wave phenomenon in dielectric. We can form a basic qualitative


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which interact with the next dipole. Radiation from this dipole adds to the previous field as

before and the process repeats from dipole to dipole. The net phase shift at each location

results in the slowing down the phase velocity of the resultant wave. Attenuation of the field

is accounted in the classical model by the partial phase cancellation between the incident and

the radiated fields.

3.2. Complex Permittivity ]13[

3.2.1. The Classical Spring Model

In the classical model, the dielectric mediums are the collection of the identical and fix

electron oscillators, in which Coulombs force of attraction is modelled as the spring between

the electronic cloud and the nuclei. Following Figure3.2.2 shows the single oscillator located

at the position “ z ” in the material and oriented in the x-axis. A uniform plane wave, assumed

to be linearly polarized along “ x ”, propagates in the material in the “ z ” direction. The

electric field in the wave displaces the electron of the oscillator in the x -direction through a

distance represented by the vector d

; a dipole moment is thus established,

( ) ( )t zd et z p ,,

×−= (3.2.1)

Where the applied force is given by

( ) ( )t z E et zF a ,,

×−= (3.2.2)


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phase differences between oscillators is accurately determined by the spatial and temporal

behaviour of ( )t z E ,

.

d k forcerestoring

vm forcedamping

E e forcent displaceme

s

d

−=←

−=←

−=→

γ

Figure-3.2.2: - Atomic dipole model with Coulomb force between

positive and negative charges modelled by that of a spring having spring

constant “ sk ”. An applied electric field displaces the electron through

distance “ d ”, resulting in a dipole moment d e p

×−=

The restoring force on the electron r F

is that produced by the spring which is assumed to obey

Hook’s Law:

( ) ( )t zd k t zF sr ,,

−= (3.2.3)

Where, “ sk = Spring constant (not the propagation constant).

The negative sign shows that the restoring force is in opposite directions to that of the

displacement of the electron from the mean position which is the nucleus in this case. If the

field is turned off the electron is released and will oscillate (as shown in the Figure3.2.3

below) about the nucleus at the resonant frequency, given by the following expression

( ) ( )

k

t zd mt zd k

maF

s

r

=⇒

=

2

0 ,, ω (3.2.4)


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( ) ( )t zvmt zF d d ,,

γ −= (3.2.5)

Where ( )t zv , is the velocity of the electron at the position “ z ” and at time “ t ”.

Dephasing is a very important process which is completely associated with the damping in the

electron of the oscillator in the system. Their relative phasing, once fixed by the applied

sinusoidal field, is destroyed through collisions and dies away exponentially until a state of

totally random phase exists between oscillators.

By applying the Newton second law of motion and write down the vector sum of all the

forces (damping, applied and restoring) equal to the product of the mass of the electron and its

acceleration,

ccsc

d c

d r a

E ed k t

d m

t

d m

F F F am

−=+∂

∂+

∂

∂⇒

++=

γ 2

2 (3.2.6)

Where “ c E

” is the complex form of the electric field at point ( )t z, which includes applied

and radiated field as discussed already. Its general form is given as

t j jkz

c ee E E ω −

= 0

(3.2.7)

As the system is being excited by complex Electric field “ c E

”, one can anticipate a

displacement wave “ cd

” of the form


34/224

( )( ) ( ) t j jkzt j jkzs

t j jkz

d

t j jkz ee E eeed k eed jmeed j jm ω ω ω ω ω γ ω ω −−−− −=++ 0000

(3.2.10)

Dividing by t jme ω , and putting jkzs ed d

−= 0

, jkzs e E E −

= 0

andm

k s=2

0ω the simplified version

of this equation is

[ ]sd s E

m

e jd

−=++−

2

0

2 ω ωγ ω (3.2.11)

Making sd

as subject, the resulting equation takes the form,

( )( )[ ] s

d

s E j

me

d

ωγ ω ω +−

−=

22

0

(3.2.12)

The dipole associated with the displacement “ sd

” is

ss d e p

×−= (3.2.13)

The polarization vector of the medium can be found by assuming that all the dipoles are

identical

sss d eN p N P

×−=×= (3.2.14)

( )[ ] sd

s E j

me N

P

ωγ ω ω +−=⇒

22

0

2

(3.2.15)

Comparing this equation with the 3 2 2 we get the susceptibility at resonance as


35/224

Where

( )( )[ ]

( )

( )[ ]22222002

222

22

00

220

2

d

d res

d

res

m

Ne

m

Ne

γ ω ω ω ε

ωγ χ

γ ω ω ω ε

ω ω χ

+−

=′′

+−

−=′

(3.2.18)

Now, the real and imaginary parts of the permittivity can be found through the real and

imaginary parts of “ res χ ”.

