modelling techniques and novel configurations for meander

Modelling Techniques and Novel Configurations

for Meander-line-coil Electromagnetic Acoustic

Transducers (EMATs)

A thesis submitted to the University of Manchester for the degree of

Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2016

By

Yuedong Xie

School of Electrical and Electronic Engineering

List of Contents

2

LIST OF CONTENTS

LIST OF CONTENTS ...................................................................................................... 2

LIST OF FIGURES .......................................................................................................... 6

LIST OF TABLES .......................................................................................................... 15

NOMENCLATURE ........................................................................................................ 16

ABSTRACT ..................................................................................................................... 18

DECLARATION ............................................................................................................. 19

COPYRIGHT STATEMENT ........................................................................................ 20

ACKNOWLEDGEMENTS ............................................................................................ 21

Chapter 1 Introduction ............................................................................................... 22

1.1 Motivation .......................................................................................................... 22

1.2 Aim and Objectives ............................................................................................ 23

1.3 Contributions ...................................................................................................... 23

1.4 Organization of thesis ......................................................................................... 24

1.5 List of Publications ............................................................................................. 25

Chapter 2 EMAT background ................................................................................... 27

2.1 Introduction ........................................................................................................ 27

2.2 Coupling Mechanisms of EMATs ...................................................................... 27

2.2.1 Lorentz Force Mechanism........................................................................... 27

2.2.2 Magnetostriction Mechanism ...................................................................... 28

2.2.3 Advantages and Disadvantages of EMATs................................................. 29

2.3 Types of EMATs ................................................................................................ 31

2.3.1 Major Types of Mechanical Waves ............................................................ 31

2.3.2 Classification of EMATs............................................................................. 34

2.4 State-of-the-art in EMAT Modelling ................................................................. 38

List of Contents

3

2.5 Conclusions ........................................................................................................ 40

Chapter 3 FDTD for Ultrasonic Modelling ............................................................... 41

3.1 Ultrasonic Testing Techniques ........................................................................... 41

3.1.1 Phased Array Techniques ............................................................................ 41

3.1.2 Ultrasonic Testing Methods ........................................................................ 44

3.2 FDTD Method for Ultrasonic Modelling ........................................................... 47

3.2.1 Elastodynamic Equations ............................................................................ 47

3.2.2 The Finite-difference time-domain (FDTD) Method .................................. 48

3.3 Ultrasonic Phased Array Modelling with FDTD ............................................... 49

3.4 Novel Radiation Pattern with Hilbert Transformation ....................................... 53

3.4.1 Hilbert Transformation................................................................................ 53

3.4.2 Novel Radiation Pattern with the Hilbert Transformation .......................... 54

3.5 Near Field and Far Field Modelling ................................................................... 55

3.5.1 Near Field Analysis ..................................................................................... 56

3.5.2 Far Field Analysis ....................................................................................... 61

3.5.3 Conclusions of Section 3.5 .......................................................................... 63

3.6 Scattering Modelling .......................................................................................... 64

3.7 Conclusions ........................................................................................................ 66

Chapter 4 Development and Validation of A Novel Method for Modelling

Meander-line-coil EMATs Operated on Lorentz Force Mechanism ......................... 68

4.1 Introduction ........................................................................................................ 68

4.1.1 Modelling Geometry ................................................................................... 68

4.2 EMAT-EM Modelling ........................................................................................ 69

4.2.1 Classic Dodd and Deeds Solutions ............................................................. 69

4.2.2 Adapted Analytical Solutions for A Straight Wire ..................................... 73

4.2.3 Validation and Comparison with FEM ....................................................... 74

List of Contents

4

4.2.4 Analytical EMAT-EM Modelling ............................................................... 80

4.3 Novel Methods for EMATs ................................................................................ 85

4.3.1 The Combination of EM and US Models ................................................... 85

4.3.2 The Propagation of Rayleigh Waves........................................................... 87

4.3.3 Displacement Calculation and Depth Profile .............................................. 88

4.3.4 The Effect of the Fractional Bandwidth ...................................................... 90

4.4 The Property of Rayleigh Waves ....................................................................... 92

4.4.1 Radiation Pattern ......................................................................................... 92

4.4.2 Beam Features ............................................................................................. 93

4.5 EMAT-receiving Mechanism ............................................................................. 95

4.6 Experimental Validations ................................................................................... 96

4.6.1 Experiments Set-up ..................................................................................... 96

4.6.2 Received Signals from Experiments ........................................................... 98

4.6.3 Validation of EMAT Models with Experiments ......................................... 99

4.7 EMAT Scattering Phenomena .......................................................................... 102

4.7.1 Modelling of Rayleigh Waves’ Scattering ................................................ 102

4.7.2 Experiments and Validations .................................................................... 106

4.8 Modelling of Unidirectional Rayleigh Waves EMATs .................................... 108

4.9 Conclusions ...................................................................................................... 113

Chapter 5 Directivity Analysis of Conventional Meander-line-coil EMATs ....... 115

5.1 Introduction ...................................................................................................... 115

5.2 The Analytical Solution to the Radiation Pattern of Rayleigh Waves on the

Surface of the Material ................................................................................................ 115

5.3 Beam Directivity Analysis of the Conventional Constant-length Meander-line-

coil (CLMLC) ............................................................................................................. 117

5.3.1 Wholly Analytical Models ........................................................................ 118

List of Contents

5

5.3.2 The Effect of the Length of the Conventional Constant-length Meander-line-

coil (CLMLC) on Radiation Pattern ....................................................................... 121

5.4 Experimental Results ........................................................................................ 123

5.5 Conclusions ...................................................................................................... 125

Chapter 6 Novel Configurations for Meander-line-coil EMATs .......................... 127

6.1 Introduction ...................................................................................................... 127

6.2 Novel Variable-length Meander-line-coil (VLMLC) EMATs ......................... 128

6.2.1 Wholly Analytical Models for the Novel Variable-length Meander-line-coil

(VLMLC) EMATs .................................................................................................. 129

6.2.2 Analysis of Beam Properties of Rayleigh Waves Generated by the Novel

Variable-length Meander-line-coil (VLMLC) EMATs .......................................... 130

6.3 Novel Multi-directional Meander-line-coil EMATs ........................................ 138

6.3.1 Introduction ............................................................................................... 138

6.3.2 Four-directional Meander-line-coil (FDMLC) EMATs............................ 138

6.3.3 Six-directional Meander-line-coil (SDMLC) EMATs .............................. 143

6.3.4 Discussion ................................................................................................. 146

6.4 Conclusions ...................................................................................................... 146

Chapter 7 Conclusions and Recommendations for Future Work ........................ 148

7.1 Conclusions ...................................................................................................... 148

7.1.1 FDTD Method for Simulating US Behaviours ......................................... 148

7.1.2 Vertical Plane Modelling for EMATs ....................................................... 149

7.1.3 Surface Plane Modelling for EMATs ....................................................... 150

7.1.4 Novel Configurations for EMATs ............................................................ 151

7.2 Recommendations for Future Work ................................................................. 152

REFERENCES .............................................................................................................. 154

List of Figures

6

LIST OF FIGURES

Figure 2-1: Schematic of the Lorentz force mechanism. From [19]. ................................... 27

Figure 2-2: Microscopic process of the field induced magnetostriction. H is the external

magnetic field; ∆l is the deformation due to the reorientation of the magnetic domain,

which is simplified represented by an elliptic shape. From [18]. ........................................ 29

Figure 2-3: EMATs operated on the magnetostriction mechanism. εd is the dynamic stress,

εs is the static stress, and εr is the resultant stress. ................................................................ 29

Figure 2-4: Longitudinal waves. The black arrow denotes the direction of the wave

propagation; red arrows denote directions of the particle motion. From [50]. .................... 31

Figure 2-5: Shear waves. The black arrow denotes the direction of the wave propagation;

red arrows denote directions of the particle motion. From [50]. ......................................... 32

Figure 2-6: Rayleigh waves. The black ellipse denotes the particle motions. From [54]. ... 32

Figure 2-7: Modes of Lamb waves. (a), symmetric mode; (b), anti-symmetric mode. The

black arrows denote the displacement of the particle; black curves denote the resulting

Lamb waves. From [58]. ...................................................................................................... 33

Figure 2-8: The cross-sectional view of a normal longitudinal wave EMAT. The white

hollow arrows denote the direction of the static magnetic field; the grey arrows denote the

direction of the Lorentz force; the solid black arrow means the direction of wave

propagation. From [19, 20]. ................................................................................................. 34

Figure 2-9: The cross-sectional view of normal shear waves EMATs. The white hollow

arrows denote the direction of the static magnetic field; the grey arrows denote the

direction of the Lorentz force; the solid black arrows mean the direction of wave

propagation. Adapted from [19, 20]. .................................................................................... 35

Figure 2-10: The structure of the periodic-permanent-magnet (PPM) EMAT to generate

SH waves. From [16]. .......................................................................................................... 35

Figure 2-11: The cross-sectional view of the PPM EMAT. From [48]. .............................. 36

List of Figures

7

Figure 2-12: The structure of the meander-line-coil EMAT to generate SH waves. From

[16]. ...................................................................................................................................... 37

Figure 2-13: The structure of a meander-line-coil EMAT to generate Rayleigh waves.

From [16, 34]. ...................................................................................................................... 37

Figure 3-1: Phased array techniques: steering and focusing [10]. ....................................... 42

Figure 3-2: A model used for time delays calculation for steering. ..................................... 43

Figure 3-3: A model for time delays calculation for focusing. ............................................ 44

Figure 3-4: Ultrasonic pulse-echo method. (a), Inspection diagram; (b) Received signals. 45

Figure 3-5: Ultrasonic through-transmission method. (a), Inspection diagram; (b) Received

signals. .................................................................................................................................. 45

Figure 3-6: Ultrasonic pitch-catch method. ......................................................................... 46

Figure 3-7: Modelling geometry for steering (a) and focusing (b). ..................................... 49

Figure 3-8: Pure sine wave; (a) the time domain signal of the pure sine wave; (b) the

magnitude of the pure sine wave’s Fourier transform [75]. ................................................. 51

Figure 3-9: Gaussian-modulated sine wave; (a) the time domain signal of the Gaussian-

modulated sine wave; (b) the magnitude of the Gaussian-modulated sine wave’s Fourier

transform [75]....................................................................................................................... 51

Figure 3-10: Steering techniques: firing elements at prescribed calculated times, the

wavefront is steered at 00, 300, 600, 900 respectively. .......................................................... 52

Figure 3-11: Focusing techniques: the wavefront is focused at the prescribed focal point. 53

Figure 3-12: Signals to indicate the arrival times of ultrasound waves. .............................. 54

Figure 3-13: (a), Radiation pattern for the beam steered at 300; (b), radiation pattern for

studying beam features. ........................................................................................................ 55

Figure 3-14: The description of the focal length and the steering angle. ............................. 57

Figure 3-15: The radiation pattern of the focusing behaviour in the near field. .................. 57

List of Figures

8

Figure 3-16: Beam features of focusing within the near filed. (a), Beam directivity of the

focusing behaviour; (b) Field distribution along the steering angle of the focusing

behaviour. ............................................................................................................................. 58

Figure 3-17: Radiation pattern of the steering behaviour with a steering angle of 300. ...... 59

Figure 3-18: Beam features of steering within the near filed. (a), Beam directivity of the

steering behaviour; (b) Field distribution along the steering angle of the steering behaviour.

.............................................................................................................................................. 59

Figure 3-19: The beam directivity of the focusing behaviour at different radial lengths. ... 61

Figure 3-20: Radiation pattern of the focusing behaviour in the far field. .......................... 61

Figure 3-21: Beam features of focusing in the far field: (a) beam directivity at a focal

length of 150 mm, (b) field distribution along the steering angle 300. ................................ 62

Figure 3-22: Beam features of steering in the far field: (a) beam directivity at a radial

length of 150 mm, (b) field distribution along the steering angle 300. ................................ 63

Figure 3-23: The geometry of scattering modelling. ........................................................... 65

Figure 3-24: Wave propagation of the scattering modelling at different times. .................. 66

Figure 3-25: The received signals from the receiving array. (a), directly transmitted signals;

(b), the scattered longitudinal waves; (c), the scattered shear waves................................... 66

Figure 4-1: The configuration of a typical meander-line-coil EMAT. ................................ 69

Figure 4-2: A model built by Dodd and Deeds [79]. ........................................................... 70

Figure 4-3: Geometry for the conductor with only one layer. ............................................ 71

Figure 4-4: For a circular coil, the distribution of the magnitude of the vector potential

within the conductor. ............................................................................................................ 72

Figure 4-5: For a circular coil, the vector potential distribution along the surface of the

conductor (𝒙=0). .................................................................................................................. 73

List of Figures

9

Figure 4-6: For a large-radius circular coil, the vector potential distribution within the

conductor. ............................................................................................................................. 74

Figure 4-7: For a large-radius circular coil, the vector potential along the surface of the

conductor (𝒙=0). .................................................................................................................. 74

Figure 4-8: (a), the model built with Maxwell Ansoft; (b), mesh of the model. ................. 75

Figure 4-9: In FEM solver, the energy error versus the number of triangles. ...................... 75

Figure 4-10: At 10 kHz, the vector potential distribution within the stainless steel plate. (a),

the analytical method; (b) the finite element method (FEM). .............................................. 76

Figure 4-11: At 10 kHz, the vector potential along the surface of the stainless steel plate.

(a), (b) and (c) denotes the magnitude, the real part, and the imaginary part of the vector

potential respectively. .......................................................................................................... 77

Figure 4-12: At 1 MHz, the vector potential distribution within the stainless steel plate. (a),

the analytical method; (b) the finite element method (FEM). .............................................. 78

Figure 4-13: At 1 MHz, the vector potential along the surface of the stainless steel plate.

(a), (b) and (c) denotes the magnitude, the real part, and the imaginary part of the vector

potential respectively. .......................................................................................................... 78

Figure 4-14: With various lift-offs, the distribution of the real part of the vector potential

along the surface of the stainless steel plate......................................................................... 79

Figure 4-15: 2D model of the EMAT-EM simulation. ........................................................ 81

Figure 4-16: The real part of the vector potential produced by a meander-line-coil. .......... 81

Figure 4-17: The real part of the induced eddy current produced by a meander-line-coil. . 82

Figure 4-18: The eddy current distribution along the surface of the stainless steel plate

(x=0). .................................................................................................................................... 82

Figure 4-19: The mesh of the static magnetic field modelling. ........................................... 83

Figure 4-20: The relationship between the elements number and the energy error for the

static magnetic field modelling. ........................................................................................... 83

List of Figures

10

Figure 4-21: The vector of the magnetic flux density generated by the permanent magnet.

.............................................................................................................................................. 83

Figure 4-22: The distribution of the magnitude of the magnetic flux density within the

stainless steel plate. .............................................................................................................. 84

Figure 4-23: L The distribution of the magnetic flux density along the surface of the

stainless steel plate (x=0). .................................................................................................... 84

Figure 4-24: The distribution of the Lorentz force density along the surface of the stainless

steel plate. ............................................................................................................................. 85

Figure 4-25: The combination between the EM model and the US model. ......................... 86

Figure 4-26: The excitation signal for wire 1 and wire 2. .................................................... 87

Figure 4-27: The wave propagation at 18 µs and 35 µs after firing respectively. ............... 88

Figure 4-28: The received signals; (a), signals received by the receiver R1; (b), signals

received by the receiver R2. ................................................................................................. 89

Figure 4-29: The depth profile of Rayleigh waves’ displacement. ...................................... 90

Figure 4-30: The excitation signal with various fractional bandwidths. .............................. 91

Figure 4-31: The received signals with excitation signals at various fractional bandwidths.

.............................................................................................................................................. 92

Figure 4-32: (a), the radiation pattern of the EMAT-Rayleigh waves; (b), the radiation

pattern used for the analysis of beam features. .................................................................... 93

Figure 4-33: Beam directivity of Rayleigh waves. .............................................................. 94

Figure 4-34: Field distribution along the steering angle 00 of Rayleigh waves. .................. 94

Figure 4-35: The model used for calculating the induced voltage in the receiving coil. ..... 95

Figure 4-36: The schematic diagram of the experimental system. ...................................... 97

Figure 4-37: Set-up of the experimental system. ................................................................. 98

List of Figures

11

Figure 4-38: The frequency domain of the experimentally received signals. ...................... 98

Figure 4-39: The received signal from experiments. ........................................................... 99

Figure 4-40: The comparison between the simulation and the experiment. ...................... 100

Figure 4-41: The maximum amplitude of the induced voltage with various distances

between the transmitter and the receiver. ........................................................................... 101

Figure 4-42: The received signals with a meander-line-coil as the transmitter. ................ 102

Figure 4-43: The geometry of Rayleigh waves’ scattering simulation. ............................. 103

Figure 4-44: Scattering behaviours of Rayleigh waves. .................................................... 104

Figure 4-45: Received signals from R1. ............................................................................ 105

Figure 4-46: Received signals from R2. ............................................................................ 105

Figure 4-47: The comparison of the received signals from the receivers R1 and R2. ....... 106

Figure 4-48: The experimentally received signal from the receiver R1. ........................... 107

Figure 4-49: The amplitude comparison between the simulation and the experiment. ..... 108

Figure 4-50: The envelop comparison between the simulation and the experiment.......... 108

Figure 4-51: The configuration of the URW EMAT. From [23]. ...................................... 109

Figure 4-52: The wave superposition between the source 1 and the source 2. From [80]. 110

Figure 4-53: The excitation signal for the coil A and the coil B........................................ 110

Figure 4-54: The wave propagation of Rayleigh waves generated by the URW-EMAT. . 111

Figure 4-55: The received signal from the URW-EMAT. ................................................. 112

Figure 4-56: The received signal from the BRW-EMAT. ................................................. 112

Figure 4-57: The comparison between the URW and the BRW. ....................................... 113

Figure 5-1: Surface waves generated by the point source.................................................. 117

List of Figures

12

Figure 5-2: The transformation between the analytical EM model and the analytical US

model. ................................................................................................................................. 118

Figure 5-3: The Rayleigh waves’ radiation pattern on the surface of the aluminium plate.

............................................................................................................................................ 120

Figure 5-4: The model used to study the beam directivity. ............................................... 120

Figure 5-5: The beam directivity of Rayleigh waves generated by a 30mm-length meander-

line-coil EMAT. (a), the curve of the beam directivity; (b) the curve used for describing

HPBW and SLL. ................................................................................................................ 121

Figure 5-6: The beam directivity of the meander-line-coil with various lengths. ............. 122

Figure 5-7: (a), experimental set-up; (b), the scan path of the receiver; Tx means the

transmitter and Rx means the receiver. .............................................................................. 123

Figure 5-8: The measured beam directivity from experiments. ......................................... 124

Figure 5-9: Comparison between the simulated and measured results for the meander-line-

coil with a length of 10 mm (a), 20 mm (b), 30 mm (c) and 40 mm (d) respectively. ...... 125

Figure 6-1: The configuration of the variable-length meander-line-coil (VLMLC). (a), the

schematic diagram; (b), the fabricated variable-length meander-line-coil. ....................... 128


model. ................................................................................................................................. 130

Figure 6-3: The radiation pattern of the variable-length meander-line-coil (VLMLC). .... 131

Figure 6-4: The beam directivity of the 50 mm variable-length meander-line-coil (VLMLC)

with a step of 8 mm. ........................................................................................................... 132

Figure 6-5: The beam directivity comparison between the conventional constant-length

meander-line-coil (CLMLC) and the novel variable-length meander-line-coil (VLMLC).

............................................................................................................................................ 132

Figure 6-6: The beam directivity of a 50 mm variable-length meander-line-coil (VLMLC).

............................................................................................................................................ 133

List of Figures

13

Figure 6-7: The comparison between the 40 mm VLMLC and the 50 mm VLMLC at

different steps. .................................................................................................................... 135

Figure 6-8: The measured beam directivity of the 50 mm VLMLC with a step of 8 mm. 136

Figure 6-9: Measured beam directivity from experiments. ................................................ 136

Figure 6-10: For 50 mm VLMLC with various steps, the comparison between the

simulated beam directivity and the measured beam directivity. ........................................ 137

Figure 6-11: Four-directional meander-line-coil (FDMLC). (a), the schematic diagram of

FDMLC; (b) the fabricated FDMLC.................................................................................. 139

Figure 6-12: The approximated configuration of the four-directional meander-line-coil

(FDMLC). .......................................................................................................................... 140

Figure 6-13: The simulated beam directivity of the four-directional meander-line-coil

(FDMLC) EMAT. .............................................................................................................. 141

Figure 6-14: The magnitude of the received Rayleigh waves............................................ 141

Figure 6-15: (a), The scan path of the receiver; (b), the experimental beam directivity. .. 142

Figure 6-16: The simulated beam directivity and the measured beam directivity of FDMLC.

............................................................................................................................................ 142

Figure 6-17: Six-directional meander-line-coil (SDMLC). (a), the schematic diagram of

SDMLC; (b) the fabricated SDMLC.................................................................................. 143

Figure 6-18: The approximated model for the six-directional meander-line-coil (SDMLC).

............................................................................................................................................ 144

Figure 6-19: The simulated beam directivity of the six-directional meander-line-coil

(SDMLC) EMAT ............................................................................................................... 144

Figure 6-20: The magnitude of the received Rayleigh waves............................................ 145

Figure 6-21: (a), The scan path of the receiver; (b), the experimental beam directivity. .. 145

List of Figures

14

Figure 6-22: The simulated beam directivity and the measured beam directivity of SDMLC.

............................................................................................................................................ 146

List of Tables

15

LIST OF TABLES

Table 2-1: The state-of-the-art in EMAT modelling............................................................ 39

Table 3-1: Parameters used for modelling steering and focusing ........................................ 50

Table 3-2: Detailed parameters used for near and far fields modelling. .............................. 56

Table 4-1: Detailed parameters used for studying analytical solutions proposed by Dodd

and Deeds. ............................................................................................................................ 72

Table 5-1: Detailed parameters used for the analytical US model..................................... 119

Table 5-2: HPBW and SSL for the meander-line-coil with various lengths...................... 123

Table 6-1: Detailed parameters used for fabricating the variable-length meander-line-coil

(VLMLC). .......................................................................................................................... 129

Table 6-2: Detailed parameters used for the EMAT-US modelling. ................................. 130

Table 6-3: Comparison: Beamwidth and the Sidelobe Level. ........................................... 133

Table 6-4: HPBW and SLL at various steps. ..................................................................... 134

Table 6-5: Detailed parameters used for the four-directional meander-line-coil (FDMLC).

............................................................................................................................................ 139

Nomenclature

16

NOMENCLATURE

Abbreviations and Acronyms

EMATs Electromagnetic Acoustic Transducers

EM Electromagnetic

US Ultrasound

FEM Finite Element Method

FDTD Finite-Difference Time-Domain

VLMLC Variable-Length Meander-Line-Coil

FDMLC Four-Directional Meander-Line-Coil

SDMLC Six-Directional Meander-Line-Coil

DC Direct Current

AC Alternating Current

SH Shear Horizontal

SV Shear Vertical

PPM Periodic-Permanent-Magnet

CFL Courant–Friedrichs–Lewy

PML Perfectly Matched Layer

FFT Fast Fourier Transform

RMSE Root-Mean-Square Error

L waves Longitudinal Waves

S waves Shear Waves

RRW Reflected Rayleigh Waves

SRW Scattered Rayleigh Waves

DRW Directly transmitted Rayleigh Waves

BRW Bidirectional Rayleigh Waves

Nomenclature

17

URW Unidirectional Rayleigh Waves

IEEE Institute of Electrical and Electronics Engineers

RL Receiver on the Left

RR Receiver on the Right

HPBW Half Power Beamwidth

SLL Sidelobe Level

CLMLC Constant-Length Meander-Line-Coil

Abstract

18

ABSTRACT

Name of University: The University of Manchester

Candidate’s Name: Yuedong Xie

Degree Title: Doctor of Philosophy

Thesis Title: Modelling Techniques and Novel Configurations for Meander-line-coil

Electromagnetic Acoustic Transducers (EMATs)

Date: July 2016

Electromagnetic acoustic transducers (EMATs) are increasingly used in industries due to

their attractive features of being non-contact, cost-effective and the fact that a variety of

wave modes can be generated, etc. There are two major EMATs coupling mechanisms: the

Lorentz force mechanism for conductive materials and the magnetostriction mechanism for

ferromagnetic materials; EMATs operated on Lorentz force mechanism are the focus of

this study.

This work aims to investigate novel efficient modelling techniques for EMATs, in order to

gain further knowledge and understanding of EMATs wave pattern, how design parameters

affect its wave pattern and based on above propose and optimise novel sensor structures.

In this study, two novel modelling methods were proposed: one is the method combining

the analytical method for EM simulation and the finite-difference time-domain (FDTD)

method for US simulation for studying the Rayleigh waves’ properties on the vertical plane

of the material; the other one is the method utilizing a wholly analytical model to explore

the directivity of surface waves. Both simulations models have been validated

experimentally. The wholly analytical model generates the radiation pattern of surface

waves, which lays a solid foundation for the optimum design of such sensors. The beam

directivity of surface waves was investigated experimentally, and results showed the length

of wires has a significant effect on the beam directivity of Rayleigh waves.

A novel configuration of EMATs, variable-length meander-line-coil (VLMLC), was

proposed and designed. The beam directivity of surface waves generated by such novel

EMATs were analytically investigated. Experiments were conducted to validate such novel

EMATs models, and results indicated that such EMATs are capable of supressing side

lobes, and therefore resulting in a more concentrated surface waves in the desired direction.

Further, another two novel configuration of EMATs, the four-directional meander-line-coil

(FDMLC) and the six-directional meander-line-coil (SDMLC), were proposed and

designed; results showed these EMATs are capable of generating Rayleigh waves in

multiple directions and at the same time suppressing side lobes.

Declaration

19

DECLARATION

No portion of the work referred to in this thesis has been submitted in support of an

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

institution of learning.

