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Microwave detection and characterization of sub-surface defect properties in composites using an open ended rectangular waveguide

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D ISSE R T A T IO N
MICROWAVE DETECTION AND CHARACTERIZATION
OF SUB-SURFACE DEFECT PROPERTIES IN COMPOSITES
USING AN OPEN ENDED RECTANGULAR WAVEGUIDE
Submitted by
Nasser Nidal Qaddoumi
Department of Electrical Engineering
In partial fulfillment of the requirements
for the degree of Doctor of Philosophy
Colorado State University
Fort Collins, Colorado
Spring 1998
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UMI Number: 9 835027
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COLORADO STATE UNIVERSITY
February 17, 1998
WE THEREBY RECOMMEND THAT THE DISSERTATION PREPARED
UNDER OUR SUPERVISION BY NASSER NIDAL QADDOUMI ENTITLED
MICROWAVE DETECTION AND
DEFECT
PROPERTIES
IN
CHARACTERIZATION OF
COMPOSITES
USING
AN
SUB-SURFACE
OPEN
RECTANGULAR WAVEGUIDE BE ACCEPTED AS FULFILLING
ENDED
IN PART
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY.
Committee on Graduate Work
Adviser
Department Head
ii
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ABSTRACT OF DISSERTATION
MICROWAVE DETECTION AND CHARACTERIZATION
OF SUB-SURFACE DEFECT PROPERTIES IN COMPOSITES
USING AN OPEN ENDED RECTANGULAR WAVEGUIDE
Near-field microwave imaging of dielectric composite structures, using open-ended
rectangular waveguides, has shown to be a promising and powerful nondestructive testing
(NDT) tool for the evaluation of these structures. Experimentally obtained raw images
provide a great deal of detailed information about the properties of a specimen. To better
interpret the information contained in such images it is important to develop theoretical
models that explain the behavior of microwave energy inside a specimen under inspection.
This will aid in the development of a methodology to obtain information about the shape
and dimensions o f a defect or inclusion from such image.
Near-field microwave imaging is based on transmitting a high frequency wave into
a dielectric structure, which is located in the near-field of a sensor, and using a signal
proportional to the magnitude or phase of the transmitted or reflected wave to create a two
or three dimensional image of the structure under investigation. To analyze the features and
properties of an image, it is important to understand the mechanism by which the incident
electric and magnetic fields interact with the structure.
In chapter 2 a theoretical study is conducted to expand on and demonstrate the
ability of utilizing an open-ended rectangular waveguide probe to monitor the existence and
iii
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to determine the position of a disbond in a layered composite structure. The analyses and
procedures applied in detecting and locating layers o f air (disbonds) can be applied to detect
any defective dielectric layer. The transverse to the direction of propagation extent of the
disbond was assumed to be large enough to consider the disbond a layer. In practice, the
extent of a defect is not always larger than the aperture size in addition to the fact that large
defects have edges which may significantly contribute to the scattering and diffraction from
these defects. In chapter 3, near-field microwave imaging of dielectric composite structures
using open-ended rectangular waveguides is studied experimentally. Experimental setups
are presented and their operations are discussed.
The utility of applying near-field
microwave techniques to inspect a wide variety of composite structures with different types
of defects is demonstrated, and several experimental results are presented.
The
experimentally obtained raw images provide a great deal of detailed information about the
structure under inspection.
A near-field microwave image is the result of several factors such as the probe type
(for example a rectangular waveguide, a circular waveguide, a coaxial line, etc.), field
properties (i.e. main lobe, sidelobes, and half-power beamwidth, etc.), geometrical and
physical properties of both the defect and the material under inspection.
Therefore, it is
important to develop theoretical models that explain the behavior of microwave energy
inside the structure under inspection.
Chapter 4 will be devoted to study the field
properties in the near-field region of an open-ended rectangular waveguide and its
interaction with a dielectric material. This study will include investigating the influences of
frequency and dielectric properties on the radiation pattern.
In chapter 5 a study of the
mechanism by which the fields interact with an inclusion will be presented. An effective
dielectric constant formula will be used to model the reflection properties of dielectric
structures.
The influence of the non-uniformity associated with the electric field
iv
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distribution at the aperture of the waveguide will be investigated and incorporated in
calculating the effective dielectric constant.
Nasser Nidal Qaddoumi
Department of Electrical Engineering
Colorado State University
Fort Collins, CO 80523
Spring 1998
V
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ACKNOWLEDGMENTS
First and foremost I would like to thank God for all that I am and all that I have. I
would like to offer my sincere appreciation to those who supported me during this
endeavor.
I am very grateful for the patient guidance and the endless support and
encouragement given by my advisor and mentor Dr. R. Zoughi. I would also like to
express my gratitude to Dr. D. Lile, Dr. C. Menoni, and Dr. M. Peterson for their input
and serving on my committee. I would like to thank in particular Dr. H. Abiri, Dr. S .
Bakhtiari, Dr. S. Ganchev, Mr. K. Bois, Mr. E. Ranu, and everyone at the Applied
Microwave NDT Laboratory for their help and encouragement during this work. I would
also like to express my gratitude to the staff of the Electrical Engineering Department for
their encouragement and support.
I would like to thank my parents for their guidance and continuing support in every
aspect of my life.
To my wife Rania and my daughter Lana I must express my sincere appreciation for
their continuing encouragement and love.
vi
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TABLE OF CONTENTS
CHAPTER I
Introduction.............................................................................................................
1
1.1 Background................................................................................................
I
1.2 Motivation....................................................................................
2
1.3 Previous Research Findings...................................................................
3
1.4 Potential Impact and Benefits...............................................................
4
15 Methodology.............................................................................................
5
CHAPTER II
M icrow ave D isbond D etection an d D epth D eterm ination
in Thick Layered Structures..............................................................................
11.1 Variational Formulation o f Aperture Admittance............................
8
10
11.2 Radiation From Rectangular Waveguide into Stratified
Composite Media...........................................................................................
16
11.2.1 Theoretical Formulation..........................................................
16
11.2.2 Termination o f Layered Media into an Infinite
II5
Half-Space..................................................................................
19
Theoretical Results...............................................................................
21
II.3 .1 Sample Description..................................................................
21
11.32 Standoff Distance and Frequency Analyses........................
23
11.3.2.1 Ku-Band (12-18 GHz) Results.......................................
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24
11.3 .2 2 K-Band (18-265 GHz) Results.....................................
29
11.3 .2 3 Ka-Band (263-40 GHz) Results...................................
36
11.3.3 Potential o f Disbond Depth Determination.......................
45
11.33.1 Ka-band...........................................................................
45
11.33.2 K-band..............................................................................
48
113.4 Real Part o f the Reflection Coefficient................................
51
II.3 3 Ka-Band, 1 mm Thick Disbond...............................................
52
II.4 Summary and Remarks..........................................................................
53
CHAPTER III
N ear-F ield R ectan gu lar Waveguide Probes Used f o r Imaging....
56
III.I Imaging Setups and Techniques.........................................................
58
111.2 Optimizing Scan Parameters.............................................................
64
111.3 Applications and Experimental Results..........................................
<56
III3 .1 Near-Field Imaging o f Thick Composites With
Metallic Defects.....................................................................
66
111.3.1.1 Measurement Results and Discussion.....................
67
III.3.2 Near-Field Imaging o f Rust Under Paint and
Dielectric Laminates.............................................................
71
111.3.2.1 Measurement Results and Discussion.....................
71
111.3.3 Near-Field Imaging o f Composites With Porosity
Defects....................................................................................
75
111.3.3.1 Measurement Results and Discussion......................
76
111.3.4 Near-Field Imaging o f Fiberglass Composites With
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Variable Binder Percentage and Cure State......................
78
111.3.4.1 Measurement Results and Discussion..................
79
III.3 5 Near-Field Imaging o f Composites With Impact
Damage Defects................................................................................
82
111.3 S . 1 Measurement Results and Discussion...................
82
III.4 Summary and Remarks................................................................................
84
CHAPTER IV
Analysis o f th e F ield Properties in the Near-Field o f
Rectangular W aveguide Probes U sed f o r Imaging......................................
86
IV. I
Radiation Pattern Theoretical Modeling...................................................
86
IV.2
Fields in an Infinite Half-Space o f a Dielectric Material.......................
91
IV.3
Normalized Power Patterns in Different Planes.....................................
102
IV.3.1 Influence o f Frequency on the Radiation Pattern.........................
106
IV.3.1.1 Influence o f the Waveguide Dimensions.............................
106
IV.3 .1 2 Influence of The Frequency Within The Same Band
112
TV.3.2 Influence o f the Dielectric Properties on the
Radiation Pattern.............................................................................
114
IV.3 2 .1 Influence o f Permitivitty......................................................
114
IV.3.2 2 Influence o f Loss Factor.....................................................
116
IV.4 Summary and Remarks.................................................................................
720
CHAPTER V
Influences o f The Effective D ielectric Constant and N onL in ear Probe A pertu re Field D istribu tion on Near-Field
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M icrowave Images...................................................................................................
122
V.l The Effective Dielectric Constant o f A Medium.......................................
123
V.2 Effective Dielectric Constant Formulae.....................................................
124
V 2.1
Far-Field vs. Near-Field...............................................................
124
V 2.2
Volume Fraction Calculation.........................................................
126
V.2.3 Effective Dielecrtic Constant Formulae.......................................
130
V.2.3.1 Simple Average Effective Dielectric
Constant Formula...........................................................
130
V.2.3 2 Rayleigh Effective Dielectric Constant Formula
131
V .2 3 3 K.Wakin’s Effective Dielectric Constant Formula...
131
V2.3.4 Comparison.....................................................................
132
V 3 Results............................................................................................................
138
V3 . 1 Two Infinite Half-Spaces...............................................................
139
V32
Surface Slab in An Infinite Half-Space.........................................
143
V33
Deep Slab in An Infinite Half-Space............................................
146
V 3.4
Deep Slab in A Multi-Layered Structure.....................................
149
V.4 Summary and Remarks.................................................................................
152
CH APTER VI
Conclusions.............................................................................................................
154
BIBLIOGRAPHY.........................................................................................................
160
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LIST OF TABLES
2.1
Phase differences at 25.3 GHz and at I mm standoff distance for all disbonds... 49
2.2
Phase differences at 30.3 GHz and at 1 mm standoff distance for all disbonds... 52
4.1
The ratios of the a-dimensions of x, ku- and k-bands and the half­
power widths associated with different frequencies in these bands..................
4.2
108
The ratios of the a-dimensions of the x, ku- and k-bands and the half­
power widths associated with the same frequencies in these bands.................. 110
4.3
The half-power widths for materials with different permittivities at 24 GHz
4.4
The distances at which the power density drops to 32.8% of its original value for
115
materials with different permittivities at 24 GHz............................................. 116
4.5
The half-power widths for materials with different loss factors at24 GHz
4.6
The distances at which the power density drops to 32.8%of its original value for
117
materials with different loss factors at 24 GHz................................................ 117
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LIST OF FIGURES
2.1
Aperture of an arbitrary cross-section opening in a perfectly conducting ground
plane of infinite extent radiating into an infinite half-space................................
10
2.2a Cross-section o f a rectangular waveguide radiating into a layered media terminated
into an infinite half-space...................................................................................
16
2.2b Cross-section of a rectangular waveguide radiating into a layered media terminated
into a conducting sheet........................................................................................
17
2.3
Schematic o f the sandwich com posite...............................................................
22
2.4
Phase of the reflection coefficient as a function of frequency at various standoff
distances for the case of no disbond in the composite.......................................
2.5
Phase of the reflection coefficient as a function of frequency at various standoff
distances for the case of a disbond under the skin laminate (1st disbond)
2.6
27
Phase difference between no disbond and the third disbond as a function of
frequency at various standoff distances............................................................
2.9
26
Phase difference between no disbond and the second disbond as a function of
frequency at various standoff distances............................................................
2.8
25
Phase difference between no disbond and the first disbond as a function of
frequency at various standoff distances............................................................
2.7
25
27
Phase difference between no disbond and the forth disbond as a function of
frequency at various standoff distances..............................................................
28
2.10 Phase difference between no disbond and the fifth disbond as a function of
frequency at various standoff distances..............................................................
28
2.11 Phase difference between no disbond and the sixth disbond as a function of
frequency at various standoff distances..............................................................
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29
2.12 Phase of the reflection coefficient as a function of frequency at various
standoff distances for the case of no disbond in the composite...........................
31
2.13 Phase of the reflection coefficient as a function of frequency at various standoff
distances for the case of a disbond under the skin laminate (1st disbond).............
31
2.14 Phase difference between no disbond and the first disbond as a function o f
frequency at various standoff distances..............................................................
32
2.15 Phase difference between no disbond and the second disbond as a function of
frequency at various standoff distances..............................................................
32
2.16 Phase difference between no disbond and the third disbond as a function of
frequency at various standoff distances..............................................................
33
2.17 Phase difference between no disbond and the forth disbond as a function of
frequency at various standoff distances............................................................
33
2.18 Phase difference between no disbond and the fifth disbond as a function of
frequency at various standoff distances............................................................
34
2.19 Phase difference as a function of frequency at a standoff distance of 1 mm
due to the first disbond......................................................................................
35
2.20 Phase difference as a function of frequency at a standoff distance of 1 mm
due to all other disbonds.....................................................................................
35
2.21 Phase of the reflection coefficient as a function of frequency at various
standoff distances for the case of no disbond in the composite...........................
37
2.22 Phase of the reflection coefficient as a function of frequency at various standoff
distances for the case of a disbond under the skin laminate (1st disbond).............
38
2.23 Phase difference as a function of frequency at various standoff distances for the
case o f a disbond under the skin laminate (1st disbond)....................................
38
2.24 Percent magnitude change as a function of frequency at various standoff distances
for the case of a disbond under the skin laminate (1st disbond)..........................
2.25 Phase difference as a function of frequency at various standoff distances
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39
for the second disbond.........................................................................................
39
2.26 Percent magnitude change as a function of frequency at various standoff
distances for the second disbond.......................................................................
40
2.27 Phase difference as a function of frequency at various standoff distances
for
the third d isb o n d ............................................................................................
40
2.28 Percent magnitude change as a function of frequency at various standoff
distances
for the third disbond...........................................................................
41
2.29 Phase difference as a function of frequency at various standoff distances
for
the forth d isb o n d .............................................................................................
41
2.30 Percent magnitude change as a function of frequency at various standoff
distances
for the forth disbond...........................................................................
42
2.31 Phase difference as a function of frequency at various standoff distances
for
the fifth d isb o n d.............................................................................................
42
2.32 Percent magnitude change as a function of frequency at various standoff
distances
for the fifth disbond............................................................................
43
2.33 Phase difference as a function of frequency at various standoff distances
for
the sixth d isb o n d .............................................................................................
43
2.34 Percent magnitude change as a function of frequency at various standoff
distances
for the sixth disbond............................................................................
44
2.35 Phase of the reflection coefficient as a function of frequency for all disbonds
,
in
co n tact...............................................................................................................
46
2.36 Phase of the reflection coefficient as a function of frequency for all disbonds
at 2 mm standoff distance..................................................................................
47
2.37 Phase difference as a function of frequency for the second, third, forth,
fifth and sixth disbond at 2mm standoff distance.............................................
47
2.38 Phase difference as a function of frequency at a standoff distance of 1 mm
due to the first disbond.......................................................................................
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50
2.39 Phase difference as a function of frequency at a standoff distance of 1 mm
due to all other disbonds........................................................................................
50
2.40 Percentage change in the real part of the reflection coefficient as a function
of frequency for the second through the sixth disbond.........................................
52
2.41 Phase difference as a function of frequency at various standoff distances
for the second disbond with a thickness of 1 mm................................................
3.1
53
Relative Geometry of an open-ended rectangular waveguide sensor and a thick
composite panel with a defect: (a) side view, (b) plan view..................................
59
3.2
A general near-field microwave imaging experimental setup................................
60
3.3
A single reflectometer module used to produce magnitude and phase images
3.4
A single reflectometer module used to produce magnitude images........................
63
3.5
A single reflectometer module used to produce phase images...............................
63
3.6
Standing wave patterns in a waveguide produced with and without a defect
62
65
3.7 Change in voltage across three separate diode detectors at 10 GHz with
respect to
3.8
input power to the diode.....................................................................
66
Descriptive geometry of a thick composite panel with an aluminum
inclusion embedded at the center of the panel......................................................
68
3.9 An in contact phase scan of the composite shown in Figure 3.8 at a
frequency
of
10.5 G H z............................................................................................
69
3.10 The voltage (related to the phase) with and without a defect as a function
of the
stan d o ff distance...........................................................................................
70
3.11 A phase scan of the composite shown in Figure 3.8 at a standoff distance
of 9 mm and a frequency o f 10.5 GHz................................................................
70
3.12 A 40 mm by 40 mm area of rust on a steel plate................................................
72
3.13 Image of rust under 0.145 mm paint at 24 GHz and a standoff distance of 4 mm....
3.14 Image o f rust under 0.60 mm paint at 24 GHz and a standoff distance of 4 mm
3.15 Image of rust under 25.4 mm laminate at 24 GHz and a standoff distance
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73
73
of 4
74
mm
3.16 Image o f rust under 25.4 mm laminate at 10 GHz and a standoff distance
of
4
m m ....................................................................................................................