( ) ( ) ( ) resresresresres j j j χ ε χ ε χ χ ε χ ε ε ε ε ′′−′+=′′−′+=+=′′−′= 0000 111 (3.2.19)

Thus it is clear that

( )

res

res

χ ε ε

χ ε ε

′′=′′

′+=′

0

0 1 (3.2.20)

The plot of the real and the imaginary parts of complex susceptibility is shown in Figure

3.2.3.


36/224

The important points to note in the plot are the symmetric behaviour of the “ res χ ′′ ” about

“ 0ω ω = ” whose full- width at its half-maximum amplitude is “ d γ ”. Near the resonant

frequency where res χ ′′ become maximum, so the wave attenuation is at the peak. There is a

still significant variation of “ res χ ′ ” with frequency away from resonance, which leads to the

frequency.

3.2.2. Limitation of the Classical Model

Here it is important to mention that this model which is based on the “Classical Physics”

provide the very accurate prediction of the dielectric constant behaviour with respect to

frequency (particularly off-resonance) and can be use to a certain extent to model absorption

properties. The model is incomplete, specifically, it assumes that the oscillating electron can

assume any one of continuum, of energy state, when infect the energy state in the in any

atomic system are quantized. As a result, the important effects arising from transitions

between discrete energy levels, such as spontaneous and stimulated absorption and emission,

are not included in this classical spring system. Quantum mechanical model must be used to

fully describe the medium polarization properties, but the results of such studies often reduce

to those of the spring model when field amplitude is very small.

3.4. Waveguides

“A id i t t th h hi h l t ti b t itt d f


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In 1897, Lord Rayleigh (John Willim Stutt) mathematically proved that wave propagation is

possible in waveguides both for circular and rectangular. In this project rectangular wave

guides are given more importance because they are easily available in the labs also the

mathematical analysis for rectangular shaped waveguides are simple than that of the circular

waveguides. In addition to this the experimental results with rectangular wave guide are

available. The method developed here is by using rectangular waveguide but this can easily be

generalized for other shapes of waveguides.

“Early microwave systems relied on waveguide and coaxial lines for transmission line media.

Waveguide has the advantage of high power-handling capability and low loss but is bulky and

expensive” ]14[ .

Wave guides, often consisting of simple conductors, support transverse electric (TE) and/or

transverse magnetic (TM) waves, characterized by the presence of longitudinal magnetic or

electric field components respectively.

3.4.1. General Solution for TEM, TE and TM Waves

Pozar in his book ]14[ briefly derived the expressions by using the point form of Maxwell’s

equations for the transverse and longitudinal components of electric and magnetic fields as a

general case, means expressions which are independent of the geometry of the waveguide it

may be parallel plate, rectangular, cylindrical or may be of any shape.

Let’s review these mathematical derivations to understand the working and response of the


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Where

( ) y xe,

= Transverse vector component of electric field.

( ) y xh ,

= Transverse vector component of magnetic field.

( ) y xe z , = Longitudinal component of electric field.

( ) y xh z , = Longitudinal component of electric field.

β = Propagation constant, because attanuaton is zero( 0=α .

ẑ = unit vector in z-direction.

1−= j

For the free space case that is by assuming that wave is travelling in free space, now one can

use the free space Maxwell’s equations in point form which are

H j E

ωµ −=×∇ (3.4.3)

E j H

ωε =×∇ (3.4.4)

Where

ω = Angular frequency of the electromagnetic wave.

ε = Permittivity of the material through which wave is propagating.

µ = Permeability of the material through which wave is propagating.

By inserting equations (3.4.1) and (3.4.2) in the above equations and using the definition of

vector product to convert these vector equations in to simple algebraic equations on can get


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x y z E j H j

y

H ωε β =+

∂

∂ (3.4.8)

y z

x E j x

H H j ωε β =

∂

∂−− (3.4.9)

z x y E j

y

H

x

H ωε =

∂

∂−

∂

∂ (3.4.10)

These six equations can be solved for the four transverse field components in terms of the

longitudinal components of electric and magnetic field.