Copyright Statement

20

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Acknowledgements

21

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my supervisor Dr. Wuliang Yin and co-

supervisor Prof. Anthony Peyton, for their continuous support of my Ph.D study and

research, for their patience, motivation and encouragement. Their guidance helped me in

all the time of research and writing of this thesis.

Besides my supervisors, I would like to thank Prof. Zenghua Liu in Beijing University of

Technology, for providing the experimental instrument, which is really helpful for me to

carry out the experimental study. Also, I would like to thank Mr Peng Deng, Mr Yanan Hu

and Miss Muwen Xie, for setting up the experiments in Beijing University of Technology.

My thanks go to Professor Emmanuel Bossy in Langevin Institute, for his contributions to

the open source FDTD solver, SimSonic, from which some of the simulations contained in

this work were carried out.

I would like to express my appreciation to my colleagues in SISP group in University of

Manchester, for the research discussions and communications, and for all the fun we have

had in the last four years.

Last but not the least, I would like to thank my family: my parents Jianzhi Xie and

Zhuanmei Zhao, my sister Miaoling Xie, my brother-in-law Xiaobo Huo, my niece

Mengxuan Huo, and my girlfriend Dr. Weiwei An, for supporting me spiritually

throughout my life.

CHAPTER 1 Introduction

22

Chapter 1 Introduction

In this chapter, the motivation, aim, objectives and contributions of this study are

introduced, followed by the organisation of the thesis.

1.1 Motivation

Ultrasonic non-destructive testing, which normally operates at a frequency with a range

from 20 kHz to 100 MHz, is a branch of non-destructive testing techniques. This technique

is based on the ultrasound waves’ propagation within the test piece: ultrasound waves are

generated into the test piece; when ultrasound waves encounter any discontinues or

boundaries of the test piece, they are scattered and picked up by the transducer. Hence

ultrasonic non-destructive testing is able to perform thickness measurement, crack

detection and material characterisation [1-6].

The transducer frequently used for the ultrasonic non-destructive testing is piezoelectric

ceramics or crystals [5-7]. The piezoelectric transducer offers several advantages, such as

good penetration depth, mechanical flexibility, insensitive to electromagnetic fields and

radiation, ease of use and relatively low cost, etc. [8-10]. However, one primary

disadvantage of the piezoelectric ultrasonic testing is the need to have good sonic contact

between the piezoelectric transducer and the test piece, typically by means of a couplant

for acoustic impedance matching [9, 10]. This drawback places limits on piezoelectric

transducers in several applications, such as high temperature detecting, low temperature

detecting, and moving samples detecting, etc. [9, 11].

There are mainly two non-contact ultrasonic techniques, laser-based ultrasonic techniques

and Electromagnetic Acoustic Transducers (EMATs) techniques; while the former is

relatively more expensive [12]. EMATs techniques are the focus of this study due to their

attractive features of being non-contact, cost-effective and the fact that a variety of wave

modes can be generated, etc. Although considerable works have been reported on the

study of EMATs, there are still many important issues which need further investigation,

especially advanced and efficient modelling methods are needed to fully explore the wave

phenomenon, the effects of the design parameters and how new EMAT can be designed

and further optimised.


23

1.2 Aim and Objectives

The aim of this study is to investigate novel efficient modelling techniques for EMATs, in

order to gain further knowledge and understanding of EMATs wave pattern, how design

parameters affect its wave pattern and based on above propose and optimise novel sensor

structures. The objectives of this study include:

1, To seek novel modelling methods for simulating EMATs. Currently, many

modelling methods focus on the vertical plane. This thesis intends to expand this 2D

capability to pseudo – 3D cases, where the surface plane is also taken into

consideration.

2, To analyse beam directivity and radiation pattern of Rayleigh waves generated by

meander-line-coil EMATs; to investigate how design parameters such as the length

of the wire affect the Rayleigh waves’ beam directivity; and to perform quantitative

analysis of the beam directivity of Rayleigh waves and provide useful information

for the optimal design of such EMATs.

3, To propose and design novel EMATs which produce superior performance than

conventional meander-line-coil EMATs.

This study mainly focuses on the meander-line-coil EMATs operated on Lorentz force

mechanism for Rayleigh wave generation. However, the methodology for sensor analysis

and design can be extended to other types of EMATs.

1.3 Contributions

This thesis has made significant and novel contributions in several areas of EMATs.

1, Proposed a novel modelling method on the vertical plane, which combines an

analytical method for EM simulation and the finite-difference time-domain (FDTD)

method for US simulation to produce EMATs simulation models. The simulation

methodology and results have been experimentally validated.

2, Proposed a novel modelling method which is based on a wholly analytical

approach. This method is suitable for investigating surface waves and extends the

modelling method for EMATs’ simulation from 2-D to 3-D.


24

3, Beam directivity of the conventional meander-line-coil EMATs were

quantitatively analysed with simulations and experiments. There has been little

research on the analysis of beam directivities of Rayleigh waves generated by

EMATs, hence this work has significantly filled the knowledge gap.

4, Proposed and designed a novel meander-line-coil with variable-length wires,

termed as variable-length meander-line-coil (VLMLC). The VLMLC EMAT is

capable of suppressing the side lobes of the Rayleigh waves’ beam, and therefore

makes Rayleigh waves more concentrated in desired directions.

5, Two novel EMATs, the four-directional meander-line-coil (FDMLC) and the six-

directional meander-line-coil (SDMLC), to generate multiple-directional Rayleigh

waves have been proposed and designed. These multiple-directional Rayleigh waves’

EMATs can be viewed as a combination of several sets of variable-length meander-

line-coils (VLMLC); they are capable of generating Rayleigh waves in four or six

directions and at the same time suppressing side lobes. These multiple-directional

Rayleigh waves EMATs are especially useful for large specimen inspections.

1.4 Organization of thesis

Chapter 1 states the motivation, aim and the objectives of this study, highlighting the major

contribution and novelties of this study. In addition, the thesis outlines are presented.

Chapter 2 presents the background of EMATs, including the basic coupling mechanisms of

EMATs, the advantages and limitations of EMATs, and their applications. Followed by the

introduction of wave modes, some of the popular EMATs for generating different wave

modes are presented and discussed. In addition, the state-of-the-art modelling methods for

EMATs operated on Lorentz force mechanism are summarized, highlighting the novelty of

the modelling methods proposed by the author.

Chapter 3 introduces the finite-difference time-domain (FDTD) method, and uses FDTD

method to model several behaviours of ultrasound waves, such as steering, focusing and

scattering. In addition, the combination of the FDTD method and the Hilbert

transformation to generate the radiation pattern is introduced, followed by the quantitative

analysis of beam features by means of the radiation pattern. The study on ultrasonic

modelling with the FDTD method is one important part of the EMAT modelling, which is

introduced in Chapter 4.


25

Chapter 4 presents a novel modelling method combining the analytical method and the

FDTD method to model EMATs operated on Lorentz force mechanism to generate

Rayleigh waves; this novel modelling method is a 2D modelling method focusing on the

vertical plane of the test piece. The analytical method is adapted from classic Dodd and

Deeds solutions to calculate eddy current phenomena; the FDTD method, as described in

Chapter 3, is used to model ultrasound waves’ propagation within the test sample.

Experiments were conducted to validate the proposed modelling methods; this novel

modelling method presented in this chapter and related works have been published in

Ultrasonics, Journal of Sensors, and International Journal of Applied Electromagnetics and

Mechanics[10, 13, 14].

Chapter 5 focuses on the directivity analysis of Rayleigh waves generated by conventional

meander-line-coil EMATs; a novel 2-D modelling method to model Rayleigh waves’

distribution on the surface plane of the test piece is proposed; this work, focusing on the

surface plane of the test piece, is an extension of the work contained in Chapter 4. The

effect of the length of the conventional meander-line-coil on the radiation pattern was

studied analytically and experimentally; the work contained in this chapter has been

submitted to Ultrasonics and is under revision.

Chapter 6 illustrates several novel EMATs configurations proposed by the author,

including the variable-length meander-line-coil (VLMLC), the four-directional meander-

line-coil (FDMLC), and the six-directional meander-line-coil (SDMLC). These novel

EMATs are capable of suppressing the side lobes of the Rayleigh waves’ beam /

generating multiple-directional Rayleigh waves. A paper based on part of the work in this

chapter has been accepted by IEEE Sensors Journal [15].

Chapter 7 provides a summary of this thesis work, followed by the discussions of future

work.

1.5 List of Publications

Journal Papers:

1. Y. Xie, W. Yin, Z. Liu, and A. Peyton, "Simulation of ultrasonic and EMAT arrays

using FEM and FDTD," Ultrasonics, vol. 66, pp. 154-165, 2016.


26

2. Y. Xie, S. Rodriguez, W. Zhang, Z. Liu, and W. Yin, "Simulation of an

Electromagnetic Acoustic Transducer Array by using Analytical method and FDTD,"

Journal of Sensors, vol. 501, p. 5451821, 2016.

3. Y. Xie, L. Yin, R. G. Sergio, T. Yang, Z. Liu, and W. Yin, "A wholly analytical

method for the simulation of an electromagnetic acoustic transducer array,"

International Journal of Applied Electromagnetics and Mechanics, pp. 1-15, 2016.

4. Y. Xie, L. Yin, Z. Liu, P. Deng, and W. Yin, "A Novel Variable-Length Meander-line-

Coil EMAT for Side Lobe Suppression," IEEE Sensors Journal, vol. PP, 2016.

5. Y. Xie, Z. Liu, P. Deng, and W. Yin, "Directivity analysis of Meander-Line-Coil

EMATs with a wholly analytical method," Ultrasonics, under revision.

6. Y. Xie, R. G. Sergio, Z. Liu, Q. Zhao, M, He, B. Wang, A. Peyton and W. Yin, "A

pseudo – 3D model for meander-line-coil Electromagnetic Acoustic Transducers

(EMATs)," IEEE Transactions on Instrumentation and Measurement, under review.

Conference Papers:

1. Y. Xie, W. Yin, and A. Peyton, "Quantitative Simulation of Ultrasonic and EMAT

Arrays Using FEM and FDTD," presented at the 11th European Conference on Non-

Destructive Testing (ECNDT 2014), Prague, Czech Republic, 2014.

2. Q. Zhao, K. Xu, Y. Xie, and W. Yin, "Measurement of liquid level with a small surface

area using high frequency electromagnetic sensing technique," in 2015 IEEE

International Instrumentation and Measurement Technology Conference (I2MTC)

Proceedings, 2015, pp. 1414-1419.

3. Y. Xie, S. RODRIGUEZ, Z. Liu, Q. Zhao, J. Hao, B. Wang, W. Yin, and A. Peyton,

"Simulation and experimental verification of a meander-line-coil Electromagnetic

Acoustic Transducers (EMATs)," accepted by the 2016 IEEE International

Instrumentation and Measurement Technology Conference (I2MTC), Taipei, Taiwan,

2016.

CHAPTER 2 EMAT background

27

Chapter 2 EMAT background

2.1 Introduction

In this chapter, EMATs’ coupling mechanisms are introduced first. Based on the EMATs’

coupling mechanisms, the advantages and disadvantages of EMATs and EMATs’

applications are discussed. The classification of waves and their features are described,

followed by the classification of EMATs. At the end, the state-of-the-art EMAT modelling

methods are summarized and compared.

2.2 Coupling Mechanisms of EMATs

A typical EMAT sensor contains a test piece, a coil to induce dynamic electromagnetic

fields and a permanent magnet to produce biasing magnetic fields [16]. There are mainly

two coupling mechanisms, the Lorentz force mechanism and the magnetostriction

mechanism, which are operated on different materials [16-19].

2.2.1 Lorentz Force Mechanism

Lorentz force mechanism is operated on non-ferromagnetic conductive materials, such as

aluminium, copper and stainless steel [19, 20]. Figure 2-1 shows the elementary structure

of an EMAT; a wire carrying an alternating current (I) at a desired frequency is placed

above the test piece; the wire induces eddy currents (Je) into the near surface region of the

test piece. The eddy currents (Je) interact with the static magnet field (B), which is

generated by the permanent magnet, produces a body force per unit volume, which is the

Lorentz force density (f), as shown in Equation 2-1. The Lorentz force density acts as a

driving force to generate ultrasound waves.

Figure 2-1: Schematic of the Lorentz force mechanism. From [19].


28

Equation 2-1

𝒇 = 𝑱𝒆 × 𝑩

The receiving process of the Lorentz force EMAT is: the ultrasound waves produce

deformations and particle vibrations within the material; in the presence of the biasing

magnetic field, induced currents are generated in the near surface of the material; induced

currents in turn generate dynamic magnetic fields which can be picked up by the receiving

EMATs [16, 21]; the explicit governing equations for the EMAT transduction will be

detailed in Chapter 4. Practical coils used in EMATs are a combination of wires; with

different layouts of each wire, ultrasound waves can be steered to a specific angle or

focused on a specific point [22, 23].

2.2.2 Magnetostriction Mechanism

Magnetostriction mechanism is operated on ferromagnetic materials, such as iron, nickel

and their alloys [16, 20, 24]. One point should be noted is that some ferromagnetic

materials, such as iron and nickel, are conductive as well; both the Lorentz force

mechanism and the magnetostriction mechanism operated on such materials; the strength

of both of mechanisms in such materials is deserved to study in the future.

There are two types of magnetostriction mechanisms, spontaneous magnetostriction and

field induced magnetostriction. From a Microscopic view, above the Curie temperature,

each magnetic dipole in a ferromagnetic material has a particular orientation, providing a

net magnetic dipole of zero; when the ferromagnetic material is dropped below its Curie

temperature, the close-by magnetic dipoles are aligned with one another and are reoriented

to the same direction, forming the magnetic domain [19]. The alignment of the magnetic

dipoles within a domain results in a spontaneous magnetisation of the domain along a

certain direction and this is associated with a spontaneous strain; the average deformation

of the ferromagnetic material is named spontaneous magnetostriction [17-19, 24].

The field induced magnetostriction is: when an external magnetic field (H) is applied to the

ferromagnetic materials, the magnetic domain tend to towards the direction of the external

magnetic field; the reorientation of the magnetic domain results in a deformation (∆l), as

shown in Figure 2-2 [18, 19].


29

Figure 2-2: Microscopic process of the field induced magnetostriction. H is the external

magnetic field; ∆l is the deformation due to the reorientation of the magnetic domain, which

is simplified represented by an elliptic shape. From [18].

EMATs exploits field induced magnetostriction [16, 17, 24]. As shown in Figure 2-3, a

coil and a magnet are placed above a ferromagnetic material; the alternating current in the

coil induces the dynamic magnetic field, which in turn generates the dynamic stress εd. The

permanent magnet provides the static magnetic field, which in turn produces the static

stress εs. The resultant stress εr causes volume changes in the form of contracting and

stretching, which in turn produces ultrasound waves within the test object [17, 19]. The

receiving mechanism is based on the inverse magnetostriction effect: the elastic

deformation produces a magnetic flux density, which can be converted into a voltage

signal detected by the receiving coil [16, 25].

Figure 2-3: EMATs operated on the magnetostriction mechanism. εd is the dynamic stress, εs

is the static stress, and εr is the resultant stress.

2.2.3 Advantages and Disadvantages of EMATs

Compared to conventional piezoelectric transducers, EMATs have several advantages. The

first attractive feature is the non-contact nature [26-28]. Because EMATs generate

ultrasound waves directly into the test piece instead of coupling through the transducer, no

couplant is needed between the EMAT sensor and the test object. Hence, EMATs have

advantages in applications where surface contact is not possible or not desirable, such as


30

the high temperature testing, low temperature testing and moving samples testing, etc. [10,

11]. Since couplant is not needed for the EMAT operation, EMATs simplify the operation

and eliminate the possible errors arises from the couplant during the testing. Moreover,

because no couplant is needed, EMAT inspection is a dry inspection with no chemicals or

hazardous materials involved in this inspection [11, 29, 30] [31]. In addition, the EMAT

inspection is less sensitive to surface conditions; due to this nature, EMATs are capable of

inspecting rough, dirty oxidized or uneven surfaces [32].

Another attractive feature of EMATs is that a variety of wave modes can be generated.

With different combinations of coils and magnets, multiple wave modes, including

longitudinal waves, shear waves, Rayleigh waves and Lamb waves, can be produced [10,

11, 22, 29, 30, 33-39]. Especially, EMATs are capable of efficiently generating shear

horizontal (SH) waves, which do not present mode conversion at the structure boundaries

[11, 32]. Since wedges, as well as Snell’s Law of refraction, are not applied into EMATs

operation, the deployment of EMATs is easier. Due to these attractive features, EMATs are

widely used in industries for depth measurement, crack detection and material

characterisation [40-42].

However, EMATs have some limitations. Low transduction efficiencies are always

observed; the efficiency of the EMATs’ transduction decays exponentially with the lift-off

distance, limiting the practical lift-off to only a few millimetres [16, 18, 25, 31]. Hence, the

most important problem of EMATs is improving the transduction efficiency and the signal-

to-noise ratio [18, 25]; special electronics are required to overcome the low transduction

efficiency and the low signal-to-noise ratio [31]. Some works have been reported on

improving the transduction efficiency of EMATs, such as the optimal design of EMATs

and using a ferrite back-plate, etc. [28, 43-46]. Moreover, typical EMATs generate

multiple wave modes within the material simultaneously; that means the interpretation of

signals is difficult [13, 16, 18]. Hence, generating and receiving ultrasound waves by

EMATs with purer wave modes are another problems of concern. Another limitation of

EMATs is material dependent, that is, due to EMATs’ coupling mechanisms, only

conductive materials and ferromagnetic materials can be detected [16, 25, 31]. For other

important industrial materials, such as plastic, composites, and ceramics, EMATs are not

desirable methods to detect such materials [31].


31

2.3 Types of EMATs

There are a variety of EMAT configurations to generate various wave modes. In this

section, four ultrasound wave types, longitudinal waves, shear waves, Rayleigh waves and

Lamb waves, are introduced at first. Based on these wave modes, some of most popular

configurations of EMATs are presented.

2.3.1 Major Types of Mechanical Waves

2.3.1.1 Longitudinal Waves

A longitudinal wave is a type of body waves travelling within the material. As shown in

Figure 2-4, for longitudinal waves, the particle motion is parallel to the direction of the

wave propagation; a longitudinal wave is transmitted by particle movements in back and

forth forms [47]. Longitudinal waves are also known as compressional waves, because

they involve compression and rarefaction when travelling through a material (Figure 2-4).

Longitudinal waves are able to inspect defects within the material, however, longitudinal

waves experience strong mode conversion at structural and weld boundaries [11, 48, 49].

Figure 2-4: Longitudinal waves. The black arrow denotes the direction of the wave

propagation; red arrows denote directions of the particle motion. From [50].

2.3.1.2 Shear Waves

Another type of body waves is a shear wave, which is slower than a longitudinal wave

within the same medium; the particle motion of a shear wave is perpendicular to the

direction of wave propagation as shown in Figure 2-5. Based on the particle vibration

plane, there are two types of shear waves, shear vertical waves and shear horizontal waves.

Shear waves polarized in the horizontal plane are classified as shear horizontal (SH) waves;


32

shear vertical (SV) waves, on the other hand, are polarized in the vertical plane of the

material [51]. Shear horizontal (SH) waves do not present mode conversion at the

structural and weld interferences, hence, shear horizontal (SH) waves are promising

methods to inspect welds conditions [47, 48]. Shear horizontal (SH) waves cannot be

excited easily with conventional piezoelectric transducers while they can be excited easily

by EMATs [11, 49].

Figure 2-5: Shear waves. The black arrow denotes the direction of the wave propagation; red

arrows denote directions of the particle motion. From [50].

2.3.1.3 Rayleigh Waves

A Rayleigh wave is a type of surface waves travelling near the surface of the material

whose depth is comparable to the Rayleigh waves’ wavelength [52]. Rayleigh waves

include longitudinal and transverse motions; an arbitrary particle in Rayleigh waves moves

in an elliptical path, and the major axis of the ellipse is perpendicular to the surface of the

test object, as shown in Figure 2-6; at the near surface of the material, the elliptical path is

in a counter-clockwise direction [52, 53]. Rayleigh waves are mainly concentrated within a

depth of Rayleigh waves’ wavelength and decay significantly as the depth increases [13,

53].

Figure 2-6: Rayleigh waves. The black ellipse denotes the particle motions. From [54].


33

Rayleigh waves are slower than body waves, however, they have a larger amplitude and a

long duration [52]. Because Rayleigh waves are concentrated on the near surface of the

material, they are sensitive to near surface defects [55]. In addition, the decay of Rayleigh

waves along the surface direction is smaller compared to that of body waves, hence,

Rayleigh waves are capable of long distance detections for locating and sizing defects in

materials with a time delay measurement [56, 57]. However, Rayleigh waves’ accuracy is

limited by the orientation of the crack; that is, the crack parallel to the surface is difficult to

detect. In addition, the generation of Rayleigh waves are normally accomplished with the

generation of longitudinal waves and shear waves; multiple wave modes make the signal

complicated [10, 13].

2.3.1.4 Lamb Waves

A Lamb wave is a kind of plate waves, which can only be generated in materials with a

few wavelengths’ thick; Lamb waves are guided by the free upper and lower surfaces of

the plate-like structures [52, 58]. Because Lamb waves are capable of propagating over a

long distance of several meters, Lamb waves are able to inspect a large structure in a short

time [58, 59]. Another attractive feature of Lamb waves is the capability of inspecting both

the surface and the internal damages; that is because Lamb waves propagate parallel to the

surface of the material throughout the thickness of the material [52, 58, 59].

However, Lamb waves’ detection is complicated due to the dispersive nature of Lamb

waves. Another problem is that more than one wave mode exist at a specific frequency [59,

60]. Figure 2-7 shows two main modes of Lamb waves, symmetric and anti-symmetric

modes; with the frequency increasing, more wave modes are presented and signals’

interpretation is difficult. In addition, Lamb waves have the problem of mode conversion

when they encounter any discontinuities [59].

Figure 2-7: Modes of Lamb waves. (a), symmetric mode; (b), anti-symmetric mode. The black

arrows denote the displacement of the particle; black curves denote the resulting Lamb

waves. From [58].

(b) (a)


34

2.3.2 Classification of EMATs

In this section, a variety of most popular configurations of EMATs operated on Lorentz

force mechanism are mainly described; with different combinations of the magnet and the

coil, EMATs are capable of generating various wave modes.

2.3.2.1 Longitudinal waves EMATs

Figure 2-8 shows an EMAT to generate longitudinal waves normal to the surface of the

material; the magnet provides a tangential magnetic field; the coil induces eddy currents in

the surface layer of the material. Based on Lorentz force mechanism, Lorentz forces are

generated normal to the surface of the material, and longitudinal waves are generated into

the material with a propagation direction normal to the surface of the material.

Figure 2-8: The cross-sectional view of a normal longitudinal wave EMAT. The white hollow

arrows denote the direction of the static magnetic field; the grey arrows denote the direction

of the Lorentz force; the solid black arrow means the direction of wave propagation. From

[19, 20].

In addition, angled longitudinal waves can be generated by EMATs. EMATs to generate

angled longitudinal waves will be introduced in section 2.3.2.3, because the Rayleigh

waves EMAT in section 2.3.2.3 is capable of generating both angled longitudinal waves

and Rayleigh waves.

2.3.2.2 Shear waves EMATs

Figure 2-9 shows two types of EMATs, which are operated on Lorentz force mechanism,

to generate radially polarized shear waves and linearly polarized shear waves respectively.

In Figure 2-9 (a), a spiral coil and a cylindrical magnet are used; eddy currents are induced

by the spiral coil and the biasing magnetic field are produced by the magnet; the

interaction between the induced eddy currents and the biasing magnetic field produces


35

Lorentz forces which are parallel to the surface of the material; radially polarized shear

waves are generated normal to the surface of the specimen [19, 38]. In Figure 2-9 (b), two

rectangular magnets are used to provide reverse magnetic fields; the coil typically used is a

racetrack coil or a rectangular coil; linearly polarized shear waves are generated due to this

EMAT configuration [16, 38].

Figure 2-9: The cross-sectional view of normal shear waves EMATs. The white hollow arrows

denote the direction of the static magnetic field; the grey arrows denote the direction of the

Lorentz force; the solid black arrows mean the direction of wave propagation. Adapted from

[19, 20].

Especially, EMATs are capable of generating shear horizontal (SH) waves. Figure 2-10

shows one kind of SH waves EMATs, which is named the periodic-permanent-magnet

(PPM) EMAT operated on Lorentz force mechanism; it consists of an array of permanent

magnets and an elongated spiral coil. The array of permanent magnets provides alternative

magnetic fields normal to the surface of the material; the elongated spiral coil induces eddy

currents into the near surface of the material; due to the Lorentz force mechanism, the

interaction between the alternative magnetic fields and eddy currents generates tangentially

polarized forces, which in turn generate shear horizontal (SH) waves not only along the

surface but also into the material [16, 17].

Figure 2-10: The structure of the periodic-permanent-magnet (PPM) EMAT to generate SH

waves. From [16].

(a) (b)


36

The cross-sectional view of the PPM EMAT is shown in Figure 2-11; the SH wave’s

propagation angle can be controlled by Equation 2-2 [37, 49].

Equation 2-2

𝑠𝑖𝑛 𝜃 =(2𝑛 + 1)𝜆

𝐷=(2𝑛 + 1)𝑣

𝑓𝐷

where θ denotes the angle at which the SH waves are steered, n denotes the order of the

interference, v is the velocity of the shear waves within the material, λ is the wavelength of

shear waves within the material, f is the operational frequency, D is the centre-to-centre

distance between two permanent magnets with the same magnetic polarities.

Figure 2-11: The cross-sectional view of the PPM EMAT. From [48].

There is another SH wave EMAT operated on the magnetostriction mechanism. As shown

in Figure 2-12, a permanent magnet provides a tangentially biasing magnetic field; the

meander-line-coil produces the dynamic magnetic field normal to the static magnetic field.