75
3.17 The schematic of an epoxy resin sample with three different levels of
local
p o ro sity ...........................................................................................................
77
3.18 Image o f the sample shown in Figure 3.17 at a frequency of 34.8 GHz:
(a) plan view, (b) signal intensity.....................................................................
78
3.19 Schematic o f the multibinder fiberglass sample................................................
80
3.20 Voltage difference as a function of the standoff distance for all the
local defects at 24 GHz....................................................................................
81
3.21 Image o f the sample shown in Figure 3.19 at a frequency of 24 GHz
and a standoff distance of 4 m m ........................................................................
81
3.22 Image o f the 4 ply disk at a frequency of 34.8 GHz after 500 impact cycles
83
3.23 Image o f the 8 ply disk at a frequency of 34.8 GHz after 3000 impact cycles
83
3.24 Image o f the 8 ply disk at a frequency of 34.8 GHz after 4000 impact cycles
84
4.1
92
An open-ended rectangular waveguide aperture radiating into an infinite half-space.
4.2 The normalized power pattern in the xz-plane ( y = ^ plane) at 24 GHz
inside a material with er = 2 J - j 0 J ...........................................................................
104
4.3 The normalized power pattern in the yz-plane ( x = j plane) at 24 GHz
inside a material with £r = 2.5 - j 0 5 ...........................................................................
104
4.4 The normalized power pattern in the xy-plane (z=l mm plane) at 24 GHz
inside a material with £r = 2.5 - j 0 5 ...........................................................................
4.5
105
Plan view image of a phase scan at a standoff distance of 3.8 mm at 9.2 GHz,
a) E-field is parallel to vertical axis, b) E-field is parallel to the horizontal axis
4.6 The normalized power patterns in the x-direction at 10, 14 and 22 GHz
xv i
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105
inside a material with e r = 5 - jO .l...............................................................................
4.7
The normalized power patterns in the x-direction at 12.4 GHz in the xand ku-bands inside a material with er =5 —jO .l.......................................................
4.8
109
The normalized power patterns in the x-direction at 18 GHz in the ku- and
k-bands inside a material with er = 5 - jO .l................................................................
4.9
108
109
The normalized power patterns in the z-direction at 10, 14 and 22 GHz
inside a material with eT — 5 - jO .l...............................................................................
Ill
4.10 The normalized power patterns in the z-direction at 12.4 GHz in the
x- and ku-bands inside a material with er = 5 - jO .l...................................................
Ill
4.11 The normalized power patterns in the z-direction at 18 GHz in the
ku- and k-bands inside a material with er = 5 —jO .l...................................................
112
4.12 The normalized power patterns in the x-direction at 18, 22 and 26 GHz
inside a material with er = 5 - jO .l...............................................................................
113
4.13 The normalized power patterns in the z-direction 18, 22 and 26 GHz
inside a material with er = 5 —jO .l...............................................................................
114
4.14 The normalized power patterns in the x-direction at 24 GHz inside
dielectric materials with constant loss factor.....................................................
115
4.15 The normalized power patterns in the z-direction at 24 GHz in side
dielectric
materials with constant loss factor.......................................................
116
4.16 The normalized power patterns in the x-direction at 24 GHz inside
dielectric
materials with constant permittivity.....................................................
117
4.17 The normalized power patterns in the z-direction at 24 GHz inside
dielectric
materials with constant permittivity....................................................
118
4.18 The normalized power patterns in the x-direction at 24 GHz inside
dielectric
materials with constant loss tangent...................................................
4.19 The normalized power patterns in the z-direction at 24 GHz inside
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119
dielectric m aterials with constant loss tangent....................................................
119
5.1 Dominant mode electric field distribution at a waveguide aperture.......................
125
5.2
The volume
fraction as a function of scan position as a slab of
10 % the
area of the aperture is being seen by the aperture...............................................
5.3
The volume
fraction as a function of scan position as a slab o f 30 %
the area of the aperture is being seen by the aperture.........................................
5.4
The volume
128
129
fraction as a function of scan position as a slab o f 50 %
the area o f the aperture is being seen by the aperture.........................................
129
5.5 Two infinite half-spaces of cement paste and EPDM arranged side-by-side
to model a large defect........................................................................................
134
5.6 Real part o f the effective dielectric constant obtained using the K.Wakin formula...
134
5.7 Real part o f the effective dielectric constant obtained using the Rayleigh formula...
135
5.8 Real part o f the effective dielectric constant obtained using the average formula
135
5.9
Imaginary part of the effective dielectric constant obtained using
the
K .W akin
form ula............................................................................................
136
5.10 Imaginary part of the effective dielectric constant obtained using
the R ayleigh
form ula...........................................................................................
136
5.11 Imaginary part of the effective dielectric constant obtained using the
average
fo rm u la .........................................................................................................
137
5.12 Real part o f the effective dielectric constant obtained using the
three formulae with the non-linear approach.......................................................
137
5.13 Imaginary part of the effective dielectric constant obtained using the three
formulae with
the non-linear approach..............................................................
138
5.14 a) The phase and b) magnitude of the reflection coefficient at the aperture
of the waveguide using the linear modelfor the specimen shown in Figure 5.5.....
5.15 a) The phase and b) magnitude of the reflection coefficient at the
aperture o f the waveguide using the non-linear model for the specimen
x v ii i
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141
shown
in
Figure
5.5.................................................................................................
142
5.16 A thin Plexiglas slab inserted in between an infinite half-space of cement
paste to m odel a one dimensional defect..............................................................
144
5.17 The magnitude of the reflection coefficient for 4 mm-thick Plexiglas in
an infinite half-space of cement paste...................................................................
145
5.18 The magnitude of the reflection coefficient for 5.8 mm-thick Plexiglas in
an infinite half-space of cem ent paste...................................................................
145
5.19 A thin Plexiglas slab inserted at depth d in an infinite half-space of cement
paste to model a one dimensional defect under a layer of material..........................
5.20 The normalized YZ-plane field pattern at 7 GHz in an infinite half space of cement.
147
147
5.21 The magnitude of the reflection coefficient for a 5.8 mm-thick Plexiglas
slab 8 mm deep in an infinite half-space of cement paste......................................
148
5.22 The magnitude of the reflection coefficient for a 5.8 mm-thick Plexiglas
slab 4.8 mm deep in an infinite half-space of cement paste...................................
148
5.23 The magnitude of the reflection coefficient for a 8.8 mm-thick Plexiglas
slab 4.8 mm deep in an infinite half-space of cement paste...................................
149
5.24 A thin Plexiglas slab inserted at depth d in an infinite half-space of cement
paste under a 3.17 mm thick layer of synthetic rubber to model a one
dim ensional defect in a multi-layered structure....................................................
150
5.25 The magnitude of the reflection coefficient for a 5.8 mm-thick Plexiglas
slab 8 mm deep in an infinite half-space of cement paste......................................
150
5.26 The magnitude of the reflection coefficient for a 5.8 mm-thick Plexiglas
slab 4.8 mm deep in an infinite half-space of cement paste...................................
151
5.27 The magnitude of the reflection coefficient for a 8.8 mm-thick Plexiglas
slab 4.8 mm deep in an infinite half-space of cement paste...................................
xix
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151
CHAPTER I
In tro d u ctio n
1.1
Background
Nondestructive inspection methods utilizing microwave radiators for the purpose of
material examination, either in a contact or a non-contact manner, began years ago and have
been applied to various media [Alt.63] [Arc.88] [Ash.81] [Bah.82] [Bak.94] [Bel.90]
[Dec.74] [Gho.89] [Lun.72] [Qad.95] [Ven.86] [Zou.90]. The increased use of composite
materials both for industrial and military applications presents quite a challenge to the
standard nondestructive testing (NDT) methods [Car.94].
Difficulties arise from the
inherent anisotropy and physical property inhomogeneities of these materials, as well as the
relative high absorption and scattering of the radiated signals. The ability of microwaves to
penetrate deeply inside dielectric materials makes microwave nondestructive testing and
evaluation (NDT&E) techniques very attractive for interrogating such materials [Lav.67]
[Bah.82] [Zou.94]. The same applies to structures made of dielectric composites. The
sensitivity of microwaves to the presence of dissimilar layers in these materials allows for
accurate thickness variation measurement in the range of a few micrometers at 10 GHz
[Zou.90] [Bak.94] [Edw.93].
Microwave NDT techniques, applied to composites, are
performed on a contact or non-contact basis.
In addition, these measurements may be
conducted from only one side o f the sample (reflection techniques).
Microwave NDT
techniques are also capable of material characteristic identification and classification
[Gan.94],
It has also been shown that microwaves have the ability to detect voids,
t
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delaminations and porosity variations in a variety of dielectric materials and composite
structures. The polarizability of microwave signals enables the study of fiber bundle
orientation or misalignment during manufacturing [Qad.95]. This same feature may also
provide information about cut or broken fiber bundles inside a thick composite member.
Microwave NDT techniques have also shown tremendous potential for detecting impact
fatigue/damage in composites. This is shown to be true for both non-graphite and graphite
composites [Rad.94].
1.2 Motivation
The accumulation of flaws or defects (and damage) in a composite structure is
closely tied to its loaded physical and mechanical properties such as strength, durability,
stiffness, etc. It is imperative to have a good knowledge of the integrity of a composite
structure before and during use.
In composite media, defects may be divided into two
groups (depending on the size with respect to the footprint, sensing area, of the microwave
sensor), namely: large and small defects. A large defect, is a defect whose area is several
times larger than the footprint (e.g. a disbond or a delamination), and it can be considered
as an extra layer in a structure. This layer can be detected and characterized using the
measured reflection coefficient. If the defect is small (i.e. its extent is smaller than the
footprint) it will have a different interaction with the fields, due to the boundaries and
edges, and this interaction will influence the refection coefficient and consequently the
signal measured by the microwave sensor. Characterization of defects (determining their
sizes, locations and properties), after they are detected, is a very important part of any
nondestructive testing technique. That is why it is of great importance to understand the
interaction of the measurement probe with the structure under inspection.
2
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1.3
P revious Research F indings
Since 1990, several approaches of applying microwaves for NDT purposes of sub­
surface defect detection have been conducted at the Applied Microwave Nondestructive
Testing Laboratory at the Electrical Engineering Department at Colorado State University.
It has been established that an open-ended waveguide probe operating at a certain
frequency, polarization and excitation mode can be effectively used for defect detection and
classification. A homogeneous dielectric panel without a defect provides a certain reflection
coefficient at the aperture of the waveguide. The presence of a defect (i.e. inhomogenity)
will cause the reflection coefficient to change. Thus, the study of the reflection properties,
as a defective structure is being scanned by an open-ended waveguide, renders information
about the existence of a defect (detection) and its characteristics (dimensions, type, etc.).
The ability of microwaves to penetrate inside dielectric materials and interact with
their inner structure makes them an excellent candidate for nondestructively inspecting
dielectric media for defects and material property characterization.
Microwave
nondestructive evaluation techniques offer novel solutions when inspecting composites for
the purpose of detecting various types of defects such as inclusions, voids, delaminations,
etc. [Qad.95]. Microwave techniques do not require a couplant, and can effectively scan
samples in contact and non-contact fashions (i.e. at some stand-off distance) [Bak.93]
[Zou.90] [Zou.94], Furthermore, the use of standoff distance has been theoretically and
experimentally shown to significantly improve measurement resolution when detecting
coating thickness variations and delaminations [Zou.94]. Real-time imaging systems using
battery operated, simple and inexpensive hardware used in an on-line fashion can be
relatively easily designed and built. Several of these systems have been built and employed
3
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at the Applied Microwaves Nondestructive Testing Laboratory during the past few years,
yielding impressive results [Qad.95] [Gra.95].
The use of an open-ended rectangular waveguide as a sensor for measuring material
properties at microwave frequencies has received considerable attention [Com.64]
[Bov.89] [Nik.89] [Zou.90] [Bak.95] [Gan.95]. The use of an open-ended rectangular
waveguide is very attractive, since it offers advantages such as a relatively small footprint
(sensing area) compared with an antenna operating from the far-field.
The problem of
open-ended rectangular waveguides radiating into a layered dielectric composite media has
been addressed by many investigators, including investigators at the Applied Microwaves
Nondestructive Testing Laboratory.
1.4
P oten tia l Im pact an d Benefits
As mentioned earlier, microwave NDT techniques offer several unique advantages
over other techniques for inspecting thick composites. Microwave imaging is a way by
which a composite structure can be interrogated.
Microwave imaging is based on
transmitting a microwave signal into a dielectric material and using the magnitude and/or
phase information of the transmitted and/or the reflected signal to create a two or three
dimensional image of the object. This can be done in contact, non-contact and either in the
near- or the far-field.
Near-field imaging uses simple probes such as open-ended
waveguides and coaxial lines, whereas far-field imaging requires an antenna for focusing
the microwave energy.
Furthermore, far-field imaging approaches do not offer good
spatial resolution since the footprint is relatively large. Therefore, focusing lenses are often
used to remedy this problem [Gop.941.
Images can be obtained using either phase or
amplitude information. Far-field techniques generally use amplitude information. Near-
4
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field imaging is more versatile since phase images are easily produced and often contain
more or complimentary information to amplitude imaging. Transmission type microwave
approaches require access to both sides of the sample [Gop.94] [d’Amb.93]. Near-field
approaches have recently been used to image a variety of thick composites with defects and
other types of anomalies [Gra.95] [Qad.95]. These investigations have shown the potential
of obtaining high spatial resolutions without using any special antenna or image processing
of any kind. Near-field microwave imaging systems can be made simple, handheld, and
battery operated at a relatively low cost. The output signals of these microwave sensors are
easy to interpret and minimal operator skills are required.
1.5
Methodology
A rigorous study o f the interaction of microwave signals radiating out of an openended rectangular waveguide with stratified dielectric media has resulted in optimization
techniques with which dielectric slab thickness variation detection, in the order of a few
microns has become possible at a relatively low frequency (10 GHz, wavelength of 3 cm).
Therefore, an extensive theoretical code for this purpose has been developed in a previous
research [Bak.94],
This code can be used to optimize the measurement parameters
(frequency of operation and standoff distance) to monitor variations in any layer in a
stratified dielectric media made of up to 5 layers. Recently, we have been faced with the
problem of detecting disbonds in a 16 or more layers composite structures (generally n
layers). That is why it was of great importance to upgrade the existing code to analyze
structures made of any number of layers.
Near-field microwave imaging of composite structures with small defects has
received considerable attention recently. The success achieved on the experimental level
5
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motivated the development of a theoretical model to describe the high quality images
obtained using near-field microwave imaging, and to explain some of the features
associated with these images [Qad.951 [Rad.94], This theoretical model should also help
in building an intuitive understanding of the behavior of the fields inside dielectric materials
in the near-field o f an open-ended rectangular waveguide probe. As mentioned earlier near­
field microwave image is the result of several factors such as probe type (example
rectangular waveguide, circular waveguide or coaxial line), field properties (i.e. main lobe,
sidelobes and half power beam width, etc.), geometrical and physical properties o f both the
defect and the material under inspection. Thus, in order to characterize a defect, the effect
of all non-defect factors needs to be taken out of an image. This requires understanding the
interaction of the fields with the structure and the defects.
In chapter 2 a theoretical study is conducted to expand on and demonstrate the
ability of utilizing an open-ended rectangular waveguide probe to monitor the existence and
to determine the position of a disbond in a layered composite structure. In chapter 3, nearfield microwave imaging of dielectric composite structures using open-ended rectangular
waveguides is studied experimentally.
operations are discussed.
Experimental setups are presented and their
The utility of applying near-field microwave techniques to
inspect a wide variety of composite structures with different types of defects is
demonstrated, and several experimental results are presented in this chapter. Chapter 4 will
be devoted to study the field properties in the near-field region of an open-ended
rectangular waveguide and its interaction with a dielectric material. This study will include
investigating the influences of frequency and dielectric properties on the radiation pattern.
In chapter 5 a study of the mechanism by which the fields, radiated out of an open-ended
rectangular waveguide probe, interact with an inclusion will be presented. An effective
dielectric constant formula will be used to model the reflection properties of dielectric
structures.
The influence of the non-uniformity associated with the electric field
6
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distribution at the aperture of the waveguide will be investigated and incorporated
calculating the effective dielectric constant.
7
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CHAPTER II
M icrowave D isb o n d Detection a n d Depth D eterm ination in Thick
L ayered S tructures
Rectangular waveguides were one of the earliest types of transmission lines used
to transport microwave signals.
Because o f recent trend toward minimization and
integration, most microwave circuitry is currently fabricated using planner transmission
lines, such as microstrip and striplines, rather than waveguides. But there is still a need
for waveguides in many applications such as high-power systems, millimeter wave
systems, and in some precision test applications. Open-ended rectangular waveguides are
by far the most commonly used transducers in microwave NDT applications.
A large
variety of waveguide components such as couplers, detectors, isolators, attenuators, and
slotted lines are commercially available for various standard waveguide bands from 1 GHz
to over 220 GHz.
One o f the most important features of utilizing waveguides as
measurement sensors is that for the dominant TEi0 mode of propagation, the field
throughout the guide may be conveniently monitored, using a small probe, with minimum
perturbation of the field distribution inside the waveguide. This fact is utilized in making
slotted lines where a slot, cut axially in the center of the guide, is used for inserting a
probe to accurately monitor variations of the standing wave in the guide.