∂

∂−

∂

∂=

x

H

y

E

k

j H z z

c

x β ωε 2 (3.4.11)

∂

∂+

∂

∂−=

y

H

x

E

k

j H z z

c

y β ωε 2 (3.4.12)

∂

∂+

∂

∂−=

y

H

x

E

k

j E z z

c

x ωε β 2 (3.4.13)

∂∂+

∂∂−=

x H

y E

k j E z z

c

z ωε β 2 (3.4.14)

Where222 β += k k c (3.4.15)

“ ck ”Is defined as the cut-off wave number.

Equations (3.4.11) to (3.4.15) are derived from the six fundamental equations (3.4.5) to

(3.4.10) which are relatively more general. TEM waves cannot be derived from equations

(3 4 11) to (3 4 15) but they can be explained completely using equations (3 4 5) to (3 4 10)


40/224

very small. The Figure-3.5.1 shows some of the standard rectangular wave guides

components that are available

Figure 3.5.1: - Standard rectangular wave guides

The hollow rectangular waveguide can support TM and TE modes, but not TEM mode of

electromagnetic waves because TEM mode need more then one conductor while rectangular

waveguide consists of only single conductor.

Rectangular waveguide has a cut-off frequency below which propagation is not possible for

TM and TE mode.

3.5.1. TE Mode

TE (Transverse Electric) wave are the waves in which the component of electric field in the

direction of propagation of the wave is zero. It may have the non-zero component of magnetic

field in the direction of propagation of the wave that is the reason it is also known as the H-

waves. In this analysis as the direction of propagation of wave is +z-axis thus mathematically


41/224

( ) a xat y xe y ,00, == (3.5.2)

3.5.2. TM Mode

TM (Transverse Magnetic) wave are the waves in which the component of magnetic field in

the direction of propagation of the wave is zero. It may have the non-zero component of

electric field in the direction of propagation of the wave that is the reason it is also known as

the E-waves. In this analysis as the direction of propagation of wave is +z-axis thus

mathematically TM mode can be defined as 0≠ z E and 0= z H . As far as the boundary

conditions are concern they are same as that in TE-Mode. That is with reference to Figure-

3.5.2.

( ) b yat y xe x ,00, == (3.5.1)

( ) a xat y xe y ,00, == (3.5.2)

The following Table-3.5.1 ]14[ will show the summary of the results of the study of rectangular

wave guide. In this table time-harmonic field with a t je ω dependence and wave propagation

along the z-axis and the electric and magnetic fields of the form of Equation (3.4.1) and

(3.4.2) are assumed.

Considering

k = wave number.

ck = Cut off wave number.


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d α = attenuation constant.

δtan = loss tangent.

QUANTIT

Y mnTE MODE mnTM MODE

k εµ ω εµ ω

ck ( ) ( )22 bnam π π + ( ) ( )22 bnam π π +

β 22ck k −

22

ck k −

cλ ck

2

ck

π 2

gλ

β

2

β

π 2

pv β

ω

β

d α β

tanδk 2

2

β

tanδk 2

2

Z

β

η k

β

η k

z E 0 z j β

mn eb

ynπ sin

a

xmπ sin B −

z H z j β

mn eb

ynπ cos

a

xmπ cos A −

0

x E z j β

mn2

c

e

b

ynπ sin

a

xmπ cos A

bk

nπ jω −

µ

z j β

mn2

c

e

b

ynπ sin

a

xmπ cos B

ak

mπ j β −

−

y E z j β

mn2

c

eb

ynπ cos

a

xmπ sin A

ak

mπ jωω −

−

z j β

mn2

c

eb

ynπ cos

a

xmπ sin B

bk

nπ j β −

−

ynπxmπmπjβ ynπxmπnπjω ε


43/224

3.6. Scattering by Conducting Wedge ]10[

Scattering of electromagnetic waves from some regular shaped objects are studied in which

scattering from the conducting wedge is the most important one because the wedge is a

canonical problem that can be used to represent locally (near the edge) the scattering of more

complex structures, asymptotic forms of its solution have been utilized to solve numerous

practical problems. The asymptotic forms of its solution are obtained by taking the infinite

series modal solution and first transforming it into an integral by the so-called Watson

transformation]21,20[ . The integral is then evaluated by the method of steepest descent (saddle

point method) ]22[ . The resulting terms of the integral evaluation can be recognized to

represent the geometrical optics fields, both incident and reflected geometrical optics fields,

and the diffracted fields, both incident and reflected diffracted fields. These forms of the

solution have received considerable attention in the geometrical theory of diffraction (GTD)

which has become a generic name in the area of antennas and scattering.


