The Positioning of Electromagnetic Near Field Hotspots
within a Resonant Cavity for Applications in Microwave

Thermal Ablation

Author:

Graeme R. Young

A dissertation submitted to the Faculty of Engineering and the Built

Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the

requirements for the degree of Master of Science in Engineering.

Johannesburg, 2021



Declaration

I declare that this dissertation is my own unaided work. It is being submitted to the

Degree of Master of Science to the University of the Witwatersrand, Johannesburg. It

has not been submitted before for any degree or examination to any other University.

day of ,

i

6th December 2021



Abstract

The investigation into moving electromagnetic near field hotspots inside a resonant cav-

ity is presented. The investigation is focused on providing an alternative approach to

thermal ablation of tumours, by inducing a hotspot over a tumour instead of using an

interstitial antenna. The methodology comprised comparing various electromagnetic

solvers, verifying the simulation techniques, characterising the resonance within a rect-

angular resonant cavity, and attempting to control the movement of its hotspots by

introducing a phase shift between its sources and modifying their frequency. The effects

of dielectric media of the field were also investigated. It was determined that incre-

mental frequency shifts only progressively moved the system’s hotspots between 2.6 and

2.7 GHz and phase shifting only worked between 2.55 and 2.7 GHz when the feeds were

on opposite walls. At the system’s eigenfrequencies, no pattern change was evident,

indicating that when the chamber was resonating, the field pattern was set. Further,

it was determined that the bandwidth of the characteristic modes of the system were

very narrow, such that the addition of dielectric material completely altered the reso-

nance of the system and the eigenfrequencies shifted. Therefore, the application of this

method to thermal ablation, which requires high precision, accuracy and control, was

deemed impractical. Future recommendations include using adjustable cavity geometry

and directive microwave sources to design for specific field patterns. Additionally, it is

recommended to investigate the validity of the ‘reverse problem’ to create a specific cur-

rent distribution around the resonant cavity. This is reminiscent of the three-dimensional

Green’s Theorem, which would induce the desired hotspot pattern from the surrounding

current distribution.

ii



Acknowledgements

I wish to gratefully acknowledge:

My supervisors, Prof. Alan Clark and Prof. David Rubin, who were always available for

consultation, able to keep me focused, who called a spade a spade when necessary and

who provided invaluable feedback and encouragement.

The University of the Witwatersrand for supplying me with the resources I needed to

conduct this investigation, particularly the FEKO license, which was paramount to the

research.

To Dassault Systèmes for providing a student edition of CST Studio Suite, an integral

program throughout the investigation; and EMWorks for providing me with a student

evaluation of HFWorks.

My friends and family, for supporting me through this process.

iii



Contents

Declaration i

Abstract ii

Acknowledgements iii

Contents vii

List of Figures x

List of Tables xi

List of Symbols xii

List of Abbreviations xiv

1 Introduction 1

1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Research Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Methodology Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 4

2.1 Thermal Ablation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Biophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Current Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Relevant Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Electromagnetic Waves, Fields and Radiation . . . . . . . . . . . . 7

2.2.2 Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.4 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

iv



Contents v

2.2.5 Near and Far Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.6 Array Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.7 Resonance and Resonant Cavities . . . . . . . . . . . . . . . . . . . 14

2.3 Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Microwave Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Dielectric Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Microwave Ovens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Electromagnetic Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.3 Fast Multipole Method . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.4 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.5 Finite Difference Time Domain . . . . . . . . . . . . . . . . . . . . 20

2.4.6 Finite Integration Technique . . . . . . . . . . . . . . . . . . . . . 21

2.4.7 Eigenmode Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.8 Modelling Software . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Methodology 26

3.1 Software Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.1 Benchmark Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.2 Results and Deliverables . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Microwave Oven Model . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 Actual Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Operating Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.4 Simulated Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.5 Results Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Cavity Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Ideal Rectangular Cavity . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.2 Waveguide Resonant Frequency . . . . . . . . . . . . . . . . . . . . 33

3.3.3 Cavity Resonant Frequency . . . . . . . . . . . . . . . . . . . . . . 33

3.3.4 Dielectric Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Field Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.2 Phase Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.3 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Software Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

v



Contents vi

3.5.1 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.3 Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5.4 Frequency Sweeping . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5.5 Phase Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5.6 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Results 39

4.1 Software Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Microwave Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.2 Microwave Oven Fields . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.3 Operating Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.4 Simulated Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.5 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.6 Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Cavity Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.1 Waveguide Resonator . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.2 Cavity Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.3 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.4 Dielectric Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Hotspot Manipulation: Frequency Shifting . . . . . . . . . . . . . . . . . . 60

4.4.1 2.6 - 2.7 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Hotspot Manipulation: Phase Shifting . . . . . . . . . . . . . . . . . . . . 63

4.5.1 Same Face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5.2 Adjacent Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5.3 Opposite Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Conclusion 74

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 Future Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

References 78

vi



Contents vii

A NEC2 Hotspot Plotter 83

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.2 NEC2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.3 Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.4 Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.5 I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.6 Future Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

B Comparison of Electromagnetic Field Solvers 87

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B.2 Licences and Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B.3 Solver Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

B.4 Computer Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.5 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

B.6 Comparison Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

B.7 Solution Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

B.7.1 Frequency Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

B.7.2 Characteristic modes . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B.8 Simulation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Bibliography 101

vii



List of Figures

2.1 Electromagnetic Wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Electromagnetic Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Near Field vs Far Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Near (left) and Far (right) field radiation patterns. . . . . . . . . . . . . . 13

2.5 Radiation Pattern of Broadside and End-fire arrays. . . . . . . . . . . . . 14

2.6 The Hybridisation of the FEKO Solvers for their Respective Applications. 23

3.1 Waveguide Orientations. Images adapted from CST Studio. . . . . . . . . 35

4.1 Comparison of the Centre Horizontal Plane of the Benchmark Model. . . 39

4.2 Comparison of the 3D Iso-Surface of the Benchmark Model. . . . . . . . . 41

4.3 The Microwave Geometry and Simulated Hotspots at 2.45 GHz in FEKO.

Images exported by FEKO. . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 The near fields inside the microwave oven as shown on thermal paper. . . 43

4.5 Process Binary Images of the Measured Near Field. . . . . . . . . . . . . . 43

4.6 Logical Operations on the Binary Images. . . . . . . . . . . . . . . . . . . 43

4.7 Microwave Leakage with a Glass of Water. . . . . . . . . . . . . . . . . . . 44

4.8 Microwave Leakage with a Sheet of Polystyrene. . . . . . . . . . . . . . . 45

4.9 Simulated Operational Fields inside the Microwave Oven for 2.4–2.5 GHz

with Styrofoam as a Dielectric. . . . . . . . . . . . . . . . . . . . . . . . . 46

4.10 Simulated Operational Fields inside the Microwave Oven for 2.33–2.35 GHz

with Polystyrene as a Dielectric. . . . . . . . . . . . . . . . . . . . . . . . 46

4.11 Comparison of the Simulated and Measured Binary Images. . . . . . . . . 47

4.12 Electric Field from the Resonant Study of the Bounded WR340 Waveguide. 50

4.13 Excitation of a Bounded WR340 Waveguide with Different Frequencies. . 50

4.14 E-Field Pattern of the Cavity at 2.386 GHz, exhibiting Mode TE104. . . . 52

4.15 E-Field Pattern of the Cavity exhibiting Mode TE303. . . . . . . . . . . . 52

4.16 E-Field Pattern of the Cavity at 2.651 GHz, exhibiting Mode TE122. . . . 53

4.17 Maximum E-field value across the Frequency Range in CST Studio. . . . 54

viii



List of Figures ix

4.18 Accepted Power across the Frequency Range in CST Studio. . . . . . . . . 54

4.19 Radiated Power across the Frequency Range in FEKO. . . . . . . . . . . . 54

4.20 Radiated Power at the Characteristic Modes in FEKO. . . . . . . . . . . . 55

4.21 VSWR of Cavity with Water in CST Studio. . . . . . . . . . . . . . . . . 56

4.22 Accepted Power of Cavity with Water in CST Studio. . . . . . . . . . . . 57

4.23 Maximum Electric Field of Cavity with Water in CST Studio. . . . . . . . 57

4.24 Comparison of Near Fields With and Without Water in the Cavity. . . . . 59

4.25 Longitudinal Contour of Near Field Pattern at 2.6 and 2.7 GHz. . . . . . 60

4.26 Plot of Field Magnitude through Top Hotspot from 2.6 GHz to 2.7 GHz. . 61

4.27 Position of Maximum along Z-Axis from Waveguide against Frequency. . . 61

4.28 Position of First Null along Z-Axis from Waveguide against Frequency. . . 62

4.29 Position of Second Null along Z-Axis from Waveguide against Frequency. 62

4.30 The Effect of Feed Phase at 2.3861 GHz for the Same Face Model. . . . . 64

4.31 The Effect of Feed Phase at 2.4478 GHz for the Same Face Model. . . . . 64

4.32 The Effect of Feed Phase at 2.5889 GHz for the Same Face Model. . . . . 65

4.33 The Effect of Feed Phase at 2.6513 GHz for the Same Face Model. . . . . 65

4.34 The Effect of Feed Phase at 2.419 GHz for the Same Face Model. . . . . . 65

4.35 The Effect of Feed Phase at 2.501 GHz for the Same Face Model. . . . . . 66

4.36 The Effect of Feed Phase at 2.635 GHz for the Same Face Model. . . . . . 66

4.37 The Effect of Feed Phase at 2.387 GHz for the Adjacent Face Model. . . . 67

4.38 The Effect of Feed Phase at 2.455 GHz for the Adjacent Face Model. . . . 67

4.39 The Effect of Feed Phase at 2.588 GHz for the Adjacent Face Model. . . . 67

4.40 The Effect of Feed Phase at 2.644 GHz for the Adjacent Face Model. . . . 67

4.41 The Effect of Feed Phase at 2.385 GHz for the Adjacent Face Model. . . . 68

4.42 The Effect of Feed Phase at 2.421 GHz for the Adjacent Face Model. . . . 68

4.43 The Effect of Feed Phase at 2.635 GHz for the Adjacent Face Model. . . . 69

4.44 The Effect of Feed Phase at 2.387 GHz for the Opposite Face Model. . . . 70

4.45 The Effect of Feed Phase at 2.4558 GHz for the Opposite Face Model. . . 70

4.46 The Effect of Feed Phase at 2.6535 GHz for the Opposite Face Model. . . 70

4.47 The Effect of Feed Phase at 2.588 GHz for the Opposite Face Model. . . . 70

4.48 The Effect of Feed Phase at 2.421 GHz for the Opposite Face Model. . . . 71

4.49 The Effect of Feed Phase at 2.499 GHz for the Opposite Face Model. . . . 71

4.50 The Effect of Feed Phase at 2.635 GHz for the Opposite Face Model. . . . 72

A.1 Application Interface with Loaded Model and Plotted Data. . . . . . . . . 85

B.1 Comparison of CST Studio and FEKO at 2.2 GHz. . . . . . . . . . . . . . 92

B.2 Comparison of CST Studio and FEKO at 2.45 GHz. . . . . . . . . . . . . 92

ix



List of Figures x

B.3 Comparison of CST Studio and FEKO at 2.7 GHz. . . . . . . . . . . . . . 93

B.4 Radiated Power for Mode 1 in FEKO . . . . . . . . . . . . . . . . . . . . 93

B.5 Radiated Power for Mode 2 in FEKO . . . . . . . . . . . . . . . . . . . . 94

B.6 Accepted Power for Mode 1 in CST . . . . . . . . . . . . . . . . . . . . . 95