8
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The problem of radiation out of a waveguide has been addressed by many
investigators. Most of the earlier research have addressed the problem of plasma covered
aperture antennas [Gal.64] [Vil.65] [Com.64][Cro.67,68].
More recent analyses o f the
problem have been in conjunction with applications such as the measurements of
dielectric properties of materials [Teo.85] [Jam.77] [Dec.74] [Mac.80] and field
interactions with biological tissue layers [Nik.89],
The foundation of the theoretical
implementation adopted here is parallel with the work done by Compton in application
to radiation from plasma covered aperture antennas [Com.64], Bakhtiari expanded and
modified
Compton’s formulation
for application
to
microwave
nondestructive
examination o f layered lossy dielectric composite media [Bak.92] (structures made of 5
layers were analyzed).
The formulation is further expanded to analyze any layered
composite structure (up to n-layers).
To construct a stationary expression o f the
aperture admittance of general cylindrical waveguides of arbitrary cross-section radiating
into an infinite half-space, a variational formulation was evoked [Bak 92], This result was
then used for the analysis of multi-layered composite structures backed by free-space or a
conductor. Fourier transform boundary matching technique is applied in the medium in
front o f the waveguide aperture to solve for the appropriate set of field components
pertaining to the inspected geometry. This solution is then coupled with the admittance
expression to calculate the reflection coefficient at the aperture of the waveguide.
9
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//./
Variational Form ulation o f A perture Adm ittance
The theoretical derivations of this part of the work were originated by S. Bakhtiari
as part of his Ph.D. dissertation [Bak.92]. The limitation of Bakhtiari’s work was that
althoughthe modeling was for an n layered medium, the numerical simulation was limited
to structures made of up to 5 layers. So, I repeated the work and expanded it to be able
to analyze structures made o f any number of layers. The basis of his formulations are
given here. The geometry of an open-ended cylindrical waveguide o f arbitrary cross
section, opening onto a perfectly conducting infinite ground plane is shown in Figure 2.1.
To construct a general expression for the admittance at the waveguide aperture, variational
formulation is implemented first.
Figure 2.1: Aperture of an arbitrary cross-section opening in a perfectly conducting ground
plane of infinite extent radiating into an infinite half-space [Bak.93].
Figure 2.1 shows the waveguide’s cross-sectional symmetry orthogonal to the
direction of propagation, z. In the waveguide the electric field (E ) and the magnetic field
10
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iff) components arc orthogonal, similarly we may introduce transverse mode functions
e(x,y) and h(x,y), along with the mode currents and voltages V(z) and f(z), respectively
[Bak.93] [Col.66] [Har.61]. Then one may write the following simple relations
E(x,y,z) = e(x, y) V(z)
(2.1)
H(x,y, z) = h(x,y)I (z)
where the orthogonality and normalization properties over the waveguide cross-section and
the transverse aperture plane of the vector mode functions are as follows
ei =hi '<at
(2.2a)
^ = a, x ^
(2 .2b)
\ \ ^ d S = \ j ] ^ d S = \ 0t
\* J .
(2.2d)
Each mode vector is orthogonal to all the other mode vectors. To verify this, one can use
Green’s first and second identities and the divergence theorem [Har.61].
The construction of the aperture admittance for the geometry of Figure 2.1 will be
conducted by evoking a variational formulation method [Bak.93]. In the waveguide, the
11
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electric and magnetic fields may be written as a summation of infinite number of discrete
modes. With the dominant TEi0 mode being incident on the aperture, these fields can be
constructed in terms o f the vector mode functions as
E(x,y,0) = V ^ i x . y ) +
Vr en(x,y)
(2.3a)
*=0
<2 -3b)
H{x,y,0) = YytJ ^ { x , y ) - ^ Y RVrM x^
«=0
where V and V are the magnitudes of n* incident and reflected modes respectively, and
Yn is the admittance of the ntb mode. The normalized aperture admittance in terms of these
mode voltages may be written as
Y
Y=
/- £
^
vT
( 2 -4>
where Yq is the characteristic admittance of the fundamental mode. The above expression
(2.4) may be rewritten as
Y.(V,-V,) N
Y = - llJ z
sr l = 1 _
(2 .5)
D
where N and D denote the numerator and denominator of the admittance function
respectively, and will be evaluated separately [Bak.93]. In order to construct D, both sides
of (2.3a) are dot producted by eQand integrated over the aperture area as
12
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JJ E(x,y,0).etdxdy = V0j j e o.eod x d y + j j ^ V r eH.e0dxdy
s
S
(2.6)
S «=0
which with the aid of (2.2a) simplifies to
Vio + VTm= \ \ E (x , y, 0).e0(x, y)dxdy
(2.7)
s
Equation (2.7) represents the denominator of the admittance function. To construct N,
(2.3b) is first rearranged by taking the dominant mode out of the summation and rewriting
the equation as
Yo { \ - K.)R(x,y) = H (x,y , 0 ) + J Y X M x ^
n-i
(2-8)
which may be written as
Nh0(x,y) = W
(2.9)
So, to extract N from this equation the aperture electric field is cross producted by both
sides of (2.9) and the z-component is integrated over the aperture surface resulting in
^ N [ E ( x , y , 0 ) x h o(x,yj\.dt = ^ [ E ( x , y , 0 ) x W ( x , y ) \ d t dxdy
s
(2.10)
s
Substituting (2.3a) in the left hand side of the above equation and using the orthogonal
properties of the vector mode functions in the following manner
13
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results in the above simplified relation and leads to the following form of (2.10)
+K.)
\l[E(.x,y,0)xWU,y)\a,dxdy=Y,{v,'
(212)
S
The complete expression for W may be simply expanded by dot producting both sides of
(2.3a) by an orthogonal vector mode function em and integrating over the aperture cross
section, resulting in
V% = ^ E ( x , y , 0 ) . e n(x,y)dxdy
s
(2.13)
Once the above is substituted back into (2.8) it results in the following
W(x,y)= H(x,y,0)+ j ^ A C ^ J j Edi,Z,0).en(Ti,£)dridZ
n~o
(2.14)
5
To be able to incorporate the above results in the admittance expression, (2.5) can be
manipulated to take the following form
14
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(2.15)
Y=—
D
which can be constructed using the results of (2.7) and (2.12). Consequently, substituting
the former equation for the denominator and the later for numerator respectively, renders
the variational form of the complex admittance function which can be expressed as
JJ [E (x, y, 0 ) x W (x, y, 0)].az dxdy
(2.16)
Y = G + jB = - ~
JJ E(x,y,0).ea(x,y)dxdy
where W is given by (2.14). The conductance G and susceptance B represent the real and
imaginary parts of the admittance, respectively. The above expression is stationary since
small variations of the approximate aperture electric field distribution about its exact value
would not effect the determined parameter Y.
A typical stationary expression may be
recognized from its general form of containing the square of the trial function in both the
numerator and denominator. This in effect implies that, for a reasonable estimate of this
function the resulting calculated parameter will not deviate form its actual value. Proof of
stationarity of such an admittance expression of (2.16) has been established by several
authors [Sax.68] [Har.61] [Com.64] [Bak.93].
15
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II.2
Radiation
From Rectangular W aveguide into S tra tifie d
C o m p o site
M e d ia
The general problem of aperture admittance radiating into a stratified dielectric
media is addressed in this section. Fourier transform boundary matching procedure is used
to construct a complete set of solutions for the external medium, z > 0. Consequently, the
solutions are used in the stationary admittance expression to achieve the desired solution.
The versatility o f such a model for near-field in-situ interrogation of stratified composite
media arises from the fact that it allows addressing non-contact as well as contact type
measurements.
11.2.1
T h eoretical Formulation
The cross section of an open-ended rectangular waveguide radiating into a layered
medium which is terminated into an infinite half-space and a perfectly conducting sheet are
shown in Figures 2.2a and 2.2b, respectively. Each layer is assumed to be homogeneous
and nonmagnetic with relative complex dielectric constant of em.
Waveguide
Figure 2.2a: Cross-section of a rectangular waveguide radiating into a layered media
terminated into an infinite half-space [Bak.93].
16
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Waveguide
Figure 2.2b : Cross-section of a rectangular waveguide radiating into a layered media
terminated into a conducting sheet [Bak.93].
With the TE10 mode incident on the aperture, a symmetrical electric field distribution
over the large dimension can be written as
Over Aperture
(2.17)
Elsewhere
where a and b are the broad and small dimensions of the waveguide cross-section,
respectively.
The electric field distribution is normalized so that a Fourier transform
boundary matching technique is used to construct the field solutions in an N-layer stratified
generally lossy dielectric medium. The transverse field components are expanded in each
layer in terms of Fourier integrals. Subsequently, appropriate boundary conditions across
each interface are enforced to solve for the unknown field coefficients in each medium.
The fields outside the waveguide may be constructed using a single vector potential. In
each layer, denoted by layer number n, fields must satisfy the source-free wave equation
En(x,y,z) = - V x n„
(2.18)
17
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The vector potential II satisfies the field conditions for 0 < z < d and can be decomposed
into two orthogonal components
n = O al + 4 'a y
(2.20)
General solutions of (2.17) and (2.18) may be expressed in terms of integrations
over the entire mode space as
A
E n (x,y ,z) = — A - f f
j*Jv '
(2;r)2 J J
-+Aa*
e ~ikznZj.An ±A
ft!
.
ft!
xe
-j(kxx+kyy)
t
H n (x,y,z) =
{;}
j
J hJ<Wo
- k xky A
—
e
jk 7
e jkZnz
(2.21 )
.
dkxdky
\
+ A'
k2
n-k 2
0 ,
Z
Jk z z
n +A
J kZnZ
•„r
>e J(kxX+kyy)>dkzdky
(2.22)
•-J
where kn is the complex propagation constant in each layer, and
18
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which is chosen such that
R e{^}>0,
lm { ^ } < 0
(2.23b)
to comply with the appropriate direction of propagation.
[1.2.2
Termination o f L ayered M edia into an Infinite Half-Space
Referring to Figure 2.2a, only positively traveling waves exist in the last (N ^ )
layer which is unbounded in the +z direction. Thus, for the field components in this
region only those terms in (2.21 and 2.22) which are associated with positive direction o f
propagation remain. Using Fourier properties of the field components in (2.21 and 2.22)
and enforcing the continuity o f the tangential field components at each interface, the
unknown field coefficients for each layer can be determined. The admittance expression
normalized with respect to the guide admittance, can be expressed as
19
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x j J ||^£r<—(%.cosfl)2]f2 ^ ^ + 1 9 ^ ® } —2‘R j s i n d c o s 6 ^ t
x.-oo-o L
J
V
x g fa d ) itdedn.
where Kj and g(%0) are given by (2.25) and (2.26) respectively.
(2.25)
K' = ^ £rl - ^
.. ,i am
. .'b’X.sinO')
( d ^cosd}
o s |----------|---j~ , i(4 ;r)sin
----------- ctua
-------3( X0) = r J
V
2
)
y
2
V b'
(^.sin 0 )|tf2 - { d f£.cos0)2]
(2.26)
where R and 6 are the new variables of integration, and a’ and b ’ are the normalized
waveguide dimensions (with respect to the wavelength A.). When dealing with lossless
media (2.24) has some singularities. This can be resolved by contour integration around
singular points of the integrand located on the real axis and take into account the residue.
In application to a composite medium which contains generally lossy layers, which is the
case with most materials, the poles of the integrand move off of the real axis and the
integrand becomes smooth. This allows quick and efficient numerical integration schemes
such as Gauss quadrature method (used here) to be applied.
20
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11.3
Theoretical Results
The complex reflection coefficient at the aperture of the waveguide T is related to
the normalized complex admittance, y,, by
r = |r y #= - ^ i+y,
(2.27)
which is a complex quantity whose phase and magnitude can be calculated and measured
for comparisons. The theoretical formulation for an open-ended rectangular waveguide
radiating into a multi-layer dielectric composite was expanded to include a fifteen-layer
composite structure (a thirteen-layer sandwich composite, a standoff distance and a freespace backing) as shown in Figure 2.3.
The output of this model is the reflection
coefficient properties (magnitude, phase, real part and imaginary part) calculated at the
waveguide aperture for a given composite geometry, standoff distance and operating
frequency.
II.3.1
Sample Description
This work was a part o f a funded research in which the sponsor supplied us with
a description of the composite structure for the theoretical study. The dimensions and
the dielectric properties (£,■) of the constituents o f the sandwich composite are as follows:
21
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Adhesive
Foam Core
W aiveguidej
v e g u id e
A ir B a c k in g
n
/
S ta n d o ff D ista n c e
S ubstrate,
S k in L a m in ate
Figure 2.3: Schematic of the sandwich composite [Qad.96].
1. standoff distance - variable size
2. skin laminate - 2.54 mm, £r = 4.5 - j0.045
3. adhesive (glue line) - 0.28 mm, ^ = 3.1 -jO.Ol
4. outer core - 45 mm, ^ = 1 . 1 - j0.0026
5. adhesive (glue line) - 0.28 mm,
= 3.1 - jO.Ol
6. substrate - 0.14 mm, £,. = 4.5 - j0.045
7. adhesive (glue line) - 0.28 mm, £r = 3.1 - jO.Ol
8. inner core - 40.64 mm, £r = 1.1 -j0.0026
9. adhesive (glue line) - 0.28 mm, £r = 3.1 - jO.Ol
10. substrate - 0.14 mm, £,. = 4.5 - j0.045
22
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11. adhesive (glue line) - 0.28 mm, £,. = 3.1 - jO.01
12. outer core - 45 mm, £,. = 1.1 - j0.0026
13. adhesive (glue line) - 0.28 mm, ^ = 3.1 - jO.Ol
14. skin laminate - 2.54 mm, £,. = 4.5 - j0.045
15. air backing (infinite half-space o f free-space)
The constituents used in this composite structure are considered to be in the
family of low permittivity and low loss dielectric materials. The dielectric properties (£,.)
o f the constituents were provided to us by the sponsor at a frequency of 3 GHz.
Since
these are low permittivity and low loss dielectric materials their dielectric properties
remain fairly constant as a function of frequency. Thus, for all frequencies used in this
study the values shown earlier were used. However, the dielectric properties of the foam
core were measured, at 8.5, 10 and 12 GHz, to be (£r = 1.085 - j0.002) in average which is
very close to the provided values.
11.3.2
S tan doff Distance an d Frequency A nalyses
For the schematic of the composite shown in Figure 2.3 there are two parameters
which may be used for measurement optimization o f disbond detection and depth
determination. These two are the standoff distance and the frequency of operation. For
all cases described in this section a disbond is assumed to replace an adhesive layer (0.28
23
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mm thick). Therefor, there are six possible disbond locations in the sandwich composite
shown in Figure 2.3. Furthermore, the spatial extent o f a disbond is assumed to be much
larger than the area o f the sensing waveguide aperture (greater than a few centimeters
squared). In this section the results of the combined standoff distance and frequency
analyses are presented. Unless otherwise specified the standoff distance varies between 0
mm (contact) to 6 mm at one millimeter intervals. The frequency range o f 12-40 GHz
(Ku-, K- and Ka-bands) was used for these analyses. The waveguide dimensions for Kuband (12-18 GHz) are 15.8 mm x 7.9 mm, for K-band (18-26.5 GHz) are 10.67 mm x 4.32
mm and for Ka-band (26.5-40 GHz) are 7.11 mm x 3.56 mm.
II.3.2.1
Ku-Band (12-18 GHz) Results
Figure 2.4 shows the phase of the reflection coefficient as a function o f frequency
at all discrete standoff distances for the case of no disbond. Figure 2.5 shows the same
results except for when there is a disbond under the skin laminate (1st disbond). For the
1 mm standoff distance case there were some calculation difficulties (division by very
small number), and thus those results are not shown.
Figure 2.6 shows the phase
difference between the no disbond and the disbonded case at
24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
contact
3 mm
5 mm
2 mm
4 mm
6 mm
200
150
100
50
0
-50
-100
-150
-200
12
13
14
16
15
17
18
F re q u e n c y ( G H z )
Figure 2.4: Phase o f the reflection coefficient as a function of frequency at various
standoff distances for the case of no disbond in the composite.
contact
3 mm
5 mm
2 mm
4 mm
6 mm
200
150
100
50
0
-50
-100
-150
-200
12
13
14
15
16
17
18
F re q u e n c y ( G H z )
Figure 2.5: Phase o f the reflection coefficient as a function of frequency at various
standoff distances for the case of a disbond under the skin laminate (1st disbond).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
contact
3 mm
5 mm
2 mm
4 mm
6 mm
o
-10
12
13
14
15
16
17
18
F re q u e n c y (G H z )
Figure 2.6: Phase difference between no disbond and the first disbond as a function of
frequency at various standoff distances [Qad.96],
each standoff distance as a function of frequency.
The results show maximum phase
change o f =9 degrees at 18 GHz at a standoff distance of 3 mm.
For this standoff
distance the phase variation is relatively constant as a function of frequency. Relatively
constant phase variation is a desirable feature from the practical point o f view since no
frequency selectivity is required in such cases. Collectively, the results shown in Figure
2.6 are encouraging since not only there is a great deal of frequency independence but also
the level o f the phase change due to the first disbond is adequately large (=10 degrees).