44/224

Balanis, used the incident field on the conducting wedge from the infinite electric line source

in which the currente

I is following and is of the form,

( ) ( )( ) ( )

( ) ( )( ) ( )

′≥′

′≤′

−=

∑

∑∞

−∞=

′−

∞

−∞=

′−

m

jm

mm

m

jm

mm

ei

z

e H J

e H J I

E

ρ ρ βρ ρ β

ρ ρ ρ β βρ

ωε

β

φ φ

φ φ

2

2

2

4 (3.6.1)

The z-component of the total electric field because of the incident (Electric field of infinite

Electric Line Source as mentioned above, in equation (3.6.1)) and scattered electric field from

the wedge in circular cylindrical co-ordinate system with total internal wedge angle WA= 2

is calculated in reference ]10[ by considering the reciprocity and is given by

( ) ( )( ) ( )[ ] ( )[ ]

( ) ( )( ) ( )[ ] ( )[ ]

′≥−×−′′

′≤−×−′′

=+=∑

∑

v

vvv

v

vvvs

z

i

z

t

zvv H J a

vv H J a

E E E ρ ρ α φ α φ βρ ρ β

ρ ρ α φ α φ ρ β βρ

,sinsin

,sinsin

2

2

(3.6.2)

The corresponding magnetic field components can be obtained by using Maxwell’s Equations

as

φ ρ ωµ ρ

∂

∂−=

t

zt E

j H

11 (3.6.3)

ρ ωµ φ

∂

∂=

t zt E

j H

1 (3.6.4)

3.7. Maxwell- Garnett Mixing Theory


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was published in 1904. It basically treated the impurities as spherical shaped object with

different complex permittivity was present in the material.

The two other papers which were studies are ]3,2[ , they basically based on the application of

the Maxwell-Garnett theory in special cases. In the reference ]2[ Maxwell Garnett theory was

proved mathematically and then checked by the three experiments only by taking the

homogenous medium in which the shape of the impurity was considered spherical.

In the reference]3[ Maxwell Garnett Theory was applied for the mixtures of anisotropic

inclusions with conducting polymers. The effective dielectric function eff ε for a medium of

anisotropic inclusions embedded in an isotropic host is calculated using the Maxwell Garnett

approximation. For uniaxial inclusions,eff ε depends on how well the inclusions are aligned.

Then the approximation to study eff ε for a model of quasi-one-dimensional organic polymers

was performed. The polymer is assumed to be made up of small single crystals embedded in

an isotropic host of randomly oriented polymer chains. The host dielectric function is

calculated using the effective-medium approximation (EMA) The resulting frequency-

dependent ( )ω ε eff closely resembles experiment. The formula used to approximate the total or

effective permittivity is

( )2121

2122 2

ee f ee

eee f eeeff

−−+

−×+= (3.7.1)


46/224

with unique voltages and currents defined at any point in the circuit. In this situation the

circuit dimensions are small enough so that there is negligible phase change from one point in

the circuit to another. In addition, the fields can be considered as TEM fields supported by

two or more conductors. “This leads to a quasi-static type of solution to Maxwell's equations,

and to the well-known Kirchhoff voltage and current laws and impedance concepts of circuit

theory”]8[

. Simply there are very familiar helpful and easy laws like Kirchhoff voltage and

current laws governs the low frequency circuits and the theory used for their analysis is

known as circuit theory. But this theory or these laws are not applicable for high frequency

circuits like at microwave frequency, or one can say that microwave circuits are not in the

domain of circuit theory because of their less generality.

To solve high frequency circuits (with frequency greater than 1GHz) scientist and engineers

usually use Maxwell’s Equations which are the complete classical description and solution of

any electromagnetic phenomenon at any frequency. But the problem with this method is the

difficulty level and the mathematic involved in it. The analysis of any electromagnetic

phenomenon using Maxwell’s Equations is known as field analysis. One more problem with

field analysis is that it gives much more information about the particular problem under

consideration then one really wants or need. That is because the solution to Maxwell equation

for a given problem is complete; it gives the electric and magnetic field at all points in space

and at all times.


47/224

elements together and find the response without the field analysis. Field analysis using

Maxwell’s Equations for such problems be hopelessly difficult.