B.7 Accepted Power for Mode 2 in CST . . . . . . . . . . . . . . . . . . . . . 95

B.8 Accepted Power for Time Domain Solver in CST . . . . . . . . . . . . . . 96

B.9 Near Field Pattern for Mode 1 in FEKO. . . . . . . . . . . . . . . . . . . 97

B.10 Near Field Pattern for Mode 1 in CST. . . . . . . . . . . . . . . . . . . . 97

B.11 Near Field Pattern for Mode 2 in FEKO. . . . . . . . . . . . . . . . . . . 97

B.12 Near Field Pattern for Mode 2 in CST. . . . . . . . . . . . . . . . . . . . 98

x



List of Tables

2.1 Characterisation of some of the ionising and non-ionising regions of the

electromagnetic spectrum and their process of interacting with matter. . . 9

2.2 Dissipation factors of various solvents and materials at 3 GHz and 25◦ C. 16

2.3 Maxwell’s Equations in Differential and Integral Form . . . . . . . . . . . 18

3.1 Dimensions of the basic rectangular cavity. . . . . . . . . . . . . . . . . . 27

3.2 Characteristics of the WR340 waveguide. . . . . . . . . . . . . . . . . . . 32

3.3 Dimensions of the ideal rectangular cavity. . . . . . . . . . . . . . . . . . . 33

3.4 Summary of the Software Used for the Benchmark Section. . . . . . . . . 37

3.5 Summary of the Software Used for the Verification Section. . . . . . . . . 37

3.6 Summary of the Software Used for the Characterisation Section. . . . . . 37

3.7 Summary of the Software Used for the Frequency Sweeping Section. . . . 38

3.8 Summary of the Software Used for the Phase Shifting Section. . . . . . . 38

3.9 Summary of the Software Used for the Software Comparison. . . . . . . . 38

4.1 Radiated Power and Mean Ẽ Field Comparison. . . . . . . . . . . . . . . 40

4.2 Resonant Modes of the WR340 Waveguide. . . . . . . . . . . . . . . . . . 49

4.3 Resonant Modes of Cavity between 2.2 and 2.7 GHz . . . . . . . . . . . . 51

4.4 Frequencies at which Phase Shift was Examined. . . . . . . . . . . . . . . 63

B.1 Dimensions of ideal rectangular cavity. . . . . . . . . . . . . . . . . . . . . 90

B.2 Solver Times of CST Studio and FEKO. . . . . . . . . . . . . . . . . . . . 99

xi



List of Symbols

ρ tissue density [kg·m−3]

Cp tissue specific heat capacity [J·kg−1·K−1]

T temperature [K]

k thermal conductivity [W·m−1·K−1]

Qh heat flux of the source of ablation [W·m−1]

Qm metabolic heat flux [W·m−1]

Qp′ blood perfusion heat flux [W·m−1]

λ wavelength [m]

f frequency [Hz]

h Planck’s constant

G antenna gain [dBi]

D antenna directivity, essentially gain [dBi]

η antenna efficiency

Prad power radiated by the antenna [W]

Pin power input to the antenna [W]

θ◦HP half power beam angle in the θ direction [degrees]

ϕ◦
HP half power beam angle in the ϕ direction [degrees]

Ae effective aperture [m2]

R radial distance from the source [m]

L largest physical dimension of antenna [m]

E electric field strength [V·m−1]

J current density [A·m−1]

σ electric conductivity [S·m−1]

µ0 magnetic permeability of free space

ϵ′′ dielectric loss factor

ϵ0 permittivity of free space

ϵ′ dielectric constant

ϵ∗ relative complex permittivity

xii



List of Symbols xiii

E electric field intensity vector [V·m−1]

B magnetic flux density vector [Wb·m−2 or T]

H magnetic field intensity vector [A·m−1]

D electric flux density vector [C·m−2]

J electric current density vector [A·m−2]

ρv electric charge density [C·m−2]

fcmn cut-off frequency of mode mn [Hz]

kcmn wavenumber corresponding to mode mn

a length of waveguide in H direction, or width [m]

b length of waveguide in E direction, or height [m]

c speed of light

fmnl cavity resonant frequency for mode TEmnl [Hz]

kmnl wave number corresponding to mode TEmnl

xiii



List of Abbreviations

AMP Antenna Modelling Program

BEM Boundary Element Method

BEP Bidirectional Eigenmode Propagation

CAD Computer Aided Design

CMA Characteristic Mode Analysis

CPU Central Processing Unit

EM Electromagnetic

EME Eigenmode Expansion

FDTD Finite Difference Time Domain

FEM Finite Element Method

FIT Finite Integration Technique

FMM Fast Multipole Method

FT Fourier Transform

GPU Graphics Processing Unit

HF High Frequency

HIFU High Intensity Focused Ultrasound

HPBW Half Power Bandwidth

ISM Industrial, Scientific and Medical

LA Laser Ablation

LE-PO Large Element Physical Optics

LPDA Log Periodic Dipole Array

MLFMM Multilevel Fast Multipole Method

MOD Model Order Reduction

MoM Method of Moments

MRI Magnetic Resonance Imaging

MWA Microwave Ablation

NEC Numerical Electromagnetic Code

PEC Perfect Electric Conductor

xiv



List of Abbreviations xv

PO Physical Optics

RAM Random Access Memory

RF Radio Frequency

RFA Radio Frequency Ablation

RL-GO Ray Launching Geometric Optics

SAR Specific Absorption Rate

TE Transverse Electric

TEM Transverse Electromagnetic

TLM Transmission Line Matrix

UTD Uniform Theory of Diffraction

VNA Vector Network Analyser

VSWR Voltage Standing Wave Ratio

xv



Chapter 1

Introduction

Thermal ablation of a tumour is a clinical method of neoplastic tissue destruction by

inducing extreme local hyperthermia or hypothermia [1]. The procedure is like surgery

in the sense that it targets the whole tumour and a small margin of regular tissue

surrounding the tumour. Instead of the tissue being surgically removed, requiring cutting

the patient open, the tissue is killed and then reabsorbed by the body over some time

following the procedure [1].

Some sources of clinical hyperthermic ablation, such as microwave and radio frequency,

use needle-like interstitial devices, which are physically inserted into the tumour, as the

heat delivery mechanism. For microwave ablation, microwaves radiate from the needle-

like antenna. Microwaves propagate through all types of human tissue and induce the

most heat in tissues rich with polar molecules such as water [1]. Because tumours are

usually quite water-dense, microwave ablation can be quite effective at destroying them.

The inserted interstitial antenna, when energised, creates an electromagnetic field at

microwave frequency surrounding it, which induces heat in the tissue over time. From

an electromagnetic perspective, the antenna is fed with a signal at microwave frequency.

The antenna resonates and radiates electromagnetic radiation according to the specific

antenna’s far-field radiation pattern. A simple monopole antenna is often used, resulting

in a largely omnidirectional pattern, hence inducing a relatively even pocket of heat or

hotspot over the desired location.

1.1 Problem Statement

The current microwave ablation method, while being less invasive than open surgery to

remove tumours, is still an invasive procedure, as a device must be inserted into the body

1



Chapter 1. Introduction 2

to deliver the microwave effect to the needed area. As such, the procedure is usually

relatively superficial in areas that are easily accessible percutaneously, such as the liver,

the most common site in the human anatomy for microwave thermal ablation [2].

The efficacy of the procedure is therefore limited to the location and percutaneous ac-

cessibility of the patient’s tumour. If it is hidden behind organs or other structures,

the procedure becomes difficult and ineffective. Additionally, concerns about general

surgery are not circumvented, for instance, poor clotting factors or allergies to anaes-

thesia etc. on the patient’s behalf. Some methods of ablation, such as High-Intensity

Focused Ultrasound (HIFU) eliminate these issues, however, HIFU has been shown to

have various problems associated with trying to ablate tumours, such as skin burns,

acoustic impedance from the lungs and heart, particularly in the thoracic area, and

inefficient ultrasound transduction through bone [3].

The problem with microwave ablation arises from the need to insert an interstitial an-

tenna into the desired tumour. The problem would be mitigated if an electromagnetic

hotspot could be induced over a tumour without the need for the inserted antenna. The

process could be accomplished using electromagnetic fields at microwave frequencies. At

these frequencies, the radiation isn’t as easily steered as with the much higher frequency

X-band radiation therapy. Additionally, the size of the antenna required to narrow a

beam towards a point as small as a tumour becomes impractically large. Therefore, this

investigation is a resonance study, looking at the electromagnetic near fields inside a res-

onant cavity. Currently, there is no method for controlling the position of a microwave

frequency hotspot within a resonant cavity.

1.2 Research Question

The above problem statement leads to the following research question.

“Can the position of an electromagnetic hotspot at microwave frequencies be controlled

consistently and feasibly within a resonant cavity?”

This research question can be broken down into a series of sub-questions as follows:

• Can a microwave frequency electromagnetic hotspot be effectively modelled in a

resonant cavity?

• Can the resonance and field pattern of the model be changed by adding another

microwave source?

• Can altering the phase and frequency of each source change the electromagnetic

2



Chapter 1. Introduction 3

field pattern within the resonant cavity and hence, change the hotspot positions?

• Is the change of hotspot position consistent and feasible for the application of

clinical microwave thermal ablation?

1.3 Methodology Description

The methodology used to answer the research questions is divided into four sections.

Software testing, where the software packages used are tested against each other for

unanimity and for their ability to effectively handle resonance problems; software veri-

fication, where simulated data is compared to data acquired through real-world testing

to establish accuracy in resonance problems; characterisation of the established field in

simulation to quantify characteristic modes and resonant frequencies; and field manip-

ulation, where feed frequency and phase differences are introduced in an attempt to

control the position of the hotspots inherent to a field pattern.