Figures 2.7-2.11 show phase difference as function o f frequency for all standoff distances
for all cases (when the second, third, forth, fifth and sixth adhesive layer is replaced by a
disbond, respectively). The results are not as encouraging as those shown in Figure 2.6.
In particular, the level o f phase change for all these cases is very small (less than one
26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
degree) which may not be adequate for detection. Therefore, one may conclude that this
frequency band may not be optimum for detecting disbonds at all possible locations using
standoff distances in the 0-6 mm range.
contact
1.5
1
• * *•
mmm
3 mm
4 mm
— — 2 mm
- • - • 5 mm
— i— |— i— r - i -----1----- 1---- t
6 mm
~ r — 1—
-
0.5
-
0
-
-0.5
-
-e-
<
12
13
14
15
16
17
18
F re q u e n c y ( G H z )
Figure 2.7: Phase difference between no disbond and the second disbond as a function of
frequency at various standoff distances.
contact
1 mm
2 mm
3 mm
4 mm
5 mm
6 mm
2
1
0
1
12
13
14
15
16
17
18
F re q u e n c y ( G H z )
Figure 2.8: Phase difference between no disbond and the third disbond as a function of
frequency at various standoff distances.
27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
contact
1 mm
4 mm
5 mm
2 nun
3 mm
1
0
-0.5
1
12
13
14
15
16
17
18
F re q u en c y (G H z )
Figure 2.9: Phase difference between no disbond and the forth disbond as a function of
frequency at various standoff distances.
1
1 mm
3 mm
5 mm
2 mm
4 mm
6 mm
0
\
.
1
12
13
14
15
16
17
18
F re q u e n c y (G H z )
Figure 2.10: Phase difference between no disbond and the fifth disbond as a function of
frequency at various standoff distances.
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
contact
1 mm
2 mm
- 3 mm
’■ 4 mm
* 5 mm
0.15
0.1
0.05
o
-e -
<
-0.05
-
0.1
-0.15
-
0.2
12
13
14
15
16
17
18
F r e q u e n c y (G H z)
Figure 2.11: Phase difference between no disbond and the sixth disbond as a function of
frequency at various standoff distances.
11.3.2.2
K -B and (18-26.5 GHz) R esults
Figure 2.12 shows the phase o f the reflection coefficient as a function of
frequency at all standoff distances for the case of no disbond.
Figure 2.13 shows the
same results except for when there is a disbond under the skin laminate (1st disbond).
The frequency range shown here is 18-24 GHz. We had some calculation difficulties for
the 24-26.5 GHz frequency range. However in a couple o f subsequent figures sample
results in this frequency range will be shown as later. The calculation difficulty arises
from the fact that there are singularities in some of the integral operations which cause
this problem. Figure 2.14 shows the phase difference between the no disbond and the
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
first disbond at each standoff distance as a function o f frequency. The results show
maximum phase change o f =22 degrees at 24 GHz and at a standoff distance of 1 mm. As
in the previous frequency band, the phase variation is relatively constant for all standoff
distances in the 18-23 GHz frequency range. The results shown in Figure 2.14 are even
more encouraging than those for the previous frequency band since the phase change due
to the first disbond is in excess of 20 degrees.
Figures 2.15-2.18 show the phase
difference as function o f frequency for all standoff distances for when the second, third,
forth and fifth adhesive layer is replaced by a disbond, respectively.
For the sixth
disbond the program did not work correctly, however similar results as those of the fifth
disbond would have been produced. These results are not as nice as those shown in
Figure 2.14. In particular, the level of phase change for all these cases is relatively small
(with the exception of a few instances where the phase difference is as high as 5 degrees).
Therefore, one may conclude that this frequency band may offer enhanced disbond
detection possibilities. However, beyond the first disbond the overall phase differences
may still be considered small.
30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
contact
1 mm
2 mm
6 mm
3 mm
4 mm
5 mm
200
150
100
50
0
-50
-100
-150
18
19
20
21
22
F re q u e n c y (G H z)
24
23
Figure 2.12: Phase of the reflection coefficient as a function o f frequency at various
standoff distances for the case of no disbond in the composite.
contact
1 mm
2 mm
3 mm
4 mm
5 mm
6 mm
200
150
100
50
0
-50
-100
-150
-200
18
19
20
21
22
23
24
F re q u e n c y (G H z)
Figure 2.13: Phase of the reflection coefficient as a function o f frequency at various
standoff distances for the case of a disbond under the skin laminate (1st disbond).
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6 mm
3 mm
4 mm
5 mm
contact
1 mm
2 mm
o
-10
18
19
20
22
21
24
23
F re q u e n c y (G H z )
Figure 2.14: Phase difference between no disbond and the first disbond as a function of
frequency at various standoff distances.
contact
1 mm
2 mm
3 mm
4 mm
5 mm
6 mm
5
4
3
2
0
1
■2
■3
18
19
21
22
F re q u e n c y (G H z )
20
23
24
Figure 2.15: Phase difference between no disbond and the second disbond as a function of
frequency at various standoff distances.
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3 mm
4 mm
5 mm
contact
1 mm
2 mm
6 mm
6
4
2
0
2
A
18
19
20
21
23
22
24
F re q u e n c y (G H z )
Figure 2.16: Phase difference between no disbond and the third disbond as a function of
frequency at various standoff distances.
— —
-------- 3 mm
• 4 mm
—• — 5 mm
contact
1 mm
2 mm
—
§ mm
-e-
<
18
19
20
21
22
23
24
F re q u e n c y (G H z )
Figure 2.17: Phase difference between no disbond and the forth disbond as a function of
frequency at various standoff distances.
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3 mm
4 mm
5 mm
contact
1 mm
2 mm
“ ■ 6 mm
15
05
a
<
18
19
21
20
22
23
24
F re q u e n c y (G H z )
Figure 2.18: Phase difference between no disbond and the fifth disbond as a function of
frequency at various standoff distances.
To investigate the phase difference characteristics in the frequency band of 2426.5 GHz the following was performed. Since the standoff distance of 1 mm seemed to
have been the most sensitive standoff distance in the K-band frequency range, the phase
difference between no disbond and all disbonded cases at 1 mm standoff distance was
calculated in the 23-26.5 GHz frequency range. Figure 2.19 shows the results of these
calculations for the first disbond, while Figure 2.20 shows the results for all other
disbonds, respectively. For the first disbond there is a large phase difference which is
more than sufficient for its detection. For the other disbonds and at around 25.5 GHz the
phase difference is more than those shown in the 18-24 GHz region. Furthermore, the
sign of the phase difference may be used as an indicator of the location o f disbond. For
example at 25.5 GHz, if a phase difference of 5 degrees is measured and if its sign is
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
negative then it is due the fifth disbond, whereas if its sign is positive then it is due to the
second disbond. This issue will be discussed further in the subsequent sections.
140
120
100
80
60
40
20
0
23
23.5
24
24.5
25
25.526
26.5
27
F re q u e n c y (G H z )
Figure 2.19: Phase difference as a function of frequency at a standoff distance of 1 mm
due to the first disbond [Qad.96].
2nd
4th
3rd
5th
6th
8
6
4
2
0
•2
■4
6
23
23.5
24
24.5
25
25.5
26
26.5
27
F re q u e n c y (G H z )
Figure 2.20: Phase difference as a function of frequency at a standoff distance of 1 mm
due to all other disbonds [Qad.96].
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
II.3.2.3
Ka-Band (26.5-40 GHz) Results
Figure 2.21 shows the phase variation as a function o f frequency and standoff
distance for the sandwich composite with no disbond.
Figure 2.22 shows the same
results except for when the first disbond exists. Figure 2.23 shows the phase difference
between these two cases. From the results shown in this figure it is clear that the phase
difference necessary for detecting the first disbond is sufficiendy large in this frequency
band. For example a phase difference of =50 degrees is measured at a standoff distance of
2 mm and at a frequency of 33 GHz. Furthermore, there are several regions in which the
phase difference is not only large but fairly constant (1 mm standoff distance between 34
and 40 GHz and also 2 mm standoff distance between 31.8 to 33.5 GHz). These results
are by far more encouraging than those reported earlier.
Since this band seems to render better overall detection results we decided to look
into the variation of the magnitude o f the reflection coefficient as well. This information
may be used in tandem with phase difference information for not only enhanced detection
but also possibly for disbond depth discrimination.
Figure 2.24 shows the percent
magnitude change as a function o f frequency and at different standoff distances for the
first disbond.
The results show that there is a substantial percentage change in the
magnitude o f the reflection coefficient for the contact case and at a frequency range of 3238 GHz. Results of several standoff distances also show =10% change for a wide range
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o f frequencies.
It must be noted that the magnitude of the reflection coefficient is
relatively easy to measure. Therefore, these results are very important from a practical
point of view. Figures 2.25-2.34 show similar results to Figures 2.23 and 2.24 except for
second through sixth disbonds, respectively.
“ ™ “ contact
■ — — I mm
— — 2 mm
6 mm
3 nun
4 mm
200
150
100
O
-50
-100
-150
-200
26
28
30
32
34
36
38
40
F re q u e n c y (G H z )
Figure 2.21: Phase of the reflection coefficient as a function of frequency at various
standoff distances for the case of no disbond in the composite.
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
contact
1 mm
2 mm
3 mm
4 mm
5 mm
6 mm
200
150
100
0
-e-50
-100
-150
-200
26
28
30
32
34
36
38
40
F re q u e n c y (G H z )
Figure 2.22: Phase o f the reflection coefficient as a function of frequency at various
standoff distances for the case of a disbond under the skin laminate (1st disbond).
contact
1 mm
2 mm
6 mm
3 mm
4 mm
5 mm
o
■e<3
r—
-20
-40
26
28
30
32
34
36
38
40
F re q u e n c y (G H z )
Figure 2.23: Phase difference as a function of frequency at various standoff distances for
the case of a disbond under the skin laminate (1st disbond) [Qad.96].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
contact
1 mm
2 mm
3 mm
4 mm
5 mm
■ 6 mm
< 10
-20
26
30
28
32
34
36
40
38
F re q u e n c y (G H z)
Figure 2.24: Percent magnitude change as a function of frequency at various standoff
distances for the case of a disbond under the skin laminate (1st disbond).
contact
1 nun
2 mm
— —
'
-6 L
26
.
I
28
-------- 3 mm
■ ™ - 4 mm
— — 5 mm
i ■ i -t— |---- 1—_
■ 6 mm
r_
r - -i—
T .
—
\
.
I
30
.
I
32
.
i
.
34
i
36
. I
38
.
40
F re q u e n c y (G H z )
Figure 2.25: Phase difference as a function of frequency at various standoff distances for
the second disbond [Qad.96],
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
contact
2 mm
4 mm
1 mm
3 mm
3 mm
5
4
3
_
2
L_
< 1
0
-1
-2
-3
26
28
30
32
34
36
38
40
F re q u e n c y (G H z )
Figure 2.26: Percent magnitude change as a function of frequency at various standoff
distances for the second disbond.
contact
1 mm
2 mm
3 mm
4 mm
* — - 5 mm
6 mm
-© -
<
26
28
30
32
34
36
38
40
F r e q u e n c y (G H z )
Figure 2.27: Phase difference as a function o f frequency at various standoff distances for
the third disbond.
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6 mm
4
3
2
L 1
<
e£ 0
-1
-2
-3
26
28
30
32
34
36
38
40
F re q u e n c y (G H z )
Figure 2.28: Percent magnitude change as a function of frequency at various standoff
distances for the third disbond.
— —
5
■
1
contact
1 mm
2 mm
1/ 1
--------3 mm
4 mm
—• —* 5 mm
■ 1 i 1 i
6 mm
1
'
1
'
i
.
i
.
M
/ \
I I
4
3
2
o
■e- i
<
1
0
-1
-2
■ * * * ■ * •
-3
26
28
30
32
i
.
34
36
38
40
F re q u e n c y (G H z )
Figure 2.29: Phase difference as a function of frequency at various standoff distances for
the forth disbond.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3
---------3 mm
“ “ “ ■ 6 mm
**“ “ ■ 4 mm
-------- 2 mm
—■— 5 mm
— '---- 1-----1— 1— r - -T- - i — |---- 1- - I -----1— | - - T -
2
< -
contact
_
1
u
<
* 0
-1
1 * * ■ 1 • 1 •
-2
26
28
30
32
34
1
i
36
38
40
F re q u e n c y (G H z )
Figure 2.30: Percent magnitude change as a function of frequency at various standoff
distances for the forth disbond.
“ “ “ ““ “ contact
----------1 mm
--------- 2 mm
-------- 3 mm
4 mm
—' — 5 mm
—
6 mm
6
4
o
2
<
0
-2
* ■ * ■ 1 ■ 1 •
-4
26
28
30
32
34
* ■ 1
36
38
40
F re q u e n c y (G H z )
Figure 2.31: Phase difference as a function of frequency at various standoff distances for
the fifth disbond.
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6 mm
-------- 3 mm
■■■ 4 mm
— — 2 mm
— — 5 mm
— '---- 1-----'---- 1” - i ---- 1---- 1— |---- 1—n — i— i— 1—
contact
26
28
30
32
34
36
38
40
F re q u e n c y (G H z)
Figure 2.32: Percent magnitude change as a function of frequency at various standoff
distances for the fifth disbond.
contact
1 mm
2 mm
6 mm
3 mm
4 mm
5 mm
6
5
4
3
2
1
0
•2
26
28
30
32
34
36
38
40
F re q u e n c y (G H z)
Figure 2.33: Phase difference as a function of frequency at various standoff distances for
the sixth disbond.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-------- 3 mm
“ “ “ * 6 mm
■■ " 4 mm
— — 5 mm
— — 2 mm
----1-----1-----r - T " -i---- 1---- 1---- 1---- 1-“ 1— i— T —' ■
contact
.1
I__ I__ ___I__ I__I__ .__ I__ • I .
26
28
30
32
34
36
I
38
.__
40
F re q u e n c y (G H z )
Figure 2.34: Percent magnitude change as a function of frequency at various standoff
distances for the sixth disbond.
The phase difference results indicate that at a standoff distance o f 2 mm and an
operating frequency of =29 GHz (and to a lesser extent at =30.5 GHz) the second
through the sixth disbond may be consistently detected. Thus, for the sole purpose of
detection it is clear that this frequency band in general and =29 GHz in particular is the
optimum operating frequency. The percent magnitude changes for these disbonds are not
as high as those for the first disbond and may not be considered always useful (although
in some cases changes of up to 5% are detected).
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
II.3 .3
P o ten tia l o f D isbond D epth D eterm ination
Another goal of this research project was to investigate the potential of this
particular microwave nondestructive testing technique to determine the depth of a
disbond in a sandwich composite as well. There are two parameters that may be used for
this purpose, namely the standoff distance and the operating frequency. Thus far we
have established that disbonds replacing all adhesive layers may be detected at certain
standoff distances and operating frequencies.
The depth determination could be
performed by using several frequencies at a given standoff distance, several standoff
distances at a given frequency or a combination of these. In a sandwich composite of the
type addressed here there are six discrete depths at which a disbond could be present.
The results o f the potential of this microwave method for disbond depth determination
are shown next.
I I.3.3.1
K a-band
Figure 35 shows the phase of the reflection coefficient, in contact (0 standoff
distance), as a function of frequency for the six different disbond locations/depths. The
frequency at which the phase transition occurs may be used for disbond depth
determination. The first disbond is unambiguously located in the range of =28-32.5 GHz.
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The second and third disbond (not uniquely) can be detected at 28.4 GHz since at this
frequency there is a phase reversal between the
1
no disbond
1st
2nd
3rd
4th
5th
6th
200
150
100
50
V
0
-50
-100
-150
-200
26
28
30
32
34
36
38
40
F re q u e n c y ( G H z )
Figure 2.35: Phase of the reflection coefficient as a function of frequency for all disbonds,
in contact.
phase of these two disbonds and that of the no disbond case. However, beyond
this information not much more can be said about this case. Figure 36 shows the same
results except at a standoff distance of 2 mm.
Once again, the first disbond is
unambiguously located almost at all frequencies. Not considering the first disbond, Figure
37 shows the phase difference between no disbond and all other disbonds. Operating at
=28.9-29.5 GHz the second and the third disbond cluster together while the forth, fifth
and the sixth disbonds cluster together. The phase difference between the two clusters is
=2 degrees which theoretically should be enough for unambiguous detection. However,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
once again in this case the distinction between the second and the third or the forth, fifth
and the sixth disbond is not uniquely possible.
3rd
4th
5th
no-disbond
1st
2nd
6th
-20
-60
-80
-100
26
28
30
32
34
36
38
40
F r e q u e n c y (G H z )
Figure 2.36: Phase o f the reflection coefficient as a function of frequency for all disbonds
at 2 mm standoff distance.
2nd
3rd
4th
5th
" 6th
6
4
2
o
<
0
2
-4
6
8
26
28
30
32
34
36
38
40
F r e q u e n c y (G H z )
Figure 2.37: Phase difference as a function o f frequency for the second, third, forth, fifth
and sixth disbond at 2 mm standoff distance [Qad.96],
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11.3.3.2
K-band
The overall results so far indicate that the first disbond is the easiest to detect and
locate. Furthermore, the phase change due to the first disbond is much more than all
other. This is due to the fact that at this depth (under the skin laminate) the disbond’s
influence is more than at other depths since at these depths the disbond is closer to the
foam core (or under it) and hence its influence is reduced.