S-parameters are basically the relation between incident voltage wave and reflected voltage

wave from the ports. For some components and circuits, the scattering parameters can be

calculated using network analysis technique. Otherwise the scattering parameters can be

measured directly with the help of Vector Network Analyzer (VNA).

Pozar ]14[ has defined the S-parameters by considering the N-Port network shown in Figure-

3.7.1. where +n

V is the amplitude of the voltage wave incident on port “n” and −n

V is the

amplitude of the voltage wave reflected from port “n”. The scattering matrix or [ ]S matrix is

defined in relative to these incident and reflected voltage wave as

=

+

+

+

n

2

1

nnn2n1

1n2221

1n1211

-

n

-

2

-

1

V

V

V

SSS

SSS

SSS

V

V

V

⋮

⋯

⋮⋮⋮⋮

⋯

⋯

⋮

[ ] += VSV -

A specific element of the [ ]S -matrix can be determined as

jk ≠∀=

++

=

0V1

-

1ij

k

V

VS

Extraction of Complex Permittivity from S-Parameters


48/224

CHAPTER No 4

4. EXTRACTION OF COMPLEX PERMITTIVITY FROM

S-PARAMETERS

This chapter includes complete mathematical derivation to calculate the complex permittivity

from the S-parameters of the multilayer problem. Firstly S-parameters for all layers are

converted into ABCD matrix then by knowing this and the ABCD matrices of the known

layers, ABCD matrix of the profiled layer is calculated using a mathematical technique called

“Deembedding” , and finally mathematical formulation is given to derive the complex

permittivity of that layer from this known characteristic matrix.

4.1. Two Independent Ways of Obtaining the S-Parameters

In this project S-parameters of the samples are taken from following two different

independent sources

1. HFSS Simulations, which will be discussed in Chapter No 5 of this report.

2. Experimental Measurements using Vector Network Analyzer (VNA), which will

be discussed in Chapter No 6 of this report.

After measuring the S-parameters from the experimental setup one has to convert it into the

normal rectangular form, because Vector Network Analyser (VNA) gives output magnitude



49/224

4.2. Conversion of Scattering Matrix to ABCD Matrix

At low frequency the behaviour of two-port network can easily be characterise by the,

z, y, h or ABCD parameters because at low frequency it is relatively easy to do short and open

circuit experimentally. But at higher frequencies like at microwave frequencies these

parameters cannot be measured accurately because the required short circuit and open circuit

tests are difficult to achieve over a broadband range of microwave frequencies.

A set of parameters which are then useful at higher frequencies are Scattering parameters

known as S-parameters, which deals with the incident and reflected voltage waves at a

particular node or port rather then input port impedance, admittance, total current or voltage.

According to Pozar ]14[

“The scattering matrix relates the voltage wave’s incident on the ports to those reflected from

the ports”

These are mathematically defined in article 3.7 at page 40 of this report.

If the network is the cascaded version of two or more then two two-port networks then

another set of parameters known as ABCD-parameters are very useful. For two-port network

as shown in figure 4.2.1 ABCD parameters can be written as 22× matrix of the form

[ ]

=

d c

ba A` (4.2.1)



50/224

The well-defined relationship between ABCD matrix and Scattering matrix derived in

Pozar [ ]14 , given by

( )( )

( )( )

( )( )

( )( )

21

21122211

021

21122211

0

21

21122211

21

21122211

2

11

1

2

11

2

11

2

11

S

S S S S d

Z S

S S S S

c

Z S

S S S S b

S

S S S S a

×

++−=

×

−−−

=

×

−++=

×

+−+=

(4.2.2)

For the ABCD matrix mentioned above.

4.3. Example of 4-Layer Problem (Calculation of S-Parameters of 4-Layers)

Consider a four layer problem as shown in the two dimensional figure 4.3.1 below,



51/224

In case of simulation in HFSS the length of layer-1 and layer-4 (see figure 4.3.1) which are

basically filled with air plays an important role and one cannot neglect this because these

layers makes the problem as 4-layer problem. So the S-parameters taken from the simulation

data are the S-parameters of the 4-layers.

After taking the S-parameters from simulation data these are converted in transmission

parameters (ABCD-parameters) for the four layers using the relations in equations (4.2.2)

(with 10 = Z ) and the matrix obtained is named as [ ]5A .