Since the static magnetic field is parallel to the induced eddy current, there is no Lorentz

force generated [31]. The resultant magnetic field, based on the magnetostriction

mechanism, results in a deformation, which in turn generates SH waves.


37

Figure 2-12: The structure of the meander-line-coil EMAT to generate SH waves. From [16].

2.3.2.3 Rayleigh waves EMATs

For EMATs to generate Rayleigh waves, typically used coils are meander-line-coils [16,

21, 28, 34, 38]. As shown in Figure 2-13, a rectangular magnet and a meander-line-coil are

place above the conductive materials. The biasing magnetic field produced by the

permanent magnet interacts with the eddy currents induced by the meander-line-coil,

producing alternating Lorentz forces parallel to the surface of the material; which in turn

generates bidirectional Rayleigh waves travelling along the surface of the material. The

spacing intervals between two adjacent wires of the meander-line-coil equals to one half of

the Rayleigh waves’ wavelength to form the constructive interference [10, 28, 39].

Figure 2-13: The structure of a meander-line-coil EMAT to generate Rayleigh waves. From

[16, 34].

Based on the EMAT configuration shown in Figure 2-13, longitudinal waves and shear

waves are generated as well; the longitudinal waves and shear vertical (SV) waves are


38

travelling obliquely into the material. The propagation angle of SV waves can be

controlled by Equation 2-3.

Equation 2-3

𝑠𝑖𝑛 𝜃 =𝜆/2

𝑑

where λ is the wavelength of SV waves and d is the spacing intervals between two adjacent

wires of the meander-line-coil [22, 23]. The propagation angle of the longitudinal waves

can be determined in the same manner.

[23] proposed an unidirectional Rayleigh waves EMAT using two identical coils with a

distance of a quarter of Rayleigh waves’ wavelength and a phase difference of 900; this

unidirectional Rayleigh waves will be detailed in section 4.8 with simulations. In addition,

omni-directional Rayleigh waves can be generated by a contra-flexure coil [17, 33]. When

the test sample is a thin plate, the EMAT configuration shown in Figure 2-13 can be used

to generate Lamb waves [61, 62].

2.4 State-of-the-art in EMAT Modelling

Since the 1970s, the study on EMATs developed rapidly; considerable works were

reported on the study of EMATs, including the theoretical and experimental research [16,

18, 19]. After around 50 years’ improvement, modelling methods of EMAT are

increasingly complete. Because this work is focusing on Lorentz force mechanism; some

important modelling methods for Lorentz force EMATs within the past ten years are listed

in Table 2-1.

Due to the Lorentz force coupling mechanism, the EMAT model contains the

electromagnetic model and the ultrasonic model [28, 45, 63, 64]. Electromagnetic

simulation can be achieved by the finite element method (FEM) and the analytical method;

ultrasonic simulation can be carried out with the finite element method (FEM), finite-

difference time-domain (FDTD) method, and the analytical method. Some of papers

combined the finite element method (FEM) and the analytical method to model EMATs,

that is, the finite element method (FEM) for electromagnetic simulations and the analytical

method for ultrasonic simulations [28, 45, 63, 64]. Others used the finite element method

(FEM) for both the electromagnetic simulation and the ultrasonic simulation, that is, the

implicit finite element software COMSOL for the electromagnetic simulation and the


39

explicit finite element software ABAQUS for the ultrasonic simulation [65, 66]; [17, 19]

exploited the FE software COMSOL for both the electromagnetic and ultrasonic

simulations, because COMSOL allows the coupling between several different physical

fields. In addition, Kundu used the Distributed Point Source Method (DPSM), which is

considered as a semi-analytical method based on the analytical solutions of basic point

source problems, to calculate the EM phenomena [67, 68].

Table 2-1: The state-of-the-art in EMAT modelling.

Paper

Electromagnetic simulation Ultrasonic simulation

FEM Semi-analytical Analytical FEM FDTD Analytical

[28, 45, 63, 64]

[17, 65, 66]

[67]

Author[10]

Author[13]

Author[14, 15]

As inherently a time domain solver, FDTD technique is well suited to simulate the

ultrasound wave propagation for our purposes, because the measured response in our

experimental setup is also a time sequence signal. ABAQUS has been reported in other

people’s reports to work well for ultrasonic simulation as it is an explicit FEM solver, but

does not deal with EM simulation [65, 66]. COMSOL is a multiphysics solver, but our

experience is that it needs careful setup in order to make it converge (even at very slow

speed) when simulating EM and ultrasonic coupled phenomena together. Thus, FDTD is

employed to model ultrasound waves’ propagation in this work. In addition, the finite

element method (FEM) solver in the frequency domain deals with the electromagnetic

induction efficiently. Hence, authors proposed a method combining the finite element

method, FEM, and the finite-difference time-domain, FDTD, to model EMATs; the

frequency domain simulation (FEM) and the time domain simulation (FDTD) are linked

together [10]; this work was extended to combine the analytical method and the FDTD

method, that is, the analytical method for the eddy current calculation and FDTD for the

ultrasonic simulation; this novel method will be detailed in Chapter 4. The semi-analytical


40

method for the electromagnetic calculation is not adopted because only a certain number of

passive sources on the interface between two media are selected, resulting the calculation

results not as accurate as the analytical methods [67].

Because 3-D modelling has a high demand of the computer capacity and requires

significant running time, most of the previous work were 2-D simulation focusing on the

vertical plane of the material [10, 11, 13, 21]. There has been little research on the

Rayleigh waves’ beam directivity on the surface plane of the material; a wholly analytical

method to analyse the Rayleigh waves’ beam directivity is proposed by the author and will

be detailed in Chapter 5.

2.5 Conclusions

In this chapter, the basic operational principles of EMATs, the Lorentz force mechanism

and the magnetostriction mechanism, are introduced. Followed by the description of

EMATs’ benefits over the conventional piezoelectric transducer, the applications of

EMATs are introduced. Some of popular EMATs with different configurations to generate

various wave modes are presented. At the end, the state-of-the-art modelling methods for

Lorentz force EMATs are summarized, highlighting the novelty of the modelling methods

proposed by the author.

CHAPTER 3 FDTD for ultrasonic modelling

41

Chapter 3 FDTD for Ultrasonic Modelling

In this chapter, conventional ultrasonic phased array techniques, such as steering and

focusing, are studied. The finite-difference time-domain (FDTD) method is introduced and

is exploited for modelling ultrasonic phased arrays. The combination of the finite-

difference time-domain (FDTD) method and Hilbert Transformation is able to generate

radiation pattern which is the foundation for quantitative analysis of beam features. In

addition, a model with a crack is built to simulate ultrasound scattering phenomena.

3.1 Ultrasonic Testing Techniques

For conventional ultrasonic testing, transducers typically used are made of piezoelectric

ceramics or crystals. Depending on the number of elements contained in the piezoelectric

transducer, there are two types of transducers: non-phased array transducers with a single

element and phased array transducers with multiple elements. Non-phased array

transducers are limited because only ultrasound waves in fixed directions can be generated;

phased array (multiple elements) transducers, on the other hand, are capable of controlling

the beam to sweep without moving the transducer.

3.1.1 Phased Array Techniques

Ultrasonic phased array techniques are based on the interference of ultrasound waves: by

firing elements at prescribed time delays, multiple ultrasound waves with various phases

are generated; the resultant wave is the sum of multiple ultrasound waves generated by

each element and the wavefront travels along the prescribed path.

Steering and focusing are two widely used phased array techniques as shown in Figure 3-1;

eight elements are contained in a transducer; with prescribed time delays, the wavefront is

steered to an arbitrary angle or focused to a specific point. Due to attractive features of

eliminating the mechanical scanning, enlarging the scanning area, and the optimum beam

shape, ultrasonic phased array techniques are widely applied in industrial and medical

fields [69].


42

Figure 3-1: Phased array techniques: steering and focusing [10].

For phased array techniques, firing times are critical because they determine the mode of

waves’ interference and in turn determine the beam path. Time delays are calculated based

on the propagating velocity and centre-to-centre distance between adjacent elements; the

model for time delays’ calculation for steering and focusing are shown in Figure 3-2 and

Figure 3-3 respectively.

The steering behaviour is shown in Figure 3-2; an 8-elements array is placed on the top of

the test piece; define 900 along the positive y-axis and 00 along positive x-axis. The centre-

to-centre distance between two adjacent elements in this array is constant, 𝑑, which results

in the time delays for beam steering between two adjacent elements are constant as well;

the propagating velocity of ultrasound waves is 𝑣; 𝜃 is the steering angle. The relationship

between the steering angle 𝜃 and the time delay between two adjacent elements 𝑡𝑑 is

described in Equation 3-1.

Equation 3-1

𝑠𝑖𝑛 𝜃 = 𝑣 ∙ 𝑡𝑑𝑑

Assuming the firing time of the element 1 is 𝑡1, the firing times of other elements are,

Equation 3-2

𝑡𝑖 = 𝑡1 + (𝑖 − 1) ∙ 𝑡𝑑 , 𝑖 = 2,3…8

where 𝑡𝑖 is the firing times for elements 2-8; the subscript, i, indicates the sequence number

of the element.

Combining Equation 3-1 and Equation 3-2, for steering angles between 00 and 900,

elements are fired sequentially from the element 1 to the element 8; for steering angles


43

between -900 and 00, elements are fired in a reversed order, which is from the element 8 to

the element 1. For the steering angle 00, elements are fired simultaneously.

Figure 3-2: A model used for time delays calculation for steering.

The focusing behaviour is shown in Figure 3-3; the wavefront is concentrated on the focal

point. 𝑟𝑖 is the travel distance from each element to the focal point; the subscript 𝑖 is the

sequence number of the element. For an arbitrary element in the array, the travelling

distance 𝑟𝑖 and the travelling time 𝑇𝑖 are calculated with Equation 3-3 and Equation 3-4, and

the firing times for each element, 𝑡𝑖, are determined by Equation 3-5.

Equation 3-3

𝑟𝑖 = √(𝑥𝑓2 − 𝑥𝑖2) + (𝑦𝑓2 − 𝑦𝑖2)

Equation 3-4

𝑇𝑖 = 𝑟𝑖/𝑣

Equation 3-5

𝑡𝑖 = 𝑚𝑎𝑥(𝑇𝑖) − (𝑇𝑖 −𝑚𝑖𝑛(𝑇𝑖))

where (𝑥𝑓 , 𝑦𝑓) is the coordinate of the focal point, and (𝑥𝑖 , 𝑦𝑖) is the coordinate of the

element.


44

Figure 3-3: A model for time delays calculation for focusing.

Based on Equation 3-1 to Equation 3-5, for a transducer with constant centre-to-centre

distances between adjacent elements, elements are fired sequentially for steering while

disorderly for focusing. In addition, the centre-to-centre distance between two adjacent

elements may be not constant; but the calculation process is the same.

3.1.2 Ultrasonic Testing Methods

Based on whether transmitted waves or reflected waves are used, there are three principal

ultrasound inspection methods: the pulse-echo method, the through-transmission method,

and the pitch-catch method [70]; these three ultrasonic inspection methods are not only

used for piezoelectric ultrasound testing, but also for electromagnetic acoustic transducers

(EMATs) testing.

3.1.2.1 Pulse-echo Method

For the pulse-echo method, only a single transducer is employed not only as a transmitter

but also as a receiver, as shown in Figure 3-4. In Figure 3-4 (a), a transducer is placed on

the top of the test piece to transmit and receive ultrasound waves; the presence of

discontinues and boundaries of the test piece is the source to generate reflected ultrasound

waves. The amplitude of the echo is the indicator of the discontinues; echoes (reflected

waves) generated by cracks have a different amplitude from those generated by the bottom

boundary of the material, as shown in Figure 3-4 (b). The depth of the crack can be

deducted from the travelling times of these two echoes.


45

Figure 3-4: Ultrasonic pulse-echo method. (a), Inspection diagram; (b) Received signals.

The pulse-echo method can be used in applications where only one surface of the material

can be accessed. However, the performance of the pulse-echo method is limited by the

orientation of the crack; only the crack with proper orientations can reflect transmitted

waves into the transducer. In addition, the main bang signal, received by the receiver due

to the high voltage excitation, limits the near surface inspection [47].

3.1.2.2 Through-transmission Method

Another ultrasonic testing method is the through-transmission method with two transducers

being placed on the opposite surfaces of the material, as show in Figure 3-5. In Figure 3-5

(a), the transmitter is placed on the top of the material and the receiver is placed on the

bottom of the material; the transmitted waves are received by the receiver. With the

presence of the crack, parts of the ultrasound waves are reflected by the crack, the rest of

transmitted waves can be received by the receiver. Receiving signals are shown in Figure

3-5 (b), the amplitude of receiving signals is smaller with the presence of discontinues.

However, the through-transmission method cannot determine the depth of the crack.

Figure 3-5: Ultrasonic through-transmission method. (a), Inspection diagram; (b) Received

signals.


46

By comparing Figure 3-4 with Figure 3-5, to determine cracks, reflected waves are used in

the pulse-echo method while transmitted waves are used in the through-transmission

method; the through-transmission method shortens one half of the propagation path, so it is

widely used for high-attenuation materials detection. The transmitter and the receiver for

the through-transmission method are electronically separated; that reduces the complex of

electronic instrument. However, the through-transmission method is limited in applications

where the surface is not flat or only one surface of the material can be accessed. In addition,

the through-transmission method cannot provide the depth information about the crack.

3.1.2.3 Pitch-catch Method

The third ultrasonic testing method is the pitch-catch method, which uses reflected waves

to detect as well; two separate transducers placed on the same surface of the material as

shown in Figure 3-6. The pitch-catch method creates a V-shaped acoustic path in the test

piece, which allows a complete inspection. This pitch-catch method has the advantage of

near surface inspection, because main bang signals are removed in the pitch-catch method

[47, 70]. In addition, pitch-catch method can be used for single-sided access detection.

However, the experimental set-up is slightly complicated, because these two separate

transducers need to be aligned and the receiver should be well positioned to catch the

receiving signals.

Figure 3-6: Ultrasonic pitch-catch method.

Each of these three ultrasonic testing methods has its own strength and weakness and can

be used in various applications. A good consideration of the beam path, the crack

orientation, the surface access, and the attenuation of the material, etc, is helpful to

determine which ultrasonic testing method should be used.


47

3.2 FDTD Method for Ultrasonic Modelling

In section 3.1, ultrasonic phased array techniques and ultrasonic testing methods are

described. The finite-element time-domain (FDTD) method is capable of modelling

ultrasound propagation phenomena; in this section, the elastodynamic equations and the

finite-element time-domain (FDTD) are introduced.

3.2.1 Elastodynamic Equations

Elastodynamic equations are a set of partial differential equations describing how linearly

elastic material deforms and becomes internally stressed as shown in Equation 3-6 and

Equation 3-7 [71, 72].

Equation 3-6

𝜌(𝑥)𝜕𝒗𝒊𝜕𝑡

(𝑥, 𝑡) =∑𝜕𝑻𝒊𝒋

𝜕𝑥𝑗(𝑥, 𝑡)

𝑑

𝑗=1

+ 𝒇𝒊 (𝑥, 𝑡)

Equation 3-7

𝜕𝑻𝒊𝒋

𝜕𝑡(𝑥, 𝑡) =∑∑𝑐𝑖𝑗𝑘𝑙(𝑥)

𝑑

𝑖=1

𝜕𝒗𝑘𝜕𝑥𝑙

𝑑

𝑗=1

(𝑥, 𝑡)

where 𝜌 is the mass density of the material and 𝑐𝑖𝑗𝑘𝑙 is a 4th-order stiffness tensor of the

material, 𝒇𝒊(𝑥, 𝑡) is the force source; 𝒗𝒊(𝑥, 𝑡) is the velocity component and 𝑻𝒊𝒋(𝑥, 𝑡) is the

stress tensor component.

Equation 3-6 is Newton’s Second Law: when a force is applied to a testing sample, stress

and deformation are generated, as well as the particle displacement. The force per unit

volume on the infinitesimal of sample, 𝒇, is,

Equation 3-8

𝒇 = 𝑚 ∙ 𝒂 = 𝜌 ∙𝜕𝒗

𝜕𝑡

where 𝑚 is the mass of the infinitesimal, 𝒂 is the acceleration of infinitesimal; 𝒇 contains

the force source per unit volume (𝒇𝒊) and the internal stress term (𝒇𝒊𝒔). The internal stress

term can be described as,


48

Equation 3-9

𝒇𝒊𝒔 = 𝜕𝑻

𝜕𝑥

where 𝑻 is a 2nd-order stress tensor.

Equation 3-7 describes the relationship of the stress tensor rate and the strain tensor rate

based on Hooke’s Law; Hooke’s Law is shown in Equation 3-10.

Equation 3-10

𝑻 = 𝑐 ∙ 𝜺 = 𝑐 ∙𝜕𝒖

𝜕𝑥

where 𝑻 and 𝜺 are the 2nd-order stress tensor and 2nd-order strain tensor respectively, 𝒖 is

the displacement of the infinitesimal. The relationship between the stress tensor rate and

the strain tensor rate is,

Equation 3-11

𝜕𝑻

𝜕𝑡= 𝑐 ∙

𝜕𝒖

𝜕𝑥𝜕𝑡= 𝑐 ∙

𝜕𝒗

𝜕𝑥

3.2.2 The Finite-difference time-domain (FDTD) Method

From Equation 3-6 and Equation 3-7, the parameters 𝒗𝒊 and 𝑻𝒊𝒋 are to be calculated; the

finite-element time-domain (FDTD) method is a reliable method to solve these partial

differential equations. The finite-element time-domain (FDTD) method employs partial

derivatives approximations, which including forward difference approximations, backward

difference approximations, and central difference approximations; in this work, central

difference approximations (as shown in Equation 3-12) are used.

Equation 3-12

𝜕𝑓(𝑎)

𝜕𝑎≈𝑓 (𝑎 +

∆𝑎2 ) − 𝑓 (𝑎 −

∆𝑎2 )

∆𝑎

Applying Equation 3-12 to Equation 3-6 and Equation 3-7, both the time derivatives 𝜕𝒗𝒊

𝜕t,

𝜕𝑻𝒊𝒋

𝜕t and the spatial derivatives

𝜕𝑻𝒊𝒋

𝜕𝑥𝑗, 𝜕𝒗𝑘

𝜕𝑥𝑙 can be approximated with a proper time step ∆𝑡 and

a proper spatial step ∆𝑥. The time step ∆𝑡 and the spatial step ∆𝑥 should be small enough

to maintain the calculation accuracy [72].


49

Typically, the spatial step ∆𝑥 should be as small as one tenth of the wavelength, 𝜆 10⁄ ; for

some simulations with a propagating distance over tens of wavelengths, the spatial step ∆𝑥

should be smaller as 𝜆 20⁄ [72]. The time step ∆𝑡 is determined by the Courant–Friedrichs–

Lewy (CFL) condition,

Equation 3-13

∆𝑡 ≤1

√𝑑∙∆𝑥

𝑣𝑚𝑎𝑥

where 𝑑 is the space dimension, e.g., 𝑑 = 2 for 2D simulations; 𝑣𝑚𝑎𝑥 is the maximum

velocity within the modelling material. The CFL condition is to choose the maximum time

step based on the spatial step, in other words, the maximum time step is limited by or

convergence to the minimum of spatial step [72, 73].

3.3 Ultrasonic Phased Array Modelling with FDTD

In this section, the finite-difference time-domain (FDTD) method is used to model

ultrasonic phased array techniques; this is a 2D simulation, and the modelling geometry is

shown in Figure 3-7 (a) and Figure 3-7 (b), where the test piece used is a steel plate. The

steel plate has a dimension of 100mm×100mm; a transducer with 8 elements is used; the

focal point is placed in the centre of the modelling geometry. Perfectly matched layer

(PML) conditions are applied to the surround of the modelling geometry to absorb

reflections. Detailed parameters are shown in Table 3-1; the pitch means the centre-to-

centre distance between two adjacent elements.

Figure 3-7: Modelling geometry for steering (a) and focusing (b).


50

Table 3-1: Parameters used for modelling steering and focusing

Test piece

(Steel plate)

Parameters

Length Height Density

100 mm 100 mm 7.8 g/cm3

Elements

Elements

number Height Length Pitch

8 0.1 mm 0.1 mm 4 mm

FDTD setup Spatial step Time step Frequency Maximum Velocity

0.1 mm 0.0119 𝜇s 2 MHz 5.9 mm/𝜇s

Boundary

conditions

Boundary type Thickness of the boundary

Perfectly matched layer (PML) 4 mm

The excitation source used is an explosive source, which is the same with the one used in

[8]. Various sources can be added, but the explosive source is easily modelled by adding

the same excitation source, such as Gaussian pulse, to T11 and T22 at the same source point.

The pulse used in this work is a Gaussian-modulated sinusoidal wave, which can be

expressed as [74],

Equation 3-14

𝑠(𝑡) = 𝐴𝑒−𝛽(𝑡−𝑡0)2∙ 𝑠𝑖𝑛 [2𝜋𝑓(𝑡 − 𝑡0)]

where 𝐴 is the amplitude, 𝑡 is the excitation duration, 𝑡0 determines the time delay of the

waveform, 𝛽 is the bandwidth factor, 𝑓 is the central frequency.

Gaussian-modulated sinusoidal wave has various advantages over the pure sine wave, 1),

better time and frequency localization, in other words, signal arriving times of the

Gaussian-modulated sinusoidal wave are more clearly in the time domain and the energy

is more concentrated on the central frequency in the frequency domain as shown in Figure

3-8 and Figure 3-9 [75]; 2), the Gaussian-modulated sinusoidal wave covers a range of

frequencies of interest, and the waveform can be adjusted by changing the central

frequency and the bandwidth; both of these properties provide a better simulation

environment.


51

Figure 3-8: Pure sine wave; (a) the time

domain signal of the pure sine wave; (b)

the magnitude of the pure sine wave’s

Fourier transform [75].

Figure 3-9: Gaussian-modulated sine wave;

(a) the time domain signal of the Gaussian-

modulated sine wave; (b) the magnitude of

the Gaussian-modulated sine wave’s

Fourier transform [75].

The firing time’s calculation of the steering technique has been described in section 3.1.1;

firing elements with prescribed calculated times, the wavefront can be controlled to steer at

an arbitrary angle, as shown in Figure 3-10. Because perfectly matched layer (PML)

conditions are applied to the boundary of the simulated geometry, no reflections are

generated from the boundary. From Figure 3-10, a transducer with multiple elements is

capable of sweeping beam without mechanically moving the transducer, so it enlarges the

inspecting area and improving the scanning time.


52

Figure 3-10: Steering techniques: firing elements at prescribed calculated times, the

wavefront is steered at 00, 300, 600, 900 respectively.

Figure 3-11 describes the focusing behaviour; after firing elements at prescribed times, the

wavefront is becoming smaller before arriving at the prescribed focal point and becoming

larger when beyond the focal point. Because focusing allows the beam shape to be

controlled and the acoustic energy to be concentrated on the expected defect location, it

further optimizes the detection capability.


53

Figure 3-11: Focusing techniques: the wavefront is focused at the prescribed focal point.

Both steering and focusing are valuable for ultrasonic inspections; in order to further study

the properties of steering and focusing, radiation pattern is introduced in section 3.4;

section 3.5 combines radiation pattern, near field and far field together to perform

quantitative analysis of beam properties.

3.4 Novel Radiation Pattern with Hilbert Transformation

Radiation pattern defines the strength or the power of the ultrasound waves generated by

the transducer; in other words, the strength of the velocity field or the strength of the stress

field within the test piece forms the radiation pattern. Radiation pattern is the foundation of

studying beam features and further facilitates the optimal design of the transducer. In this

work, a novel radiation pattern with Hilbert transformation is proposed.

3.4.1 Hilbert Transformation

Hilbert transformation, which can extend the real signal of time-domain into the analytical

signal of the same domain, is used to return the envelope of a time series signal, as shown

in Equation 3-15, Equation 3-16 and Equation 3-17.

Equation 3-15

ℎ(𝑡) = 𝑓(𝑡) ∗1

𝜋𝑡

Equation 3-16

𝑧(𝑡) = 𝑓(𝑡) + 𝑗ℎ(𝑡)

Equation 3-17

𝑒(𝑡) = √𝑓(𝑡)2 + ℎ(𝑡)2


54

where 𝑓(𝑡) is the original time series signal, ℎ(𝑡) is the signal with Hilbert transformation,

𝑧(𝑡) is the analytical signal, and 𝑒(𝑡) is the envelope of the analytical signal 𝑧(𝑡).

As described in section 3.1.2, the depth of the defect can be deducted from the travelling

time of echoes, so how to get the arriving time of echoes is paramount. The maximum

amplitude of the time series signal indicates the arrival time of the signal, as shown in

Figure 3-12(a). In order to further improve the arrival time identification, Hilbert

transformation is employed; the envelope of the signal, returned by Hilbert transformation,

results in the signal arrival times can be identified more clearly as shown in Figure 3-12(b).

It can be seen using the Hilbert transformation to return the analytical signal is similar to

the process of amplitude modulation.

Figure 3-12: Signals to indicate the arrival times of ultrasound waves.

3.4.2 Novel Radiation Pattern with the Hilbert Transformation

Traditional radiation pattern for ultrasonic phased arrays is obtained with combining the

analytical method and the numerical method, that is, the analytical method for the pressure

distribution of focusing and the numerical method for the pressure distribution of steering

[76, 77].

In this work, a wholly numerical model is built; the finite-difference time domain (FDTD)

method and the Hilbert Transformation are combined together to produce the radiation

pattern. The finite-difference time domain (FDTD) method is used to solve the

elastodynamic equations and generate the velocity fields or pressure fields; Hilbert

transformation is used to obtain the envelope fields. Taking the steering model used in

Figure 3-7 (a) as an example, the modelling geometry is 100 mm×100 mm and the spatial

step is 0.1 mm, hence there are 1001×1001 spatial points in this model. With FDTD


55

calculation, each spatial point on the modelling geometry has a time series signal related to

the velocity or the pressure. The maximum value of the envelope of each time series signal

forms a 1001×1001 envelope fields, which is referred as the radiation pattern in this work.