Since a standoff distance o f 1 mm was shown to be the most sensitive standoff
distance in K-band, the phase difference between no disbond and all disbonded cases, at I
mm standoff distance was calculated in the 23-26.5 GHz frequency. Figure 38 (same as
Figure 19) shows the results o f these calculations for the first disbond, while Figure 39
(same as Figure 20) shows the results for all other disbonds, respectively. For the first
disbond there is a great deal o f phase difference (as large as 130 degrees) which is more
than sufficient for its detection.
For the other disbonds and at =25.5 GHz the phase
difference is more than those shown in the 18-24 GHz frequency range. Furthermore, the
sign of the phase difference may be used as an indicator of the location of disbond. For
example at 25.3 GHz the following phase differences are calculated and shown in Table 1.
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 2.1: Phase differences at 25.3 GHz and at 1 mm standoff distance for all disbonds.
1st
2nd
3rd
4th
5th
6th
115.8°
3.6°
6.3°
0.9°
-2.6°
-0.5°
It is clear that the first disbond causes significantly more phase difference than the
rest, thus its depth is unambiguously determined. If the phase difference has a positive
sign then it is due to either the second, the third or the forth disbond. If the sign of the
phase difference is negative then it is due to the fifth or the sixth disbond.
Since a
commercial network analyzer is capable of measuring phase within =0.5 degrees and a
custom designed phase detector is capable of measuring phase within =1 degree, then the
magnitude of the phase difference may be used to distinguish among the second, third,
forth or fifth and sixth disbond. The ramification of this result is quite significant since
via operating at a single frequency and at 1 mm standoff distance the depth of the
disbonds may be unambiguously determined.
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
140
120
100
60
23
23.5
24
24.5
25
25.5 26
26.5
27
F re q u e n c y (G H z )
Figure 2.38: Phase difference as a function of frequency at a standoff distance of 1 mm
due to the first disbond [Qad.96].
8
2nd
4th
3rd
5th
6th
6
4
2
0
2
■A
■6
23
23.5
24
24.5
25
755
26
26.5
27
F re q u e n c y (G H z )
Figure 2.39: Phase difference as a function of frequency at a standoff distance of 1 mm
due to all other disbonds [Qad.96].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
It must also be mentioned that, for example, the forth disbond becomes the third
disbond from the opposite side of the composite. Thus, such information may also be
utilized to unambiguously determine the depth of a disbond.
A combination of the results at Ka-band and at K-band, as described in the last
two sections may also be used to unambiguously determine the depth of the forth
disbond. Other such combinations may also be possible. However, this would require
operating at two different frequencies and as many standoff distances compared to the
results shown in Table 1.
11.3.4
R eal P art o f the Reflection C oefficient
Since the magnitude and the phase of the reflection coefficient are calculated we
may use this information to obtain the real and/or the imaginary part o f the reflection
coefficient as well. To illustrate this point Figure 40 shows the percentage change in the
real part of the reflection coefficient for the second, third, forth, fifth and the sixth
disbond (compared to the no disbond case) at a standoff distance o f 2 mm in the 27-31
GHz frequency range. The results indicate that at a frequency of 30.3 GHz it is not only
possible to detect all disbonds, but it is also possible to determine their depths
unambiguously. Table 2 shows the percentage changes at this frequency. This approach
seems very successful for disbond depth determination for this case.
51
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Table 2.2: Phase differences at 30.3 GHz and at 1 mm standoff distance for all disbonds.
disbond
2nd
3rd
4th
5th
6th
% Areal part
44.7
51.5
19.7
25.1
0.1
2nd
3rd
4th
5th
/■—V
L.
o03
06
<
-20
-40
27
27.5
28
285
29
29_5
30
30.5
31
F re q u e n c y (G H z)
Figure 2.40: Percentage change in the real part of the reflection coefficient as a function of
frequency for the second through the sixth disbond [Qad.96].
II.3 .5
K a-Band, / mm T h ick Disbond
To illustrate the effect o f disbond thickness, Figure 41 shows the phase difference
as a function of frequency for all standoff distances calculated for the second disbond
(similar to the results of Figure 25). For this case, and at a wide range of frequencies,
52
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phase difference levels in excess of 30 degrees were calculated at 1 mm standoff distance.
For other standoff distances the phase difference was also relatively large. In contrast, in
Figure 25 (0.28 mm thick disbond) at 2 mm standoff distance the maximum phase
difference was calculated to be only =8 degrees.
contact
50
40
30
o
-©<
20
10
0
-10
-20
-30
26
28
30
32
34
36
Frequency (G H z)
38
40
Figure 2.41: Phase difference as a function of frequency at various standoff distances for
the second disbond with a thickness of 1 mm [Qad.96],
11.4
Summary and Remarks
Theoretical analysis of radiation from a rectangular waveguide into layered dielectric
composite media was presented in this chapter.
Initially, variational formulation was
evoked to come up with a stationary expression for the terminating aperture admittance of
general cylindrical waveguides with arbitrary cross section. The formulation was then
53
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expanded to take into account general N-layer media terminated into an infinite half-space
or a perfectly conducting sheet. The results o f a theoretical study, using an open-ended
rectangular waveguide radiating into a multi-layered structure, for detecting disbonds in
thick sandwich composite structures and determining their depths unambiguously were
presented in this chapter.
The results indicated that disbond detection at depths is
possible at a number of frequencies and standoff distances. Ka-band was shown to be the
most optimum frequency band to operate in. K-band also showed promise for not only
disbond detection but also for depth determination. It was shown that several frequencies
and/or standoff distances may be used for unambiguous depth determination. All of these
results involved the calculation of the phase o f the reflection coefficient at the waveguide
aperture. It turns out that other related parameters such as the magnitude and/or the real
part o f the reflection coefficient may also be used in conjunction (or individually) with
other parameters to determine disbond depth unambiguously.
In practice, when using a network analyzer or a custom designed phase detector,
the microwave characteristics (scattering parameters) of the microwave hardware must be
measured and taken into account. It has been shown that the scattering parameters of a
microwave system may be used as an optimization tool in certain applications. Such an
optimization may also be applied to the calculated results presented here for enhanced
detection and depth determination.
54
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The disbond thickness used in this study was assumed to be equal to the
thickness o f the adhesive layer (0.28 mm). In practice however, disbond thickness may
be in excess o f 0.5 mm to a few millimeters. In such cases the phase difference values that
were used for detection will increase, rendering the disbond much easier to detect.
Additionally, disbond depth determination in such cases will be easier as well.
Since
disbond thickness influences the phase o f the reflection coefficient, it is very likely to not
only be able to determine its depth but also its thickness (within a given range) as well.
Multiple disbonds may exist in a sandwich composite as well. This microwave
nondestructive testing method is (should be) capable of detecting multiple disbonds as
well (as part o f a future investigation).
The formulation used in this general theoretical model is quite complicated.
However, in the future, it may be worth looking into the possibility of developing an
analytical inverse model to determine disbond properties such as existence, depth and
thickness.
Furthermore, in the study conducted here the effect of dielectric property
variations in each layer of the composite was not investigated.
In a future study the
influence o f such variations (including layer thickness variations) should also be taken into
account as well.
55
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CHAPTER III
N ear-F ield Rectangular W aveguide Probes Used f o r Im aging
The increased use of light weight, durable and strong dielectric composites for both
industrial and military applications presents quite a challenge to the field of nondestructive
testing and evaluation (NDT&E). Due to the inherent anisotropy and physical property
inhomogeneities o f these materials, many techniques have been shown to be ineffective
when inspecting these materials. Other techniques that can still be used for inspecting these
materials have the limitation that they can not be employed in an on-line fashion in addition
to the high cost associated with using some o f them (x-ray, proton, etc.). The ability of
microwaves to penetrate deeply inside dielectric materials and composites makes
microwave NDT techniques very attractive for interrogating such materials [Lav.671
[Bah.82] [Zou.94] [Qad.94], Additionally, the sensitivity of microwaves to the presence
of dissimilar layers in these materials allows for accurate thickness variation measurement
in the range of a few micrometers at frequencies as low as 10 GHz [Zou.90] [Zou.94]
[Bak.94] [Bak.93] [Edw.93]. Microwave NDT hardware systems may be inexpensive,
simple in design, hand held, battery operated, operator friendly and easily incorporated into
on-line inspection systems.
In Chapter 2 a theoretical study was conducted to expand on and demonstrate the
ability of utilizing an open-ended rectangular waveguide probe to monitor the existence and
to determine the position of a disbond in a layered composite structure. The analyses and
procedures applied in detecting and locating air layers (disbonds) can be applied to detect
any defective dielectric layer. The transverse to the direction of propagation extent of the
56
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disbond was assumed to be large enough to consider the disbond a layer. In practice, the
extent of a defect is not always larger than the aperture size in addition to the fact that large
defects have edges which may significantly contribute to the scattering and diffraction from
these defects.
In this chapter, near-field microwave imaging of dielectric composite
structures using open-ended rectangular waveguides is studied experimentally. In the next
section experimental setups are presented and their operations are discussed. The utility of
applying near-field microwave techniques to inspect a wide variety of composite structures
with different types of defects is demonstrated, and several experimental results will be
presented in this chapter.
The experimentally obtained raw images provide a great deal of detailed information
about the structure under inspection [Qad.95]. To interpret the information contained in
such images, it is important to understand the mechanism by which an image is formed. A
near-field microwave image is the result of several factors such as the probe type (for
example a rectangular waveguide, a circular waveguide, a coaxial line, etc.), field
properties (i.e. main lobe, sidelobes, and half-power beamwidth, etc.), geometrical and
physical properties of both the defect and the material under inspection. That is why it is
important to develop theoretical models that explain the behavior of microwave energy
inside the structure under inspection.
Chapter 4 will be devoted to study the field
properties in the near-field region of an open-ended rectangular waveguide and its
interaction with a dielectric material. This study will include investigating the influences of
frequency and dielectric properties on the radiation pattern.
In chapter 5 a study of the
mechanism by which the fields interact with an inclusion will be presented. An effective
dielectric constant formula will be used to model the reflection properties of dielectric
structures.
The influence of the non-uniformity associated with the electric field
distribution at the aperture of the waveguide will be investigated and incorporated in
calculating the effective dielectric constant
57
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The application o f microwave and millimeter-wave NDT to the inspection of
dielectric composites will be discussed in this chapter. Structures with various defects and
inhomogenities such as metallic inclusions, corrosion under paint, porosity, resin binder
volume content variation and binder cure state, impact damage and voids are experimentally
inspected to evaluate the utility of this microwave technique for experimental defect
detection.
/ / / . / Imaging S etu ps an d Techniques
A microwave image is obtained by arranging detected microwave signals/data,
gathered by performing a raster scan over a structure, to produce a visual impression of the
presence of defects or the structural geometry.
The microwave data may include
information such as the phase and/or the magnitude of either the reflection coefficient or the
transmission coefficient.
Also, attenuation information can be used to produce a
microwave image of a structure as well as any combination of all of the above. The general
geometry of a composite panel with an embedded defect is shown in Figures 3.1a and
3.1b. The waveguide operates at a certain frequency and at a certain standoff distance.
The side view of the geometry is shown in Figure 3.1a, while Figure 3.1b shows the plan
view of the geometry and the scan directions.
Microwave imaging is based on transmitting a wave into a dielectric specimen and
using the magnitude and/or phase information of the transmitted and/or the reflected waves
to create a two or three dimensional image of the specimen [d'Amb.93] [Bol.90] [Gop.94].
58
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W a v e g u id e
A p ew" ~
T h ic k C o m p o s ite
Panel
F lan g e
Scan
D ire c tio n s
O p e n -E n d e d
W a v e g u id e
I
Defect
S ta n d o ff
D ista n c e
(Side View)
(Plan View)
(a)
(b)
Figure 3.1: Relative Geometry of an open-ended rectangular waveguide sensor and a thick
composite panel with a defect: (a) side view, (b) plan view [Qad.94].
Transmission type microwave approaches require access to both sides of the sample
[Bol.90]. To achieve fine spatial resolution (detection of small defects) high frequency
transmission techniques have been used [Gop.94],
A general near-field microwave NDT measurement setup is shown in Figure 3.2.
A single frequency transmitter (sweep oscillator or a Gunn-diode) generates a microwave
signal that is transmitted through an open-ended rectangular waveguide probe which is
terminated into a large metallic flange. As a standard practice, a square flange with sides
grater than A.0 is used to terminate the aperture of the waveguide to approximate an infinite
ground plane [Cro.67]. As the signal reaches the aperture, part of it gets reflected back into
the waveguide depending on the effective dielectric properties of the medium in front of the
aperture. A receiver (a detector diode or a mixer) is used to measure a voltage that is related
to the properties of the reflected signal. As the scan progresses the measured voltage
values are recorded in a matrix form. Assuming that the scan starts at a spot devoid of any
59
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defect, the detector measures a certain, almost constant, voltage. When a defect is “seen”
(i.e. within the sensing range o f the probe) by the aperture, the effective dielectric
properties of the structure in front of the aperture varies. This results in a change in the
reflected signal, and consequently the measured voltage. Images produced in this fashion
are referred to as contrast images (i.e. the presence of an inhomogenity is indicated by a
different color, or intensity level, on the image).
In this work reflection type measurements are employed.
Microwave probes
operating at a certain frequency and standoff distance are employed to obtain microwave
images o f defective samples.
The microwave probes were mounted on a computer
controlled 2-D scanning table to scan over dielectric composite samples with embedded
defects. As mentioned earlier, a voltage that is related to either the phase of the reflection
Probe
'
O p e n -E n d e d R e c ta n g u la r W a v e g u id e ,
C ir c u la r W a v e g u id e o r C o a x ia l L in e
Transmitter
S w e e p O s c illa to r o r a G u n n Diode,
Receiver
D io d e D etec to r o r a M ix e r
Display
Figure 3.2: A general near-field microwave imaging experimental setup.
60
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coefficient or the magnitude of the reflection coefficient or a combination of both can be
used to create microwave images. To show how either one or both can be used, three
microwave systems were built. The first system considers both properties (phase and
magnitude) o f the reflection coefficient and is similar to the setup shown in Figure 3.3. As
a signal is generated, by a Gunn diode or a sweep oscillator, it travels through the
waveguide towards the medium in front o f i t Depending on the effective properties of that
medium, a certain part of the signal is reflected back into the waveguide. This reflected
signal forms a standing wave pattern with the transmitted signal inside the waveguide. The
properties o f the standing wave pattern depend on the phase (which determines the
positions of maxima and minima) and the magnitude (which determines the ratios of
maximum voltage to minimum voltage, i.e. the standing wave ratio) of the effective
reflection coefficient at the aperture of the waveguide. A detector diode is used to monitor
the properties of the standing wave pattern inside the waveguide.
The voltage of the
detector diode is related to both the magnitude and the phase of the reflection coefficient.
When a defect is present, the effective dielectric properties of the medium, in front of the
aperture, changes and so does the properties of the reflected wave.
This changes the
standing wave pattern in the waveguide and consequently the diode will read a different
value. By recording these values (the output of the diode), as a function of scan position in
a matrix form, and plotting them a contrast image of the sample is created.
Images
produced with this setup depend on the phase and magnitude of the reflection coefficient.
The second microwave system employs a reflectometer unit operating at a certain
frequency band to scan over composite structures, as shown in Figure 3.4. This unit uses
a digital sweep oscillator or a Gunn diode to generate the microwave signal. The power
level and frequency may be adjusted by the oscillator. A circulator is used to separate the
transmitted and reflected signals. A diode detector monitors the reflected signal and records
61
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Digital Voltmeter
A d ju sta b le D io d e M o u n t
T o S w e e p O s c illa to r
A perture
Figure 3.3: A single reflectometer module used to produce magnitude and phase images.
a voltage that is related to the magnitude of the reflection coefficient. Images created using
similar setups will be referred to as magnitude images.
The third microwave system employs a similar method, a reflectometert unit that
operates at a certain frequency band can also be used to scan over composite structures as
shown in Figure 3.5. A directional coupler is used to split the signal into two portions.
The first portion is used as a reference signal and is fed to a mixer. The second portion is
fed through a circulator and then transmitted through the waveguide aperture to the media in
front. The circulator is used to separate the transmitted and reflected signals. The reflected
signal is then mixed with the reference signal and a voltage that is related to the phase of the
reflection coefficient is measured and recorded. Images created using similar setups will be
referred to as phase images.
62
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D ig ita l V o ltm e te r
D e tec to r D io d e
...
Ill 1
*
D ig ita l S w e e p O sc illa to r
C irc u la to r
W a v e g u id e A p e rtu re
Figure 3.4: A single reflectometer module used to produce magnitude images.
DOD
D ig ita l V o ltm e ter
L O -P o rt
R F -P o rt
© ■■■ ■ o
■■■ sss i
D ig ita l S w e e p O s c illa to r
M atch ed
D irectio n al C o u p le r
C irc u la to r
W a v e g u id e A p e rtu r e
Figure 3.5: A single reflectometer module used to produce phase images.