4.4. Deembedding of ABCD Matrix of Profiled Layer

Characteristic matrices for the layers 1, 2 and 4 are also calculated which in this report are

denoted by [ ] [ ]21 A,A and [ ]4A respectively. The general form of these matrices is

[ ] ( ) ( )

( ) ( )

×=

mmmmm

mmmmm

mlγcosh zlγsinh

zlγsinhlγcoshA (4.4.1)

Where

ml = Physical length of layer-m

2

*2

−=

c

mm jγλ

λ ε

λ

π

= Complex propagation constant in the waveguide with cut-off wavelength

cλ filled with the material of complex permittivity*

mε


“Th h i i i f ll f l i l h d f h h i i


52/224

“The characteristic matrix of all four layers is equal to the product of the characteristic

matrices of the individual layers in the order of their physical existence.”

[ ][ ][ ][ ] [ ]54321 AAAAA = (4.4.2)

Where [ ]3A is the characteristic matrix of the layer-3 (profiled layer) whose complex

permittivity is to be determined. [ ]5A is determined experimentally or from the simulation

results and the other three matrices [ ] [ ]21 A,A and [ ]4A are calculated from the physical

length, the known complex permittivities of the respective materials, free space wave length

and cut-off wave length for a particular wave guide which in this case is WG-14.

4.5. Calculation of Complex Permittivity of Profiled Layer from its ABCD

Matrix

The extraction of [ ]3A from equation-1 is known as Deembedding of [ ]3A . It consists of 3-

steps.

Step-1: -

Multiplying with the multiplicative inverse of [ ]4A that is [ ]1

4A −

on the right side of the

equation-1 to get [ ]6A

[ ] [ ][ ][ ][ ][ ] [ ][ ]

[ ][ ][ ] [ ][ ]==

==

−

−−

1

1

45

1

443216

AAAAA

AAAAAAAA

(4 5 1)


[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 111


53/224

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ][ ] [ ] [ ]

−

−

=

==

==

−

−−−

66

66

11

11

77

77

6

1

132

1

45

1

1321

1

17

AAAA

AAAAAAAA

d c

ba

d c

ba

d c

ba

(4.5.2)

Step-3: -

Multiplying with the multiplicative inverse of matrix [ ]2A that is [ ]1

2A −

to the left side of

equation-3 to get [ ]8A

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ]

−

−

=

==

==

−

−−−−

77

77

22

22

88

88

7

1

23

1

45

1

1

1

232

1

28

AAA

AAAAAAAA

d c

ba

d c

ba

d c

ba

(4.5.3)

Finally [ ]8A is obtained which is the characteristic matrix of the layer-3 (profiled layer) from

which the complex permittivity can be calculated using the following steps

[ ]

( ) ( )( ) ( )( )[ ]

[ ]8 8 8 10

3

0010

3

33310

3

cbalogl

1

Z γlsinh Z γlsinhγlcoshlogl

1

cbalogl

13layer for constant n propagatio

×+=

+=

×+=−=3γ

Where 3l = physical length of the profiled layer, and finally the following equation helps us to


4 6 Soft are Used for the Calc lation of Comple Permitti it


54/224

4.6. Software Used for the Calculation of Complex Permittivity

The complete algorithm for the extraction of the complex permittivity from the ABCD matrix

of the 4-layers in the case of the simulation results or 2-layers in the case of the experimental

results is given above. The calculations are very lengthy so in order to do all these calculation

manually four programs are written C++. These programs are included as the appendix-D to

this dissertation report. C++ language is selected because it can handle the complex numbers

easily and it easy to program. Two of them are for 4-layers and rest of 2 are for the 2-layers

one. Out of two programs for 4 layers one takes complex permittivity and lengths of each

layers as an input and gives the result in the form of S-parameters and the second one is the

reverse of this, it takes S-parameters and the physical lengths as an input and results in

complex permittivity of the layer whose complex permittivity is to be determined. Same is the

case of the two programs for 2-layers.

HFSS Simulation and its Results


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CHAPTER No 5

5. HFSS- SIMULATION AND ITS RESULTS

This chapter deals with the simulations perform to check the experimental data. The software

used for simulation in this project is three dimensional electromagnetic simulation tool named

HFSS which is the abbreviation of “High Frequency Structure Simulator”. Method of

developing a model and ways of getting results are also discussed. This software uses

meshing technique for FEM method and the effect of meshing on simulation results is also

discussed. Finally the results in the form of complex permittivity of the four layer problem for

different shapes and sizes of discontinuities in the Perspex layer are given. For the results of

simulation in the form of S-parameters see Appendix-E.