1.4 Organization of Dissertation

The rest of the dissertation is organized as follows. Chapter 2 pertains to the literature

review and background information relevant to this investigation. Chapter 3 discusses

the detailed methodology used to answer the research questions. Chapter 4 contains the

results obtained and conclusion hence drawn. The conclusion is discussed in Chapter 5,

along with future recommendations and a review of the research questions.

3



Chapter 2

Background

This chapter discusses the types of thermal ablation and their underlying physics. Elec-

tromagnetic theory on near field problems, the microwave frequency spectrum and its

interactions with dielectric tissue are discussed, along with field manipulation methods

including antenna and array theory. Additionally, the solver methods and modelling

software to be used in the investigation, including background into how each solver

methods works and their respective benefits and shortcomings, is presented.

2.1 Thermal Ablation

2.1.1 Biophysics

Thermal ablation is a method of destroying tissue by inducing hyper or hypothermia

in the tissue, as a substitute for open surgery. This is a transient thermal heat trans-

fer problem that can be described using Fourier’s heat equation. Penne produced a

biothermal formulation of Fourier’s heat equation [4] as follows:

ρCp
∂T

∂t
= k∇2T +Qh +Qm +Qp′ (2.1)

Where:
ρ = tissue density [kg·m−3]

Cp = tissue specific heat capacity [J·kg−1·K−1]

T = temperature [K]

k = thermal conductivity [W·m−1·K−1]

Qh = heat flux of the source of ablation [W·m−1]

Qm = metabolic heat flux [W·m−1]

Qp′ = blood perfusion heat flux [W·m−1]

4



Chapter 2. Background 5

High volume blood vessels in the target area can create a heat sink for the thermal

energy, which Equation (2.1) does not account for, however, for nearby vessels with a

diameter of approximately 3 mm or less, the formulation holds [5].

2.1.2 Current Procedures

There are currently many implementations of clinical thermal ablation, the most com-

mon of which being Radio Frequency Ablation (RFA), Microwave Ablation (MWA),

Cryoablation, Laser Ablation (LA) and High Intensity Focused Ultrasound (HIFU). All

except HIFU use interstitial antenna devices to deliver the thermal energy and all are

often accompanied by an image guidance technique, such as magnetic resonance imaging

(MRI) [6].

Radiofrequency

RFA, one of the most used techniques, utilises an electrode; which is connected to a

radio wave generator, operating at radio frequency, and inserted directly into the target

tumour, while a reference electrode is often placed on the subject’s skin [1]. This is

known as unipolar operation. The bipolar operation is sometimes used, where the two

electrodes are placed on either side of the target tissue. When an electric current at

radiofrequency is applied through the electrode, the surrounding tissue heats up as a

result of resistive losses as current travels towards the reference electrode, which excites

the ionic molecules in the tissue through which it passes [6].

Cryoablation

Cryoablation uses the Joule-Thomson effect, which describes gas under high pressure

being forced through a porous plug or valve into an area of low pressure, causing the gas

to expand and cool down [7]. This method requires a probe, which carries the required

gas internally, to be inserted into the target tissue. The standard procedure utilises

argon gas to cool the tumour down (to as low as -160◦ C) by forming an ice ball, which

also clearly defines the ablation margins, and helium gas to warm the probe up again.

Typically, the procedure comprises multiple cycles of freeze and thaw in the area, after

which the probe is removed [6].

Laser

LA is performed by directly exposing the tissue to laser energy via a transport medium.

The laser provides a monochromatic light, the wavelength of which determines the laser

properties and tissue interactions, which is transported, usually through an optical fibre,

5



Chapter 2. Background 6

directly to the ablation site. The absorption of the laser energy by the tissue is converted

to heat, which creates a temperature gradient through the surrounding tissue. The

treatment can be administered in continuous mode, where low laser power is used over a

longer period, or pulsed mode, where high laser power is administered intermittently [8].

Focused Ultrasound

Ultrasound energy can be delivered both internally, using interstitial devices, like RFA

and MWA, or using external transducers. This makes ultrasound a very promising field.

HIFU is a process that requires only external transducers, lending the ability to ablate

tissue truly non-invasively. It relies on multiple ultrasonic beams converging to a focal

point of heating [1]. The ablation zone is usually tiny, about the size of a grain of rice,

and the process for destroying a full tumour involves overlapping many of these areas of

focus to cover the entire volume of the tumour [1]. Because of this, HIFU can often take

a matter of hours to complete, instead of the handful of minutes needed by the other

invasive techniques, and hence, the ablation site must generally be in easily accessible

areas that are not subject to any substantial movement over time; from breathing, for

instance. HIFU has also been shown to have other associated problems in practice, such

as skin burns and inefficient signal transduction through dense bone [3].

Microwave

MWA uses the microwave effect to induce heat in the target tissue. Like RFA, a device

is inserted into the target tumour, however, in this instance, it is a microwave antenna.

Electromagnetic microwaves radiate out from the interstitial antenna, creating a hotspot

around the device and interacting with the surrounding tissue. Tissue containing polar

molecules such as water are most affected by this method of heating, as the microwave

heating effect causes the molecules to vibrate [6]. Additionally, microwave frequency

electromagnetic waves propagate through all types of human tissue, in contrast to RF

ablation, which relies on the impedance of the tissue for current propagation. Therefore,

the entire ablation zone undergoes heating if it contains polar molecules. The main

difference between RFA and MWA is the frequency. RFA operates at radio frequency

and MWA operates at microwave frequency, which is much higher. Therefore, the tissues

of the human body are much larger in terms of wavelength for RFA when compared to

MWA. Because of this, RFA relies largely on current flow between the electrodes, whereas

the probe used in MWA becomes an actual antenna instead of an electrode.

6



Chapter 2. Background 7

Figure 2.1: Electromagnetic Wave. Adapted from [11].

2.2 Relevant Electromagnetics

2.2.1 Electromagnetic Waves, Fields and Radiation

Microwaves are a form of electromagnetic radiation. Unlike mechanical waves, like sound,

they can radiate through a vacuum, not requiring a medium to carry them, and through

transmission lines, as a Transverse Electromagnetic (TEM) wave [9]. Electromagnetic

waves have two components, Electric field (E) and Magnetic field (H), which act per-

pendicular to each other, as shown in Figure 2.1. These waves are characterised by their

main three properties, being frequency, velocity and electric field strength. The velocity

at which electromagnetic waves propagate in a vacuum, or in ‘free space’, is denoted by

the symbol c, which is ≈ 3× 108 m·s−1, also called the speed of light in free space [10].

A fundamental factor of electromagnetic waves is its frequency and wavelength, which

are related by the velocity of the wave, given in Equation (2.2) and the energy it carries,

given by the Planck relation, Equation (2.3). Electric or magnetic field strength simply

refers to the magnitude of the field, often given in volts per meter (V/m) and amperes

per meter (A/m).

λ =
c

f
(2.2)

E = hf (2.3)

Where:
λ = wavelength [m]

f = frequency [Hz]

h = Planck’s constant ≈ 6.63× 10−34 m2·kg·s−1

Electromagnetic fields are simply the combination of interconnected electric and mag-

netic fields. Radiation is simply the fluctuation of these fields in space [12].

7



Chapter 2. Background 8

Figure 2.2: Electromagnetic Spectrum. Adapted from [11].

2.2.2 Electromagnetic Spectrum

The frequencies at which electromagnetic waves can propagate are limitless. The electro-

magnetic spectrum refers to the phenomena that occur as a result of an electromagnetic

field oscillating within specific ranges of frequencies across the continuum of frequen-

cies [10]. Figure 2.2 shows some phenomena that occur at various wavelengths. It is

noteworthy that as the frequency of the wave increases, so too does the wave energy.

The microwave spectrum is generally accepted as occurring between the wavelengths of

30 cm to 3 mm, or frequencies of 1 GHz to 100 GHz. The typical frequency at which mi-

crowave ovens and, incidentally, Industrial, Scientific and Medical (ISM) devices operate

is 2.45 GHz, corresponding to a wavelength of ≈ 122 mm.

Most of the electromagnetic spectrum that we experience in our daily lives, including

irradiation from the sun, forms part of the non-ionising spectra of frequencies, those

below the X-ray frequencies, which are usually regarded as ‘harmless’. Radiation asso-

ciated with nuclear fission is ionising radiation, consisting of electromagnetic waves at

a very high frequency (X-ray and above) and alpha and beta particles, resulting from

the decay of radioactive material. These waves are generally highly penetrating and can

cause tissue damage when absorbed [13]. It is important to note that microwave spectral

frequencies are not considered ionising. Table 2.1 shows the spectral characteristics of

some types of ionising and non-ionising radiation [11].

2.2.3 Antennas

An antenna is a structure that radiates electromagnetic energy. An isotropic antenna

radiates electromagnetic energy equally in all directions. A useful property of antennas

is directivity, which means the field is directed. Directivity, D is generally analogous with

8



Chapter 2. Background 9

Table 2.1: Characterisation of some of the ionising and non-ionising regions of the elec-
tromagnetic spectrum and their process of interacting with matter.

Radiation Type Frequency (Hz) Energy (J) Process

Ionising

γ-ray > 1019 > 6.6 · 10−15 Ionisation and
nuclear effects

X-ray 1016 − 1019 6.6 · 10−18 to 6.6 · 10−15 Ionisation

Non-ionising

Ultraviolet 1014 to 1016 6.6 · 10−20 to 6.6 · 10−18 Electronic
transitions

Visible light 1014 6.6 · 10−20 Electronic
transitions

Infrared 1011 to 1014 6.6 · 10−23 to 6.6 · 10−20 Rotation of
molecules and
vibration of
flexible bonds

Microwaves 109 to 1011 6.6 · 10−25 to 6.6 · 10−23 Rotation of
molecules

antenna gain, G, being dependant on antenna efficiency, as shown in Equation (2.4) [14].

For practical purposes, the efficiency, η, of an antenna is usually assumed to be 1,

meaning Gain and Directivity are equivalent. This is a general assumption for high

frequency (HF) and up.