63
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III.2
Optimizing Scan Parameters
Optimization of the measurement parameters enhances defect detection sensitivity
and increases the dynamic range of contrast due to the presence of a defect
As was
mentioned earlier a contrast image can be obtained by monitoring a certain location on the
standing wave pattern. Figure 3.6 shows the standing wave pattern in a setup without and
with a defect. To improve the contrast and to obtain a relatively large dynamic range, the
diode detector must be positioned such that maximum difference on the standing wave
pattern is measured as shown in Figure 3.6. If the diode is set at position (a), maximum
sensitivity is achieved, meanwhile if the diode is set at position (b) no detection is achieved.
In many cases the position of the detector is fixed. To improve the detection o f defects
(e.g. image quality), the two parameters that can be used to enhance the sensitivity are the
standoff distance and the frequency of operation. The reflection coefficient depends on the
effective impedance of the medium seen by the aperture [Chapter 2], as the standoff
distance (or frequency of operation) changes, the effective impedance seen by the aperture
of the waveguide changes. This changes the reflection coefficient at the aperture, and
consequently the standing wave pattern. So, by comparing the voltage at several standoff
distances (frequency is fixed) when there is no defect and when a defect is present,
maximum contrast can be achieved. The frequency can be used in a similar fashion by
comparing the voltages measured while sweeping the frequency at a certain standoff when
there is no defect and when a defect is present. By repeating this process several times
optimal measurement parameters can be obtained.
Another influential measurement parameter is the sensitivity of the diode detector.
Sensitivity is high if as the input to the diode changes slightly the output of the diode
changes drastically. To demonstrate this different diodes were independently connected to
64
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Without Defect
M a x im u m D iffe re n c e (a)
L oad End
Source End
W ith D e fe c t
M in im u m D iff e re n c e (b )
Figure 3.6: Standing wave patterns in a waveguide produced with and without a defect
a sweep oscillator and the diode output voltage was recorded as the power on the oscillator
was increased in 0.5 dBm steps at a constant frequency of 10.0 GHz. Figure 3.7 shows
the output produced by each diode as the input signal increases.
The curve that
corresponds to diode 2 has a higher slope than the other two indicating it is the most
sensitive to small changes in input signal power (in this range of input signal power). A
sensitive diode is much more likely to detect slight changes in the effective dielectric
properties o f a medium which is essential for defect detection.
In all of the imaging results shown in this work, the measured voltage for each
image is normalized with respect to its maximum value and then different colors ate
assigned to different normalized voltages (maximum is red and minimum is lavender) to
produce an image. Consequently, these images must be viewed from a qualitative point of
view since only detection of defects is the goal of these images, although a very good
impression o f the size of a defect can be obtained as well.
65
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12
diode 1
• diode 2
1.65
diode 3
,= k .
■5
•3
1
3
1
5
7
9
11
I n p u t S ig n a l P o w e r (d B m )
Figure 3.7: Change in voltage across three separate diode detectors at 10 GHz with respect
to input power to the diode.
III.3
Applications and Experim ental Results
Experimental results obtained using the near-field microwave imaging setups
presented in the previous section to inspect a variety of dielectric composite structures with
different types of defects are presented in the following sections.
III.3.1
Near-Field Imaging o f Thick Composites With M etallic Defects
The results of an experimental study investigating the use of microwaves to inspect
the presence o f metallic inclusions in thick composite structures are presented in this
section.
Specially fabricated thick glass reinforced polymer composite panel with
embedded metallic inclusion is inspected using an open-ended rectangular waveguide
66
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sensor. Phase images of the sample under inspection are created. For optimal detection,
the influence of standoff distance between the sensor and the panel is also studied. As will
be shown later, results indicate that the proper choice of standoff distance may significandy
enhance defect detection capability [Qad.94].
The sample material used in this study was fabricated from S-2 glass reinforced
polyester composite. This sample (among others) was developed by the Army Research
Laboratory for a series of NDT tests; it was not specifically intended for microwave
measurements. However, it does contain intentionally introduced defect and it has been
tested and characterized using a variety of NDE techniques [Car.94]. A metallic inclusion
was embedded in the sample during hand lay-up prior to cure. The nominal size of the
panel is 190 mm x 190 mm x 25 mm. The density of the panel is 1.7 g/cnA
III.3.1.1
M easurement Results a n d Discussion
The panel, shown in Figure 3.8, has an aluminum inclusion of 6.35 mm by 6.35
mm by 0.8 mm located at a distance of 12.7 mm from the surface of the sample. A phase
scan was performed at a frequency of 10.5 GHz (using a setup similar to that shown in
Figure 3.5) in a contact fashion. The scan covered an area of 85 mm by 98 mm. Figures
3.9a and 3.9b show the signal intensity and the plan view of this scan, respectively. The
contrast is very high, and the defect is clearly visible. Both figures show a very good
indication of the fiber bundle pattern associated with the opposite side of the sample. Also,
the size of the defect on the image corresponds well with the physical size of the defect.
This indicates the high spatial resolution that may be achieved with microwave near-field
imaging. Another important observation on the image is the presence of two features along
the sides of the defect parallel to the broad dimension of the waveguide. These features are
67
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due to the sidelobes of the radiator and the edge effects of the defect. These features will be
analyzed further in chapter 4. As was mentioned earlier, the measurement parameters
(frequency o f operation and standoff distance) may be varied to optimize for a given
measurement.
To demonstrate this, two measurements were performed in which the
standoff distance between the sensor and the surface of the sample was changed. In the
first measurement the sensor was pointing directly at the defect (maximum signal), and in
the second one it was pointing at a non-defect area (background or minimum signal).
Subsequently, the standoff distance was changed, and the voltage proportional to the phase
of the effective reflection coefficient at the aperture of the waveguide was recorded as
shown in Figure 3.10.
203 mm
6.35 mm
Figure 3.8: Descriptive geometry of a thick composite panel with an aluminum inclusion
embedded at the center of the panel [Qad.94].
68
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The results show that operating at a standoff distance of 0 mm (i.e. in contact) and at this
frequency (10.5 GHz) the difference between the signals due to defective and non defective
areas is maximum. On the other hand, if a standoff distance of around 5 mm or 13 mm is
used, there will be no distinction between these two areas of the panel.
Another
observation is that, operating at a standoff distance of between 5 mm to 13 mm the contrast
in the image is reversed. Also, operating in the range where the phase variation as a
function of the standoff distance is almost constant (7-10 mm) is important from a practical
point of view since slight changes in the standoff distance do not influence the outcome
significantly. To illustrate these observations, a phase scan of this sample at a standoff
distance of 9 mm was produced.
Figure 3.11 shows that the contrast is reduced (i.e. the dynamic range is less) and reversed
(compared to the results of Figure 3.9). However, at this standoff distance, the fiber
0
20
40
60
80
Figure 3.9: An in contact phase scan of the composite shown in Figure 3.8 at a frequency
of 10.5 GHz [Qad.94].
69
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0.2
0.15
With
Defect
0.1
>
Without
Defect
0.05
oso
a
o
>
0.05
-
0.1
0.15
0
5
10
15
20
S ta n d o f f D ista n c e (m m )
Figure 3.10: The voltage (related to the phase) with and without a defect as a function of
the standoff distance [Qad.94].
bundle orientation seems to be more visible than that of Figure 3.9.
This is another
indication of the sensitivity of the standoff distance to different sample characteristics (e.g.
small thickness variation),
mm
0
20
40
60
80
Figure 3.11: A phase scan of the composite shown in Figure 3.8 at a standoff distance of 9
mm and a frequency of 10.5 GHz [Qad.94].
70
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III.3.2
N ear-F ield
Imaging
of Rust
Under
Paint
and
D ie le c tr ic
L a m in a te s
In many applications the detection of rust or corrosion under paint and composite
laminate coatings is an important practical issue [Bab.95] [Fun.81] [Col.93].
When the
detection of rust is conducted accurately and in its early stages, it results in the savings of
million of dollars in maintenance costs, damage minimization and reduction in repaint
cycles in various industrial and military environments. Microwave NDT techniques posses
the ability of detecting minute thickness variations in dielectric coatings as well as slight
dielectric property variations in stratified dielectric composites [Bak.94a] [Bak.94b]
[Bak.93] [Gan.95] [Chapter 2]. The presence of rust or corrosion may be considered as an
additional new thin layer under a coating or a composite laminate.
Microwave
nondestructive methods are very well suited for inspecting this type of layered materials
[Qad.97].
In this section the utility of using open-ended rectangular waveguide sensors for
detecting rust under paint and dielectric laminate coating is presented. Several experiments
are conducted at two different frequencies on steel specimens with areas of rust in them.
III.3.2.I
M easurement Results and Discussion
A steel specimen with a 40 mm by 40 mm area of rust is shown in Figure 3.12.
This specimen was produced by acquiring a relatively flat piece of steel on which a thin
layer of rust had already been produced (naturally). A 40 mm by 40 mm area (at the center)
was then masked out by a piece of tape and the remaining surface was sand blasted. The
71
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average thickness of the rust layer was measured (using a micrometer) to be approximately
0.08 mm. Subsequently, this specimen was painted with up to 0.60 mm of common spray
paint, as uniformly as possible. 0.60 mm represents ten painting cycles. After applying
various layers of paint microwave images of the rust specimen were produced using raster
scans (every 2 mm by 2 mm) o f the specimen at 24 GHz. Measurement systems (similar to
Figures 3.3 and 3.4) were used to create contrast images of the rust area on the samples.
Figures 3.13 and 3.14 show the scans of this specimen at a standoff distance o f 4
mm and at 24 GHz when covered with a paint thickness of 0.145 mm and 0.60 mm,
respectively. The rusted/corroded area is clearly visible in the center of all of these images
corresponding to the rust region shown in Figure 3.12. There is an elongated region in the
upper right hand comer of these images which shows up as a region (e.g. color) inbetween paint and rust. The steel specimen had a very subtle indentation in this region.
Consequently, one can consider that this region has a slightly thicker paint layer than the
rest o f the painted areas. Therefore, these images not only show the clear possibility of
detecting a thin layer of rust under paint, they also illustrate the fact that paint thickness
variation can be distinguished from the presence of rust.
Figure 3.12: A 40 mm by 40 mm area of rust on a steel plate [Qad.97].
72
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f
t
Figure 3.13: Image of rust under 0.145 mm paint at 24 GHz and a standoff distance of 4
mm [Qad.97],
Figure 3.14: Image of rust under 0.60 mm paint at 24 GHz and a standoff distance of 4
mm [Qad.97].
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In many applications metals could be covered by other laminates than paint. A
rusted metallic plate (similar to the one used earlier) was prepared in a similar way, and was
covered with layers of laminate material (synthetic rubber) of thicknesses up to 25.5 mm
(1”). The sample was again scanned using the same setups used on the other sample.
Figure 3.15 shows the image obtained at a frequency of 24 GHz and a standoff distance of
4 mm. Figure 3.16 shows the image obtained at a frequency of 10 GHz and standoff
distance of 4 mm. Again the rusted area is detected even under 25 mm thick synthetic
rubber laminate. It should be noted that at a frequency of 10 GHz (waveguide dimensions
22.86 mm x 10.16 mm) the rusted area looks smaller than it does at 24 GHz (waveguide
dimensions 10.86 mm x 4.32 mm). This is as a direct result of the ratio of the dimensions
of the rusted area to those of the waveguide aperture. This also indicates that the spatial
resolution is related to the waveguide dimensions.
nun
Figure 3.15: Image of rust under 25.4 mm laminate at 24 GHz and a standoff distance of 4
mm.
74
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mm
0
20
40
60
Figure 3.16: Image of rust under 25.4 mm laminate at 10 GHz and a standoff distance of 4
mm.
III.3.3
N ear-Field Imaging o f Com posites With Porosity Defects
In many composites (e.g. polymers), the presence of porosity causes lowered
mechanical performance due to stress concentrations.
Localized porosity can be
particularly damaging to the joint strength of adhesively bonded components [Gra.95]. In
ceramics, the relative density is an important processing parameter, and again the ceramic is
extremely sensitive to stress concentration (lowered density).
ceramic is weak and has low stiffness.
If not fully densified, a
In composites, the porosity can be within the
matrix material which will affect the performance in a similar fashion to those in bulk
materials.
However, porosity often concentrates at specific locations in composite
materials (either between plies or at the fiber/matrix interface), and can dramatically lower
75
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the flexural and shear performance [Gra.95].
Increases in porosity during operation
(material under loading) may precede macroscopic damage and possibly indicate the
presence of delamination. In most practical cases porosity in a structure is clustered (local).
The results of a study on the detection of local porosity in composites using microwave
near-field imaging techniques are presented next.
III.3.3.1
M easurem ent Results an d Discussion
An epoxy resin disk with a diameter o f 76.5 mm and a thickness of 8.2 mm was
produced with three porous inclusions embedded in it. The inclusions were in the shape of
a pill with a diameter o f 6.35 mm and thickness of 4.45 mm. These inclusions were made
of air-filled microballoons providing three clustered porosity levels of 44%, 49% and 56%
as shown in Figure 3.17. The distance between the centers of each two inclusions was 19
mm, and in the thickness direction they were all located in the middle of the disk. An area
of 56 mm by 18 mm (as shown in Figure 3.17) was scanned in contact at 34.8 GHz using
an open-ended rectangular waveguide sensor. This frequency, in the 26.5-40 GHz range,
was chosen since for this range the waveguide aperture is 7.1 mm by 3.5 mm and provides
for a higher spatial resolution compared with those at lower frequencies. A magnitude scan
was performed to produce an image of the defective disk as shown in Figure 3.18.
three inclusions are clearly seen as red spots.
The
The color intensity (of the red color)
associated with each inclusion gives a qualitative measure of the difference in the porosity
levels among the three inclusions. The distance between the centers of each two inclusions
matches very well the distance in the sample (19 mm). This indicates the high spatial
resolution associated with these measurements. The image effectively shows the utility of
using microwave nondestructive techniques not only for detecting local porosity, but also
for quantitative estimate of the air content associated with it.
Such an image, once
76
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calibrated may indicate absolute porosity level, and the size of the porous inclusion. The
quality of the image as it pertains to porosity level determination may be enhanced, for a
given composite, by changing parameters such as frequency and standoff distance. Figure
3.18 shows that there are additional ring shaped dark blue areas around the inclusions.
These features are due to the side lobes along the broad dimension of the waveguide in
addition to the edge effects associated with the inclusions (similar to the features on Figure
3.9).
Epoxy Resin
Scan Area
44%
49%
56%
Air-Filled
Microballoons
Figure 3.17: The schematic of an epoxy resin sample with three different levels of local
porosity [Gra.95].
77
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Figure 3.18: Image of the sample shown in Figure 3.17 at a frequency of 34.8 GHz: (a)
plan view, (b) signal intensity [Gra.95].
7/ 7.3 .4
N ear-F ield
Imaging o f Fiberglass
Composites
With
V ariable
Binder Percentage and Cure State
Low density fiberglass composites are used in many environments for insulation
purposes. Nondestructive inspection of thick, low density, low permittivity and low loss
dielectric composites poses many challenging problems for most (NDT) techniques.
Microwave NDT methods, however, are very well suited for inspecting this type of
composite materials [Zou.94] [Qad.95] [Gra.95]. This is particularly true when accurate
thickness, thickness variation, material composition uniformity, cure state and density
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variation determination are of interest. Microwave methods have been used to characterize
material properties o f several composite mixtures in which curing takes place (a chemical
mixing process) [Liv.93] [Jow.87] [Gan.94].
There are several important issues associated with the production of fiberglass
products, namely, the uniformity by which the resin binder is applied, resin binder cure
state, and variations in the glass fiber density. The utility of applying microwave NDT
methods to distinguish among fiberglass samples with different resin binder levels, in a
non-contact fashion, using an open-ended rectangular waveguide sensor, is presented next.
III.3.4.1
M easurem ent Results a n d Discussion
To conduct this experimental study, a 25.4 mm (1”) thick conductor backed real
life fiberglass sample with 18.6% resin binder (i.e. base material) was used. Four regions
of this sample were first removed and then embedded with fiberglass with resin binder
levels (by weight) o f 0%, 9.4 %, 13.4 % and with uncured resin binder, as shown in
Figure 3.19. These inclusions had square shapes with sides of approximately 25.4 mm
(1”) and the distance between them was 50.8 mm (2”).
The influence of the standoff distance was experimentally investigated by placing
the microwave sensor in front of the sample (for all five regions) while the standoff
distance was varied from 0 to 10 mm. For each standoff distance a dc voltage related to the
effective reflection coefficient at the waveguide aperture was recorded. To be able to detect
the presence of these local inclusions and distinguish them from one another, a difference
between the voltages when the waveguide is over the inclusions and the base material must
exist. These differences, can be maximized by choosing the correct standoff distance.
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Figure 3.19: Schematic of the multibinder fiberglass sample [Qad.96].
Consequently, the voltage difference between the base material and all other local
inclusions, as a function of standoff distance was obtained at 24 GHz, as shown in Figure
3.20. From this figure it may be deduced that for some standoff distances it is difficult to
distinguish among the different local inclusions (1-3 mm), while for others (4-5 mm) there
is a considerable voltage difference between the base signal and the signal from the
inclusion regions.
Subsequently, a microwave image of this sample was made at a
standoff distance of approximately 4 mm, as shown in Figure 3.21. The different intensity
levels in the image are proportional to the dielectric properties of each region, and hence
related to the difference in their respective resin binder levels. This simple image illustrates
the potential of using such a nondestructive and non-contact testing method for
distinguishing among different resin binder contents in a 25.4 mm-thick low density
conductor backed fiberglass sheet. The voltage readings that produced these images may
be calibrated to obtain an estimate of the resin binder level associated with each inclusion
regions.