The model used to study radiation pattern is the one used in the steering modelling with a

steering angle 300 (Figure 3-10); by combining the finite-difference time-domain (FDTD)

method and Hilbert transformation, the radiation pattern of the steering with a steering

angle 300 is shown in Figure 3-13 (a). The radiation pattern indicates the strength of

ultrasound waves; from Figure 3-13 (a), the beam has the largest strength along this

prescribed steering angle 300.

Figure 3-13: (a), Radiation pattern for the beam steered at 300; (b), radiation pattern for

studying beam features.

In order to quantitatively analyse the beam features, beam directivity and field distribution

along the steering angle are introduced. Beam directivity is, at a specific distance from the

centre of the sensor, the velocity field distribution; as shown in Figure 3-13 (b), beam

directivity is the velocity distribution along the white arc. Field distribution along the

steering angle is the velocity distribution along the steering angle; in Figure 3-13 (b), field

distribution along the steering angle is the velocity distribution along the red line.

Actually, the beam features are strongly related to the near field and the far field [76, 77].

Next, the beam properties of steering and focusing behaviours in the near field and in the

far field are studied respectively in section 3.5.

3.5 Near Field and Far Field Modelling

The near field and the far field are two different regions of ultrasound waves transition,

which are defined based on Equation 3-18 [69, 78].

(a) (b)


56

Equation 3-18

𝑇 =𝐴2

4𝜆

where 𝐴 is the overall length of the ultrasonic array, and 𝜆 is the wavelength of the

ultrasound wave. The near field means the distance from the centre of the ultrasonic array

is smaller than T, while the far field is larger than 𝑇.

A model is built to study the focusing and the steering properties in the near field and in

the far field; detailed parameters are shown in Table 3-2. The operational frequency used is

2 MHz and the velocity within the steel plate is 5.9 mm/𝜇s, so the wavelength λ is close to

3 mm. The spatial step chose is 0.1 mm, which approximately equals to λ/30; The pitch

(centre-to-centre distance) of two adjacent elements equals to λ/2 in order to provide a

sharper main lobe [11]. Based on Equation 3-18, T is calculated, which approximately

equals to 43.3 mm; the near field is the zone with a radial distance from the centre of the

array smaller than 43.3 mm, while the far field is beyond that zone.

Table 3-2: Detailed parameters used for near and far fields modelling.

Test piece

(Steel plate)

Parameters

Length Height Density

200 mm 200 mm 7.8 g/cm3

Elements

Elements

number Height Length Pitch

16 0.1 mm 0.5 mm 1.5 mm

FDTD setup Spatial step Time step Frequency Maximum Velocity

0.1 mm 0.0119 𝜇s 2 MHz 5.9 mm/𝜇s

Boundary

conditions

Boundary type Thickness of the boundary

Perfectly matched layer (PML) 4 mm

3.5.1 Near Field Analysis

3.5.1.1 Focusing

For the focusing behaviour, by firing elements at different times, the beam can be focused

at a specific point, which is named the focal point; the distance between the centre of the

element and the focal point is called the focal length. There are two parameters related to

the focusing beam; as shown Figure 3-14, r denotes the focal length and θ denotes the


57

steering angle. By controlling the focal length to a specific value which is smaller than T

(Equation 3-18), the focusing behaviour in the near field can be analysed.

Figure 3-14: The description of the focal length and the steering angle.

In this model, the near field is the zone with a focal length smaller than 43.3 mm

(calculated from parameters listed in Table 3-2). Any focal points with a distance from the

centre of the elements array smaller than 43.3 mm can be chosen to study the focusing

behaviour in the near field; here a focal point with the focal length of 40 mm and the

steering angle of 300 is chosen; the radiation pattern of the focusing behaviour in the near

field is shown in Figure 3-15. From this image, the beam is focused to the specific focal

zone; the acoustic intensity of the focal zone is largest.

Figure 3-15: The radiation pattern of the focusing behaviour in the near field.


58

Beam features of the focusing behaviour within the near field are shown in Figure 16. In

Figure 3-16(a), the focusing behaviour shows a good directivity with a narrow main lobe;

the maximum magnitude occurs at the simulated steering angle 300, which is consistent

with the prescribed steering angle. Figure 3-16(b) shows the field distribution along the

steering angle 300; the maximum magnitude occurs at a focal length of 32.9 mm instead of

the prescribed focal length 40mm. This observation is consistent with the analytical results

from [69], who claims this phenomenon is due to the effect of diffraction.

Figure 3-16: Beam features of focusing within the near filed. (a), Beam directivity of the

focusing behaviour; (b) Field distribution along the steering angle of the focusing behaviour.

3.5.1.2 Steering

For the steering behaviour, by firing elements with different delays, the beam can be

steered to a specific angle. In order to compare beam features of steering with them of

focusing within the near field, the same steering angle, 300, is chosen to study the steering

behaviour. The radiation pattern of the steering behaviour with the steering angle 300 is

shown in Figure 3-17. From this image, the acoustical intensity is distributed along the

steering angle; it is hard to tell where the maximum acoustical intensity occurs.

(a) (b)


59

Figure 3-17: Radiation pattern of the steering behaviour with a steering angle of 300.

Any radial lengths smaller than 43.3 mm can be chosen to study the steering beam features

within the near field. The radial length used in the steering behaviour is the same as that

used in the focusing behaviour, 40 mm. The results of beam features of the steering

behaviour within the near field is shown in Figure 3-18.

The beam directivity of steering at the radial length of 40mm is shown in Figure 3-18(a);

the steering within the near field shows a bad directivity with a wide and irregular main

lobe, which means the acoustic intensity is distributed. In addition, the simulated steering

angle 270 is not consistent with the prescribed steering angle 300. In Figure 3-18(b), field

distribution of steering along the steering angle 300 within the near field is described; the

maximum acoustic intensity occurs at the radial length of 68.9 mm.

Figure 3-18: Beam features of steering within the near filed. (a), Beam directivity of the

steering behaviour; (b) Field distribution along the steering angle of the steering behaviour.

(a) (b)


60

3.5.1.3 Comparison and Discussion

Comparing the beam directivity between the focusing and the steering within the near field,

as shown in Figure 3-16(a) and Figure 3-18(a) respectively, the focusing behaviour within

the near field provides a better directivity over the steering behaviour within the near field;

the acoustic energy is more concentrated along the steering angle with the focusing

behaviour.

Comparing the field distribution between the focusing and the steering within the near field,

as shown in Figure 3-16(b) and Figure 3-18(b) respectively, the acoustic intensity of the

focusing is more concentrated than that of the steering. Due to the good beam directivity

and concentrated acoustic intensity in the specific area, focusing is normally employed for

near-field inspections.

One point should be noted is that the maximum acoustic intensity of the focusing does not

occur at the prescribed focal length, as shown in Figure 3-16(b); the simulated focal

length, 32.9 mm, is 17.75% ahead of the prescribed focal length 40mm. However, the

prescribed focal length shows the optimal beam directivity as shown in Figure 3-19: only

for the prescribed focal length of 40 mm, the simulated steering angle is consistent with the

prescribed steering angle. Away from this radial distance, the beam directivity has a wider

main lobe (Figure 3-19(a)).

(a) (b)


61

Figure 3-19: The beam directivity of the focusing behaviour at different radial lengths.

3.5.2 Far Field Analysis

3.5.2.1 Focusing

The model used for studying the focusing behaviour in the far field is the same with the

one used for the focusing behaviour within the near field, apart from one difference, that is,

the focal point is placed in the far field. Any focal points beyond the near field can be

chosen to study the focusing behaviour in the far field, here the focal point has a focal

depth of 150 mm and the steering angle is 300 is chosen. The radiation pattern of the

focusing behaviour in the far field is shown in Figure 3-20; the acoustic intensity is not

clearly focused on a specific zone in the far field.

Figure 3-20: Radiation pattern of the focusing behaviour in the far field.

(c) (d)


62

Figure 3-21 (a) depicts the beam directivity of the focusing behaviour in the far field; away

from the near field, focusing still shows a good directivity with a narrow main lobe; the

simulated steering angle is matched with the prescribed steering angle. However, the

maximum acoustic intensity occurs at a radial length of 55.8 mm instead of 150 mm as

shown in Figure 3-21 (b). It means the beam does not concentrate on the expected location

when the focal point is placed in the far field. Unlikely with the focusing technique in the

near field, the focusing technique in the far field lose the advantage of the concentrated

acoustic intensity.

Figure 3-21: Beam features of focusing in the far field: (a) beam directivity at a focal length

of 150 mm, (b) field distribution along the steering angle 300.

3.5.2.2 Steering

The beam features of the steering behaviour within the near field has been studied in

section 3.5.1.2; the model and the radiation pattern used to study the steering behaviour in

the far field are exact the same as ones used in the near field.

Any radial lengths larger than the near field can be picked to study the beam features of the

steering behaviour; in order to compare the steering and the focusing in the far field, the

same radial length, 150 mm, is chosen; Figure 3-22 depicts the beam features of steering in

the far field.

The beam directivity of the steering behaviour in the far field, as shown in Figure 3-22(a),

shows a better directivity than that in the near field; the acoustic intensity in the far field is

mainly along the prescribed steering angle 300. The simulated steering angle is matched

with the prescribed steering angle 300. Figure 3-22(b) shows the field distribution along the

steering angle, the maximum acoustic intensity occurs at a radial length of 68.9 mm.

(a) (b)


63

Figure 3-22: Beam features of steering in the far field: (a) beam directivity at a radial length

of 150 mm, (b) field distribution along the steering angle 300.

3.5.2.3 Comparison and Discussion

Comparing the beam directivity between the focusing and the steering in the far field, as

shown in Figure 3-21(a) and Figure 3-22(a) respectively, both the focusing and the steering

provide a good beam directivity with a narrow main lobe.

Comparing the field distribution along the steering angle, as shown in Figure 3-21(b) and

Figure 3-22(b) respectively, the focusing does not concentrated the beam to the expected

area, and the field distribution of the focusing in the far field is close to that of steering in

the far field; it indicates, in the far field, the field distribution of focusing is convergence to

that of steering.

Combining the beam directivity and field distribution along the steering angle, steering is

sufficiently reliable in the far field. In addition, because steering needs less modelling time,

such as less firing time calculation and less scanning points, the steering provides more

advantages than focusing in the far field [69]. Therefore, the steering technique is normally

used for far-field inspections.

3.5.3 Conclusions of Section 3.5

For near-field inspections, the focusing technique provides some attractive features over

the steering technique, such as a good beam directivity and a concentrated acoustic

intensity in the expected area; so focusing is normally employed for near-field inspections.

Especially, for applications that possible defect locations in the near field are known, the

focusing technique is a reliable method optimizing the detection capability. It is important

to note that the beam is focusing at a radial length slightly smaller than the focal length,


64

however, the beam directivity at the focal length is optimal among the beam directivities at

any radial lengths within the near field.

For the far-field inspection, both the focusing technique and the steering technique offer a

good beam directivity. However, for the field distribution along the steering angle, the

focusing technique does not focus the beam to the expected area, and is convergence to the

steering technique, which means the steering technique is sufficiently reliable in the far

field. The steering technique needs less modelling time than the focusing technique, which

further proves that the steering technique is an efficient and reliable method for the far-

field inspection.

3.6 Scattering Modelling

The purpose of quantitatively analysing beam features is to facilitate the optimum design

of such sensors, which are finally used in practical applications. Based on the study of

beam features, for a full-field inspection, the focusing is typically for the near-field

inspection and the steering is typically for the far-field inspection. In this section, the

scattering phenomenon of the focusing in the near field is studied.

The scattering geometry is shown in Figure 3-23; one 1 mm ×1 mm scatter is placed in the

near field to make the focal length, r, to 32.9 mm and the steering angle, θ, to 300. By

firing elements at prescribed times, the beam is controlled to focus at the scatter, which is

the reason to produce scattered waves; scattered waves can be captured by the receiving

array. The receiving array placed on the top of the test piece is symmetrical with the

transmitting array. The parameters used for the scattering modelling are the same with

those used for studying the focusing in the near field in section 3.5.1.1. Perfectly matched

layers (PML) are applied to the surround of the modelling geometry to model an

unbounded domain.


65

Figure 3-23: The geometry of scattering modelling.

The wave propagation at different times are shown in Figure 3-24; because the perfectly

matched layer is applied to the surround of the modelling geometry, there is no reflections

from boundaries. Before arriving at the scatter, the wavefront is becoming smaller, as

shown in Figure 3-24(a) and Figure 3-24(b). Ultrasound waves are scattered when

encounter the scatter, which produces the scattered longitudinal waves and the scattered

shear waves (Figure 3-24(c)). The velocity of the longitudinal wave is larger than that of

the shear wave, so longitudinal waves propagate ahead of shear waves; as shown in Figure

3-24(d), two circles are generated due to the scatter, the outside circle is the longitudinal

wave and the inside circle is the shear wave.

(a) (b)


66

Figure 3-24: Wave propagation of the scattering modelling at different times.

Both the scattered longitudinal waves and the scattered shear waves can be received by the

receiving array. Each element on this receiving array can capture the scattered signal; the

totally received signal is the sum of the scattered signal received by each element. The

receiving signal is shown in Figure 3-25; the first arrival signals are the directly transmitted

waves generated from the transmitted array (Figure 3-25(a)). The followed signals, as

shown in Figure 3-25(b) and Figure 3-25(c) respectively, are scattered longitudinal waves

and scattered shear waves.

Figure 3-25: The received signals from the receiving array. (a), directly transmitted signals;

(b), the scattered longitudinal waves; (c), the scattered shear waves.

3.7 Conclusions

Steering and focusing are two widely used phased array techniques; steering enlarges the

inspection area and focusing optimizes the detection capability. The FDTD solver,

SimSonic, was further developed / adapted by the author for simulating ultrasound waves’

behaviours; Gaussian-modulated sinusoidal wave, which provides a better simulation

(c) (d)

(a)

(b) (c)


67

environment, was used as the excitation source. The combination of the finite-difference

time-domain (FDTD) method and Hilbert transformation is capable of generating the

radiation pattern, which lays a substantial foundation for studying the beam features of

steering and focusing.

Based on the modelling results, a number of important conclusions can be drawn; the

focusing technique has some advantages in the near field, such as a concentrated acoustic

intensity and a good beam directivity; the steering technique has advantages in the far field,

such as less modelling time and a good beam directivity. For a full-field inspection, the

focusing technique is used for the near-field inspection while the steering technique is used

for the far-field inspection. Finally, the scattering phenomenon is simulated by the finite-

difference time-domain (FDTD) method. Both the directly transmitted signals and the

scattered signals can be captured by the receiving array.

All of the work included in this chapter verified the ability of FDTD to model ultrasound

waves’ propagation; this lays a foundation for the EMATs modelling, which contains

electromagnetic and ultrasonic models; the EMATs modelling based on FDTD and the

analytical method is introduced in Chapter 4.

CHAPTER4 Development and validation of a novel method for modelling meander-line-coil EMATs

operated on Lorentz force mechanism

68

Chapter 4 Development and Validation of A Novel Method

for Modelling Meander-line-coil EMATs Operated

on Lorentz Force Mechanism

In this chapter, a novel modelling method for simulating electromagnetic acoustic

transducers (EMATs) operated on Lorentz force mechanism to generate Rayleigh waves is

proposed. This novel modelling method, focusing on the vertical plane of the material,

combines the analytical method and the finite-difference time-domain (FDTD) method

together. Based on this proposed modelling method, several behaviours of Rayleigh waves,

including the bidirectional propagation (Section 4.3), the scattering behaviour (Section 4.7)

and the unidirectional propagation (Section 4.8), are studied. Related experiments were

conducted to validate the capability of this novel modelling method. The novel modelling

method presented in this chapter and related chapters haven been published in or accepted

by Ultrasonics, Journal of Sensors and International Journal of Applied Electromagnetics

and Mechanics [10, 13, 14].

4.1 Introduction

The simulation of EMATs operated on the Lorentz mechanism is normally divided into

two parts, the electromagnetic (EM) simulation and the ultrasonic (US) simulation. The

link between the EM simulation and the US simulation is the Lorentz force density. In this

work, the EM simulation is performed with an analytical method, which is adapted from

classic Dodd and Deeds solutions which were originally intended for the circular coil. The

US simulation is accomplished by the finite-difference time-domain (FDTD) method,

which has been described in Chapter 3.

4.1.1 Modelling Geometry

The test piece used is a stainless steel plate with a dimension of 600 mm×600 mm×25 mm,

the permanent magnet used is NdFeB35 whose size is 60 mm×60 mm×25 mm. The

conductivity and the permittivity of the stainless steel is 1.1×106 Siemens/m and 1.26×10−6

H/m respectively. The frequency used is 500 kHz, thus the skin depth calculated is 0.679

mm. The velocity of the longitudinal waves, the shear waves and the Rayleigh waves

within the stainless steel is 5.9 mm/µs, 3.25 mm/µs and 3.033 mm/µs respectively [39].

The coil used is a meander-line-coil carrying an alternating current (Figure 4-1); the



69

effective length of the meander-line-coil (L) is 40 mm; the width of each wire of the

meander-line-coil (w) is 0.4 mm; the height of the meander-line-coil is 0.035 mm, and the

centre-to-centre distance between two adjacent wires of the meander-line-coil (s) is 3.033

mm, which is half of the Rayleigh waves’ wavelength to achieve the constructive

interference.

Figure 4-1: The configuration of a typical meander-line-coil EMAT.

4.2 EMAT-EM Modelling

The electromagnetic simulation is accomplished with an analytical method adapted from

the classic Dodd and Deeds solution, which was originally used for circular coils [79].

Firstly, the strategy of adapting the analytical solution for a straight wire is proposed.

Finite element method (FEM) is employed to validate the adapted analytical solution. Next,

the analytical EM simulation for the meander-line-coil EMAT is described; the distribution

of Lorentz force density is obtained.

4.2.1 Classic Dodd and Deeds Solutions

The Dodd and Deeds solution for the eddy current phenomenon is employed in this work,

because the analytical solution proposed by Dodd and Deeds has a high accuracy and a

minimum model difference between the model built by Dodd and Deeds and the author.

The eddy current calculation is governed by Equation 4-1, Equation 4-2 and Equation 4-3

[79],

Equation 4-1

𝛻2𝑨 = −𝜇𝑰 + 𝜇𝜎𝜕𝑨

𝜕𝑡+ 𝜇𝜀

𝜕2𝑨

𝜕𝑡2+ 𝜇𝛻(

1

𝜇) × (𝛻 × 𝑨)



70

Equation 4-2

𝑬 = −𝑗𝜔𝑨

Equation 4-3

𝑱 = 𝜎𝑬

where 𝑨 is the vector potential generated by the coil carrying a current density 𝑰; µ, σ, and

ε are the permeability, the conductivity and the permittivity of the test piece respectively. 𝑬

is the induced electric field, and 𝑱 is the induced eddy current density.

Based on Equation 4-1, Equation 4-2 and Equation 4-3, for eddy current calculation, the

fundamental step is to calculate the vector potential. Dodd and Deeds built a model with a

rectangular cross-sectional coil placed above a two-layered conductor with an infinite

length (Figure 4-2); the length “R” as shown in Figure 4-2 is the length of the conductor to

be calculated and shown; the model is axial symmetrical along the 𝑥 axis and the direction

of the AC current is anti-clockwise. Equation 4-1 is a full wave equation; with the

quasistatic approximation, the final solution to the vector potential within Layer 1 is shown

in Equation 4-4, Equation 4-5 and Equation 4-6.

Figure 4-2: A model built by Dodd and Deeds [79].

Equation 4-4

𝑨(𝑟, 𝑥) = 𝜇0𝑰∫1

𝑎3(∫ 𝛾𝐽1(𝛾)𝑑𝛾) 𝐽1(𝑎𝑟)

𝑎𝑟2

𝑎𝑟1

(𝑒−𝑎𝑙∞

0

− 𝑒−𝑎(𝑙+ℎ))𝑎(𝑎1 + 𝑎2)𝑒

(2𝑎1𝐻1)𝑒(2𝑎1𝑥) + 𝑎(𝑎1 − 𝑎2)𝑒(−𝑎1𝑥)

(𝑎 − 𝑎1)(𝑎1 − 𝑎2) + (𝑎 + 𝑎1)(𝑎1 + 𝑎2)𝑒(2𝑎1𝐻1)𝑑𝑎



71

Equation 4-5

𝑎1 = √𝑎2 + 𝑗𝜔𝜇1𝜎1

Equation 4-6

𝑎2 = √𝑎2 + 𝑗𝜔𝜇2𝜎2

Where µ0, µ1, and µ2 are the permeability of air, Layer 1 and Layer 2 respectively; σ1 and

σ2 are the conductivity of Layer 1 and Layer 2 respectively. Assuming the conductor only

has one layer, that is, the material property of Layer 1 and Layer 2 are exactly the same

(Figure 4-3); the final solution of the vector potential within the conductor with only one

layer can be simplified, as shown in Equation 4-7 and Equation 4-8.

Figure 4-3: Geometry for the conductor with only one layer.

Equation 4-7

𝐀(𝑟, 𝑥) = 𝜇0𝐈 ∫1

𝑎2(∫ 𝛾𝐽1(𝛾)𝑑𝛾) 𝐽1(𝑎𝑟)

𝑎𝑟2

𝑎𝑟1

(𝑒−𝑎𝑙 − 𝑒−𝑎(𝑙+ℎ))𝑒(2𝑎1𝑥)

(𝑎 + 𝑎1)𝑑𝑎

∞

0

Equation 4-8

𝑎1 = √𝑎2 + 𝑗𝜔𝜇1𝜎1

where 𝐀(𝑟, 𝑥) is the vector potential within the conductor, 𝜇0 is the permeability of air, 𝐈 is

the applied current density, 𝛾 and 𝑎 are the integration variables, 𝑟1 and 𝑟2 are the inside

radius and outside radius of the coil respectively, 𝐽1(𝛾) and 𝐽1(𝑎𝑟) are the first kind Bessel

functions, 𝑙 is the lift-off, ℎ is the height of the coil, 𝜇1 and 𝜎1 are the permeability and

conductivity of the conductor respectively, and 𝜔 is the angular frequency.



72

A model is built to study the vector potential distribution on the basis of the analytical

solutions proposed by Dodd and Deeds; detailed parameters used for this modelling are

shown in Table 4-1. The test piece used is a stainless steel plate with a circular coil placed

above the test piece; the frequency used is 10 kHz. The software used to build this model is

Matlab; for the integration, a limit inferior of 1×10-9 and a limit superior of 5000 are used;

the computer used for computation is with a Random Access Memory of 8 GB and with a

dual Intel Core i5-3570 with a speed of 3.4 GHz; the computation time for this model is

roughly 20 minutes.

Table 4-1: Detailed parameters used for studying analytical solutions proposed by Dodd and

Deeds.

The distribution of the magnitude of the vector potential within the conductor is shown in

Figure 4-4, where the vector potential is concentrated on the near surface of the conductor

and under the circular coil; the unit of the vector potential is Tm. The skin depth calculated

is 4.8 mm, beyond of which the vector potential significantly decreases.

Figure 4-4: For a circular coil, the distribution of the magnitude of the vector potential

within the conductor.

Test piece

(Stainless steel

plate)

Parameters

Length (R) Height (H) Permeability (𝝁𝟏) Conductivity (𝝈𝟏)

5 mm 10 mm 1.26×10−6 H/m 1.1×106 Siemens/m

Coil used

(Circular coil)

Parameters

Inside Radius (𝒓𝟏) Outside Radius (𝒓𝟐) Mean Radius

(𝒓𝟏 + 𝒓𝟐)/2 Applied Current Density(I)

2.45 mm 2.55 mm 2.5 mm 1 A/m2

Frequency Lift-off Height (h) Air permeability (μ0)

10 kHz 1 mm 1 mm 1.257×10−6 H/m



73

Figure 4-5 shows the distribution of the magnitude of the vector potential along the surface

of the conductor (𝑥=0); the green marker in this curve states that the maximum magnitude

occurs at where the circular coil is located. The curve of the vector potential distribution is

not symmetrical with the radius because the wire of a circular coil is bent instead of

straight.

Figure 4-5: For a circular coil, the vector potential distribution along the surface of the

conductor (𝒙=0).

4.2.2 Adapted Analytical Solutions for A Straight Wire

The coil used for the EMAT system is a meander-line-coil with straight wires, so the

analytical solution for a straight wire is needed. On the basis of the calculation results from

section 4.2.1, we propose a hypothesis, that is, the analytical solution for a straight wire

may be obtained by expanding the radius of the circular coil to very large. In other words,

the bent wire may be approximated to a straight wire when the radius of the circular coil is

very large, and the distribution of the vector potential should be symmetrical with the

radius.

To verify this hypothesis, an analytical model is built with the same parameters used in

Table 4-1, except that the mean radius is set to 20.05 m and the length of the stainless steel

plate is 20.1 m. With a large-radius circular coil placed above the conductor, the

distribution of the magnitude of the vector potential within the conductor is shown in

Figure 4-6; it is clearly that the vector potential is mainly concentrated within the skin

depth, which is 4.8 mm in this model; beyond the skin depth, the vector potential is very

small.



74

Figure 4-6: For a large-radius circular coil, the vector potential distribution within the

conductor.

Figure 4-7 shows, for a large-radius circular coil, the distribution of the magnitude of the

vector potential along the surface of the conductor (𝑥=0). From this image, the vector

potential is symmetrical with the radius r=20.05 m, where the straight wire is located. This

verifies the hypothesis that, when the radius of the circular coil is very large, the bent wire

of the circular coil can be approximated to a straight wire, and the distribution of the

magnitude of the vector potential is symmetrical with the radius.