G = ηD (2.4)

η =
Prad

Pin
(2.5)

Where:
G = antenna gain [dBi]

D = antenna directivity, essentially gain [dBi]

η = antenna efficiency

Prad = power radiated by the antenna [W]

Pin = power input to the antenna [W]

The higher the gain in a particular direction, the more the field is directed in that solid

angle, instead of radiated omnidirectionally [14]. The field pattern is manipulated by the

antenna’s geometric characteristics, to radiate more energy in a certain direction than in

others. There are limitations on this, however. The approximation in Equation (2.6) [14]

9



Chapter 2. Background 10

shows the Half-Power Beamwidth (HPBW) of an antenna’s radiation pattern compared

to its gain. Here, the gain is absolute, not in logarithmic form.

G ≈ 36000

θ◦HPϕ
◦
HP

(2.6)

Where:
θ◦HP = half power beam angle in the θ direction [degrees]

ϕ◦
HP = half power beam angle in the ϕ direction [degrees]

This shows that the more concentrated the radiation must be, or the smaller the HPBW,

the higher the gain of the antenna must be. Equation (2.7) [14] shows that the effective

aperture of an antenna, or the space it occupies, is directly proportional to the gain of

the antenna. Therefore, an antenna that could produce a radiation pattern that focuses

a beam to a small enough HPBW to affect a tumour and not its surrounding tissue

would require a very large gain, requiring a very large antenna.

Ae =
Gλ2

4π
(2.7)

Where:

Ae = effective aperture [m2]

For these relationships to hold, the receiver and source must be sufficiently far away, in

the so-called far field, discussed in Section 2.2.5.

2.2.4 Impedance

Antenna impedance plays a large role in the operation of antennas and electromag-

netic devices. An impedance mismatch involving an antenna ultimately results in signal

attenuation. Therefore, antennas are ideally used in their impedance bandwidth, the

range of frequencies where their impedance is deemed acceptable for enough energy to

be radiated by the antenna instead of reflected. A common metric for quantifying this

impedance bandwidth is the voltage standing wave ratio (VSWR) of an antenna. When

impedances are not matched, reflection occurs and standing waves are set up so not all

available power is transferred [14]. The measure of the impedance mismatch is quantified

by ρ, the voltage reflection coefficient, given in Equation (2.8). The lower the reflection,

the better the power transmission.

ρ =
Z2 − Z1

Z2 + Z1
(2.8)

10



Chapter 2. Background 11

VSWR is the more common way of stating impedance mismatch. It is related to the

reflection coefficient as follows:

V SWR =
1 + |ρ|
1− |ρ|

(2.9)

In terms of the generated standing waves resulting from an impedance mismatch, VSWR

can be calculated by dividing the maximum voltage of the envelope by the minimum

voltage: Vmax/Vmin. VSWR is often used to describe the impedance bandwidth of an

antenna. By general rule, if the VSWR ≤ 2, the antenna is considered matched and the

range of frequencies where the VSWR is below 2 is the impedance bandwidth.

2.2.5 Near and Far Fields

The area surrounding an electromagnetic source, such as an antenna, is broken up into

three regions: the reactive near field, the radiative near field and the far field. For

small antennas, this is often approximated to a single near field far field transition

point at λ/2π. This approximation stems from the Eθ component of the electric field

radiated by an antenna. It contains three components based on 1/r3, 1/r2 and 1/r, which

respectively describe the electric field component of the reactive near field, radiative near

field and far field surrounding the source.

Immediately surrounding the source, the 1/r3 term dominates, but as the distance from

the source increases, or r increases, that term becomes smaller very quickly. The far

field is described by the 1/r term. At r = λ/2π, a crossing point occurs, and the far

field term becomes most dominant. Therefore, this is usually approximated as the far

field transition point.

This approximation becomes difficult for physically large antennas, where it is unclear

where to start measuring the distance from. Therefore, these regions are described

more accurately by Equations (2.10), (2.11) and (2.12) [15]. Equation (2.10) is the

transition point between the reactive and radiative near field, Equation (2.11) describes

the radiative near field region between the reactive near field and the far field, and

Equation (2.12) is the transition point at which the far field is dominant. They hold for

all antennas including large antennas.

R <

√
L3

λ
(2.10)√

L3

λ
≤ R <

2L2

λ
(2.11)

11



Chapter 2. Background 12

Figure 2.3: Near Field vs Far Field. Adapted from [18].

R ≥ 2L2

λ
(2.12)

Where:
R = radial distance from the source [m]

L = largest physical dimension of antenna [m]

λ = the wavelength of the signal frequency [m]

In the reactive near field, henceforth referred to as the near field, reactive power densities

dominate. In the far field, the field behaviour is predictable and approaches plane waves,

whereas the near field is operationally complex and reactive, with the field pattern being

far more complicated and irregular [16]. Figure 2.3 shows a basic comparison of the wave

interactions of the three regions.

In the near field, there are far more wave interactions, resulting in a non-uniform field

pattern [16]. The diffraction in this region is referred to as Fresnel diffraction [17]. If

the pattern is observed from the point of view of magnitude, as with standard radia-

tion patterns, the near field pattern has small areas of increased magnitude and areas

of decreased magnitude, resulting from the wave interactions, as shown on the left in

Figure 2.4. Near field interactions make it possible to concentrate a field on a small point

in space, such as over a tumour. In the far field, all common antenna relationships and

design criteria start becoming applicable. Diffraction in this zone is called Fraunhofer

diffraction [17].

Equation (2.12) shows that the larger the antenna becomes, the further away the near

field zone is apparent. If far field radiation is to be aimed at a tumour sized object, a

very narrow beamwidth is needed, resulting in a very large physical antenna, hence a far

field criterion that is far away from the source, meaning an increased distance between

12



Chapter 2. Background 13

Figure 2.4: Near (left) and Far (right) field radiation patterns. Adapted from [19].

the tumour and the source. It becomes infeasible to satisfy all the required relationships

while maintaining reasonable antenna size.

2.2.6 Array Theory

Array theory arises from the contribution of multiple field sources to a radiation pattern.

Vector addition is performed of the field magnitude and phase at all directional points,

or solid angles, surrounding the array to calculate the new radiation pattern [14]. This

method is very useful in that it can increase the gain of the array by adding contributions

of smaller elements instead of increasing the size of a single element. The pattern can

also be manipulated by altering the phase of the feed to each element without having

to alter the physical structure of the array and the array can be configured to create a

broadside or end-fire pattern [20]. Figure 2.5 shows the utility of array theory in creating

a broadside (left) and broadside/end-fire (right) array, both with enhanced gain using

multiple instances of a simple point source. It also allows for the forward gain, or

maximum gain, to be directed and steered without the need of changing the array’s, or

the antennas’, physical attributes.

Array theory is useful in that it supplies a method to predict a field pattern based on

the feed phase of multiple sources. It is still not feasible for tumour ablation purposes,

however, as the relationships in Equations (2.6) and (2.7) still hold, regardless of whether

the source is an array or a single antenna. The space requirements that allow for a gain

large enough to narrow the beam sufficiently are still enormous, as Array Theory is

applicable only in the far field.

13



Chapter 2. Background 14

Figure 2.5: Radiation Pattern of Broadside and Endfire arrays. Adapted from [20].

2.2.7 Resonance and Resonant Cavities

Mechanical resonance refers to the greater amplitude response of a mechanical system

when it oscillates with a frequency approaching its natural vibration frequency. It and

‘tuned circuit’ resonance demonstrates that given enough energy, even at a frequency

other than the natural frequency, the system persists in resonating at that other fre-

quency; and that the resonance bandwidth is quite narrow. In terms of antennas and

antenna theory, if an antenna is fed with an appropriate frequency that falls within its

bandwidth, it is said to be resonating, as in, radiating the power it is fed.

Resonance in electromagnetic terms, however, describes the sharp increase in ampli-

tude, or magnitude, resulting from the application of an electromagnetic force or field of

a specific frequency that approaches a natural frequency, or eigenfrequency, of the sys-

tem upon which it acts [21]. It can also be referred to as the increased absorption that

occurs when the correct combination of electromagnetic field and frequency is applied

to the structure [22]. Resonant cavities are bounded resonant structures whose dimen-

sions are comparable with, or larger than, the operating wavelength of the excitation

signal. Ideally, such cavities are operated at one of their resonant modes, to achieve

a resonant peak in terms of power delivery [23]. Because the cavity is, by definition,

enclosed in metal, it is non-radiative and highly reactive inside the cavity, meaning the

electromagnetic field patterns, or eigenfields, inside the cavity are a purely near field

problem.

14



Chapter 2. Background 15

2.3 Microwaves

Microwaves are electromagnetic waves at microwave frequencies, as described in Fig-

ure 2.2. The microwave spectra of frequencies have many applications, including the

microwave heating effect, the basis of microwave ovens, communication technologies,

such as Bluetooth and Wi-Fi and other ISM devices.

2.3.1 Microwave Heating

Microwave heating is an effect whereby a medium absorbs electromagnetic energy to

achieve rapid heating. The heating results from the conversion of electromagnetic en-

ergy at microwave frequencies to thermal energy through the agitation, or continued

realignment to the field, of polar molecules such as water [24]. The magnetic perme-

ability of most biological tissue is very similar to that of free space [25], denoted µ0,

4π × 10−7 H·m−1. Therefore, biological tissue does not interact substantially with the

magnetic component of electromagnetic fields and hence, heating of the tissue under the

influence of a microwave field is predominantly a result of the fluctuation of the electric

field.

The heating power term from Equation (2.1) that describes microwave heating is shown

below in Equation (2.13).

Qh = J ·E =
J2

2σ
=

σE2

2
(2.13)

Where:
Qh = heat flux of the source of ablation [W·m−1]

E = electric field strength [V·m−1]

J = current density [A·m−1]

And σ is the effective electric conductivity, in [S·m−1], given by

σ = 2πfϵ0ϵ
′′ (2.14)

Where:
ϵ′′ = dielectric loss factor

ϵ0 = permittivity of free space [≈ 8.85× 10−12 F·m−1]

Hence, microwave power as a function of displacement and temperature is related to the

electric field and can be described by Equation (2.15) [26].

P (x, T ) =
1

2
ωϵ0ϵ

′′E2 (2.15)

15



Chapter 2. Background 16

2.3.2 Dielectric Influence

Different materials exhibit different affinities for converting electromagnetic field energy

into heat, dependant on the material’s dielectric response, a function of its dielectric

characteristics [11], [27]. Each material’s capacity to absorb electromagnetic energy is

proportional to how well the microwaves can penetrate the material.

For reflective materials, such as metals, the penetration is zero, hence there is no ab-

sorption, whereas, for transparent materials, penetration tends to infinity [27]. When

the energy is absorbed, the energy conversion into thermal energy is dependent on the

dissipation factor (or loss tangent) of the material, as shown in Equation (2.16) [11].

tan δ =
ϵ′′

ϵ′
(2.16)

Where ϵ′ is the dielectric constant. The relative complex permittivity, ϵ∗ is given by

Equation (2.17).