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---------- No Binder
— — - Uncured Binder
9.4%
. . . . . 13.8%
4
2
0
•2
-4
6
•8
0
2
6
4
S ta n d o ff D is ta n c e (m m )
8
10
Figure 3.20: Voltage difference as a function o f the standoff distance for all defects at 24
GHz [Qad.96].
Figure 3.21: Image of the sample shown in Figure 3.19 at a frequency of 24 GHz and a
standoff distance of 4 mm [Qad.96].
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II1.3.5
Near-Field Im aging o f Composites With Impact Damage Defects
In the design of glass fiber/epoxy composites, the useful life of a composite as it is
subjected to impact fatigue loading is a very important issue.
Due to the difficulties
encountered in monitoring damage accumulation in composites, lifetime predictions of
these composites have been a problem since they are being used in a variety of applications.
A build up of micro damage, such as matrix micro-cracks and micro-delaminations, is
hypothesized to occur even though there is no apparent change in material compliance. A
critical level is finally reached at which time the properties of the composite begin to fall and
compliance change is evident [Rad.94], The utility of applying microwave NDT methods
to detect the initiation and propagation of impact damage, using an open-ended rectangular
waveguide sensor, is presented next.
III.3.5.1
Measurement Results and Discussion
To study the effect o f cyclical impact on composite materials, specific polymer
reinforced epoxy samples were subjected to cyclical impact fatigue [Rad.94]. The general
experimental approach applied to this set of composite specimens was to introduce
cumulative damage by repetitive impact [Rad.94],
Out-of-plane impact fatigue test
specimens were modeled in a disk geometry of 63.5 mm diameter and 2.5 mm thickness
(after polishing). These composite specimens were made up of either 4 or 8 layers of 10 oz
stain weave fiberglass fabric in an Epon 813 epoxy resin matrix. The plies were stacked in
a mid-plane symmetric, quasi-isotropic fashion with a fiber volume fraction of
approximately 20% for the 4 ply and 40% for the 8 ply specimens [Rad.94].
A setup
similar to that shown in Figure 3.4 was used to create magnitude images of these samples
at a frequency of 34.8 GHz. At the beginning a 4 ply disk was impacted at a load of 0.53
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KN. for 500 cycles. No visible damage was observed on the disk. However, as a scan
was performed the presence of damage at the warp/weave crossovers was seen (Figure
3.22). The same setup was used to scan an 8 ply disk after 3000 impact cycles at a load of
0.70 KN. The primary damage region is observed near the center of the image, shown in
Figure 3.23. The image shows that the region o f damage is not circular as one would
expect, but actually shows accelerated damage propagation along the fiber bundles.
The
same 8 ply disk was imaged after 4000 impact cycles. Figure 3.24 shows that the central
damage zone has grown noticeably and increased damage is apparent across the image.
Visual inspection of the specimen showed significant damage growth between 3000 and
4000 cycles.
0
13
30
30
Figure 3.22: Image of the 4 ply disk at a frequency of 34.8 GHz after 500 impact cycles
[Rad.94].
Figure 3.23: Image of the 8 ply disk at a frequency of 34.8 GHz after 3000 impact cycles
[Rad.94].
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Figure 3.24: Image o f the 8 ply disk at a frequency of 34.8 GHz after 4000 impact cycles
[Rad.94],
III.4
Summary and Remarks
The inspection of thick layered composite materials is essential in ensuring
structural integrity before and during use. Many types of defects, that cannot be visually
detected, can occur in production and in use situations weakening the structural integrity of
the composite and endangering structures employing such materials. Early detection of
defects is necessary to mitigate damage propagation.
The ability of microwaves to
penetrate inside dielectric materials makes microwave NDT techniques very suitable for
interrogating structures made of thick dielectric composites. Three experimental setups
were presented for three different types of near-field microwave imaging. The effects of
frequency of operation and the standoff distance as measurement optimization parameters to
enhance the sensitivity to a defect were studied and presented.
Experimental results
obtained from scanning a variety of composite samples with different types of embedded
defects were presented. Images of these defective samples were created using a measured
voltage that is related to the phase and/or magnitude of the effective reflection coefficient at
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the aperture of the rectangular waveguide sensor.
These images presented detailed
information about the structure and integrity of the inspected samples. On all of these
images the size of a defect matches closely its physical size, indicating the high resolution
associated with this technique.
The resolution is related to the dimensions of the
waveguide (i.e. the frequency band).
To interpret the information contained in such
images, it is important to understand the mechanism by which an image is formed. A nearfield microwave image is the result of several factors such as the probe type (example
rectangular waveguide, circular waveguide or coaxial line), field properties (i.e. main lobe,
sidelobes, and half power beam width, etc.), geometrical and physical properties of both
the defect and the material under inspection.
That is why it is important to develop
theoretical models that explain the behavior of microwave energy inside the structure under
inspection. A study o f the field properties in the near-field region of an open ended
rectangular waveguide and the fields interaction with a dielectric material will be presented
in Chapter 4. In Chapter 5 a study of the mechanism by which the fields interact with an
inclusion will be presented. An effective dielectric constant formula will be used to model
the reflection properties of dielectric structures.
The influence of the non-uniformity
associated with the electric field distribution at the aperture of the waveguide will be
investigated and incorporated in calculating the effective dielectric constant.
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CHAPTER IV
Analysis o f the Field Properties in the Near-Field o f Rectangular
Waveguide Probes Used fo r Imaging
The experimentally obtained data using waveguide probes, in Chapter 3, showed
that a great deal o f detailed information can be obtained from these near-field microwave
images. The spatial resolution associated with these images is high, and it seems to be
dependent on the dimensions of the waveguide probe not on the frequency of operation
(e.g. wavelength), as in far field imaging. Some features of these images were interpreted
to be a result of the presence of sidelobes along the broad dimension of the probing
waveguide. In order to understand the information contained in a near-field microwave
image and the image formation mechanism it is essential to formulate the properties of the
fields in the near-field of an open-ended rectangular waveguide probe. This will help in
building an intuitive understanding of the behavior of the fields inside dielectric materials
while in the near-field of a probing waveguide.
It will also aid in solving the forward
problem of imaging defective structures which can be used in solving the inverse problem
to obtain defect properties.
IV.1
Radiation Pattern Theoretical Modeling
The radiation from flange-mounted open-ended rectangular waveguides looking
into free space has been considered by several investigators in the past [Com.64],
[Lew.51].
The case of radiating apertures into a stratified medium has also been
considered by many investigators [Bak.92], [Nik.89], [Teo.85]. In this part of the work,
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theoretical formulation o f the fields radiating out of a waveguide probe into an infinite halfspace of material is presented.
The formulation is general and can account for the dominant mode as well as
higher-order modes. The analysis assumes that the waveguide aperture is mounted on an
infinite conducting flange. In the dielectric medium Fourier integrals are used to express
the solution of the wave equation, namely
V 2E(x,y,z) + K 2E(x, y,z) = 0
(4.1)
where, K 2 = K 02£fi, £ and jl are the permitivity and the permeability of the medium,
respectively. The electric field can be written using the Fourier integrals in the xy-plane as
(4.2)
By substituting Equation 4.2 into Equation 4.1, the following equation is obtained
(4.3)
which can be written as
(4.4)
where, y = ^jk2 + k 2 - K 2
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Equation (4.4) has a solution of the form
E(kx,ky,z) = A e 7t + B e 7z
(4.5)
where, A = xAz + yAy + zAz and B = xBz + yBy + zBz are unknown vector coefficients to
be determined. Using these unknown vector coefficients the electric field can be written as
E(x,y,z) =
+ Bzer‘) + y (Aye~r‘ + Byeri) +
z(A2e~rz + Bzer‘)leJ(l-x+t’y)dkxdky
(4.6)
A dielectric medium is a source-free region (V • E(x,y,z) = 0), using this property with
Equation (4.6) yields
jkz(Aze-ri + Bze rz) + jky (Aye yz + B/ 1) - y( V
ri ~ ^ e 71) = 0
(4.7)
Equation 4.7 is now used to obtain the following system of equations
(V'M* + jky A, - yA t )e~7z = 0
(4.8a)
{jkzBz + jk yBy + y B z)e7z = 0
(4.8b)
from which the values of the z-components can be expressed in terms of the x- and ycomponents as
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jk ^ + jk A
(49a)
r
B = J k*B* + j k yBy
(4.9b)
Now by substituting Equations (4.9a and 4.9b) in Equation (4.6) the electric field can be
written as
E{x,y,z) = \ ~ _ \ l f { k x,lcr z)eJ{l' " k’y)dkxdky
(4.10)
where,
F(k ,ky, z) = [ A z (x + z i ^y - ) + A ( y + z ^y) ] e ~ r' +
[Bx(x - z &y ) + B (y - z ^ y) ] e 7z
(4.11)
To obtain the magnetic field at any point, Maxwell’s equations can be used
V x E(x,y,z) = —jcofi H(x,y,z)
(4.12)
using Equation (4.10) the magnetic field can be expressed as
H(x,y,z) =
f V x F(k ,k ,z)eJ(k,x+i’y)dk dk
(O f!
*
(4.13)
y
To calculate the magnetic field, the coefficients for each of the unknowns A z, B x, Ay, and
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By must be calculated first. To find the coefficients of the unknowns Equations (4.10,
4.11 and 4.13) are used, after simplification, the magnetic field can be expressed as
H(x,y,z) = -j— f f [(c,Ax + c2A )e 71 + (c3Bz + c4B )en }
(
O
f
J
.
1
1
el{k'x+t’y)dkxdky
(4.14)
where,
c, =
+ y —---- — - z(jky)
(4- 15a)
tc2 —lr2
k k
c2 = - x - -----^ + y - ^ - + z(jk )
7
7
(4.15b)
kk
K 2 —k2
c3 = x y^ - y ---------7 - z(jk )
y
(4.15c)
„ K 2 —k 2 . k k
.
C4 = x
-’ y - ^ + zUk')
(4.15d)
The derivations outlined above are applicable to structures made of several layers of
material and terminated either by an infinite half-space of a material or by a conducting
layer. The electric and magnetic fields at any point in space are expressed using the
unknowns A z, B z, A y, and B y. These unknowns are obtained by applying the boundary
conditions corresponding to a given configuration.
In solving for the unknowns, the
starting point is the last layer of the structure. This helps in reducing the number of the
unknowns in that layer. The unknowns in the layer before the last are then expressed in
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terms o f the unknowns of the last layer. By back propagating the unknowns of the last
layer to express the unknowns in each layer in terms of them, the number of unknowns in
the structure is minimized. To solve for the unknowns, the boundary conditions at the
aperture of the waveguide are applied. Once the unknowns of the last layer are calculated
the fields everywhere are determined, and consequently field intensity and power density
patterns can be obtained. To understand the properties of the fields in the near-field region
of an open-ended rectangular waveguide and the influences of the waveguide dimensions,
frequency of operation and the dielectric properties of the medium on an image, the fields in
an infinite half-space of a dielectric material will be formulated and their properties will be
studied extensively.
IV.2
Fields in an Infinite Half-Space o f a Dielectric M aterial
The geometry of an open-ended rectangular waveguide radiating into an infinite
half-space o f dielectric material with respect to the coordinate system is shown in Figure
4.1. In an infinite half-space of a dielectric material the electric field components should
vanish as the distance z increases (z —>°°).
From Equations 4.10 and 4.11, the x-
component of the electric field can be written as
(4.16)
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a
b
w a v e g u id e
Infinite Half-Space
z=0
Figure 4.1: An open-ended rectangular waveguide aperture radiating into an infinite half­
space.
To satisfy the boundary condition, Ez = 0, the following equation is obtained
(4.17)
(Aze~r t +Bzerz)t^ = 0
which indicates that
B. =0
(4.18)
The y-component of the field can be expressed as
Ey = j ^ j y A y e - ri + Byert)ejik-x+k^dkxdky
(4.19)
To satisfy the boundary condition, Ey = 0, the following equation is obtained
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(Aye~r' + B yeri)l^ = 0
(4.20)
which indicates that
(4.21)
By = 0
So, Equation (4.11) reduces to
F(kI ,k ,z) = [Ax(x + z ^ - ) + A ( y + z ^ L)]e rz
r
r
(4.22)
Now there are only two coefficients to be calculated, namely, Ax and Ay. At the aperture of
the waveguide the tangential components of the electric field inside and outside the
waveguide must be continuous. Thus, at z = 0 the tangential electric field reduces to
=E
E
(4.23)
A
where,
/a \
(4.24)
/ = xAx + yA =
By applying Fourier transform to Equation (4.23) and equating it to the transform of the
aperture fields, the following is obtained
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/=ft]=
j j E a r i x ^ e - ’^ ^ d
\ *J
k ' d
(4.25)
k ,
aperture
where Ev (x,y) is the field at the aperture o f the waveguide. Therefore, the unknown A x
and Ay coefficients can be calculated using the aperture fields.
At the aperture of the waveguide the tangential components of the magnetic field
inside and outside the waveguide must be continuous.
Thus, at z = 0 the tangential
magnetic field is given by
(4.26)
Y
Again, once the unknown coefficients A x and A y are expressed in terms of the aperture
fields, the magnetic field can be calculated.
At this point, the unknowns needed to calculate the field at any point in space are
given in terms o f the aperture fields.
Now, one may either use the dominant mode to
describe the incident and reflected fields at the aperture, or also include higher-order
modes.
In this work the dominant mode TE10 is considered. To include higher-order
modes the aperture fields must be expanded in terms of the dominant and higher-order
modes.
The aperture field Ev due to the dominant mode and higher-order modes is given
by
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where, kCi is the cutoff wavenumber of the dominant mode and T is the reflection
coefficient at the aperture of the waveguide. The first term in Equation (4.27) is due to the
incident and reflected dominant mode components, while the second term is due the
reflected higher-order modes for both the TE and the TM waves.
In this work only the
dominant mode is considered, higher-order modes can be added at this point without
changing the derivations. In Equation (4.27) e™ is given by
a
(4.28)
a
where a is the broad dimension of the waveguide aperture. The aperture electric field can
be written as
(4.29)
Hence, the unknown coefficients Ax and Ay can now be calculated from the aperture fields.
First, for simplicity the aperture field expression is rewritten as
Eap= y S s i n ( - x )
a
(4.30)
where,
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s = m ± L i± £
<
(4.31)
a
To calculate the unknown coefficients, Equation (4.25) is used along with Equations (4.30
and 4.31)
(A.
7 = a \ = Q{ y [ \ asin— e-jk'xdkx ]e~jk’ydky
lA J
where, Q =
\ Ay)
{
Jo
(4.32)
a
S_
-2 , by performing the integration over the apenure
(2 n Y
n_
a
=Qy
[e~jk‘a + 7]
e jk'b - l
(4.33)
-jk,
from Equation (4.33) it is evident that there is no x-component, thus,
A=0
(4.34a)
K_
a
Ay = N = Q
{ - ) 2 - k 2x
[e~jk-a + 71
£f; V - 7
(4.34b)
-jky
a
Now substituting the expression for the unknown coefficients in Equation 4.22 yields
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F(kx,kx,z) = [N(y + z ^ - ) ) e ~ rx
(4.35)
and consequently, the electric field (Equation 4.10) can be written as
E(x,y,z) = L L ^ C y + z — =
L)}e rte
Y
(4.36)
Equation (4.36) indicates that the electric field has two components namely, E y and E t.
These two components are
(4.37)
(4.38)
The electric field component in the x-direction ( Ez) does not exist because only the
dominant mode was used in the derivations.
Thus far, the electric field components can be calculated from the last two integrals.
The magnetic field components can now be calculated using Maxwell’s equations (Equation
4.12) which yields the following
x
cofi dy
(4.39)
dz
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.= U L
(Ofj. dx
(4.40)
/ dE
H = - i ----(Ofl dx
(4.41)
h,
In order to calculate the two infinite integrals associated with each field component,
a polar coordinate transform of the following form is applied
kx = ficosa,
k = /Jsin a
(4.42)
having a Jacobian of the form
J =
dkt
dp
dkz
da
cos a
—J3 sin a
dky
dky
sin a
(3 cos a
d/3
da
= f3 [cos2a + sin2 a} = (3
(4.43)
The only unknown that is left is the reflection coefficient. The reflection coefficient
can be calculated using the derivations outlined in chapter 2, or it can be calculated from the
current derivations in the following fashion. The tangential magnetic field in the waveguide
at the aperture is given by
(4.44)
Ke,
i= l
which when using the dominant mode only reduces to
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(4.45)
where, h f = [x—(sin—*)] for the dominant mode and
a
a
Y . = ^ - = ^ - 1 ------------------------------------------------------------------------------(4.46)
*e.
con
The magnetic field outside the waveguide is given by Equation 4.13 where,
F(kz,ky,z) is as given by Equation 4.35.