Figure 4-7: For a large-radius circular coil, the vector potential along the surface of the

conductor (𝒙=0).

4.2.3 Validation and Comparison with FEM

In order to analyse the accuracy of the adapted analytical solutions, the finite element

method (FEM) is employed to compare the results between the analytical and the

numerical methods. Maxwell Ansoft based on the finite element method (FEM) is used to

solve the vector potential; the model built with Maxwell Ansoft is the same as the one built

in section 4.2.2. This is a 2D model with a rectangular cross-sectional coil located above



75

the cross-sectional stainless steel plate as shown in Figure 4-8(a); the vacuum region to be

solved is four times as larger as the stainless steel plate. The finite element method (FEM)

subdivides the large model to smaller elements, as shown in Figure 4-8(b); the mesh

quality determines the accuracy of the computed results. However, the very fine mesh /

high mesh density is limited by the capacity of the computer and requires more running

times. The computation of the FEM solver is based on minimising the energy error; in

Figure 4-9, when the elements number is beyond 20000, the energy error is as low as

0.05%, which is sufficiently accurate for the FEM computation. In this work, the mesh

number used is 20395. In section 4.2.3.1 and section 4.2.3.2, the comparison between the

analytical method and the numerical method at a low frequency and a high frequency are

described respectively.

Figure 4-8: (a), the model built with Maxwell Ansoft; (b), mesh of the model.

Figure 4-9: In FEM solver, the energy error versus the number of triangles.

Coil

Stainless steel

plate

Vacuum

(a)

(b)



76

4.2.3.1 Comparison with FEM at 10 kHz

At a frequency of 10 kHz, the magnitude of the vector potential within the conductor

(stainless steel plate) from the analytical method and from the numerical method are shown

in Figure 4-10(a) and Figure 4-10(b) respectively; the analytical and numerical methods

show a good agreement about the peak magnitude, which is close to 3.5 × 10−7 Tm. From

these images, the distribution of the vector potential based on the adapted analytical

method is smoother than that based on the finite element method.

Figure 4-10: At 10 kHz, the vector potential distribution within the stainless steel plate. (a),

the analytical method; (b) the finite element method (FEM).

The vector potential along the surface of the stainless steel plate is shown in Figure 4-11

(𝑥=0); where the distribution of the magnitude, the real part and the imaginary part of the

vector potential are shown in Figure 4-11(a), Figure 4-11(b) and Figure 4-11(c)

respectively. From these images, at 10 kHz, the adapted analytical method has a good

agreement with the finite element method (FEM). One point should be noted is, the

distribution of the imaginary part of the vector potential based on the analytical method is

more accurate than FEM because the vector potential from the FEM does not approach to

zero when it is far away from the wire.

(a)

(b)

20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 21

y (m)

0

-5

x (

mm

)



77

Figure 4-11: At 10 kHz, the vector potential along the surface of the stainless steel plate. (a),

(b) and (c) denotes the magnitude, the real part, and the imaginary part of the vector

potential respectively.

4.2.3.2 Comparison with FEM at 1 MHz

In this part, the comparison between the adapted analytical method and FEM at a higher

frequency is studied; this is because EMAT normally operates at high frequencies. The

geometry used for modelling has been described in section 4.2.2; the operational frequency

used is 1 MHz. The skin depth calculated is 0.48mm, which means the vector potential

should be concentrated on the region very close to the surface of the material.

The magnitude of the vector potential within the cross-section of the stainless steel plate is

shown in Figure 4-12; as expected, the vector potential is concentrated on the surface of

the stainless steel plate due to the high frequency used. From Figure 4-12(a) and Figure

4-12(b), the range of the vector potential magnitude shows a good agreement between the

adapted method and FEM; the range of the vector potential is from 0 to 8 × 10−8 Tm.

(a) (b)

(c)



78

Figure 4-12: At 1 MHz, the vector potential distribution within the stainless steel plate. (a),

the analytical method; (b) the finite element method (FEM).

Figure 4-13: At 1 MHz, the vector potential along the surface of the stainless steel plate. (a),

(b) and (c) denotes the magnitude, the real part, and the imaginary part of the vector

potential respectively.

The vector potential along the surface of the stainless steel plate (𝑥=0) is shown in Figure

4-13, where the magnitude, the real part and the imaginary part of the vector potential are

shown in Figure 4-13(a), Figure 4-13(b) and Figure 4-13(c) respectively. From these

(a)

(b)

(a) (b)

(c)

20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 21

y (m)

0

-5

x (

mm

)



79

images, the curve of the real part of the vector potential based on FEM is not smooth, that

is due to the inevitable errors produced by the numerical approximation.

4.2.3.3 Comparison at Different Lift-offs

In practical applications, the lift-off is a critical parameter that determines the

electromagnetic induction efficiency; this section compares the numerical results and the

analytical results when the sensor is placed at various lift-offs.

Figure 4-14: With various lift-offs, the distribution of the real part of the vector potential

along the surface of the stainless steel plate.

The model used is the same as the one used in section 4.2.3.2, the operational frequency is

1 MHz. The lift-off varies from 2 mm to 6 mm with a step of 2 mm; the distribution of the

real part of the vector potential based on FEM and the adapted analytical method is shown

in Figure 4-14. From these images, the adapted analytical method exhibits smoothness both

at a small lift-off (2 mm) and at a high lift-off (6 mm). On the other hand, the finite

element method (FEM) does not guarantee the calculation accuracy since the curve of the

vector potential becomes increasingly distorted with the lift-off increasing. Results show a



80

benefit of the adapted analytical solution over FEM, that is, the adapted analytical method

guarantees the accuracy even if the lift-off is large.

4.2.3.4 Conclusions of Section 4.2.3

In this study, the coil used for EMATs is a meander-line-coil contains several straight

wires, thus the electromagnetic induction solution to the straight wire is needed. Classic

Dodd and Deeds solution, which was originally for the circular coil, is adapted for the

straight wire by setting the radius of the circular coil to a large value, and results show that

the bent wire can be approximated to a straight wire. The adapted analytical solution for

the straight wire is numerically verified by the finite element method (FEM). The results

show that the adapted analytical solution is desirable for the electromagnetic induction

phenomenon calculation over the finite element method (FEM).

4.2.4 Analytical EMAT-EM Modelling

The EMAT-EM modelling is divided into two parts, one part is to calculate eddy currents

produced by the meander-line-coil and the other part is to calculate the static magnetic

field generated by the permanent magnet. The eddy current is calculated analytically with

the adapted analytical method as described in section 4.2.2; the static magnetic field is

modelled with the finite element method (FEM).

4.2.4.1 Eddy Currents Calculation

As described previously, the analytical solution to the vector potential with a straight wire

located above the stainless steel plate has been studied. For an EMAT transducer

containing a meander-line-coil, the total vector potential is the sum of the vector potential

generated by each straight wire segments in the meander coil. The model is shown in

Figure 4-15; 12 straight wire segments are placed above the stainless steel plate. Detailed

parameters for the EMAT system have been introduced in section 4.1.1; the frequency

used is 500 kHz. The meander coil has a dimension of 33.763 mm×40 mm×0.035 mm,

which is very small compared to the dimension of the stainless steel plate 600 mm×600

mm×25 mm. In order to reduce the modelling dimension and improve the modelling time,

only the 2D area 100 mm×3 mm, where the meander-line-coil has a major effect on, is

chosen to perform the EM modelling; the lift-off of the meander coil is 0.1 mm. Because

the length and the width of the straight wire used in this work is 40 mm and 0.2 mm



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respectively, the ratio of the length to the width is 200, which means the edge effect of the

finite length straight wire can be neglected.

Figure 4-15: 2D model of the EMAT-EM simulation.

By summing all the vector potential generated by each wire segment, the total vector

potential within the x-y plane of the stainless steel plate is shown in Figure 4-16; the unit of

the vector potential is Tm. Due to the AC directions between two adjacent wires of the

meander-line-coil are opposite, the vector potential produced by two adjacent wires are

opposite as well. The frequency used is 500 kHz, which results in a skin depth of 0.679

mm, thus the vector potential is mainly concentrated on the depth smaller than 0.679 mm

as shown in Figure 4-16.

Figure 4-16: The real part of the vector potential produced by a meander-line-coil.

The eddy current distribution can be obtained based on the vector potential calculation

(Equation 4-2 and Equation 4-3); the distribution of the real part of the eddy current

generated by the meander-line-coil is shown in Figure 4-17. The eddy current is

concentrated on the subsurface of the stainless steel plate; the unit of the induced eddy

current density is ampere per square meter. From Figure 4-17, the maximum amplitude of

the eddy current occurs at the leftmost side and the rightmost side of the meander-line-coil;

this is because that the outmost wires are only affected by the fields on one side.

Figure 4-18 shows the eddy current distribution along the surface of the stainless steel plate

(x=0); this curve further confirms the observation that the value of the eddy current under

the outmost wires are largest.

Stainless steel plate

Straight wire segments



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Figure 4-17: The real part of the induced eddy current produced by a meander-line-coil.

Figure 4-18: The eddy current distribution along the surface of the stainless steel plate (x=0).

4.2.4.2 Static Magnetic Field Calculation

The other part of the EMAT-EM modelling is to calculate the static magnetic field

generated by the permanent magnet. The permanent magnet located above the stainless

steel plate is modelled with the finite element method (FEM). The permanent magnet used

is made of NdFeB35 with a relative permeability of 1.1 and a magnetic coercivity of -

890000 ampere per meter. The dimension of the permanent magnet is 60 mm×60 mm×25

mm which is slightly smaller compared to the dimension of the stainless steel plate, 600

mm×600 mm×25 mm; the region of the stainless steel plate, 100 mm×100 mm×3 mm,

where the permanent magnet has a major effect on, is picked to study the behaviour of the

permanent magnet. In this section, only the 2D behaviour of the permanent magnet is taken

into account; the modelling geometry of the stainless steel plate is 100 mm×3 mm. The

vacuum region to be solved is two times as larger as the stainless steel plate; the mesh for

this magnetostatic modelling is shown in Figure 4-19. As mentioned before, the

computation of the FEM solver is based on minimising the energy error; as shown in

Figure 4-20, the energy error is as low as 0.1% when the elements number is beyond 2000.

In this work, the mesh number used is 3599, which is sufficiently accurate for the FEM

calculation.



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Figure 4-19: The mesh of the static magnetic field modelling.

Figure 4-20: The relationship between the elements number and the energy error for the

static magnetic field modelling.

Figure 4-21 illustrates the vector of the magnetic flux density; from this figure, the

magnetic flux density is generated from the permanent magnet and is mainly normal to the

stainless steel plate. Under the edge of the permanent magnet, the magnetic flux density

goes obliquely even parallel to the surface of the stainless steel plate due to the edge effect.

Figure 4-21: The vector of the magnetic flux density generated by the permanent magnet.

The distribution of the magnitude of the magnetic flux density within the stainless steel

plate is shown in Figure 4-22, where the magnetic flux density is mainly concentrated

under the permanent magnet and arrives to the largest under the edge of the permanent

magnet. The magnetic flux density decays with the depth of the material increasing.

NdFeB35

Vaccum



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Figure 4-22: The distribution of the magnitude of the magnetic flux density within the

stainless steel plate.

Because the static magnetic flux density is mainly distributed along the surface of the

stainless steel plate, the analysis of the static magnetic flux density along the surface of the

specimen is needed. Figure 4-23 presents the magnetic flux density along the surface of the

specimen (x=0); the blue curve and the red curve denote the x components and y

components of the magnetic flux density respectively. Because the permanent magnet has

a length of 60 mm, this curve further states that the largest magnetic flux density occurs

under the edge of the permanent magnet.

Figure 4-23: L The distribution of the magnetic flux density along the surface of the stainless

steel plate (x=0).

4.2.4.3 Lorentz Force Density

For the EMATs operated on the Lorentz force mechanism, the Lorentz force density is

obtained based on the calculation of the static magnetic field and the induced eddy current.

The induced eddy current and the static magnetic field are obtained in section 4.2.4.1 and

section 4.2.4.2 respectively; in this section, the distribution of the Lorentz force density is

presented.



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Based on Equation 2-1, the distribution of the Lorentz force density is obtained; Figure

4-24 states the Lorentz force density distribution along the surface of the stainless steel

plate; the unit of the Lorentz force density is Newton per cubic meter. Eddy currents under

any two adjacent wires of this meander-line-coil have opposite directions and the static

magnetic field are mainly normal to the surface of the material; that leads to the directions

of the Lorentz force density between two adjacent wires are opposite.

Because there are 6 pairs of adjacent wires with opposite AC directions among the used

meander-line-coil, there are 6 crests and 6 troughs in the curve of the Lorentz force density.

In addition, the value of the Lorentz force density under the outmost of this EMAT sensor

is largest.

Figure 4-24: The distribution of the Lorentz force density along the surface of the stainless

steel plate.

4.3 Novel Methods for EMATs

In this section, a novel modelling method combining the analytical method and the finite-

difference time-domain (FDTD) method is proposed. The adapted analytical method, as

described in section 4.2, is used to perform the EMAT-EM simulation, which aims to

obtain the Lorentz force density. The finite-difference time-domain (FDTD) method, as

described in Chapter 3, is used to model the wave propagation driven by a force source,

which is the Lorentz force density in this work.

4.3.1 The Combination of EM and US Models

The combination of the EMAT-EM model and the EMAT-US model is shown in Figure

4-25, the 12 peak values of the Lorentz force densities calculated from the EM modelling



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are imported into the US model as the driving source; these 12 driving sources are located

at the surface of the stainless steel plate. The top of the stainless steel plate is set to the free

surface to model a complete real circumstance; the perfectly matched layers (PMLs) are

applied to the left, right, and the bottom boundaries of the stainless steel plate to model an

unbounded domain. In addition, please note that, for a complete model, the EMATs result

in the volume force for ultrasound generation; in this work, we use an approximated model

with only the surface point source to generate Rayleigh waves, and this treatment will be

validated with experiments later.

Only the 2D behaviour in the vertical plane (i.e. x-y plane) of the ultrasound waves is taken

into account in this chapter; the dimension of the x-y plane of the stainless steel plate is 600

mm ×25 mm. In order to reduce the modelling dimension and improve the modelling time,

only the area 400 mm ×25 mm, which the EMAT sensor has a main effect on, is picked for

the ultrasonic modelling. In addition, in order to clearly identify all of the wave modes

generated by such EMAT sensors, the modelling depth is set to 80 mm; so the final

modelling geometry has a dimension of 400 mm ×80 mm (as shown in Figure 4-25). It

should be noted that, the effect of the dynamic magnetic field is neglected due to the small

excitation AC current used in this work; the reason for neglecting the dynamic magnetic

field will be detailed with experiments as described in section 4.6.3.

Figure 4-25: The combination between the EM model and the US model.

4.3.1.1 Excitation Signals

Regarding the FDTD setup in the ultrasonic model, the spatial step is 0.2 mm, which

approximately equals to one thirtieth of the Rayleigh waves’ length. The time step

calculated based on the Courant–Friedrichs–Lewy (CFL) condition is 0.024 µs. Because

the Lorentz force density is calculated in the frequency domain and the FDTD method is a



87

the time domain solver, the excitation signal for EMAT-US modelling is a time sequence

signal with the peak equalling to the value of the crests or the troughs of the curve of the

Lorentz force density distribution.

As described in Chapter 3, Gaussian-modulated sinusoidal wave is a desirable excitation

signal in the time domain. In this section, the excitation signal used is Gaussian-modulated

sinusoidal wave with a central frequency of 500 kHz, which is the same as the frequency

used in the EMAT-EM modelling; the central time of the pulse is 8 µs, and the fractional

bandwidth of the excitation signal is 0.3 (the effect of the fractional bandwidth will be

detailed in section 4.3.4); the fractional bandwidth is the passband frequency range divided

by its central frequency. Numbering the wire of the meander-line-coil from the left to the

right in order as wire 1, wire 2… wire 12; the excitation signal for wire 1 and wire 2 are

shown in Figure 4-26. The peak of the excitation signal for wire 1 equals to first crest of

the curve of the Lorentz force density; accordingly, the peak of the excitation signal for

wire 2 equals to the first trough of the curve of the Lorentz force density. Due to the

Lorentz force generated by two adjacent wires have opposite directions, the excitation

signal for two adjacent wires have opposite directions as well, as shown in Figure 4-26.

The excitation signal for other wires is applied in the same manner.

Figure 4-26: The excitation signal for wire 1 and wire 2.

4.3.2 The Propagation of Rayleigh Waves

With the FDTD calculation, the velocity field of ultrasound waves is obtained. The wave

propagation at 18 µs after firing is shown in Figure 4-27(a); four wave modes, including

longitudinal (L) waves, shear (S) waves, head waves and surface waves, can be identified



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clearly. The head wave connects the longitudinal wave and the shear wave together.

Longitudinal waves and shear waves obliquely propagate within the material; the shear

waves, more specifically, the shear vertical waves, have a larger intensity than the

longitudinal waves. The velocity of longitudinal waves is larger than that of shear waves

within the same medium, and therefore longitudinal waves arrive earlier than shear waves.

Surface waves, which are Rayleigh waves in this work, propagate along the surface of the

material. Due to the nature of the long propagation distance and low attenuation, Rayleigh

waves are increasingly used in industries for surface and subsurface detections. Normally

the velocity of the Rayleigh waves is 90 percent of that of the shear waves, so Rayleigh

waves travels slightly slower than shear waves; this phenomenon can be identified clearly

in Figure 4-27(b). Because the perfectly-matched layer (PML) is applied to the left, the

right and the bottom of the material, no waves are reflected from the boundary.

Figure 4-27: The wave propagation at 18 µs and 35 µs after firing respectively.

4.3.3 Displacement Calculation and Depth Profile

Two virtual receivers are placed within the material to inspect the arrival ultrasound waves.

Taking Figure 4-27(a) as a background, one receiver, defined as R1, is placed on x=79 mm

and y=50 mm; the other receiver, defined as R2, is place on x=75 mm and y=50 mm. The

received velocity field from the receiver R1 and R2 are shown in Figure 4-28.

(a)

(b)

L waves SV waves

Rayleigh waves

Head waves

Rayleigh waves



89

Figure 4-28: The received signals; (a), signals received by the receiver R1; (b), signals

received by the receiver R2.

For the stainless steel plate, the velocity of the longitudinal wave, the shear wave and the

Rayleigh wave is 5.9 mm/µs, 3.25 mm/µs and 3.033 mm/µs respectively. Since the

velocity of the longitudinal wave is largest, the 1st arrival signal is the longitudinal wave,

as shown in Figure 4-28(a). The longitudinal wave arrives at around 25.4 µs, so the

calculated flight distance of the longitudinal wave is 149.8 mm, which is very close to the

distance between the centre of the transducer and the receiver R1. The 2nd arrival signal is

the Rayleigh wave, which arrives at around 49.8 µs; so the flight distance of the Rayleigh

wave is 151.04 mm, which is very close to the distance between the centre of the

transmitter and R1.

Comparing the received signals from R1 and from R2, as shown in Figure 4-28(a) and

Figure 4-28(b) respectively, the amplitude of the received signal from R1 is larger than that

from R2 due to the attenuation of ultrasound waves; that means the Rayleigh wave decays

with the depth.

Figure 4-29 shows the received signals by receivers located at various depths. The

amplitude of the displacement received is normalised; this figure indicates that the

amplitude of the Rayleigh wave is larger when the receiver is close to the free surface (the

top boundary) of the stainless steel plate. When the depth is larger than the wavelength of

the Rayleigh wave, the amplitude of the Rayleigh wave is very small. More specifically,

the amplitude of the Rayleigh wave at a depth equalling to the Rayleigh waves’

(a)

(b)

Longitudinal waves

Longitudinal waves

Rayleigh

waves

Rayleigh

waves



90

wavelength, 6 mm, is 32.5 percent of that at a depth of 1 mm. At the depth 8 mm, 10 mm,

and 14 mm, the amplitude decreases to 15.63 percent, 13.43 percent, and 10.41 percent of

that at the depth of 1 mm respectively. This observation further confirms that the Rayleigh

waves are concentrated on the surface of the material within one wavelength of Rayleigh

waves.

Figure 4-29: The depth profile of Rayleigh waves’ displacement.

4.3.4 The Effect of the Fractional Bandwidth

As described in section 4.3.1.1, the excitation signal used is a Gaussian-modulated

sinusoidal wave with a central frequency of 500 kHz and a fractional bandwidth of 0.3; the

value of the fractional bandwidth has a significant impact on the interference behaviour of

the Rayleigh waves. In this section, the effect of the fractional bandwidth on the waves’

interference is studied.

Figure 4-30 illustrates the excitation signals with various fractional bandwidths, including

1.5, 1.0, 0.5 and 0.3; both of these excitation signals have a central frequency of 500 kHz.

The fractional bandwidth is the passband frequency range divided by its central frequency,

which means, with a small bandwidth, the signal energy is mainly concentrated around the

central frequency. However, with a large bandwidth, the signal energy is more distributed;

in other words, the proportion of other frequency components is large.



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Figure 4-30: The excitation signal with various fractional bandwidths.

By exciting signals at various fractional bandwidths, the received signals from the receiver

R1 are shown in Figure 4-31: with a smaller bandwidth, the interference of Rayleigh waves

is better. That is because, with a smaller bandwidth, the major frequency components are

concentrated on the central frequency and the other frequency components are very small,

so the interference effect is stronger. On the other hand, with a wider bandwidth, there are

many other frequency components, which have a damping effect on the interference of

Rayleigh waves.



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Figure 4-31: The received signals with excitation signals at various fractional bandwidths.

4.4 The Property of Rayleigh Waves

As described in the section 4.3, the novel modelling method, which combines the

analytical method and the finite-difference time-domain (FDTD) method, can generate the

ultrasound waves’ velocity fields. In this section, the radiation pattern of Rayleigh waves is

described and their beam features are analysed.

4.4.1 Radiation Pattern

As described in Chapter 3, radiation pattern is useful for the analysis of beam features of

ultrasound waves. The radiation pattern of Rayleigh waves generated by the meander-line-

coil EMAT is shown in Figure 4-32(a); the colour bar denotes the normalised intensity of

the ultrasound waves. From this image, both the surface waves and the body waves can be

identified; the body waves include the longitudinal waves and the shear waves; the

intensity of surface waves and shear waves are larger than that of the longitudinal waves.

From Figure 4-32(a), Rayleigh waves are mainly distributed along the surface and sub-

surface of the material; longitudinal and shear waves propagate obliquely into the material.

In this section, only Rayleigh waves are of interest; the beam features of Rayleigh waves

are studied by means of Figure 4-32(b).

L waves

wavesw

aves SV waves

waves

Rayleigh waves

(a)



93

Figure 4-32: (a), the radiation pattern of the EMAT-Rayleigh waves; (b), the radiation

pattern used for the analysis of beam features.

There are two beam features, beam directivity and field distribution along the steering

angle, to be analysed. In Figure 4-32(b), r denotes the radial distance from the centre of the

sensor, 𝜃 denotes the steering angle; define the angle parallel to the surface is 00; in order

to minimise the effects of the longitudinal waves and shear waves, keep the radial distance

r to 140 mm and the steering angle 𝜃 from 00 to 200.

4.4.2 Beam Features

The beam directivity calculated is shown in Figure 4-33, which shows the relationship

between the steering angle and the magnitude of the velocity fields; the magnitude of the

velocity fields is normalised. From this figure, Rayleigh waves are mainly concentrated at

angles from 00 to 2.50, and the peak occurs at the angle 00; when the steering angle is

beyond the range from 00 to 2.50, the magnitude of Rayleigh waves decays to a level lower

than 20 percent of the peak value; this further verifies the Rayleigh waves are concentrated

on the surface and the subsurface of the material.

Because the radial distance r used is 140 mm and the steering angle range of Rayleigh

waves is from 00 to 2.50, Rayleigh waves are concentrated on a depth calculated by

Equation 4-9.

Equation 4-9

𝑑 = 𝑡𝑎𝑛(𝜃) ∗ 𝑟

The depth d calculated is 6.1 mm, and the wavelength of the Rayleigh waves used in this

work is 6.066 mm, which indicates that the Rayleigh waves are mainly distributed within

one wavelength of the Rayleigh waves. In addition, when choose another value of the

radial distance r, the angle 𝜃 changes correspondingly, but the depth d of the Rayleigh

waves’ distribution is a constant.

𝜃 𝑟 (b)



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Figure 4-33: Beam directivity of Rayleigh waves.

Another beam feature, field distribution along the steering angle 00, is studied as well. The

radial distance r used is 180 mm from the centre of EMAT arrays, and the steering angle

used is 00. The field distribution along the steering angle 00 is shown in Figure 4-34, which

shows the relationship between the radial distance from the centre of the EMAT sensor and

the magnitude of the velocity field. From Figure 4-34, the length period of the maximum

magnitudes, which is marked as “A” in this figure, is about 18 mm, which is consistent

with the modelling geometry, where the distance from the centre to the end of the EMAT

sensor is 17 mm. Along with the ultrasound waves travelling, the interference between the

ultrasound waves is in process, as shown in “B” in this figure; during the interference, the

magnitude of the wavefront is changing due to the constructive and destructive interference.

After the interference, the wavefront is almost a constant; in other words, the attenuation of

Rayleigh waves along the propagating direction is very small. As a result, the maximum

magnitude occurs at the places where the sensor is located, and due to the very small

attenuation, Rayleigh waves are increasingly used for long distance detections.

Figure 4-34: Field distribution along the steering angle 00 of Rayleigh waves.

A B



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4.5 EMAT-receiving Mechanism

As described in section 4.3, the velocity fields can be obtained by combining the EMAT-

EM modelling and the EMAT-US modelling. In the presence of a static magnetic field,

such velocity fields can induce the time varying magnetic field, which can be picked up by

the receiving coil.