ϵ∗ = ϵ′ − jϵ′′ (2.17)

Higher values for tan δ translate to a higher heating capacity when microwaves are ab-

sorbed [28]. Table 2.2, data acquired from [11], shows the dielectric dissipation factor of

some materials and solvents.

All the listed solvents have a higher affinity for microwave absorption than do the ma-

terials, hence materials are sometimes mixed with solvents to be heated more efficiently

by microwave heating [11]. For all the listed substances, the dissipation factor was cal-

culated at a frequency of 3 GHz and at 25◦ C, except for acetic acid and toluene, which

were calculated at 2.45 GHz and 20◦ C.

Table 2.2: Dissipation factors of various solvents and materials at 3 GHz and 25◦ C.

Solvent tan δ Material tan δ

Ethylene glycol 1 Nylon 6,6 12.8 · 10−3

Ethanol 0.25 Poly(vinyl chloride) 5.5 · 10−3

Methanol 0.64 Porcelain 4462 1.0 · 10−3

Acetic acid 0.174 Borosilicate glass 1.06 · 10−3

Water 0.157 Ceramic F66 0.55 · 10−3

0.1 mol/l NaCl 0.24 Polyethylene 0.31 · 10−3

0.5 mol/l NaCl 0.63 PTFE-PFA 0.15 · 10−3

Toluene 0.040 Fused quartz 0.06 · 10−3

16



Chapter 2. Background 17

In cases where the dielectric microwave absorption properties are not linear and are

dependent on the temperature of the dielectric, thermal runaway can easily occur. The

dielectric heats up, its affinity to absorb microwaves increases, further heating it up in

a positive feedback loop.

2.3.3 Microwave Ovens

Microwave ovens are common household devices that utilise the microwave heating effect

to heat food. A drawback of these ovens is that heating is non-uniform, as a result of

the resonance within the cavity that causes areas of high electromagnetic field intensity

and low electromagnetic intensity, or hotspots and cold spots [29], as is characteristic of

near field patterns. To account for this non-uniformity, historical microwave ovens had a

‘stirrer’, a piece of metal that rotated, thus actively changing the near field pattern and

the position of the hotspots. These days, microwave ovens often have a glass turntable

at their base upon which the food is placed. The objective is to rotate the food through

a non-uniform field so that different sections of the food are sporadically acted upon by

a hotspot or collection of hotspots at a point in space, distributing the heating amongst

the food, instead of only a small section of food continuously heating up while the rest

remains cold.

The resonant near field pattern of the field inside the oven is dictated by the cavity geom-

etry, dielectric properties of the material inside the chamber and the size and geometry

of the dielectric material [24]. A microwave oven essentially describes the ability to heat

a certain part of a dielectric and not others, as often happens when food is heated up. It

provides an unrefined, nonspecific and imprecise mechanism for microwave thermal ab-

lation of tissue. With microwave ovens, efforts are often made to provide wide, uniform

heating of the entire dielectric, such as the addition of the turntable, instead of precise,

localised heating which is needed for clinical thermal ablation in humans.

2.4 Electromagnetic Solvers

This section discusses the solution to electromagnetic problems, the various solver meth-

ods used to solve the applicable models and the simulation packages used in this inves-

tigation.

2.4.1 Maxwell’s Equations

Maxwell’s equations are four coupled partial differential equations, developed by James

Clerk Maxwell, which form, along with the Lorentz Force Law, the basis of computational

17



Chapter 2. Background 18

electromagnetism. The equations provide a mathematical model for electromagnetic

systems, describing how the electric and magnetic components of electromagnetic fields

are generated by changing charges and currents [30]. In computational electromagnetic

problems, the electric and magnetic fields must satisfy the time dependant Maxwell

equations, given below:

∇×E = −∂B

∂t
(2.18)

∇×H = J+
∂D

∂t
(2.19)

∇ ·D = ρv (2.20)

∇ ·B = 0 (2.21)

Where:
E = electric field intensity vector [V·m−1]

B = magnetic flux density vector [Wb·m−2 or T]

H = magnetic field intensity vector [A·m−1]

D = electric flux density vector [C·m−2]

J = electric current density vector [A·m−2]

ρv = electric charge density [C·m−2]

These equations are interdependent. B and D are essential for calculating E and H.

These equations are usually represented in differential form; however, their integral form

can also be useful in certain circumstances. Both forms of the equations are shown in

Table 2.3 [30].

Table 2.3: Maxwell’s Equations in Differential and Integral Form

Differential form Integral form Name

∇×E = −∂B

∂t

˛
c
E · d̃l = −

¨
s

∂B

∂t
· ds̃ Faraday’s law

∇×H = J+
∂D

∂t

˛
c
H · d̃l = I +

¨
s

∂D

∂t
· ds̃ Ampere’s law

∇ ·D = ρv

‹
s
D · ds̃ = Q Gauss’s law for electric charge

∇ ·B = 0

‹
s
B · ds̃ = 0 Gauss’s law for magnetic charge

18



Chapter 2. Background 19

Essentially, electromagnetic solvers or simulation packages calculate the electric and

magnetic components of the electromagnetic field by solving Maxwell’s equations, which

generally cannot be solved analytically owing to the inherent interdependency.

2.4.2 Method of Moments

Maxwell’s equations, discussed in Section 2.4.1, are very difficult to solve analytically.

When dealing with complex antennas, three-dimensional space, large structures, or most

practical purposes, they cannot be solved analytically, but they can be solved numeri-

cally. The Method of Moments (MoM) technique converts the integral equations into a

linear system which can be solved numerically using a computer [31].

Essentially, the MoM solves the problem by converting integral equations into matrix

equations. A noteworthy feature of the MoM is that it is a purely surface-based method,

meaning that only the surface of the physical structure of the object being analysed is

considered [32]. This makes the MoM very effective at dealing with perfectly conductive

wire segment structures without the presence of penetrable absorbing bodies.

Additionally, the MoM is a purely frequency domain based solver method, making it

efficient at solving single frequency or narrow band problems. However, for wideband

problems, a fully new solution must be generated for each instance of frequency, which

significantly increases solver time [32].

The procedure comprises four steps. Firstly, the structure to be modelled must be dis-

cretised into a collection of small wire segments that make up geometric primitives.

The Numerical Electromagnetic Code (NEC) generally uses a grid pattern for wire seg-

ments, however, programs such as FEKO and CST Studio use triangles, because they

can conform to almost any shapes’ curvature and because of the very efficient triangle

integration rules [33]. Secondly, expansion functions that represent the unknown current

distribution along the structure and weighting functions for the segments must be cho-

sen. Thirdly, filling the matrix using the chosen functions and solving for the unknown

current distribution along the surface of the structure. Lastly, processing the calculated

current distribution of the structure to find the desired characteristics of the system,

including near field, far field, impedance and whatever else is required [32].

2.4.3 Fast Multipole Method

Iterative methods for solving the matrices involved in MoM computation can shorten

compute time by starting with an approximation of the solution vector and attempting

to minimise residuals for each iteration until the solution is sufficiently correct [31].

19



Chapter 2. Background 20

The Fast Multipole Method (FMM) is a method of speeding up MoM solutions instead of

being its own technique. It is usually implemented as an iterative numerical algorithm

that reduces the complexity of the problem. FMM uses an error-controlled approxi-

mation of the system. This allows the field interactions of a collection of particles to

be evaluated as if they were a single particle [34]. The interactions between separated

groups of functions can then be evaluated, allowing for the calculation of the matrix

product iteratively without having to store many elements, leading to faster computing

time. However, the runtime may still be high for complicated problems owing to a higher

number of matrix-vector products and the fact that the solution may not converge on a

correct solution easily [34].

2.4.4 Finite Element Method

The Finite Element Method (FEM) is a numerical method used to solve partial differ-

ential equations, like the Maxwell equations, by converting them into matrix equations.

This process is similar to that used by the MoM, however, the partial differential form

of the equations is used, instead of the integral form [35]. One of the primary features

of FEM is that it can describe the essence of the problem, its geometry, dielectric bodies

etc., with a degree of flexibility. This is owed to the discretisation of the domain of the

problem into its so-called finite elements, which use non-uniform segments of elements,

which can easily describe complex structures [36].

FEM resembles how MoM solves the problem in that they both convert a differential

or integral equation into algebraic matrix equations. However, a key difference is that

FEM is based on the principle of minimising the energy of the system [36] and uses

sparse matrices which are solved iteratively, instead of the MoM’s full matrices which

are solved traditionally, using Gaussian elimination or LU decomposition. This makes

FEM a slightly faster solver method and is more suited to 3D complex structures and

the introduction of penetrable bodies, such as dielectrics. Essentially, FEM is a bet-

ter method for solving volumetric problems, whereas MoM is ideally used for skeletal

situations.

2.4.5 Finite Difference Time Domain

The Finite Difference Time Domain (FDTD) method of electromagnetic simulation pro-

vides a direct integration of Maxwell’s time dependant differential equations, discussed

in Section 2.4.1. Since FDTD uses volumetric data instead of surface data, it is very

effective at modelling complex 3D structures [37]. The main difference between this

method and MoM and FEM is the fact that it is a time domain technique instead of a

20



Chapter 2. Background 21

frequency domain technique [37]. This makes large frequency ranges very easy to han-

dle, as, in a single time domain simulation, a response of the system to a wide range of

frequencies can be decomposed using Fourier Transform (FT) techniques.

With FDTD, the space of the solution is divided into uniform mesh cells, comprising 3D

voxels instead of surface triangles or tetrahedrons. For each mesh cell, the electric and

magnetic field components are defined, as with FEM, however, instead of developing a

matrix equation, the fields are staggered in space and moved through time, allowing for

a direct field solution in time to be calculated [37]. In other words, for each iteration in

time, the field components are determined and stored for that instance of time. After

which, the frequency domain’s spectral components can be calculated using FT.

2.4.6 Finite Integration Technique

The Finite Integration Technique (FIT) is another spatial discretisation method to nu-

merically solve electromagnetic problems. It is very similar to FDTD in terms of how

the field is solved. The main difference is that the FIT solves the integral form of time

dependant differential equations, whereas FDTD solves the differential form. Hence,

FIT can yield results in both the time and frequency spectral domains [38].