5.1. Introduction to the Software

HFSS is a commercial solver for electromagnetic structures from Ansoft Corporation. The

acronym originally stood for high f requency structural simulator. It is one of the most popular

and powerful applications used for antenna design and the design of complex RF electronic

circuit elements including filters, transmission lines, and packaging. It is also very good to

handle scattering problems.

5.2. Technique Used

The simulation technique used in HFSS software to calculate the full 3 D electromagnetic


smaller regions and represents the field in each sub-region (called element) with a local


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smaller regions and represents the field in each sub region (called element) with a local

function.

In HFSS, the geometric model is automatically divided into a large number of tetrahedral,

where a single tetrahedron is a four-sided pyramid. The collection of tetrahedral is refined as

the finite mesh.

The value of a vector field like H-field or E-field at points inside each tetrahedron is

interpolated from the vertices of the tetrahedron.

5.2.2. Size of Mesh vs. Accuracy

There is a trade-off among the size of the mesh, the desired level of accuracy, and the amount

of available computer resources.

The accuracy of the solution depends on the size of each of the individual elements

(tetrahedron). Generally one can say that the solution considering thousands of elements are

more accurate then the solution based on the mesh using relatively few numbers of elements.

To generate the precise description of a field quantity, each element must occupy a region that

is small enough for the field to be adequately interpolated from the base-function.

5.3. Meshing and its effect on simulation results

To produce the optimal mesh, HFSS uses an iterative process, called an adaptive analysis, in

which the mesh is automatically refined in critical or in sharp end regions. First, it generates a

solution based on a coarse initial mesh. Then, it refines the mesh in area of high density and


days etc. but it is found that it has no important effect on the results of simulations”.


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days etc. but it is found that it has no important effect on the results of simulations .

This observation is made number of times during the project time and at last it is

concluded that the meshing technique used in the software take care of the error in the

final result and reduces the mesh size automatically where it is needed like on some

sharp corners and edges where there is a chance of having high rate of change of fields

with respect to space.

2. It is also observed that the time taken for simulation in the case of slots is small

compared to the time taken in the cases of angled grooves.

3. Maximum time taken for simulation is in the case of 60 Degree and 120 Degree

grooves.

5.4. Developing the Model in HFSS and Results

Rectangular slots are made by defining the 3-D box objects at the specific location. Assign the

material which is Perspex with the following electrical properties

( ) 00763358.062.2

02.0tan

02.062.2*

==⇒

−=

δ

ε j

Then the port are defined and the boundary conditions which is perfect-E because of the very

good electrical conductivity of the boundary of the rectangular waveguide which is made of

copper. 90 degree, 120 degrees and 60 degrees grooves were obtained from the regular

polygons like hexagon or octagon etc.


The two dimensional picture of a four layer problem which are made in HFSS for simulation


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p y p

with all the lengths mentioned is shown in the Figure-5.4.1. In this figure the layer-3 is the

profiled layer and layer 2 is the layer of Perspex having thickness 4.8mm.

Three dimensional figure from HFSS model is shown below I figure-5.4.2.

Figure-5.4.3. Three Dimensional model developed in HFSS for simulation.

The dimensions and the position of each object in the model was really very important, hence

lot of attention was paid on that.

5.4.1. Slots

The simulation for the case of rectangular slots for different cases gave the results in the form

f h S Th S d l l h l i i i f h



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Rectangular Slots with pitch of 1.4mm, width of 1mm, depth of 1mm

and with air fill factor (AFF)=0.714285

Frequency Complex Permittivity

6GHz 1.408-j0.0284

7 GHz 1.344-j0.0283

8 GHz 1.285-j0.0226

Table-5.4.1. Complex permittivity of slots with 1.4mm pitch

Rectangular Slots with pitch of 1.7mm, width of 1mm,depth of 1mm


Frequency Complex Permittivity

6GHz 1.613-j0.024

7 GHz 1.55-j0.0365

8 GHz 1.487-j0.0189

Table-5.4.2. Complex permittivity of slots with 1.7mm pitch.

Rectangular Slots with pitch of 2.0mm, width of 1mm, depth of 1mm


HFSS Simulation and

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MUHAMMAD OMER FAROOQ, MSc Dissertation, Communication-Engineering, 2009, Student ID 7367100 Complete

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