In this part, the induced voltage calculation is introduced based on the solution proposed

by [21]. The model used for studying the receiving process is shown in Figure 4-35, where

the modelling geometry is divided into four zones. Zone 0 is the region where the

permanent magnet located at; zone 1 contains the meander-line-coil and the surrounded air;

zone 2 and zone 3 are the material with the same property. The permanent magnet and the

meander-line-coil used in the receiving transducer is exactly the same as those used in the

transmitting transducer; the permanent magnet is accurately placed on the top of the coil.

Figure 4-35: The model used for calculating the induced voltage in the receiving coil.

The ultrasonic waves, in the presence of a static magnetic field, induce the eddy current in

the material as shown in Equation 4-10; ved is the velocity field at the depth of xed; Eed and

Jed are the induced electric field and the induced current at the depth of xed respectively. B

is the static magnetic field, σ is the conductivity of the material,

Equation 4-10

{𝑬𝑒𝑑 = 𝒗𝒆𝒅 × 𝑩𝑱𝑒𝑑 = 𝜎 ∙ 𝑬𝑒𝑑

Due to the induced current 𝑱𝒆𝒅 at the depth of xed, the vector potential is generated into the

permanent magnet (zone 1), the air area with the meander-line-coil (zone 2), and the



96

conductor (zone 3 and zone 4). Based on Maxwell’s Equations, the vector potentials in

different zones are described in Equation 4-11, where Ai, σi and µi are the vector potential,

the conductivity and the permeability respectively; the subscript i corresponds to different

zones. ω is the angular frequency; j is √−1.

Equation 4-11

{

𝛻2𝑨0 = 𝑗𝜔𝜎0𝜇0𝑨0 𝛻2𝑨1 = −𝜔

2𝜀1𝜇1𝑨1𝛻2𝑨2 = 𝑗𝜔𝜎2𝜇2𝑨2𝛻2𝑨3 = 𝑗𝜔𝜎3𝜇3𝑨3

The total vector potential in zone 1 generated by the entire eddy current in the material can

be obtained by integrating over the depth of zone 2, 𝑥𝑒𝑑, (Equation 4-12), where 𝑨 denotes

the entirely induced vector potential in the region of air (zone 1). The entire vector

potential in zone 1 produces the induced voltage which can be picked up by the receiving

EMAT (Equation 4-13). Er and Vr are the induced electric field and the induced voltage

respectively; 𝑁and ℓ are the turn number and the length of the receiving coil respectively.

Equation 4-12

𝑨 = ∫ 𝑨1𝑑 (𝑥𝑒𝑑)0

−∞

Equation 4-13

{𝑬𝑟 = −𝜕𝑨

𝜕𝑡𝑽𝑟 = 𝑁ℓ𝑬𝑟

4.6 Experimental Validations

4.6.1 Experiments Set-up

In this section, experiments were conducted to validate the proposed modelling methods as

described in section 4.3. A high power tone burst pulser / receiver, RITEC RPR4000, was

used. RITEC RPR4000 is capable of pulsing up to 8 kW, hence it is desirable for

inefficient transducers, such as electromagnetic acoustic transducers (EMATs); the

standard operational frequency range of RITEC RPR4000 is from 250 kHz to 5 MHz. The

gain of the receiver is from 20 dB to 100 dB; the filter has a passband from 50 kHz to 20

MHz; in this work, the gain of the receiver used was 80 dB, and the passband filter used



97

was from 200 kHz to 2.5 MHz, which means only the signal at the frequency range from

200 kHz to 2.5 MHz can be received. RITEC RPR4000 can be connected to an external

computer via the serial interface RS-232.

The schematic diagram of the experimental system is shown in Figure 4-36; the high

power tone burst pulser and receiver, RITEC RPR4000, was used to excite and receive

EMAT signals; the impedance matching box was used to match the impedance between

the power amplifier and the coil to maximize power transfer; the transmitter and receiver

consist of the permanent magnet and the meander coil as described in the part of EMAT

simulation; oscilloscope was used to display and record signals. Data can be sent to the

external computer to process either by RITEM RPR4000 or by the oscilloscope.

Figure 4-36: The schematic diagram of the experimental system.

Figure 4-37 shows the set-up of the experimental system; the stainless-steel plate, the

meander-line-coil and the permanent magnet used had the same dimension and shapes as

those clarified in section 4.1.1. The transmitter and the receiver were located on the top of

the stainless steel plate with a centre-to-centre distance of 150 mm. The effect of the

fractional bandwidth has been described in section 4.3.4, which states a small factional

bandwidth, 0.3, for the central frequency 500 kHz is sufficiently reliable for Rayleigh

waves’ interference. For the experiments, the excitation voltage used is a Gaussian-

modulated sinusoidal wave with a peak voltage of 140 V, and the fractional width used is

0.18, which can provide a good signal resolution. The sampling frequency is 100 MHz

from the oscilloscope.



98

Figure 4-37: Set-up of the experimental system.

4.6.2 Received Signals from Experiments

Because the EMAT transmitter is operated at 500 kHz, the received signal is mainly

concentrated on 500 kHz as well. The output of the fast Fourier transform of the originally

received signal from experiments is shown in Figure 4-38(a); as expected, the major

frequency component is 500 kHz. However, there were some other frequency components,

such as 1 MHz and 1.5 MHz, which are not desirable and should be cut-off. By processing

signals, a band-pass filter is employed to allow the frequency from 480 kHz to 520 kHz to

pass. The filtered signal is shown in Figure 4-38(b); the non-desirable signal are cut-off,

and the signal are mainly concentrated on the operational frequency, 500 kHz.

Figure 4-38: The frequency domain of the experimentally received signals.

Oscilloscope RPR-4000

Receiver

Transmitter

(a) (b)

Impedance matching box



99

The time period of the oscilloscope sampling is 1 ms; in order to avoid the reflection from

the boundary, only the directly transmitted Rayleigh waves, which are within the time

period of 80 µs, are picked to verify the simulations. The received signal from experiments

is shown in Figure 4-39, where the blue curve means the amplitude of the received signal

and the red curve denotes the envelope of the received signal; the envelope of the signal

makes the identification of the arrival time of Rayleigh waves more clearly. From

experimental results, the directly transmitted Rayleigh waves arrives later than the main

bang signal, which is the overloading of the EMAT receiver by the electrical interference

produced by the high power transmitter excitation. The arrival time of Rayleigh waves

occurs at 50.3 µs after firing; based on the flight distance and the arrival time, the

calculated velocity of Rayleigh waves from experiments is 2.98 mm/µs. The velocity of

Rayleigh waves used in this work for simulations is 3.033 mm/µs. The relative error is

1.68%, which is within the noise and error tolerances of the experiments.

Figure 4-39: The received signal from experiments.

4.6.3 Validation of EMAT Models with Experiments

As described before, the arrival time of the directly transmitted Rayleigh waves from the

simulation is 49.8 µs while the arrival time from the experiment is 50.3 µs; possible

reasons are, 1), the Rayleigh waves velocity used in the simulation is slightly different with

that used in the experiment; 2) the experimental noises. In order to compare the simulated

signals and the experimental signals clearly, the arrival time of the simulated signals is

shifted with 0.5 µs offset to make the arrival time between these two signals are the same.

The comparison between the simulation and the experiment is shown in Figure 4-40.



100

By multiplying a scaling factor, simulation signals have the same peak value with the

experimental signals as shown in Figure 4-40, where the simulation shows a good

agreement with the experiment. In this section, root-mean-square error (RMSE) and the

correlation coefficient are employed to describe the difference and the linear correlation

between the simulated and experimental models. The time period of the received Rayleigh

waves is from 30 µs to 70 µs as shown in Figure 4-40; the root-mean-square error (RMSE)

within this time period calculated is 0.0834, which shows a very small difference between

the simulated model and the experiment model. The correlation coefficient calculated is

0.9927, which shows the simulated model and the experimental model are highly

correlated.

However, at the beginning and the end of Rayleigh waves (as shown in the “Note Area” in

Figure 4-40), there is a slight difference between the simulation and the experiment. More

specifically, the phases of the signals between the simulation and the experiment are not

consistent in the “Note Area”. Possible reasons are: 1), in this work, the simulated model is

a simplified model with only surface sources; 2), the numerical nature of FDTD: numerical

approximations are inevitable; 3), the inevitable noises of the experiment.

Figure 4-40: The comparison between the simulation and the experiment.

Changing the centre-to-centre distance between the transmitter and the receiver from 140

mm to 190 mm with a step of 10 mm, the maximum amplitude of the directly transmitted

Rayleigh waves from the simulation and the experiment are shown in Figure 4-41, which

show a good agreement. From Figure 4-41, the induced voltage decreases with the distance

Note Area



101

between the transmitter and the receiver increasing; that is due to the attenuation of

Rayleigh waves. However, the attenuation is very small; from the simulation, the

attenuation is 0.0034% with a travelling distance of 5 cm; from the experiment, the

attenuation is 0.0039% with a travelling distance of 5 cm.

Figure 4-41: The maximum amplitude of the induced voltage with various distances between

the transmitter and the receiver.

It should be noted that, for the EMATs modelling as described in section 4.3, the dynamic

magnetic field was neglected due to the small excitation AC current used in this work; the

excitation voltage used was a Gaussian-modulated sinusoidal wave with a peak voltage of

140 V and the impedance of the meander-line-coil at 500 kHz was 27.65 Ω, hence the

excitation current approximately equals to 5.06 A. Experiments with only the meander-

line-coil to generate Rayleigh waves were carried out to study the effect of the dynamic

magnetic field; the transmitter used was a meander-line-coil without the permanent magnet;

the other set-up were the same as those clarified in section 4.6.1. The received signals with

only the meander-line-coil as the transmitter is shown in Figure 4-42; only the main bang

signal can be captured by the receiving EMAT, and therefore the effect of the dynamic

magnetic field can be neglected due to the small excitation current used in this work. The

effect of the dynamic magnetic field generated by the meander-line-coil must be

considered when the excitation current is large as it is one of the mechanisms to generate

ultrasound waves; [39] suggested, when the excitation exceeds 528.9 A, the dynamic

magnetic field plays a more significant role than the static magnetic field, which is

generated by the permanent magnet.

0.995

0.996

0.997

0.998

0.999

1

14 15 16 17 18 19

No

rma

lise

d V

olt

ag

e

Distance between the transmitter and the receiver (cm)

Induced Voltage

Experiment Simulation



102

Figure 4-42: The received signals with a meander-line-coil as the transmitter.

4.7 EMAT Scattering Phenomena

Previously, the Rayleigh waves’ property generated by meander-line-coil EMATs have

been analysed with a novel modelling method, which combines the analytical method and

the finite-difference time-domain (FDTD) method; this method has been verified with

experiments as described in section 4.6.

Rayleigh waves propagate along the surface and the subsurface of the test piece; due to the

nature of a low attenuation and a long distance propagation, Rayleigh waves are

increasingly used in applications of surface and sub-surface cracks detections. In this

section, the scattering behaviour of Rayleigh wave is studied by means of the proposed

modelling method and experiments.

4.7.1 Modelling of Rayleigh Waves’ Scattering

4.7.1.1 Scattering Models and Wave Propagations

In section 4.3.2, in order to identify the body waves clearly, a thicken model with a depth

of 80 mm is used. Because only Rayleigh waves are of interest in this section, the thickness

of the stainless steel plate is set to 25 mm. The modelling geometry is shown in Figure

4-43; where the sources are the alternating Lorentz force densities from the EM simulation

as described in section 4.2.4. Free surface is applied on the top of the modelling geometry;

perfectly-matched layers (PML) are used to absorb reflections from left, right and bottom

boundaries of the material. A crack with a width of 0.5 mm and a depth of 2 mm is placed

on x=25 mm and y=500 mm; two virtual receivers, R1 and R2, are placed on x=24 mm and

y=400 mm, and x=24 mm and y=550 mm, to inspect Rayleigh waves. The spatial step and



103

the time step, the operational frequency, the spacing between the adjacent sources, and the

Rayleigh waves’ velocity used are the same as those used in section 4.3.

Figure 4-43: The geometry of Rayleigh waves’ scattering simulation.

The Rayleigh waves’ propagation is shown in Figure 4-44, which visualizes the scattering

behaviour of the EMAT-Rayleigh waves. At 10 µs after firing, both the body waves and

the Rayleigh waves are generated into the material (Figure 4-44(a)). Due to the PML

boundary applied to the bottom of the material, obliquely propagated body waves are

absorbed; as shown in Figure 4-44(b) and Figure 4-44(c), only Rayleigh waves exist in

these images. The directly transmitted Rayleigh waves, abbreviated for “DRW”, are

received by the receiver R1. At 67 µs after firing, the directly transmitted Rayleigh waves

encounter the crack, and are reflected and scattered by the crack (Figure 4-44(d)). The

reflected Rayleigh waves are abbreviated for “RRW” and the scattered Rayleigh waves are

abbreviated for “SRW” as shown in Figure 4-44(e). The scattered Rayleigh waves (SRW)

are received by the receiver R2 (Figure 4-44(f)), and the reflected Rayleigh waves (RRW)

are received by the receiver R1 (Figure 4-44(g)).

(b) DRW

(a)



104

Figure 4-44: Scattering behaviours of Rayleigh waves.

4.7.1.2 Received Signals from Simulations

The induced voltage picked up by the receiver R1 is shown in Figure 4-45, where the blue

curve shows the normalised amplitude of the induced voltage, and the red curve shows the

normalised envelope of the received signal based on Hilbert transform, resulting in

identification of the arrival signals more clearly [7].

From Figure 4-45, two signals are received by the receiver R1; the firstly arriving signal is

the DRW followed by the RRW. The centre-to-centre distance between the transmitter and

the receiver is 100 mm, and the Rayleigh waves’ velocity used in this work is 3.033 mm/µs,

so the theoretically arrival time of the DRW is 32.97 µs. The numerally arrival time of the

DRW, as shown in Figure 4-45, is 32.94 µs, which shows a good agreement with the

theoretically arrival time 32.97 µs.

The travelling distance of the RRW is 300 mm, and the Rayleigh waves’ velocity used in

this work is 3.033 mm/µs, hence the theoretically arrival time of the RRW is 98.91 µs,

which is very close to the numerically arrival time of the RRW 100.3 µs.

RRW SRW

SRW

RRW

DRW (c)

(d)

(e)

(f)

(g)



105

Figure 4-45: Received signals from R1.

The received signals of R2 are slightly different from those of R1 (Figure 4-46). There is

no DRW, because the DRW are scattered by the crack; only the SRW are picked up by the

receiver R2; the numerically arrival time of the SRW, as shown in Figure 4-46, is 82.95 µs.

The centre-to-centre distance between the transmitter and the receiver is 250 mm, and the

Rayleigh waves’ velocity used in this work is 3.033 mm/µs, so the theoretically arrival

time of the SRW is 82.43 µs. The relative error between the theoretically arrival time and

the numerically arrival time is 0.63%, which is very small to show a good agreement.

Figure 4-46: Received signals from R2.

The comparison between the received signals from R1 and R2 is shown in Figure 4-47,

where the blue curve shows the received signal from the receiver R1 and the red curve

shows the received signal from the receiver R2; the amplitude of the received signal is

normalised. From this image, the arrival time of the SRW is smaller than that of the RRW;

0 50 100-1

-0.5

0

0.5

1

Simulated Signals: R1

time (s)

Norm

alis

ed V

oltage

Amplitude

Envelope

0 50 100-1

-0.5

0

0.5

1

Simulated Signals: R2

time (s)

Norm

alis

ed V

oltage

Amplitude

Envelope



106

that is because the travelling distance of the SRW is smaller than that of the RRW. In

addition, the amplitude of the SRW is smaller than that of the RRW; this indicates the

DRW are mainly reflected; in other words, only a small portion of the DRW propagates

forward with the specific crack used in this work.

Figure 4-47: The comparison of the received signals from the receivers R1 and R2.

4.7.2 Experiments and Validations

4.7.2.1 Received Signals from Experiments

In this section, experiments were carried out to validate the scattering simulation;

experimental system has been detailed in section 4.6.1. The transmitter, the receiver and

the material used in the experiment were the same as those used in the scattering

simulation. The excitation signal, the operational frequency and the sampling frequency

were exactly the same as those used section 4.6.1.

In the scattering modelling, two receivers, R1 and R2, were employed. For experiments,

only the received signal from the receiver R1 was used to validate the proposed modelling

method. The experimentally received signal from the receiver R1 is shown in Figure 4-48,

where the blue curve and the red curve denote the amplitude and the envelope of the

induced voltage in the pick-up coil respectively. Three signals, the main bang signal, the

DRW and the RRW were picked up by the receiving coil.

As expected, the DRW arrives earlier with an arrival time around 33 µs and the RRW

arrives latter with an arrival time around 100 µs. The theoretically arrival time of these two

RRW SRW

DRW



107

signals are 32.97 µs and 98.91 µs; there is a slight difference between the experimentally

arrival time and the theoretically arrival time due to the experimental noise.

Figure 4-48: The experimentally received signal from the receiver R1.

4.7.2.2 Validations

The comparison between the simulated signal and the experimental signal is shown in

Figure 4-49, where the blue curve and the red curve denote the received signal from

simulations and the received signal from experiments respectively. Due to the slight

difference of the arrival time between the experiment and the simulation, an arrival time

shift is applied to make these two signals synchronized. In addition, a scaling factor is used

to keep the amplitude of the DRW between the simulation and the experiment to a same

value.

From Figure 4-49, the simulated signal and the experimental signal shows a good

agreement. In order to compare these two signals quantitatively, the root-mean-square

error (RMSE) and the correlation coefficient are employed. In addition, in order to

eliminate the effect of the main bang signal, only the signal within the time period from 12

µs to 145 µs is picked. The RMSE within this time period calculated is 0.1159, which

shown a small difference between the simulated model and the experimental model. The

correlation coefficient calculated is 0.9241, which means the simulated model and the

experimental model are highly correlated.

0 50 100-3

-2

-1

0

1

2

3

Experimental Signals

time (s)

Voltage (

V)

Amplitude

Envelope



108

Figure 4-49: The amplitude comparison between the simulation and the experiment.

In addition, the envelope comparison between these two signals are studied as well, as

shown in Figure 4-50. The difference between these two signals are more clearly by

comparing the envelope; the simulated signal is slightly narrower than the experimentally

received signal, no matter in the DRW or in the RRW. For the envelope comparison, the

RMSE calculated is 0.0784 and the correlation coefficient calculated is 0.9841, which

further verifies the small difference between the simulated model and the experimental

model.

Figure 4-50: The envelope comparison between the simulation and the experiment.

4.8 Modelling of Unidirectional Rayleigh Waves EMATs

Typically, the Rayleigh waves generated by the meander-line-coil travel are symmetrically

in two directions, as shown in Figure 4-27. This type of EMAT is named the bidirectional

Rayleigh waves (BRW) EMAT; the BRW EMAT has a non-desirable feature that the

0 50 1000

0.5

1

1.5

2

2.5

Comparison: Envelope

Time (s)

Voltage (

V)

Simulations

Experiments



109

crack detection resolution is low due to the two-directionally symmetrically propagating

Rayleigh waves. In order to improve the crack detection resolution, there is a demand for

the EMAT to generate unidirectional Rayleigh waves.

In 2013, [23] reported an EMAT which is capable of generating unidirectional Rayleigh

waves with two identical meander-line-coils; this type of EMAT is named the

unidirectional Rayleigh waves (URW) EMAT. The configuration of the URW EMAT is

shown in Figure 4-51; the URW EMAT consists of a permanent magnet, a test piece and

two identical meander-line-coils. The interaction between the eddy current and the static

magnetic field generates the Lorentz force density, which in turn produces ultrasound

waves.

Figure 4-51: The configuration of the URW EMAT. From [23].

This URW EMAT has been detailed in [23]; here a brief description about the wave

superposition for the URW EMAT is presented. Define these two identical meander-line-

coils as coil A and coil B, as described in Figure 4-51; both coil A and coil B have a

centre-to-centre distance of adjacent wires equalling to a half of Rayleigh waves’

wavelength λ/2. The distance between coil A and coil B is a quarter of Rayleigh waves’

wavelength λ/4; the excitation signal for coil A and coil B has the same amplitude but a

phase difference of 900. The wave superposition is shown in Figure 4-52; the source 1 is

generated by the coil A while the source 2 is generated by the coil B; source 1 and source 2

have a distance of a quarter of Rayleigh waves’ wavelength λ/4. Based on the wave

interference, the synthetic wave is generated, with the amplitude strengthen in one

direction while weaken in the other direction [80].



110

Figure 4-52: The wave superposition between the source 1 and the source 2. From [80].

In this section, the proposed modelling method is used to study the URW EMAT

properties. The EM simulation is performed with the analytical method and the US

simulation is accomplished by the finite-difference time-domain (FDTD) method as

described in section 4.3. The operational frequency, the material, the magnet and the

related parameters are the same as those used in section 4.1.1. The excitation signals for

Coil A and Coil B are shown in Figure 4-53; the operational frequency is 500 kHz and the

phased difference between Coil A and Coil B is 900; that means the central pulse time

between Coil A and Coil B has a shift of 0.5 µs.

Figure 4-53: The excitation signal for the coil A and the coil B.

The wave propagation of Rayleigh waves generated by the URW EMAT is shown in

Figure 4-54; the image shows the distribution of the magnitude of Rayleigh waves. From



111

Figure 4-54(a), not only the Rayleigh waves, but also the shear waves are travelling in one

specific direction. From Figure 4-54(b), Rayleigh waves are more clear; the strengthen

Rayleigh waves are shown in the red ellipse and the weaken Rayleigh waves are shown in

the yellow ellipse. It is clearly that the strengthen Rayleigh waves have a larger intensity

than the weaken ones.

Figure 4-54: The wave propagation of Rayleigh waves generated by the URW-EMAT.

In order to quantitatively analyse the strengthen Rayleigh waves, two receivers are located

on the left and the right of the transmitter with a centre-to-centre distance of 145 mm from

the transmitter; specifically, the transmitter is located on x=79mm, y=200mm, and the

receiver on the left (termed as RL) is located on x=79mm, y=55 mm, and the receiver on

the right (termed as RR) is located on x=79mm, y=345 mm. The received signals from the

receivers RL and RR are shown in Figure 4-55; where the blue curve is the amplitude of

the weakened Rayleigh waves and the red curve is the amplitude of the strengthen

Rayleigh waves. The maximum amplitude of the strengthen Rayleigh waves is 8.5 times

as larger as that of the weakened Rayleigh waves.

(a)

(b)



112

Figure 4-55: The received signal from the URW-EMAT.

Figure 4-56 illustrates the received signal from the BRW EMAT, which is capable of

generating bidirectional Rayleigh waves. Two receivers, RL and RR, are located at a

distance of 145 mm from the transmitter. From this figure, the received signal from the

receiver RL and that from the receiver RR are almost the same; the ratio between the

maximum amplitude of the received signals from the receiver RL and that from the

receiver RR is 1.0002.

Figure 4-56: The received signal from the BRW-EMAT.

The comparison between the URW EMAT and the BRW EMAT is shown in Figure 4-57,

where Figure 4-57(a) shows the received signal from the receiver RL and Figure 4-57(b)

shows the received signal from the receiver RR. It can be identified that the Rayleigh

waves travelling to the left are suppressed with the URW EMAT; and Rayleigh waves

travelling to the right are amplified with the URW EMAT. The suppressed ratio is 0.2336



113

and the amplified ratio is 1.9849. All of these results are consistent with the results from

[23].

Figure 4-57: The comparison between the URW and the BRW.

4.9 Conclusions

In this chapter, a novel modelling method combining the analytical method and the finite-

difference time-domain (FDTD) method for simulating the meander-line-coil EMAT was

proposed by the author; the analytical method is used for the EM simulation and the FDTD

method is used for the US simulation.

The analytical method was adapted from the classic Dodd and Deeds solution for the eddy

current phenomena, and was verified by the finite element method (FEM). For the US

simulation, the FDTD method was used to model the ultrasound waves’ propagation,

analyse the radiation pattern and study the beam features of the generated Rayleigh waves.

The scattering behaviour of Rayleigh waves was modelled with the proposed modelling

method as well.

In addition, the novel modelling method is capable of modelling unidirectional Rayleigh

waves EMATs as well. Quantitatively analysis of the received signal was carried out; the

results showed that the unidirectional Rayleigh waves EMAT is able to strengthen the

Rayleigh waves in one direction and weaken that in the other direction.

Experiments were carried out to validate the novel modelling method; a good agreement

was observed between the experimental model and the numerical model. The RMSE and

the correlation coefficient were employed to quantitatively analyse the model difference

level and the model related level.

(a) (b)



114

Overall, the novel modelling method proposed by the author is capable of modelling

various behaviours of Rayleigh waves, such as bidirectional Rayleigh waves, scattering

Rayleigh waves, and unidirectional Rayleigh waves.

CHAPTER5 Directivity analysis of the conventional meander-line-coil EMATs

115

Chapter 5 Directivity Analysis of Conventional Meander-

line-coil EMATs

5.1 Introduction

Because 3-D modelling has a high requirement on computing power and requires

significant running time, most of the previous works were 2-D simulation focusing on the

vertical plane (i.e. x-y plane) of the material. The orientation of the coordinate system is

shown in Figure 4-1, and all of the subsequent simulations are based on this coordinate

system. Two examples of the 2-D modelling methods for the x-y plane of the material have

been detailed in [10, 13], however, there has been little research on the beam directivity of

Rayleigh waves on the surface plane (i.e. y-z plane) of the material.

This chapter presents a wholly analytical modelling method to study the beam directivity

of Rayleigh waves on the surface of the material. This wholly analytical method, which

involves the coupling of two models: an analytical EM model and an analytical US model,

has been developed to build EMAT models and analyse the beam directivity of Rayleigh

waves. Lorentz forces are calculated using the EM analytical method, which is adapted

from the classic Dodd and Deeds solution, as described in section 4.2. The calculated

Lorentz force density are imported to the analytical US model as driving point sources,

which produce the Rayleigh waves within a layered medium. Because the analytical EM

model is the same as the one used in section 4.2, this chapter only introduces the analytical

US model. The effect of the length of the meander-line-coil on the beam directivity of

Rayleigh waves is analysed quantitatively and validated experimentally.