2.4.7 Eigenmode Expansion

Eigenmode Expansion (EME) is a computational electromagnetics method, alternatively

called the Mode Matching Technique or the Bidirectional Eigenmode Propagation (BEP)

method [39], used to calculate the various resonant frequencies and corresponding elec-

tromagnetic field pattern, or eigenmodes, of a structure, without the need of an excitation

signal [40]. It is a linear frequency domain method of solving Maxwell’s equations.

The approach is ideally suited for bounded cavities and waveguides as it can deal with

cavity segments of arbitrary length without needing to reallocate memory as length

increases [39]. Additionally, mismatch and impedance are of no concern as there is no

excitation signal. This method is suited for finding resonant frequencies of bounded

cavities fed with waveguides, however, it cannot deal with phase shifting of sources.

2.4.8 Modelling Software

This section introduces the relevant software packages used in this investigation and

their utilisation of the various electromagnetic solver methods.

21



Chapter 2. Background 22

FEKO

FEKO [41] is a comprehensive electromagnetic solver that utilises multiple solution

methods, each ideally suited to different applications, for electromagnetic field analy-

sis. The University of the Witwatersrand provided a full license to access FEKO and all

its features. FEKO supports full-wave frequency domain and time domain, as well as

asymptotic solution methods [42]. Additionally, it features a characteristic mode analy-

sis (CMA) configuration, which determines the dominance of the current modes on the

structure.

• Full-wave frequency domain

– Method of moments (MoM)

– Finite element method (FEM)

– Multilevel fast multipole method (MLFMM)

• Full-wave time domain

– Finite difference time domain (FDTD)

• Asymptotic

– Physical optics (PO)

– Large element physical optics (LE-PO)

– Ray launching geometric optics (RL-GO)

– Uniform theory of diffraction (UTD)

Only, MoM, FEM, MLFMM, a type of FMM, FDTD and CMA are relevant to this

investigation. There are also options to solve problems using the hybridisation of different

solvers. Figure 2.6 shows the applications of the various solvers. FEKO comes with its

own comprehensive CAD modelling application, used to create structures, discretise

them and set up the simulation. There is also a comprehensive post-processor used to

view the results of the simulation. This all makes FEKO ideally suited to the needs of

this investigation.

FEKO can make use of a waveguide port, making the feeding of resonant cavities easier.

For this investigation, the only solutions methods that were utilised were MoM and

MLFMM. FEM was excluded as it can only be used in FEKO to solve dielectric media

and cannot handle PEC boundaries in free space. FDTD was excluded as it cannot

utilise the waveguide port method of feeding the structure.

22



Chapter 2. Background 23

Figure 2.6: The Hybridisation of the FEKO Solvers for their Respective Applications.
Image adapted from [42].

CST Studio Suite Student

CST Studio Suite [43] is a high-performance 3D electromagnetic analysis software pack-

age, developed by Dassault Systèmes, for designing, analysing and optimising electro-

magnetic fields, components and systems. CST offers a free student version with a va-

riety of limitations on mesh sizes, available solvers and export and import capabilities.

Additionally, a watermark is always present, and it cannot be used commercially.

Like FEKO, CST has the capability of utilising many individual or multiple electro-

magnetic solver methods. Unlike FEKO, where each solver method can explicitly be

selected, CST utilises many different solver platforms, based on the needed application.

The program has five main classes of solver, namely High Frequency, Low Frequency,

Multiphysics, Particles, and EMC and EDA [40]. High Frequency, the solver platform

relevant to this investigation can be broken down into a series of solver methods, as

follows:

• Asymptotic

• Eigenmode

• Filter Designer 2D

• Filter Designer 3D

• Frequency Domain

23



Chapter 2. Background 24

• Integral Equation

• Multilayer

• Time Domain

• Hybrid Solver Task

In the student edition, only Frequency Domain, Time Domain and a selection of static

and thermal solvers are available. The frequency domain solver is based on FEM, includ-

ing a model order reduction (MOD) feature to speed up the simulation time of resonant

structures. The time domain solver utilises the FIT solver and the transmission line

matrix (TLM) solver, which is not relevant to this investigation. The combination of

FEM and FIT solver methods makes CST very efficient at wideband frequency sweeps.

CST Suite Students Edition does not include an implementation of the Boundary Ele-

ment Method (BEM), so there is no MoM or FMM solver included. These implementa-

tions are part of the Multilayer and Integral Equation solvers, which are only available

with the full version of CST. However, the frequency and time domain solvers available

are sufficient for the simulation needed in this investigation.

CST comes native with comprehensive CAD and post-processing modules, making it

easy to create, mesh and simulate models, and process the results. Additionally, for

microwave problems, it features the ability to use a waveguide port, which launches a

plane wave of the appropriate frequency down the waveguide.

HFWorks

HFWorks [44] is the high frequency software solution from EMWorks. It can carry out

S-parameter, antenna and resonance studies. A research evaluation license was provided

by EMWorks for the investigation. It features three different solution studies depending

on what is being simulated: S-Parameters, Resonance and Antennas. It uses the FEM

solver model.

HFWorks was utilised solely for its resonance study feature, which uses an eigenmode

solver to determine the resonant frequencies of the system and the corresponding electric

near fields, eliminating the need to feed the structure at all, as it determines the natural

resonance of the structure. This provides a useful characterisation of the resonance of

the cavity.

24



Chapter 2. Background 25

Numerical Electromagnetic Code

The Numerical Electromagnetic Code (NEC) is based on a previous method of moments

code for thin wires, the Antenna Modelling Program (AMP). It can model antennas in

free space, over a perfectly conducting ground plane, or lossy material [45]. The program

provides a purely MoM solution to the Maxwell equations.

SuperNEC [46], developed by The University of the Witwatersrand and Poynting Soft-

ware (Pty) Ltd, provides a C++ based MoM-UTD hybrid code, based on the FORTRAN

program, NEC2 [46]. While NEC2 is simply a simulation engine, requiring a text file

input and outputting another text file, SuperNEC provides a rudimentary interface for

generating a set of predetermined structures, or assemblies. The interface uses the na-

tive Structure Interpolation and Gridding (SIG) approach to generate NEC input files.

The list of predetermined models is limited. However, new assemblies can be created

manually, or through a guide inherent to SuperNEC [46].

The NEC approach is limited, as only purely MoM solutions are available, meaning

high simulation times for broadband sweeps. Additionally, SuperNEC cannot utilise the

waveguide port structure excitation approach adopted by FEKO and CST Studio, which

launches a plane wave from a waveguide face. Instead, a single wire feed must be used.

Additionally, the complexity of structures is not easily modelled, as each model is in

essence a collection of manually placed interconnecting wires. However, for MoM based

rectangular resonance problems, SuperNEC provides a suitable simulation platform.

Matlab

Matlab is a widely used programming and numerical computing platform, developed by

MathWorks [47]. It provides an easy method of dealing with NEC problems, like creating

assemblies, creating and editing NEC input files, programming relevant post-processing

applications. Additionally, output data from FEKO and CST Studio can be further

refined into a meaningful relationship, providing a useful tool for this investigation.

25



Chapter 3

Methodology

This chapter presents the methodology used to answer the research questions in this

investigation. The methodology was divided into four sections.

Section 3.1 discusses the method used to test the various electromagnetic simulation

programs against each other for a standard rectangular resonant cavity.

Section 3.2 discusses the verification of the simulation software by creating a model of

a microwave oven and comparing its simulated fields to the fields that were inherent in

the physical oven.

Thereafter, a new resonant cavity model was created based on documented standard

waveguide behaviour. This model was characterised in Section 3.3. The characterisation

included a resonance study, which calculated the cavity’s eigenfrequencies and corre-

sponding eigenfields, an analytical calculation of its characteristic modes, and various

near field simulations. The system’s sensitivity to dielectric influence is also presented,

as well as its frequency bandwidth.

Section 3.4 pertains to the manipulation of the field pattern inside the new rectangular

cavity to move the hotspots by frequency and phase shifting. An additional section,

Section 3.5 summarises the software used for each area of the investigation.

3.1 Software Testing

An array of software packages was used to simulate the electromagnetic near field inside

a basic rectangular cavity with a rectangular waveguide input. The programs were CST

Studio Suite Student Edition, FEKO, HFWorks and SuperNEC, all of which use slightly

different solver methods to calculate the fields and offer slightly different pros and cons.

26



Chapter 3. Methodology 27

There were two reasons for this approach. Firstly, to lend some diversity relating to how

the model is solved; FEM, FMM, MoM and FDTD were all used. Secondly, numerical

instability frequently occurs when simulating resonance owing to the peaking amplitude

over a very narrow bandwidth. The various programs and techniques exhibit different

tolerances around resonance.

The various solver methods and programs were discussed in depth in Section 2.4. For this

investigation, FEKO was used to conduct MoM and FMM simulation, SuperNEC was

used for MoM simulation, CST Studio was used for FEM and FDTD, or FIT simulation

and HFWorks was used to conduct resonance studies on the structures using its EME

solver.

This section discusses the benchmark model used to initially evaluate each program and

how the programs’ outputs were compared. HFWorks was excluded from the comparison

as it was not used for any near field simulations, only to characterise the resonance of

the waveguide.

3.1.1 Benchmark Model

The benchmark model geometry was based on a standard microwave frequency feed of

2.45 GHz, giving a wavelength of ≈ 122 mm. The cavity was set to a 2λ× 2λ× 1λ rect-

angle, with no specific design procedures being employed, as the model only functioned

to compare outputs between the programs. The feed was a standard waveguide, centred

on the front wall of the cavity, designed to the common convention of 0.6λ × 0.3λ in

cross-section. The waveguide length was set to 1λ long. The metric dimensions of the

model are shown in Table 3.1. This model functioned to establish similarity between the

different simulation platforms and solver methods.

Table 3.1: Dimensions of the basic rectangular cavity.

Cavity Waveguide

Width (mm) 244.73 73.42

Length (mm) 244.73 122.37

Height (mm) 122.37 36.71

3.1.2 Results and Deliverables

The respective software packages had to essentially simulate the same benchmark to

establish a comparison between their outputs. The benchmark model was a rectangular

cavity described by the dimensions given in Table 3.1. The electric near field inside the

cavity section of the model was simulated at 2.45 GHz using the following settings.

27



Chapter 3. Methodology 28

The frequency domain solver was used in CST, standard MoM was used in FEKO, and

standard MoM was used in SuperNEC. These are the default solvers for each software.

In SuperNEC, two models were created, one with the single wavelength waveguide and

another with a waveguide two wavelengths long, to try inducing a plane wave at the

waveguide opening. For CST and FEKO, this was not necessary as they can both

employ a waveguide port as a feed, which launches a plane wave anyway.

The material and boundary conditions were set in each case such that a vacuum, free

space, was evident inside the cavity and the metallic walls of the cavity were modelled

as a perfect electric conductor (PEC). Four representations of the electric near field

data were examined to establish a comparison. These were: three 2D contour slices

of the data on the XY, XZ and YZ planes directly through the centre of the model,

and an iso-surface 3D plot of the data. In CST and FEKO, these outputs are easily

generated, however, SuperNEC has no iso-surface plotting implementation. Therefore,

an application based in Matlab was written to receive an NEC output file, extract and

plot the iso-surface data from it. Appendix A describes the working of the application.

Additionally, all programs use slightly different feed methods, so near field amplitude

comparison would not be applicable. While CST and FEKO both use a waveguide

port, CST’s port is based on scattering parameters, while FEKO’s is based on the

input voltage. SuperNEC uses a voltage source attached to a monopole. Therefore,

the patterns were compared relative to each of their respective maximum and minimum

values. Additionally, where possible, input power from each port was compared to

demonstrate the different feed methods. In each case, the frequency sweep at which

resonance was examined was 2.2 - 2.7 GHz, given a 500 MHz bandwidth centred around

2.45 GHz. Appendix B shows a comparison of all applicable solvers available to all the

programs.

3.2 Verification

This section describes the method used to verify the simulated results. The method

constituted creating a model of a physical microwave oven and comparing the electric

near field pattern of the simulated model with that of the actual oven. The method used

to obtain planar field data from within the microwave oven is discussed. In simulation,

FEKO was used to simulate the internal electric near field of the model and attempts

were made to match the simulated data with the measured data.

28



Chapter 3. Methodology 29

3.2.1 Microwave Oven Model

A microwave oven, the Defy DMO 383 Microwave Oven, was obtained and was used to

verify the simulation software where possible. The oven has internal rectangular dimen-

sions of 301 × 311 × 202 mm according to the manual. The geometry, however, is not

rectangular, containing many lofts, bevels and indents. The complex geometry was mea-

sured, where possible, and a CAD model was generated using Autodesk Inventor [48],

which makes educational licenses available for students. Instead of a standard waveg-

uide feed, the oven’s magnetron uses a monopole antenna to pump energy into a small

chamber of 85 × 70 × 30 mm which opens into the main cavity through a rectangular

hole of 69× 56 mm. This model was then imported directly into FEKO.

CST Student Edition was not used as geometry import is not supported for the student

edition. Additionally, a limitation of the student edition is a maximum mesh size of

20 000 tetrahedrons for the frequency domain solver and 100 000 voxels for the time

domain solver, both of which were exceeded by the complex geometry. SuperNEC was

not used owing to the difficulty in creating an input NEC file that described complicated

geometry accurately. A dry run of the model was simulated at 2.45 GHz using MoM in

FEKO to find the hotspots of the system.

3.2.2 Actual Field

The actual field inside the microwave was found using thermal paper, the same that is

often found in till slips. The paper changes colour from white to black in the presence

of heat. Since microwaves interact with polar molecules, like water, the paper was

dampened so the heating of the water on the paper would induce heat in the paper

itself. Strips of paper were layered on sheets of polystyrene, using masking tape as an

adhesive, to create a ‘plane’ upon which the electric field could act. Polystyrene was

selected for its lightness, rigidity, inability to absorb water and warp under its influence,

and low dielectric permittivity, meaning it will not largely interact with and alter the

electric field.

These sheets were wet and placed inside the microwave oven individually, after which

the oven was turned on for 30 seconds. The experiment was conducted 12 times. The

paper would darken in areas of high electric field intensity and remain white in low

areas, thus creating a visual pattern to compare with. Since the magnetron inside

the microwave tends to shift in frequency during use, many different field patterns were

created. Therefore, areas of consistent overlap were looked for to be used as the dominant

hotspots of the system.

29



Chapter 3. Methodology 30

3.2.3 Operating Frequency

As mentioned previously, the frequency of the energy delivered by a magnetron in a

microwave oven can vary over time. As discussed in [49], this change in operational

frequency is influenced by many factors, including the effects of ‘pulling’ and ‘pushing’

on the magnetron owing to changes in the power reflected to the magnetron and the

cathode current in the magnetron. The presence of an absorbing body could influence

the reflected power as it absorbs microwave energy and its subsequent affinity to absorb

more energy could change. Therefore, the operational frequency or frequencies of the

microwave had to be determined. This was accomplished through measurement.

A spectrum analyser, the Anritsu MS2036A VNA Master, supplied by Wits University,

which can operate as both a spectrum analyser and a Vector Network Analyser (VNA),

was used to determine the electromagnetic leakage of the microwave oven during opera-

tion. The A-Info DS-3300 Log-Periodic Dipole Array (LPDA) at Wits was used to take

measurements.

The antenna has a wide band of operation, 30–3000 MHz, which suited the desired

measurement frequency range of 2.2–2.7 GHz. The LPDA was initially calibrated using

the VNA and the VSWR was tested in the range of 2.2–2.7 GHz to ensure the antenna

was matched (VSWR≤ 2) at those frequencies. Then it was placed inside the shielded

anechoic chamber at Wits University and pointed at the microwave oven. A series of

tests were then conducted. A glass of water was placed inside the oven to measure

the frequency of the leakage with the presence of an absorbing body, and a sheet of

polystyrene was placed at the base of the oven to test the ‘empty’ conditions which were

present to measure the hotspots of the oven.

3.2.4 Simulated Field

Because polystyrene, a dielectric medium, was used to hold the thermal paper, it would

be expected to interact a little with the electromagnetic field inside the cavity. Hence,

a similar sheet of dielectric was added to the empty microwave model for simulation.

FEKO’s media library contained both polystyrene and Styrofoam, which is a trade-

marked term for a type of polystyrene. Three simulation instances were simulated: an

empty cavity, a cavity with a sheet of polystyrene inside, and a cavity with a sheet of

Styrofoam inside.

Since the microwave’s magnetron tends to frequency shift during operation and the

nature of simulation only allows the observation of a field of a single discreet frequency

at a time, a sweep of frequencies had to be simulated. Hence, the model was simulated

30



Chapter 3. Methodology 31

using a range of 500 MHz, 2.2 GHz to 2.7 GHz, allowing for a shift of 250 MHz on

either side of the central 2.45 GHz. The frequency interval was initially set to 10 MHz,

resulting in 51 different simulation iterations. Each iteration was examined for similarity

to the measured data on the thermal paper. Additionally, another frequency sweep with

a much narrowing interval was simulated according to the results of Section 3.2.3.

The simulation was carried out in FEKO, using a pure MoM solution for the cavity and

FEM for the dielectric sheet. The FEM solution relied on the direct sparse solver, as

the iterative solver would not converge. This is also why MoM was used over MLFMM.

This solver setup successfully simulated the near field data by releasing the MoM matrix

from memory before solving the FEM elements. The results of this simulation can be

found in Section 4.2.

3.2.5 Results Comparison

Owing to the nature of the experiment, the results would be largely image-driven, requir-

ing a visual determination of accuracy. To aid this process, Matlab was used to analyse

the images and provide the means to directly compare the measured and simulated data.

Initially, all the images were cropped to an exact square and then resized so they each

had an identical number of pixels. The images were then converted to double format,

resulting in a three-dimensional numerical array. Then, they were converted to grayscale,

allowing a single numerical representation of white to black colour. A threshold value

was then set. Every pixel above the threshold was set to 0, or black, and every pixel

below the threshold was set to 1, white, resulting in a binary image. Following this,

Matlab’s image processing toolbox was used to fill in the small holes in each image that

arose from the warping of the paper and to pad each array with zeros, so they would

have a thin black border. Then the logical OR and logical AND operators were applied

to the images, to generate two images; one of which showed all possible hotspots from

all the images and the other showed the hotspots that were evident in all the images and

overlapped. The same process was applied to the FEKO generated simulation images

for a variety of frequencies and a like for like comparison was possible.

3.3 Cavity Characterisation

This section discusses the formulation of a new rectangular resonant cavity model de-

signed to ‘ideal’ specifications. A standard waveguide design was used as a starting point

for the cavity. The cavity was characterised in terms of its resonance and a frequency

sweep was simulated to determine the effects of a waveguide feed.

31



Chapter 3. Methodology 32

3.3.1 Ideal Rectangular Cavity

A simple rectangular cavity was designed based on documented waveguide behaviour.

The rectangular waveguide was selected owing to its easy geometry, well-documented

characteristics and the fact that simulation of rectangular structures is memory inexpen-

sive owing to easy meshing. Standard waveguide WR340 [50] exhibits the characteristics

shown in Table 3.2. These were ideally suited for microwave problems.

Table 3.2: Characteristics of the WR340 waveguide.

Recommended frequency band (GHz) 2.20–3.30

Cut-off frequency of lowest order mode (GHz) 1.736

Cut-off frequency of highest order mode (GHz) 3.471

Dimensions [w×h] (mm) 86.36×43.18

The following equations, taken from [51], were used to verify the waveguide cut-off

frequencies for waveguide operating in mode TEmn.

fcmn =
c

2π
kcmn (3.1)

kcmn =

√(mπ

a

)2
+
(nπ

b

)2
(3.2)

Where:
fcmn = cut-off frequency of mode mn [Hz]

kcmn = wavenumber corresponding to mode mn

a = length of waveguide in H direction, or width [m]

b = length of waveguide in E direction, or height [m]

c = speed of light ≈ 2.998m× ·s−1

Given a = 86.36 mm and b = 43.18 mm, the cut-off frequency for mode TE10 was

determined to be 1.7358 GHz and the cut-off frequency for the next mode, TE20 was

3.4715 GHz, consistent with the information in Table 3.2. The cavity itself was designed

at triple the waveguide dimensions in all directions, with the width and length being

equal. The reason behind this was to aim for a starting pattern of a 3 × 3 grid of

hotspots, a TE303 mode pattern. Seeing as the waveguide excited with a TE10 wave

at the appropriate frequency exhibits a single central hotspot, those dimensions were

tripled to achieve nine local hotspots. The waveguide feed was positioned centrally on

the front wall of the cavity. Table 3.3 shows the final dimensions of the cavity.

32



Chapter 3. Methodology 33

Table 3.3: Dimensions of the ideal rectangular cavity.