5.2 The Analytical Solution to the Radiation Pattern of Rayleigh

Waves on the Surface of the Material

N. A. Haskell proposed the analytical solution to the Rayleigh waves’ radiation pattern due

to point sources in a homogenous medium and in a multi-layered medium, in 1963 and

1964 respectively [81, 82]. Elastic waves radiate in an unbounded medium expressed in

Cartesian coordinates was given by Love [83]. However, Sezawa’s cylindrical wave

functions are the most convenient ways to impose the free surface boundary conditions;

hence, the transformation between these two representations are necessary; the free surface

conditions are imposed by vanishing the stress components at the free surface. Rayleigh


116

waves’ components are separated out by calculating the residue at the Rayleigh pole [81,

82].

The articles, [81, 82], were published in the field of seismology, and some terminology in

the field of seismology, such as strike-slip faults and dip-slip faults, were used in these

articles [81, 82]. In this work, the driving source is the tangentially polarized Lorentz force,

which corresponds to the strike-slip fault; after some manipulations, the final solution to

the displacement of Rayleigh waves due to the surface point source is,

Equation 5-1

𝒖𝒓 = 𝐴(ĸ, 𝑟)𝑒−𝜋𝑖

42ĸ(𝛾−1)

𝑣𝛽∙ 𝑭 ∙ (

𝛾+1

𝛾− 1)

Equation 5-2

𝒖𝒙 =𝑖𝛾𝑣𝛼𝒖𝒓

ĸ(𝛾−1)

where

Equation 5-3

𝐴(ĸ, 𝑟) =ĸ2𝛾𝑣𝛽

4𝜌(2𝛾2𝑣𝛼𝑣𝛽

ĸ3)

√2

𝜋ĸ𝑟𝑒−𝑖ĸ𝑟

Equation 5-4

𝛾 = 𝑐𝑜𝑠 (𝜃)

Equation 5-5

𝑣𝛼 = {√ĸ2 − (𝜔/𝑐𝐿)2 ĸ > 𝜔/𝑐𝐿

𝑖√(𝜔/𝑐𝐿)2 − ĸ2 ĸ < 𝜔/𝑐𝐿

Equation 5-6

𝑣𝛽 = {√ĸ2 − (𝜔/𝑐𝑆)2 ĸ > 𝜔/𝑐𝑠

𝑖√(𝜔/𝑐𝑆)2 − ĸ2 ĸ < 𝜔/𝑐𝑠

Equation 5-7

ĸ =𝜔

𝑐𝑅


117

where 𝒖𝒓 and 𝒖𝒙 are the in-plane displacement and out-of-plane displacement respectively.

As shown in Figure 5-1, an arbitrary point on the surface of the material is defined as the

field point; r is the distance between the source point and the field point; ĸ is the

wavenumber of Rayleigh waves; F is the driving force; ρ is the density of the material; θ is

the angle between the force vector and the in-plane displacement vector; ω is the angular

frequency; CL, CS and CR are the velocity of the longitudinal wave, the shear wave and the

Rayleigh wave respectively.

Figure 5-1: Surface waves generated by the point source.

By Equation 5-1 to Equation 5-7, the Rayleigh waves’ displacement, i.e. the radiation

pattern of Rayleigh waves, can be obtained. Based on the radiation pattern of Rayleigh

waves, the beam directivity, which is used to quantitatively analyse the distribution of

Rayleigh waves, can be obtained; the analysis of the beam directivity is presented in

section 5.3.

5.3 Beam Directivity Analysis of the Conventional Constant-length

Meander-line-coil (CLMLC)

In this section, the test piece used is an aluminium plate with a dimension of 600×600×25

mm3; the coil used is a meander-line-coil with a dimension of 30×34.163×0.035 mm3; the

permanent magnet used is NdFeB35, whose size is 60×60×25 mm3. The operation

frequency is 483 kHz, and the skin depth calculated is 0.117 mm; the lift-off of the

meander-line-coil is 1 mm. The velocity of Rayleigh waves used in the aluminium plate is

2.93 mm/µs, so the spacing between two adjacent lines of the meander-line-coil is 3.03


118

mm, which equals to one half of the Rayleigh waves’ wavelength, to form the constructive

interference.

5.3.1 Wholly Analytical Models

For modelling Rayleigh waves on the surface of the material (i.e. the y-z plane of the

material), a wholly analytical method, which involves the coupling of an analytical EM

model and an analytical US model, is utilized. The analytical EM solution, which is

adapted from the classic Dodd and Deeds solution, has been introduced in section 4.2.

Lorentz force densities, which are generated from the analytical EM simulation, are

imported to the analytical US model, as shown in Figure 5-2.

There are 12 straight wires in the meander-line-coil used in this work; therefore, there are

12 negative and positive peaks in the curve of the Lorentz force density distribution. These

12 Lorentz force densities are added to the US model as driving forces to generate

Rayleigh waves. The displacement of Rayleigh waves due to a point source can be

calculated by Equation 5-1 to Equation 5-7; with multiple point sources, the displacement

of Rayleigh waves at an arbitrary field point is the sum of the displacement caused by each

point source.


model.

Table 5-1 illustrates the detailed parameters used for the analytical US model. Field spatial

step means the distance between two adjacent field points on the surface of the Aluminium

plate; the dimension of the surface of the aluminium plate is 600×600 mm2 and the field

spatial step used is 1 mm, so there are totally 601×601 field points on the surface of the


119

Aluminium plate. Source spatial step for each wire means the distance between two

adjacent source points on each wire; the length of the wire is 30 mm and the source spatial

step used is 0.2 mm, so there are 151 source points on each wire. The reason we choose

dense source points is that lots of source points guarantee the wave interference’ integrity.

Table 5-1: Detailed parameters used for the analytical US model.

Description Symbol Value

Length of the Aluminum plate Y 600 mm

Width of the Aluminum plate Z 600 mm

Field spatial step ∆xf 1 mm

Length of the meander-line-coil L 30 mm

Source spatial step for each wire ∆xs 0.2 mm

Density of the Aluminum plate ρ 2700 kg/m3

Frequency f 483 kHz

Longitudinal waves’ velocity CL 6.375 mm/µs

Shear waves’ velocity Cs 3.14 mm/µs

Rayleigh waves’ velocity CR 2.93 mm/µs

The meander-line-coil is located on the centre of the aluminium plate, so the whole

EMAT-US model is symmetrical with y=300 mm; in order to reduce the modelling time,

only the left half of the geometry is modelled. Only the area, y from 0 to 250 mm, is

presented where the signals are significant. The calculated Rayleigh waves’ radiation

pattern is shown in Figure 5-3. From this image, it can be seen that the Rayleigh waves are

mainly concentrated along the y direction. The area with larger intensities is referred to as

the main lobe while the areas with smaller intensities are referred to as the side lobes, as

shown in Figure 5-3.


120

Figure 5-3: The Rayleigh waves’ radiation pattern on the surface of the aluminium plate.

The whole radiation pattern is shown in Figure 5-4; as mentioned previously, it is

symmetrical with y=300 mm. Beam directivity, which are used to quantitatively analyse

the Rayleigh waves’ distribution, can be obtained on the basis of the radiation pattern of

Rayleigh waves. As shown in the red arc in Figure 5-4, beam directivity is, at a specific

distance (“r” in Figure 5-4) from the centre of the EMAT sensor, the displacement

distribution of Rayleigh waves. In this work, r used is 250 mm; θ1 and θ2 used are -700 and

700 respectively.

Figure 5-4: The model used to study the beam directivity.

Main Lobe

Side Lobes


121

The calculated beam directivity is shown in Figure 5-5(a); this curve states the normalised

magnitude of the displacement versus angles. As observed, there is a main lobe containing

a larger displacement magnitude and some side lobes with smaller displacement

magnitudes. The main lobe is centred at 00, and the side lobes are roughly centred at -30.50,

-18.50, 18.50 and 30.50 respectively. The side lobes are usually the radiation in undesired

directions.

In order to quantitatively analyse the beam directivity, the half power beamwidth (HPBW)

and the sidelobe level (SLL) are utilized. The half power beamwidth (HPBW) is the angle

between the half-power (-3 dB) points of the main lobe; the half-power (-3 dB) points are

the points with a magnitude equalling to a half of the peak value of the main lobe, as

shown in Figure 5-5(b).

The sidelobe level (SLL) is described in decibels relative to the peak of the main lobe, as

shown in Figure 5-5(b). The maximum magnitude of the side lobes is 0.2166, which is

21.66 percent of the peak of the main lobe. For the 30 mm constant-length meander-line-

coil (CLMLC), the HPBW calculated is 14.940; the SLL calculated is -6.6434 dB.

Figure 5-5: The beam directivity of Rayleigh waves generated by a 30mm-length meander-

line-coil EMAT. (a), the curve of the beam directivity; (b) the curve used for describing

HPBW and SLL.

5.3.2 The Effect of the Length of the Conventional Constant-length

Meander-line-coil (CLMLC) on Radiation Pattern

In this section, the effect of the length of the meander-line-coil on the Rayleigh waves’

radiation pattern is studied. Various lengths of the meander-line-coil are modelled; the

modelling parameters are the same as those used in Table 5-1 except the length of the coil.

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The length range of the meander-line-coil is from 20 mm to 65 mm with a step of 5 mm;

the beam directivity calculated is shown in Figure 5-6.

Figure 5-6(a) shows the beam directivity of the meander-line-coil with a length of 10 mm,

15 mm, 20 mm and 25 mm respectively; from this image, with a larger length, the main

lobe is narrower. It means, for the meander-line-coil with a larger length, it has a more

concentrated Rayleigh waves’ beam. The observation is further verified by Figure 5-6(b)

and Figure 5-6(c), which show the beam directivity of the meander-line-coil with a length

from 30 mm to 65 mm respectively. In addition, one point should be noted is that a larger

length results in side lobes with a larger magnitude as well.

Figure 5-6: The beam directivity of the meander-line-coil with various lengths.

The HPBW and the SSL for the meander-line-coil with various lengths are calculated, as

shown in Table 5-2; it illustrates that a larger length leads to a smaller HPBW but a larger

SSL. When the length of the meander-line-coil is enlarged from 10 mm to 65 mm, the

HPBW decreases by 82.32 percent and the SSL increases by 64.62%.

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Table 5-2: HPBW and SSL for the meander-line-coil with various lengths.

Meander-line-coil

Length HPBW SSL(dB) Length HPBW SSL(dB) Length HPBW SSL(dB)

10 mm 41.680 -13.387 30 mm 14.940 -6.6434 50 mm 9.020 -5.8843

15 mm 29.630 -11.133 35 mm 12.960 -6.4935 55 mm 8.360 -5.5666

20 mm 22.240 -7.9553 40 mm 11.320 -6.3229 60 mm 8.020 -5.1833

25 mm 17.920 -7.0527 45 mm 100 -6.1225 65 mm 7.370 -4.7357

5.4 Experimental Results

Experiments were carried out to validate the proposed modelling method; the experimental

set-up is shown in Figure 5-7(a); the aluminium plate, the meander-line-coil and the

permanent magnet used had the same dimension and shapes with the modelling geometry

as clarified in section 5.3. The high power tone burst pulser and receiver, RITEC RPR4000,

was used to excite and receive EMAT signals; an impedance matching box was used to

match the impedance between the power amplifier and the coil to maximize the power

transfer; oscilloscope was used to display and record signals. Data can be sent to the

external computer to process either by RITEM RPR4000 or by the oscilloscope.

Figure 5-7: (a), experimental set-up; (b), the scan path of the receiver; Tx means the

transmitter and Rx means the receiver.

In EMAT modelling, the length range of the meander-line-coil used is from 10 mm to 65

mm with a step of 5 mm. For experiments, in order to reduce the experimental repeatability,

only four lengths, 10 mm, 20 mm, 30 mm and 40 mm, were fabricated to validate the

EMAT modelling. The scan path of the receiver is shown in Figure 5-7(b); the centre-to-

centre distance between the transmitter and the receiver is 250 mm; the scan path is from -

400 to 400 with a step of 2.50. The receiver used was a 10 mm-length meander-line-coil; by


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moving the receiver along the scanning path, the directly transmitted Rayleigh waves can

be received by the receiver at different angles. One point should be noted is, meander-line-

coil receivers with different wire lengths have different radiation patterns and beam

directivities as transmitters; the reason we used the 10 mm-length meander-line-coil as the

receiver is that, 10 mm-length meander-line-coil is relatively small, which allow easy

positioning of the receiver along the scan path; in addition, as shown in Figure 5-7(b), the

receiver is positioned along the normal direction of the scan path to guarantee that the

Rayleigh waves received by the receiver are mainly along the normal direction of the scan

path; further, for comparing the beam directivities of Rayleigh waves generated by

transmitters with different wire lengths, the receiver is constant so that the differences

between the beam directivities are due to transmitters.

The measured beam directivity from the experiment is shown in Figure 5-8; the signals are

normalised by the peak value among all of the measured data. Because the receiver was

moved from -400 to 400 with a step of 2.50, only 33 sampling points were recorded, leading

to a coarse curve of the measured beam directivity. The experimental results suggest that

with a larger length, the Rayleigh waves’ beam has a more concentrated main lobe of

Rayleigh waves; while with a smaller length, the Rayleigh waves are distributed; these

results are consistent with the modelling results.

Figure 5-8: The measured beam directivity from experiments.

Figure 5-9 shows the comparison between the measured results and the simulated results.

From Figure 5-9(a), for the meander-line-coil with a length of 10 mm, the measured beam

directivity is consistent with the simulated beam directivity. For the meander-line-coil with

a length of 20 mm, 30 mm and 40 mm, there are some non-overlapping points between the


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measured and simulated curves; however, the trend of the measured curve is the same as

that of the simulated ones. Possible reasons for the non-consistent points are: 1), the

inevitable noises of experiments; 2), the errors due to the positioning of the receiver.

Overall, the measured beam directivity shows a good agreement with the simulated beam

directivity.

Figure 5-9: Comparison between the simulated and measured results for the meander-line-

coil with a length of 10 mm (a), 20 mm (b), 30 mm (c) and 40 mm (d) respectively.

5.5 Conclusions

There has been little research on the beam directivity of Rayleigh waves generated by

meander-line-coil EMATs. In this chapter, a wholly analytical method to build EMAT

models, which are used to study the Rayleigh waves’ beam directivity on the surface plane

of the material, is proposed by the author. The analytical EM model has been described in

section 4.2; Lorentz force densities are calculated from the analytical EM model and are

imported to the analytical US model as driving point sources to generate Rayleigh waves.

(a) (b)

(c) (d)


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Modelling results reveal that, for a meander-line-coil, the length of the meander-line-coil

has a crucial effect on the Rayleigh waves’ beam directivity; that is, a larger length results

in a narrower main lobe, which means the Rayleigh waves are more concentrated.

However, the more concentrated beam is at the cost of a larger-length meander-line-coil,

the size of which is not desirable in applications where only small sensors are accessible.

Four meander-line-coils, with a length of 10 mm, 20 mm, 30 mm and 40 mm respectively,

were fabricated and used to validate the proposed modelling method; the experimental

results showed a good agreement with the simulated results. Overall, this chapter presents

a wholly analytical model proposed by the author, and provides quantitative analysis of the

beam directivity of Rayleigh waves generated by meander-line-coil EMATs.

CHAPTER 6 Novel configurations for meander-line-coil EMATs

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Chapter 6 Novel Configurations for Meander-line-coil

EMATs

6.1 Introduction

To generate Rayleigh waves, typically used meander-line-coils have wires with a constant-

length; however, significant side lobes are observed in such designs (as described in

Chapter 5). Considerable works were reported on the conventional meander-line-coil; most

of the previous work focused on the optimal design of the EMAT sensor, i.e., to find the

optimal parameters of the meander-line-coil EMAT, such as the length and the width of the

wires, the number of wires, the interval spacing between the wires, and the size of the

permanent magnet, etc. [28, 45, 84-86]. In addition, [22] proposed a variable-spacing

meander-line-coil to focus the shear vertical (SV) wave to a specific zone. Combining the

meander-line-coil and the spiral coil, [17] proposed a key-type coil, which is similar to the

contra-flexure coil proposed by [87]. However, there has been no report so far introducing

a meander-line-coil with variable-length wires.

In this chapter, a novel meander-line-coil with variable-length wires is proposed by the

author as described in section 6.2. The novel variable-length meander-line-coil (VLMLC)

is studied by combining an analytical EM model and an analytical US model; the analytical

EM model is used to calculate Lorentz force density, which is then fed through to the

analytical US model to study the radiation pattern of Rayleigh waves. The beam directivity

of Rayleigh waves generated by such novel EMATs is quantitatively analysed, and is

compared with the beam directivity of Rayleigh waves generated by the conventional

constant-length meander-line-coil (CLMLC). Experiments were carried out to study the

beam directivity of Rayleigh waves, and were used to validate the wholly analytical

models.

In addition, based on the novel VLMLC, two multi-directional variable-length meander-

line-coils are proposed by the author; one is the four-directional meander-line-coil

(FDMLC) and the other one is the six-directional meander-line-coil (SDMLC), as

described in section 6.3. Related experiments were conducted to study the property of

Rayleigh waves produced by such coils. Part of the work in this chapter presented by the

author has been published in the IEEE Sensors Journal [15].


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6.2 Novel Variable-length Meander-line-coil (VLMLC) EMATs

The schematic of the variable-length meander-line-coil (VLMLC) is shown in Figure

6-1(a); blue and red lines denote the bottom-layered wires and the top-layered wires of a

double-layered PCB respectively. The wires on the top layer and the bottom layer are

connected by a through-hole via. There are 12 sets of wires as shown in Figure 6-1(a); the

wires 1, 2, 11 and 12 have the same length, as well as the wires 3, 4, 9, 10 and the wires 5,

6, 7, 8. In Figure 6-1(a), the longest wire of the meander-line-coil is “L”; “s” means the

length step; “L” and “s” are two parameters to define a VLMLC. In this work, “L” used is

50 mm and “s” used is 8 mm, and therefore we name this coil as 50 mm VLMLC with a

step of 8 mm.

Figure 6-1: The configuration of the variable-length meander-line-coil (VLMLC). (a), the

schematic diagram; (b), the fabricated variable-length meander-line-coil.

A 50 mm VLMLC with a step of 8 mm is fabricated with the flexible printed circuit board

(PCB) technique; the manufactured coil is shown in Figure 6-1(b). The PCBs were made

by polyimide-based flexible laminate with the copper foil; detailed parameters for the

VLMLC are shown in Table 6-1. The operational frequency used in this work is 483 kHz,

and the velocity of Rayleigh waves used in the aluminium plate is 2.93 mm/µs, hence the

wavelength of Rayleigh waves is 6.06 mm. The distance between adjacent sets of wires,

3.03 mm, which equals to half of the Rayleigh waves’ wavelength, to form the constructive

interference.


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Table 6-1: Detailed parameters used for fabricating the variable-length meander-line-coil

(VLMLC).

Description Parameters

Sets of wires 12

Distance between adjacent sets of wires 3.03 mm

Length of the longest wire 50 mm

Step of variable-length wires 8 mm

Width of the copper foil trace 0.4 mm

Thickness of the copper foil trace 36 µm

Base material Polyimide

Thickness of the base material 170 µm

Internal diameter of the through-hole via 0.4 mm

External diameter of the through-hole via 1.0 mm

6.2.1 Wholly Analytical Models for the Novel Variable-length

Meander-line-coil (VLMLC) EMATs

The VLMLC is modelled by a wholly analytical method, which contains an analytical EM

model and an analytical US model (Figure 6-2). The analytical EM model, which is

implemented by adapted Dodd and Deeds solutions, is used to calculate the

electromagnetic induction phenomenon. The calculated Lorentz force density is imported

to the analytical US model as the driving point source to generate Rayleigh waves.

Because the analytical EM model is the same as the one used in section 4.2, this chapter

only presents the analytical US model. Table 6-2 illustrates the detailed parameters used

for the analytical US modelling.


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model.

Table 6-2: Detailed parameters used for the EMAT-US modelling.

Description Symbol Value

Length of the Aluminium plate Y 600 mm

Width of the Aluminium plate Z 600 mm

Field spatial step ∆xf 1 mm

Length of the longest wire L 50 mm

Source spatial step for each wire ∆xs 0.2 mm

Length step of the wire sets s 8 mm

Density of the Aluminium plate ρ 2700 kg/m3

Frequency f 483 kHz

Longitudinal waves’ velocity CL 6.375 mm/µs

Shear waves’ velocity CS 3.14 mm/µs

Rayleigh waves’ velocity CR 2.93 mm/µs

6.2.2 Analysis of Beam Properties of Rayleigh Waves Generated by

the Novel Variable-length Meander-line-coil (VLMLC) EMATs

6.2.2.1 Radiation Pattern

The calculated Rayleigh waves’ radiation pattern is shown in Figure 6-3; the VLMLC is

located on the centre of the aluminium plate, so the radiation pattern is symmetrical with

y=300 mm. From this image, the Rayleigh waves are mainly concentrated along the y

direction. The intensity of side lobes is quite weak compared to that of the main lobe. As


131

described in section 5.3.1, beam directivity is a reliable method to analyse the Rayleigh

waves’ distribution; beam directivity, as shown in the red arc in Figure 6-3, is at a specific

distance (“r” in Figure 6-3) from the centre of the EMAT sensor, the displacement

distribution of Rayleigh waves. In this work, r used is 250 mm; θ1 and θ2 used are -700 and

700 respectively.

Figure 6-3: The radiation pattern of the variable-length meander-line-coil (VLMLC).

6.2.2.2 Beam Directivity

The calculated beam directivity of Rayleigh waves is shown in Figure 6-4; this curve

shows the normalized magnitude of the displacement versus angles. As observed, there is a

main lobe containing a larger displacement magnitude and a number of side lobes with a

smaller displacement magnitude. The main lobe is centred at 00, and the side lobes are

distributed at undesired directions. For the 50 mm variable-length meander-line-coil with a

step of 8 mm, the HPBW calculated is 11.320; the SLL calculated is -12.01 dB.


132

Figure 6-4: The beam directivity of the 50 mm variable-length meander-line-coil (VLMLC)

with a step of 8 mm.

Conventional meander-line-coils have wires with a constant-length; the comparison

between the CLMLC and the VLMLC is shown in Figure 6-5. The red curve denotes the

beam directivity of a 50 mm-length conventional meander-line-coil, i.e. the CLMLC; the

blue curve represents the beam directivity of a 50 mm VLMLC with a step of 8 mm; both

of the curves are normalized with the peak value. From this image, the main lobe of the

VLMLC is slightly wider than that of the CLMLC. However, the magnitude of side lobes

is significantly suppressed by the VLMLC.

Figure 6-5: The beam directivity comparison between the conventional constant-length

meander-line-coil (CLMLC) and the novel variable-length meander-line-coil (VLMLC).

A good directivity plot is characterized by a narrow main lobe and small-magnitude side

lobes; a narrow main lobe means a concentrated radiated power [69]. The result of the

beam directivity is shown in Table 6-3; the proposed VLMLC enlarges the HPBW of the

main lobe slightly. Considering the beamwidth of both coils is small enough, the novel

VLMLC shows a benefit of significantly suppressing the SLL than CLMLC.

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Table 6-3: Comparison: Beamwidth and the Sidelobe Level.

50 mm meander-line-coil HPBW SLL

Convetional CLMLC

Novel VLMLC, step: 8 mm

8.760 -5.88 dB

11.320 -12.01 dB

6.2.2.3 The Effect of the Step of A Variable-length Meander-line-coil

(VLMLC) on Radiation Pattern

In this section, the effect of the step of the VLMLC on the Rayleigh waves’ radiation

pattern is studied. Various steps of the VLMLC are modelled; the modelling parameters

are the same with those used in Table 6-2 except the step of the coil. For a 50 mm VLMLC

with various steps, the beam directivity is shown in Figure 6-6.

Figure 6-6: The beam directivity of a 50 mm variable-length meander-line-coil (VLMLC).

From Figure 6-6, the step has a crucial effect on the beam directivity; with a small step, the

main lobe is narrow but the magnitude of side lobes is large; whereas with a large step, the

main lobe is slightly wider but the magnitude of side lobes is very low. The HPBW and the

SLL at various steps are shown in

Table 6-4. The 50 mm VLMLC with a step of 8 mm and 10 mm are desirable because they

maintain a small beamwidth and at the same time suppress the SLL.


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Table 6-4: HPBW and SLL at various steps.

50 mm VLMLC HPBW SLL

Step: 2 mm 9.580 -6.24 dB

Step: 4 mm 10.280 -7.1 dB

Step: 6 mm 10.940 -8.729 dB

Step: 8 mm 11.320 -12.01 dB

Step: 10 mm 11.320 -12.04 dB

6.2.2.4 The Effect of the Length of A Variable-length Meander-line-

coil (VLMLC) on Radiation Pattern

In this section, the effect of the longest wire of the variable-length meander-line-coil

(VLMLC) on the Rayleigh waves’ radiation pattern is studied. Two models are built; one

is a 40 mm VLMLC and the other one is a 50 mm VLMLC; both of these coils are built

with a step of 2 mm, 4 mm, 6 mm, and 8 mm respectively. The other modelling parameters

are the same as those used in Table 6-2.

Figure 6-7 shows the beam directivity of the VLMLC with a length of the longest wire of

40 mm and 50 mm respectively. From Figure 6-7, the 50 mm VLMLC has a narrower

main lobe compared to the 40 mm VLMLC, because the 50 mm VLMLC has more

constant-length wire segments; this observation is consistent with the results obtained in

section 5.3.2, that is, a larger length results in a more concentrated Rayleigh waves’ beam.

(a) (b)


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Figure 6-7: The comparison between the 40 mm VLMLC and the 50 mm VLMLC at

different steps.