Cavity Waveguide

Width (mm) 259.08 86.36

Length (mm) 259.08 86.36

Height (mm) 129.54 43.18

3.3.2 Waveguide Resonant Frequency

Table 3.2 shows the characteristics of the standard WR340 waveguide. In HFWorks, a

resonance study was conducted on a model of the waveguide to determine its eigenfre-

quencies. Additionally, a frequency sweep between 2.2 GHz and 3.3 GHz in CST and

FEKO was conducted to observe a near field pattern that changes with frequency within

the waveguide. Since the waveguide was excited with TE10 mode, a single electric field

hotspot was expected to be evident in the waveguide in the H direction. The frequency

of excitation determines how the waveguide resonates, or the number of hotspots evident

along the length of the waveguide.

3.3.3 Cavity Resonant Frequency

Like the waveguide, a resonant cavity exhibits certain eigenfrequencies at which the

structure resonates naturally. Additionally, the electric near field characteristics within

the chamber are dependent on its operational frequency. The dominant resonant modes

of the cavity were determined analytically using Equation (3.3) [52].

fmnl =
ckmnl

2π
√
µrϵr

(3.3a)

kmnl =

√(mπ

a

)2
+

(nπ
b

)2
+

(
lπ

d

)2

(3.3b)

Where:
fmnl = cavity resonant frequency for mode TEmnl [Hz]

kmnl = wave number corresponding to mode TEmnl

c = speed of light ≈ 2.998m× ·s−1

a, b, d = dimensions of the cavity [m]

And

1
√
µrϵr

= 1

33



Chapter 3. Methodology 34

for a cavity containing free space. This equation can be used also for structures such as

waveguides, however, waveguides are often used as feed and hence, are not completely

bounded.

Only one of the mode numbers, which must be positive integers, can be set at zero at

a time [53]. Equation (3.3) was used iteratively to determine the resonant modes in the

range of 2.2 to 2.7 GHz. This range was used as microwave ovens are centred around 2.45

GHz. The range gives a 250 MHz width on either side of this centre frequency. Addi-

tionally, this range is encompassed in the recommended range of the WR340 waveguide,

being 2.2 to 3.3 GHz. These modes are descriptive of an empty resonant cavity, as the

presence of a dielectric body within the cavity detunes the resonance and coupling within

the cavity [54]. A frequency sweep, requesting the internal near fields of the cavity with

the excited waveguide as power input, was conducted in CST and FEKO.

3.3.4 Dielectric Influence

Since the presence of dielectric media influences the resonance of a system [54], the cavity

would have to be re-examined after the addition of a body of dielectric. Since microwaves

operate best on polar molecules like water, a cylinder of water was placed centrally

inside the cavity. Thereafter, another frequency sweep between 2.2 and 2.7 GHz, again

with an excited waveguide as a feed, was conducted. Because the water can absorb the

electromagnetic energy and convert it to heat, the energy generated by the source can be

absorbed instead of reflected, so a reasonable VSWR was expected from this simulation,

making it possible to obtain the operating bandwidth of the system in terms of VSWR.

3.4 Field Manipulation

This section described the methodology used to manipulate the fields within the resonant

cavity models to move the internal hotspots. The method revolved around investigating

the effects of slight changes in frequency on the hotspots, and the effect of the feed phase

on the hotspots. To investigate the feed phase, a second excited waveguide was added

and a phase difference between the sources was introduced. The second waveguide was

added in three different orientations, as shown in Figure 3.1.

The change in hotspot position was examined, instead of the complete change of field

pattern, as the investigation requires the controlled movement of a defined hotspot.

In each case, the cavity was examined empty, without any dielectric body present. If

hotspot movement could be achieved in free space, it serves as a proof of concept which

could be extended to encompassing entities of different dielectric characteristics.

34



Chapter 3. Methodology 35

(a) Same face. (b) Adjacent faces. (c) Opposite faces.

Figure 3.1: Waveguide Orientations. Images adapted from CST Studio.

3.4.1 Frequency Shifting

As discussed in Section 3.3.2, the frequency applied to a waveguide determines how it,

and subsequently the resonant cavity it feeds, resonate. The frequency of the eigenfield

determines the nature of the field pattern and the position of the hotspots. For the

model with a single excited waveguide feed, a new frequency sweep between 2.2 GHz

and 2.7 GHz was conducted with intervals of 1 MHz using the MLFMM solver in FEKO

and using the FIT and FEM solvers in CST.

The hotspot positions were observed for a range of frequency steps surrounding each of

the cavities’ dominant eigenfrequencies as well as directly between the resonant modes.

Additionally, the maximum field strength and input power were observed at each fre-

quency to determine the location of any resonant peaks. Where necessary, new simula-

tions were conducted with a finer step interval to increase precision.

3.4.2 Phase Shifting

In cases where multiple waveguide feeds were evident, the phase of each waveguide feed

could be controlled to provide a phase difference between the sources. The waveguides

could be fed in phase, exactly out of phase, or with any phase difference in between.

The geometric positions of the waveguides relative to each other could also introduce

physical phase differences. Again, a frequency shift between 2.2 and 2.7 GHz, with a

step of 1 MHz was conducted for each of the three models. Additionally, an incremental

phase difference was introduced between the waveguides with a step of 15◦, starting at

0◦ and ending at 345◦, resulting in 24 iterations, each describing a different phase shift.

The introduction of multiple phase shifts excluded the possibility of using the FDTD or

FIT solver, as only one phased feed could be simulated at a time. Therefore, only the

FEM solver in CST was used along with the MLFMM solver in FEKO. Again, each phase

35



Chapter 3. Methodology 36

iteration at each eigenfrequency was examined, along with discreet frequency points in

between each resonant mode. For this method, the feed frequency was held constant

while only the feed phases were changed, to investigate any hotspot movement as a result

of phase shifting of the feeds.

3.4.3 Bandwidth

The frequency bandwidth at which the resonant cavity resonates was investigated. A

wide bandwidth infers that small changes in frequency around the resonant frequency

will not affect the near field pattern inside the cavity and the structure will still resonate.

A narrow bandwidth implies that if the frequency changes slightly, the near field pattern

is likely to change, or the cavity may not resonate at all. Additionally, the characteristic

modes of the empty cavity were compared to that of a cavity containing dielectric. This

investigation facilitates an understanding of the effect of dielectric bodies. The presence

of dielectric matter may shift the resonant frequency of the chamber. If the cavity can

be fed by a single frequency, and the resonant frequency of the cavity is shifted outside

the bandwidth, the cavity may not resonate when the dielectric is present. This presents

major implications to the feasibility of this investigation applying to thermal ablation

of tumours, where dielectric influence is variable and extremely important.

3.5 Software Summary

This section describes the software packages and their respective solvers used in each

section of the investigation. The information is summarised for each section.

3.5.1 Benchmark

This section entailed the comparison of the simulation results of each program for a

standard benchmark model. The default solver settings were used for each program.

SuperNEC and FEKO were both utilised with their MoM solver and CST’s frequency

domain solver, which uses FEM, was utilised, as summarised in Table 3.4. HFWorks

was not used.

3.5.2 Verification

This section entailed verifying the simulation results by comparing the near field simula-

tions to a real-world microwave oven’s measured near field data. Only FEKO was used

for this method. It was used in hybrid mode, combining its MoM and FEM solvers, as

shown in Table 3.5.

36



Chapter 3. Methodology 37

Table 3.4: Summary of the Software Used for the Benchmark Section.

Solver

Software MoM FMM FEM FDTD/FIT EME

SuperNEC •
FEKO •
CST Studio •
HFWorks

Table 3.5: Summary of the Software Used for the Verification Section.

Solver

Software MoM FMM FEM FDTD/FIT EME

SuperNEC
FEKO • •
CST Studio
HFWorks

3.5.3 Characterisation

This section entailed creating a new ‘ideal’ model of a resonant cavity and characterising

it. This focused mainly on resonance. For this, FEKO, CST Studio and HFWorks were

used. FEKO’s MoM and MLFMM solvers were used to simulate the near fields inside the

waveguide and cavity. CST used its frequency domain (FEM) and time domain (FIT)

solvers on the cavity and HFWork performed a resonance study on the waveguide. This

information is summarised in Table 3.6.

Table 3.6: Summary of the Software Used for the Characterisation Section.

Solver

Software MoM FMM FEM FDTD/FIT EME

SuperNEC
FEKO • •
CST Studio • •
HFWorks •

3.5.4 Frequency Sweeping

This section entailed conducting a broadband frequency sweep of the cavity with narrow

frequency steps. Therefore, FEKO’s FMM solver, MLFMM was used as it is faster than

the MoM solver, demonstrated in Appendix B. CST’s frequency domain (FEM) and

time domain (FIT) solvers were used, as their simulation times are both low. Table 3.7

37



Chapter 3. Methodology 38

shows this data. SuperNEC and HFWorks were not used in this section.

Table 3.7: Summary of the Software Used for the Frequency Sweeping Section.

Solver

Software MoM FMM FEM FDTD/FIT EME

SuperNEC
FEKO •
CST Studio • •
HFWorks

3.5.5 Phase Shifting

This section involved introducing feed phase differences between multiple waveguide

sources. This eliminated all time domain solvers. FEKO’s MLFMM solver was used

instead of MoM, owing to the simulation speed, and the frequency domain solver was

used in CST Studio.

Table 3.8: Summary of the Software Used for the Phase Shifting Section.

Solver

Software MoM FMM FEM FDTD/FIT EME

SuperNEC
FEKO •
CST Studio •
HFWorks

3.5.6 Comparison

This section compared the results obtained by CST Studio and FEKO. Therefore, all

applicable solvers available to the two programs were used. HFWorks and SuperNEC

were not used as they were not the predominant programs used for the investigation.

Table 3.9: Summary of the Software Used for the Software Comparison.

Solver

Software MoM FMM FEM FDTD/FIT EME

SuperNEC
FEKO • •
CST Studio • •
HFWorks

38



Chapter 4

Results

The findings of the methodology, outlined in Chapter 3, are presented in this chapter,

along with conclusions linking them to the research questions.

4.1 Software Testing

This section presents the comparison of the output from simulation in the three software

packages: CST Studio Students Edition, FEKO and SuperNEC.

4.1.1 Comparison

A standard benchmark model was created in all three software packages, as discussed in

Section 3.1. Figure 4.1 shows a comparison between the results obtained by the three

software packages in the horizontal contour plane through the centre of the model.

While clearly CST Studio, Figure 4.1a, and FEKO, Figure 4.1b, yield largely identi-

cal results, the similar SuperNEC model exhibits slightly different results, as shown in

Figure 4.1c.

(a) CST Studio (b) FEKO (c) SuperNEC (a) (d) SuperNEC (b)

Figure 4.1: Comparison of the Centre Horizontal Plane of the Benchmark Model.

39



Chapter 4. Findings 40

This could be because SuperN