6.2.2.5 Experimental Results and Validations

Experiments were carried out to measure the beam directivity of the VLMLC; the

experimental set-up was the same as the one used in Figure 5-7(a). The transmitter used

was a 50mm VLMLC with a step of 8 mm, and the receiver used was a 10 mm-length

CLMLC. The scan path of the receiver is shown in Figure 5-7(b); the centre-to-centre

distance between the transmitter and the receiver is 250 mm; the scan path is from -400 to

400 with a step of 2.50.

The measured beam directivity of the 50 mm VLMLC with a step of 8 mm is shown in

Figure 6-8; because the receiver was moved from -400 to 400 with a step of 2.50, only 33

sampling points were recorded, leading to a coarse curve of the measured beam directivity.

From this curve, the side lobes, which are observed in the CLMLC, are significantly

suppressed by the 50 mm VLMLC with a step of 8 mm.

(c) (d)


136

Figure 6-8: The measured beam directivity of the 50 mm VLMLC with a step of 8 mm.

In order to experimentally compare the beam directivity of the VLMLC to the CLMLC,

experiments were conducted to measure the beam directivity of the CLMLC. Experimental

set-up is the same as shown in Figure 5-7(a), except one difference that the transmitter

used is a 50 mm CLMLC. The measured directivity for both coils are shown in Figure 6-9;

the blue curve denotes the measured beam directivity of the CLMLC while the red curve

means the measured beam directivity of the VLMLC. The experimental results reveal that

the experimental beamwidth of the novel VLMLC is slightly wider than that of the

CLMLC. In addition, the magnitude of side lobes of the novel VLMLC is smaller than that

of the CLMLC. These experimental observations are consistent with modelling results.

Figure 6-9: Measured beam directivity from experiments.

A set of experiments were carried out to validate the effect of the step of 50 mm VLMLC

on the radiation pattern of Rayleigh waves. For 50 mm variable-length meander-line-coils

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(VLMLC) with various steps, the comparison between the simulated beam directivity and

the measured beam directivity is shown in Figure 6-10; the experiments showed a good

agreement with simulations, although some differences exist between the measured and

simulated curve. Possible reasons are: 1), the errors due to the positioning tolerances of the

receiver; 2), the inevitable noises of experiments.

Figure 6-10: For 50 mm VLMLC with various steps, the comparison between the simulated

beam directivity and the measured beam directivity.

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6.2.2.6 Conclusions of Section 6.2

For EMAT-Rayleigh waves’ generation, conventionally used coils are meander-line-coils

with constant-length wires. A novel meander-line-coil with variable-length wires, i.e. the

variable-length meander-line-coil (VLMLC), is proposed by the author for the first time.

Two parameters, the length of the longest wire and the step, are able to define the VLMLC.

This novel VLMLC is studied by a wholly analytical method, which contains an analytical

EM model and an analytical US model. Modelling results reveal that, the VLMLC offers a

benefit of suppressing the side lobes, which are radiated in non-desirable directions. In

addition, the step has a crucial effect on the Rayleigh waves’ beam directivity, especially

for the SSL; a large step guarantees a lower SSL.

The 50mm VLMLC with various steps were fabricated; the measured beam directivity is

consistent with the simulated beam directivity. In addition, a 50 mm CLMLC was

fabricated to experimentally compare the beam directivity between the novel and the

conventional meander-line-coils. Overall, the novel VLMLC has advantages in

suppressing the side lobes, and at the same time maintaining a narrow main lobe of the

Rayleigh waves’ beam.

6.3 Novel Multi-directional Meander-line-coil EMATs

6.3.1 Introduction

As described in Figure 6-3, the VLMLC is capable of generating Rayleigh waves in two

directions with suppressed side lobes. In this section, on the basis of the VLMLC,

multiple-directional meander-line-coils are proposed by the author; one is the four-

directional meander-line-coil (FDMLC) and the other one is the six-directional meander-

line-coil (SDMLC). These multiple directional EMATs are able to generate Rayleigh

waves in multiple directions, and at the same time suppress the side lobes.

6.3.2 Four-directional Meander-line-coil (FDMLC) EMATs

The schematic diagram of the four-directional meander-line-coil (FDMLC) is shown in

Figure 6-11(a); this square-type coil has a longest wire with a length of “L” and a spacing

between two adjacent wires of “s”; “g” denotes the gap of the leading-out end. In order to

generate Rayleigh waves, the spacing between two adjacent wires, “s”, equals to half of the

wavelength of the Rayleigh waves; the working frequency used is 483 kHz, and the


139

velocity of Rayleigh waves used in the aluminium plate is 2.93 mm/µs, so the spacing, “s”,

is 3.033 mm. The spacing determines the length of the longest wires; in this work, “L”

equals to 11 times of the spacing “s”, so L is 33.363 mm. Figure 6-11(b) shows the

manufactured FDMLC; the detailed parameters for FDMLC fabrication are shown in Table

6-5.

Figure 6-11: Four-directional meander-line-coil (FDMLC). (a), the schematic diagram of

FDMLC; (b) the fabricated FDMLC.

Table 6-5: Detailed parameters used for the four-directional meander-line-coil (FDMLC).

Description Parameters

Spacing between adjacent sets of wires (s) 3.03 mm

Length of the longest wire (L) 33.363 mm

The gap of the leading-out end 1 mm

Width of the copper foil trace 0.4 mm

Thickness of the copper foil trace 36 µm

Base material Polyimide

Thickness of the base material 170 µm

Internal diameter of the through-hole via 0.4 mm

External diameter of the through-hole via 1.0 mm

6.3.2.1 Analytical Models

The analytical model has been described in Chapter 5; the analytical model for the

FDMLC is built with the same method as shown in Figure 5-2. In order to describe the


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FDMLC clearly, an approximated model is used as shown in Figure 6-12; this model is

symmetrical with the origin, the point O.

There are 12 wires perpendicular to the y-axis; these 12 wires form a VLMLC. Similarly,

another 12 wires, which are perpendicular to the x-axis, form another VLMLC. Each

VLMLC is capable of generating symmetrical Rayleigh waves in two directions. Hence,

the FDMLC, which can be viewed as two sets of VLMLC, is able to produce Rayleigh

waves in four directions. In addition, due to the benefit of VLMLC, the FDMLC are able

to suppress the side lobes and at the same time generate multiple-directional Rayleigh

waves.

Figure 6-12: The approximated configuration of the four-directional meander-line-coil

(FDMLC).

The calculated beam directivity along angles from -450 to 450 is shown in Figure 6-13; the

magnitude is normalised by the peak of the main lobe. As expected, the side lobes are

suppressed significantly. By calculation, for this FDMLC, the HPBW and the SSL are

18.70 and -14.62 dB respectively. Because the model is symmetrical with the origin, the

beam directivity within the range from 450 to 1350, from 1350 to -1350, and from -1350 to -

450, are the same as that within angles from -450 to 450.


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Figure 6-13: The simulated beam directivity of the four-directional meander-line-coil

(FDMLC) EMAT.

6.3.2.2 Experimental Results

To verify the capacity of generating multiple-directional Rayleigh waves of the FDMLC, a

receiver is employed to receive the directly transmitted Rayleigh waves. The receiver used

was a 10 mm CLMLC; the distance between the transmitter (FDMLC) and the receiver

(CLMLC) is 250 mm. The receiver was located at 00, 900, 1800, and 2700 respectively. The

magnitude of the received Rayleigh waves is shown in Figure 6-14; from this image, the

magnitude of the Rayleigh waves received was from 0.06 volt to 0.07 volt. Ideally, the

amplitude of the Rayleigh waves should be the same, however, the difference exists

between the measured data due to experimental errors, which are within tolerance of this

work.

Figure 6-14: The magnitude of the received Rayleigh waves.

0

0.02

0.04

0.06

0.08

0 90 180 270

Mag

nit

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e(V

)

Angle (degree)

The magnitude of the Rayleigh waves

generated by FDMLC


142

To analyse the beam directivity of the FDMLC, the receiver was located at the position

with a distance of 250 mm from the FDMLC; the scan path is shown in Figure 6-15(a), the

receiver was moved from -400 to 400 with a step of 2.50. The measured beam directivity is

shown in Figure 6-15(b); from the experimental beam directivity, the FDMLC has a

suppressed side lobe; this observation is consistent with the calculated results.

Figure 6-15: (a), The scan path of the receiver; (b), the experimental beam directivity.

The comparison between the simulated beam directivity and the measured beam directivity

is shown in Figure 6-16; the blue curve denotes the simulated beam directivity while the

red curve with markers denotes the measured beam directivity. Although some differences

exist between these two curves, the overall trend between these two curves is the same.

Overall, this FDMLC has advantages of suppressing side lobes and generating multiple

directional Rayleigh waves.

Figure 6-16: The simulated beam directivity and the measured beam directivity of FDMLC.


143

6.3.3 Six-directional Meander-line-coil (SDMLC) EMATs

The schematic diagram of the six-directional meander-line-coil (SDMLC) is shown in

Figure 6-17(a); “L” means the length of the longest wire; “s” means the spacing between

two adjacent wires; “g” means the gap of the leading-out end. In order to generate

Rayleigh waves, the spacing “s” is 3.033 mm, which is determined by the working

frequency, 483 kHz, and the Rayleigh waves’ velocity within the aluminium plate, 2.93

mm/µs. The length of the longest wire is determined; in this work, “L” is 19.261 mm. The

fabricated SDMLC is shown in Figure 6-17(b); the other parameters for the fabrication has

been clarified in Table 6-5.

Figure 6-17: Six-directional meander-line-coil (SDMLC). (a), the schematic diagram of

SDMLC; (b) the fabricated SDMLC.

6.3.3.1 Analytical Models

An approximated model for the SDMLC is shown in Figure 6-18; this model is

symmetrical with the origin, “O”. The SDMLC can be viewed as three sets of the VLMLC;

each set of the VLMLC contains 12 wires. Due to these three sets of the VLMLC, the

SDMLC is able to generate Rayleigh waves in 6 directions, which are along 00, 600, 1200,

1800, -1200, and -600 respectively.


144

Figure 6-18: The approximated model for the six-directional meander-line-coil (SDMLC).

The beam directivity of this approximated model of the SDMLC is shown in Figure 6-19,

which shows the normalised magnitude of the Rayleigh waves versus angles from -600 to

600. As expected, the maximum magnitude occurs at 00, -600 and 600 respectively. From -

300 to 300, the magnitude of Rayleigh waves is determined by the VLMLC along the angle,

00. For angles from -600 to -300 and from 300 to 600, the magnitude of Rayleigh waves is

determined by the other two sets of VLMLC which are along the angles 600 and 1200

respectively. The calculated HPBW and the SSL are 31.740 and -7.57 dB respectively.

Figure 6-19: The simulated beam directivity of the six-directional meander-line-coil (SDMLC)

EMAT

6.3.3.2 Experimental Results

Several experiments were conducted to study the beam property of the SDMLC. The

transmitter used was the SDMLC as clarified in Figure 6-17(b); the receiver used was the

-60 -40 -20 0 20 40 600

0.2

0.4

0.6

0.8

1

X: -30

Y: 0.1749

Beam directivity of SDMLC EMAT

Angle (degree)

Norm

alis

ed M

agnitude

X: 30

Y: 0.1749


145

same as that used in section 6.3.2.2. At a centre-to-centre distance of 250 mm from the

transmitter, the receiver was placed at 00, 600, 1200, 1800, 2400, and 3000 respectively to

verify the ability of generating multiple-directional Rayleigh waves. Figure 6-20 shows the

maximum magnitude of the received Rayleigh waves versus various angles; the received

amplitude was from 33 mV to 44 mV; possible reasons for the non-uniform amplitude are

the experimental noise, and possibly the errors due to the positioning of the receiver.

Figure 6-20: The magnitude of the received Rayleigh waves.

The beam directivity of the SDMLC was experimentally studied; the scan path is shown in

Figure 6-21(a). The receiver was moved from -400 to 400 at a distance of 250 mm from the

transmitter. The measured beam directivity of the SDMLC is shown in Figure 6-21(b). As

expected, the maximum magnitude of the received Rayleigh waves occurs at 00 and the

minimum magnitude occurs at -300 and 300. The amplitude of the received Rayleigh waves

increases from -300 to -400 and from 300 to 400 due to the effect of the other two sets of

VLMLC which are along the angles -600 and 600 respectively.

Figure 6-21: (a), The scan path of the receiver; (b), the experimental beam directivity.

0

0.01

0.02

0.03

0.04

0.05

0 60 120 180 240 300

Mag

nit

ud

e (V

)

Angle (degree)

The magnitude of Rayleigh waves

generated by SDMLC


146

The comparison between the calculated beam directivity and measured beam directivity of

the SDMLC is shown in Figure 6-22, where the blue curve denotes the simulated beam

directivity whereas the red curve denotes the measured beam directivity. The trend of the

simulated beam directivity shows a good agreement with that of the measured beam

directivity. Some differences exist between these two curves due to the inevitable

experimental error.

Figure 6-22: The simulated beam directivity and the measured beam directivity of SDMLC.

6.3.4 Discussion

Two novel meander-line-coil EMATs to generate multiple-directional Rayleigh waves are

proposed by the author for the first time; one is the four-directional meander-line-coil

(FDMLC) and the other one is the six-directional meander-line-coil (SDMLC). These

multiple-directional Rayleigh waves’ meander-line-coils can be viewed as several sets of

variable-length meander-line-coils (VLMLC), so multiple-directional Rayleigh waves’

meander-line-coils are capable of suppressing side lobes. The beam property of Rayleigh

waves generated by the multiple-directional meander-line-coil EMATs is studied

analytically and analysed experimentally. For the pulse-echo detection, due to the capacity

of generating multiple-directional Rayleigh waves, multiple-directional Rayleigh waves

meander-line-coil EMATs increase the monitoring area and improve the monitoring time.

6.4 Conclusions

In this chapter, several novel configurations for meander-line-coil EMATs are proposed by

the author for the first time. Firstly, a novel meander-line-coil, the VLMLC, is proposed

and studied. Both the simulated and experimental results suggest that the VLMLC has a


147

benefit of significantly suppressing side lobes of Rayleigh waves over the CLMLC. The

step of the VLMLC has a significant effect on the beam directivity of Rayleigh waves; a

larger step guarantees a lower SSL but slightly widens the HPBW of the main lobe.

On the basis of the VLMLC, two multiple-directional Rayleigh waves’ EMATs, the four-

directional meander-line-coil (FDMLC) and the six-directional meander-line-coil

(SDMLC), are proposed by the author. FDMLC and SDMLC can be viewed as several sets

of VLMLC; and therefore both of these coils are able to generate Rayleigh waves in

multiple directions and at the same time suppressing the side lobes. The multiple-

directional Rayleigh waves EMATs are advantageous in applications of inspecting large

samples. However, multiple-directional Rayleigh waves EMATs decrease the sensitivity of

flaw detection. Multiple-directional Rayleigh waves EMATs pose a possibility in the

future, that is combining multiple-directional EMATs and unidirectional EMATs together,

especially for large specimen detection; more specifically, multiple-directional EMATs can

be used to provide an initial estimation of the flaw location and unidirectional-EMATs can

be used to locate the flaw more precisely.

CHAPTER 7 Conclusions and recommendations for future work

148

Chapter 7 Conclusions and Recommendations for Future

Work

7.1 Conclusions

This thesis focuses on the meander-line-coil EMATs to generate Rayleigh waves based on

the Lorentz force mechanism. Although considerable works have been reported on the

study of meander-line-coil EMATs by means of simulations and experiments, there are

still many important issues, such as the radiation pattern and the beam features of Rayleigh

waves, which need further investigation. The overall aim of this work was to explore novel

efficient modelling techniques and novel configurations for meander-line-coil EMATs. To

achieve this aim, the author proposed two novel modelling methods for simulating

different planes of the test piece; the first method combines the analytical method and the

finite-different time-domain (FDTD) method for simulating the vertical plane of the

material (Chapter 4); the second method utilizes a wholly analytical model to simulate the

horizontal surface plane of the material (Chapter 5). The author further proposed three

novel configurations for meander-line-coil EMATs, the radiation patterns and beam

directivities of which were analytically studied with a wholly analytical model, and

validated by experiments (Chapter 6).

7.1.1 FDTD Method for Simulating US Behaviours

The FDTD solver, SimSonic [72], was further developed / adapted by the author for

simulating ultrasound behaviours, such as steering, focusing and scattering. Several

findings were made in Chapter 3:

1) The Hilbert Transformation and the FDTD method can be combined together to

produce the radiation pattern of ultrasound waves, which in turn can be used to analyse

the beam features.

2) The beam features in the full-field inspection were investigated; simulated results

revealed that, for the near field inspection, the focusing technique provides a better

beam directivity and a more concentrated beam intensity than the steering technique; for

the far field inspection, the steering technique is advantageous because the steering

technique requires less modelling time than the focusing technique, and at the same


149

time provides a good beam directivity. Therefore, for a full-field inspection, the

focusing technique is normally used for near field inspections and the steering technique

is normally used for far field inspections; these numerical results based on FDTD

method were consistent with the analytical results presented in [69].

In addition, the FDTD method was used to model the scattering behaviours of ultrasound

waves; results revealed both the directly transmitted ultrasound waves and the scattered

ultrasound waves can be captured by the receiver. Overall, the study of FDTD provided a

solid foundation for the EMAT simulations introduced in Chapter 4.

7.1.2 Vertical Plane Modelling for EMATs

A novel modelling method combining the analytical method and the FDTD method

together to build EMAT models was proposed by the author for the first time, as described

in Chapter 4. This novel modelling method was a 2D simulation focusing on the vertical

plane of the material; the EM model was built by the adapted analytical method for

calculating Lorentz force densities, and the US modelling was carried out by the FDTD

method for simulating the propagation of ultrasound waves. There are a number of

conclusions drawn from the simulated and experimental results:

1) Because the coil used in this work was a meander-line-coil, the analytical solution to a

straight wire was needed. The strategy of adapting the analytical solution for a circular

coil to that for a straight wire was proposed by the author, and was validated by the

finite element method (FEM). Results revealed that the analytical solution to the straight

wire can be achieved by enlarging the radius of the circular coil; in addition, adapted

analytical solution was advantageous over FEM; both at a low frequency and at a high

frequency.

2) For the vertical plane modelling, the excitation signal used was a Gaussian-modulated

sinusoidal wave with a specific central frequency and a specific fractional bandwidth.

The effect of the fractional bandwidth of the excitation signal on the interference

behaviours of Rayleigh waves was investigated; results showed that, with a smaller

fractional bandwidth, the interference of Rayleigh waves was better. More specifically,

for the specific configuration EMAT used in this work, Rayleigh waves showed a good

interference with a fractional bandwidth equalling to or smaller than 0.3.


150

3) The depth profile of Rayleigh waves, the radiation pattern, and the beam features of

Rayleigh waves indicated the Rayleigh waves are mainly concentrated in a depth within

one wavelength of Rayleigh waves. At a depth beyond one wavelength, Rayleigh waves

decreased by 80% or higher.

4) Experiments were conducted to validate the proposed modelling methods; by means of

the root-mean-square error (RMSE) and the correlation coefficient, experiments showed

a good agreement with simulations: the RMSE and the correlation coefficient between

experiments and simulations were 0.0834 and 0.9927 respectively.

5) Simulations and experiments showed that the attenuation of Rayleigh waves along the

surface of the material was very small; with a travelling distance of 5 cm, the

attenuation was 0.0034% from simulations and 0.0039% from experiments; this

observation further proved the capacity of Rayleigh waves for a long distance inspection.

6) The scattering behaviours of Rayleigh waves was studied by this novel modelling

method; results showed a good agreement between experiments and simulations; the

RMSE and the correlation coefficient for the scattering behaviour were 0.1159 and

0.9241 respectively. Further, unidirectional Rayleigh waves were modelled; the

Rayleigh waves signal was strengthened by 1.9849 in one direction while was

weakened by 0.2336 in the other direction; these results based on the proposed novel

modelling method were consistent with the numerical results presented in [23].

7.1.3 Surface Plane Modelling for EMATs

A novel modelling strategy based on a wholly analytical method was proposed by the

author for the first time, as described in Chapter 5. This novel modelling method was a 2D

simulation focusing on the surface plane of the material, and therefore it extended the 2D

simulations described in Chapter 4 to pseudo – 3D cases. Based on this novel modelling

method, the beam directivity of Rayleigh waves generated by meander-line-coil EMATs

were studied; some conclusions are presented below:

1) Based on the wholly analytical model, a main lobe and several side lobes of the

Rayleigh waves beam were observed for the conventional meander-line-coil EMATs;

the main lobe of the Rayleigh waves was along the normal direction of the meander-

line-coil, whereas the side lobes of Rayleigh waves radiated in undesirable directions.


151

2) The effect of the length of the meander-line-coil on the radiation pattern of Rayleigh

waves was analytically studied; results showed that, with a larger length, the half

power beamwidth (HPBW) of the main lobe was smaller but the magnitude of sidelobe

level (SLL) was larger; with a length from 10 mm to 65 mm, the HPBW decreased by

82.32% but the SSL increases by 64.62%.

3) Experiments were carried out to study the beam directivity of Rayleigh waves and to

validate the proposed wholly analytical modelling method; experimental results

showed a good agreement with the simulated results, and therefore this novel

modelling method, focusing on the surface place of the material, is capable of

analysing the beam directivity of Rayleigh waves.

7.1.4 Novel Configurations for EMATs

Several novel configurations for meander-line-coil EMATs were proposed by the author

for the first time and described in Chapter 6. These novel meander-line-coil EMATs were

studied analytically and experimentally; some findings are presented below:

1) With the novel variable-length meander-line-coil (VLMLC), the side lobes of the beam

of Rayleigh waves, observed in the conventional constant-length meander-line-coil

(CLMLC), were significantly suppressed. However, the HPBW of the main lobe

increased slightly for VLMLC.

2) The effect of the step of the VLMLC on the radiation pattern of Rayleigh waves was

investigated; results revealed that, a larger step guaranteed a lower SSL but a larger

main lobe. For the 50 mm VLMLC with a step from 2 mm to 10 mm, the HPBW of the

main lobe increased by 18.16% and the SSL decreased by 92.95%.

3) Two multiple-directional Rayleigh waves meander-line-coil EMATs were proposed by

the author, the four-directional meander-line-coil (FDMLC) and the six-directional

meander-line-coil (SDMLC). From analytical and experimental studies, these EMATs

can be viewed as a combination of several sets of VLMLC, and therefore they were

capable of suppressing side lobes. In addition, experiments showed these novel

multiple-directional Rayleigh waves were capable of simultaneously generating

Rayleigh waves in four or six directions.


152

7.2 Recommendations for Future Work

Based on the conclusions drawn from this study, future work is recommended to further

the study of EMATs.

1) For vertical plane modelling, as shown in Chapter 4, the EMAT-US model used was an

approximated model using only point sources (Lorentz force density) for generating

Rayleigh waves; a more detailed model, with volume sources (Lorentz force density)

within the skin depth, is worth considering in the future.

2) Throughout this study, the dynamic magnetic field generated by the meander-line-coil

was neglected due to the small excitation AC current. Although the effect of the

dynamic magnetic field is very small with a small excitation AC current, taking the

dynamic magnetic field into account will increase the precision of the calculation. It

should be noted that when the AC current is large, the dynamic magnetic field must be

considered as it is one of the mechanisms to generate ultrasound waves [39].

3) For EMAT scattering behaviours, as shown in section 4.7, only the scatter normal to

the surface was studied. It will be worth investigating the scattering behaviours of

scatters in other orientations as it is a variable in practical applications.

4) For the beam directivity measurement, as shown in Chapter 5 and Chapter 6, the

receiver used was a 10 mm-length meander-line-coil, which has its own radiation

pattern as a transmitter. In order to increase the spatial resolution of the received signal,

a miniature receiver should be used. [23] employed a miniature receiver, Pinducer,

which is a displacement sensor with elements of 0.053 inch diameters, to receive the

Rayleigh waves generated by the meander-line-coil EMATs; it is worthwhile trying

this miniature receiver in the future. In addition, the Pinducer is made of piezoceramic,

and therefore it poses a possibility of combining the conventional piezoelectric

transducers and the electromagnetic acoustic transducers (EMATs) together.

5) Several novel configurations for EMATs were proposed in this work; the optimal

parameters for such novel configurations of EMATs, such as the width of the wire and

the size of the permanent magnet, are worth investigating for the optimal design of

such sensors.

6) The multiple-directional Rayleigh waves EMATs proposed in section 6.3 are capable

of inspecting large specimens but with a low sensitivity of flaw detections. [23]

proposed a unidirectional Rayleigh waves EMATs as introduced in section 4.8. In


153

practical applications, the capacity of combining the multiple-directional Rayleigh

waves EMATs to provide the initial estimation of the flaw location and the

unidirectional EMATs to locate the flaw precisely is worth investigating.

7) This study mainly focuses on the meander-line-coil EMATs operated on Lorentz force

mechanism for Rayleigh wave generation. However, the methodology for sensor

analysis and design can be extended to other type of EMATs; using this methodology

to investigate the property of bulk waves EMATs and Lamb waves EMATs can be a

focus in the future.

Extending from the work included in the thesis, one promising filed is the body inspection

with SV waves. As describe previously, the meander-line-coil is capable of generating

Rayleigh waves, longitudinal waves and shear vertical (SV) waves simultaneously, from

which SV waves have a large magnitude as Rayleigh waves, and therefore SV waves are

capable for body inspections. Based on the methodology proposed by the author, as shown

in Chapter 4, the property of SV waves generated by meander-line-coil EMATs can be

investigated. In addition, SV waves can be steered or focused with prescribed parameters,

such as the operational frequency and the spacing between two adjacent wires of the

meander-line-coil [22, 37]; the steering and focusing behaviours of SV waves can be

investigated by the modelling method proposed by the author. Further, the FDMLC and the

SDMLC are capable of generating SV waves in the material in multiple directions; these

multiple directional SV EMATs can be a design starting point for producing EMATs-SV

waves tomography in the future.

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