At the aperture Equations (4.13 and 4.45)
should be equal. By multiplying (i.e. dot product) both sides by hJE and integrating over
the aperture, the following expression of f is obtained
V=
(4.47)
L +G
where,
and
G=
f f 2 F(P, a ) d a dfi
(4.49)
(0fj.on Jo Jo
99
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in which,
F(fi,cc) = K0[er - p 2 cos2« ] ( -*— ).2 V(P,a}U(P,a)
aK0
(4.50)
where,
it
~aK„
V(P ,a ) = 2
(4.51)
[ 1 + cos(aK0P cos a)]
- (p cosa)2 + ( - ^ - ) 2
aK„
and
U ( p ,a ) = (P since)'
,P sina*0
•[/-cos(6AT0)3sina)]
(4.52)
b2K2
,Ps ina = 0
Computer codes for calculating T and all the components of the electric and
magnetic fields were developed. The function F(P,a) (Equation 4.50) has a discontinuity
at P = ■Je~ which has a much higher effect when the dielectric constant is low. This is
primarily due to the fact that the function is of a damping nature, so as the dielectric
constant increases, the discontinuity becomes less significant. The final value of p (i.e.
the value at which F(P ,a ) —»0) can be determined prior to calculating the fields by
plotting the function F ( p , a ) .
The final value of p is dependent on the frequency of
operation and the dielectric properties of the medium.
The function F(P,a) must be
integrated to obtain the fields components everywhere. To avoid the discontinuity, the
100
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integration is carried from 0 to the final value of (3 excluding a very small interval around
the discontinuity.
Once the electric and magnetic field components are calculated, normalized field
intensity and power density patterns can be obtained to study the radiation properties of an
open-ended rectangular waveguide. For the power density calculations, the time average
Poynting vector P is given by
P = t [ E xH']
(4.53)
where, * denotes the complex conjugate and
E = xE z + yE y +zE2
(4.54)
77 = xHz + yHy + zHt
(4.55)
In the derivations outlined above the x-component of the electric field was found to be zero,
so the poynting vector is expressed as
P =L{x{EyH\ - £ , / / ; ) + y (£ ,//; ) + £ ( - £ / / ; ) ]
(4.56)
The real part of P represents the radiated power density and the imaginary part represents
the reactive (stored) power density.
101
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IV.3
Normalized Power Patterns in Different Planes
The derivations oudined above were used to calculate the fields radiating out o f an
open-ended rectangular waveguide at any point in space. The electric and magnetic fields
can now be calculated as a function o f location, frequency of operation and dielectric
properties of an infinite half-space o f a dielectric material. To help understanding the
near-field properties of an open-ended rectangular waveguide probe, power density
patterns in the three planes (xz-, yz-, and xy-planes) are presented next.
The power patterns for an infinite half-space of dielectric material with dielectric
constant of ( er = 2.5 - j0.5) at a frequency of 24 GHz were calculated as a function of
location inside the material. Figure 4.2 shows the normalized power pattern in the xzplane ( y = — plane). Each color level on the figure represents a 3-dB difference. The
pattern indicates that the fields remain confined within the aperture’s a-dimension for a
long distance inside the material (around 15 mm), and they only broaden after that. No
indication of the presence of sidelobes is observed on the pattern in this plane. Figure 4.3
shows the normalized power pattern for the same parameters in the yz-plane ( x = ^
plane). Again, the fields are confined within the aperture’s b dimension up to a long
distance inside the material (around 15 mm) and they only broaden after that. Figure 4.3
also indicates the presence of sidelobes along the broad dimension, a, o f the waveguide.
To obtain a better vision of the sidelobes, the power pattern in the xy-plane (at z = 1
102
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mm) is calculated for the same parameters, as shown in Figure 4.4.
Again, from the
pattern it is clear that the fields are confined within the aperture of the waveguide. Two
sidelobes forming along the broad sides o f the waveguide are observed on the pattern.
The fact that the fields in the material remain confined within the aperture dimensions of
the waveguide explains the high resolution associated with the near-field microwave
images obtained in Chapter 3. It also indicates that the spatial resolution is primarily
influenced by the waveguide dimensions. The presence of sidelobes along the broad
dimension o f the waveguide partially explains the presence of some features on near-field
microwave images (e.g. the dark blue features observed on the image shown in Figure 3.9).
To confirm that the sidelobes form along one dimension, the following experiment was
conducted. Figures 4.5a and 4.5b show two images of an 80 mm x 80 mm area on a glass
reinforced plastic composite with a 0.8 mm thick square (6.35 mm x 6.35 mm) aluminum
inclusion (the one described in Figure 3.8) [Qad.95].
These two images were obtained
using the same probe at a frequency o f 9.2 GFlz and a standoff distance of 3.8 mm. The
difference between the two images shown is the orientation of the scanning probe (i.e.
polarization o f the electric field). In the first image the electric field was parallel to the
vertical axis (i.e. the narrow dimension of the waveguide is parallel to the vertical axis).
The second image was obtained with the electric field parallel to the horizontal axis (i.e. the
narrow dimension of the waveguide is parallel to the horizontal axis). As expected, with
both alignments the aluminum defect was easily detected. In both images the presence of
these features was observed along the broad dimension of the waveguide. This indicates
that sidelobes are only present along the broad dimension of the waveguide.
103
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I
1
10
20
I
30
Figure 4.2: The normalized power pattern in the xz-plane (y = — plane) at 24 GHz inside
a material with er = 2.5 —j0.5.
mm
Figure 4.3: The normalized power pattern in the yz-plane (x
plane) at 24 GHz inside
a material with er = 2 .5 - j0.5.
104
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-3
0
5
Figure 4.4: The normalized power pattern in the xy-plane (z=l mm plane) at 24 GHz
inside a material with er = 2 .5 - j0.5.
Figure 4.5: Plan view image o f a phase scan at a standoff distance of 3.8 mm at 9.2 GHz, a)
E-field is parallel to vertical axis, b) E-field is parallel to the horizontal axis.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IV.3.1
Influence o f Frequency on the Radiation Pattern
In Chapters 2 and 3 the significance of choosing the proper frequency of operation
on the outcome of a measurement was mentioned several times.
In this section the
influence o f the frequency on the radiation pattern is studied and discussed.
Two
parameters in the x- and z-directions will be used to study the factors that influence the
radiation patterns.
The half-power beamwidth is a property associated with antenna
radiation patterns in their far-fields. In this study, when comparing radiation patterns, a
parameter called the half-power width (similar to the half-power beamwidth) will be used.
The half-power width is the length in the x-direction (for constant y and z) at which the
power reduces to half o f its maximum value. The other parameter which is used for
comparison is the distance in the z-direction at which the power density reduces to
32.8% of is maximum value.
IV .3 .1 .1 Influence o f the Waveguide D im ensions
To study the influences of waveguide dimensions and the frequency of operation on
the radiation pattern of an open-ended rectangular waveguide, power patterns (single line or
one dimensional patterns from now on) were calculated at different frequencies and in
different frequency bands (i.e. different waveguide dimensions) inside an infinite half­
space of a material.
For this case a dielectric material with dielectric constant of
( er = 5 —jO. 1) representing the infinite half-space was used.
Figure 4.6 shows three
normalized (each with respect to its maximum value) power patterns. These patterns are in
106
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the x-direction (at y = — and z = 3 mm, all x-direction patterns will be calculated at the
same values for y and z) and they start at the middle of the aperture of each waveguide and
extend to outside the a dimension of the waveguide (i.e. only half of the pattern is shown).
The patterns were calculated at three different frequencies in three different frequency
bands. These frequencies are 10 GHz (X-band, waveguide dimensions of a=22.86 mm
and b=10.16 mm), 14 GHz (Ku-band, waveguide dimensions of a= l5.8 mm and b=7.9
mm) and 22 GHz (K-band, waveguide dimensions o f a=10.67 mm and b=4.32 mm).
The half-power widths (in mm) for these three patterns are 10.8 mm at 10 GHz, 7.4 mm at
14 GHz and 4.9 mm at 22 GHz. This indicates that the higher the frequency band (i.e. the
smaller the waveguide dimensions) the narrower the half-power width gets (i.e. the higher
the spatial resolution). It was observed that the spatial resolution in the experimental
images depends on the dimensions of the waveguide [Chapter 3]. To investigate this, the
ratios of the broad dimensions of the waveguides, a, were calculated and compared to the
ratios of the half-power widths for the different frequencies in different bands as shown in
Table 4.1. As the table indicates the ratios of the broad dimensions of the waveguides for
the different bands match closely the ratios of the half-power widths at these frequencies.
The slight differences are mainly due to the influence of different frequencies as will be
shown later.
Two other calculations were performed to study the influence o f the
waveguide dimensions even further. The power patterns in the x-direction were calculated
at frequencies corresponding to the upper end frequency in a band and the lower end
frequency of the next band (e.g. X-band 8.2 GHz-12.4 GHz and Ku-band 12.4 GHz-18
GHz). Again the dielectric constant of the infinite half-space layer is (er = 5 - j0 .1 ) .
Figure 4.7 shows the patterns obtained at a frequency of 12.4 GHz in the X- and Kubands. The half-power widths for these patterns are 10.7 mm for X-band and 7.4 mm for
Ku-band. Figure 4.8 shows the patterns calculated at a frequency of 18 GHz in the Kuand K-bands. The half-power widths for these patterns are 7.55 mm for Ku-band and 5.1
107
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mm for K-band. In both Figures the half-power width gets narrower as the dimensions of
the waveguide get smaller (i.e. higher frequency band). The ratios of the a-dimensions of
the bands and the ratios of the half-power widths are illustrated in Table 4.2. The ratios of
the half-power widths match very closely the ratios of the a-dimensions of the waveguides
in these bands.
1
10 GHz
— - 14 GHz
- - - 22 GHz
0.8
0.6
0
0
2
4
6
x (m m )
8
10
12
Figure 4.6: The normalized power patterns in the x-direction at 10, 14 and 22 GHz inside a
material with £r = 5 - jO. 1.
Table 4.1: The ratios of the a-dimensions of X, Ku- and K-bands and the half-power
___________ widths associated with different frequencies in these bands.___________
frequency and band
a dimension ratios
half-power width ratios
10 GHz X-band
14 GHz Ku-band
1.447
1.46
2.142
2.204
1.481
1.51
10 GHz X-band
22 GHz K-band
14 GHz Ku-band
22 GHz K-band
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
—
12.4 GHz x-band
— - 12.4 GHz ku-band
>*
*35
c
a
o
*
£
■S
.a
■a
0.6
0.4
o
Z
0
0
2
4
6
x (m m )
8
10
12
Figure 4.7: The normalized power patterns in the x-direction at 12.4 GHz in the X- and
Ku-bands inside a material with er = 5 —jO. 1.
■
18 GHz ku-band
— - 18 GHz k-band
0.8
C
aha
<D
s
S.
■oo
N
0.4
o
0.2
z
0.6
0
0
2
4
6
8
10
12
x (m m )
Figure 4.8: The normalized power patterns in the x-direction at 18 GHz in the Ku- and Kbands inside a material with er = 5 —jO. 1.
109
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Table 4.2: The ratios of the a-dimensions of the X, Ku- and K-bands and the half-power
____________ widths associated with the same frequencies in these bands.____________
half-power width ratios
bands
a-dimension ratios
frequency
12.4 GHz
X- and Ku-bands
1.447
1.446
lSGHz
Ku- and K-bands
1.4$ 1
1.48
The power density patterns in the z-direction (all z-direction patterns were calculated
at x = y mm and y = -j mm) were also calculated at the same frequencies for the same
dielectric material. The distance at which the power density reduces to 32.8% of its
maximum value was calculated for the cases mentioned above.
Figure 4.9 shows the
patterns calculated at three different frequencies in three different frequency bands. These
frequencies are 10 GHz, 14 GHz and 22 GHz. The distances (in mm) at which the power
density drops to 32.8% of its original value, from these three patterns, are calculated to be
13.7 mm at 10 GHz, 10.8 mm at 14 GHz and 6 mm at 22 GHz. This indicates that the
lower the frequency band the higher the distances. To assure that this distance depends on
the frequency band as well as the frequency of operation, the patterns at frequencies
corresponding to the end of a frequency band and the beginning of the next frequency band
were calculated.
Again the dielectric constant of the infinite half-space layer is
(e r = 5 - jO. 1). Figure 4.10 shows the patterns obtained at a frequency of 12.4 GHz in the
X- and Ku-bands. The distances at which the power density drops to 32.8% of its original
value for these patterns are 15.7 mm for X-band and 9.6 mm for Ku-band. Figure 4.11
shows the patterns calculated at a frequency of 18 GHz in the Ku- and the K-bands. The
distances at which the power density drops to 32.8% of its original value for these patterns
are 12.5 mm for Ku-band and 5.3 mm for K-band. In both Figures (4.10 and 4.11) this
distance increases as the dimensions of the waveguide get larger. Intuitively, this behavior
should be explained by attempting to isolate the components of equation 4.36 that varies
with z.
This dependence can only be observed in the exponential term involving z.
Clearly, this expression is not dependent on the waveguide dimensions, however, the
denominator of Equation 4.34b includes a singularity which involves the broad dimension
110
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of the waveguide. Although a clear correlation between the presence of the singularity and
the results obtained can not be shown due to the complexity o f the integral, it is clear that
for a constant frequency and a constant permittivity of the infinite half-space the only
varying components are the waveguide dimensions.
10 GHz
— - 14 GHz
- - - 22 GHz
0.8
2
&
U
0.6
S,
■8
"5
0.4
o
0.2
Z
0
0
10
20
z (m m )
30
40
50
Figure 4.9: The normalized power patterns in the z-direction at 10, 14 and 22 GHz inside a
material with er = 5 - jO. 1.
—- 12.4 GHz x-band
— - 12.4 GHz ku-band
0.8
C
o
*
o
0.6
CL
■s3
'■a
o
Z
0.2
**" I
0
0
10
20
z (m m )
30
T — t—
40
50
Figure 4.10: The normalized power patterns in the z-direction at 12.4 GHz in the X- and
Ku-bands inside a material with er =5 —jO. 1.
Ill
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1
18 GHz ku-band
— * 18 GHz k-band
0.8
c
a
0w
1
a.
■8
.a
•a
o
0.6
0.4
Z
0
10
20
,
, 30
z (mm)
40
50
Figure 4.11: The normalized power patterns in the z-direction at 18 GHz in the Ku- and Kbands inside a material with er = 5 —jO. 1.
IV.3.1.2
I n fl u e n c e o f The Frequency Within The S a m e B a n d
In this section the influence of the frequency in the same band (i.e. the waveguide
dimensions are constant) is investigated. The power density patterns in an infinite half­
space of material with similar properties to that used in the last section ( er - 5 - jO. 1) were
calculated. Figure 4.12 shows power patterns in the x-direction (at y = — and z = 3 mm),
calculated at frequencies of 18 GHz, 22 GHz, and 26 GHz in the k-band. The half-power
widths associated with these frequencies are 5.1 mm at 18 GHz, 4.9 mm at 22 GHz and
4.8 mm at 26 GHz.
It is clear that as the frequency increases the half-power width
decreases slightly. The influence of the frequency on the half-power width is minimal
when compared to that of the waveguide dimensions.
112
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The power density patterns in the z-direction were also calculated at the same
frequencies for the same dielectric material. Figure 4.13 shows the patterns corresponding
to different frequencies. The figure shows that within the same frequency band the higher
the frequency the larger the distances at which the power density drops to 32.8% of its
original value. The distances calculated for these patterns are 5.3 mm at 18 GHz, 6 mm at
22 GHz and 7.25 mm at 26 GHz. So, given a frequency band, higher frequencies in the
band have a slightly higher distances and higher resolution.
i
18 GHz
0.8
22 GHz
26 GHz
0.6
0.4
0.2
0
0
2
3
x (m m )
4
5
6
Figure 4.12: The normalized power patterns in the x-direction at 18, 22 and 26 GHz inside
a material with er = 5 —jO. 1.
113
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------------18 GHz k-band
-2 2 GHz
............... 26 GHz
0.8
0.6
0.4
0.2
0
10
20
z (m m )
30
40
50
Figure 4.13: The normalized power patterns in the z-direction 18, 22 and 26 GHz inside a
material with £r = 5 - jO. 1.
IV.3.2
In flu en c e o f the Dielectric Properties on the R a diation Pattern
In this section the influence of the dielectric properties on the radiation pattern is
investigated. In the next sub-sections, the influence of the real part (permittivity) of the
dielectric constant is studied first, then the influence of the imaginary part (loss factor) is
investigated.
Finally, both portions of the dielectric constant will be varied while
maintaining a constant loss tangent (i.e. the ratio of the loss factor to the permittivity).
IV .3 .2 .1
Influence o f Perm itivitty
The real part of a relative dielectric constant is known as the relative permittivity and
it describes the ability o f the material to store microwave energy. All calculations in this
114
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section were performed at a frequency of 24 GHz in the K-band. Figure 4.14 shows the
power patterns in the x-direction calculated for materials with dielectric constants of (3jO.l, 5-jO.l, 7-jO.l and 9-jO.l). The figure indicates that as the permittivity increases, the
half-power width varies minimally. Table 4.3 shows the half-power widths obtained for
each dielectric material.
The patterns in the z-direction were also calculated at the same frequency for the
same dielectric materials. Figure 4.15 shows the patterns corresponding to the different
dielectric materials. The figure indicates that as the permittivity increases, the distance at
which the power density drops to 32.8% of its original value increases as well. Table 4.4
shows the distance obtained for each dielectric material.
3-jO.l
5-jO.l
7-jO.I
9-jO.l
0.8
oc
Q
u.
O
£
£
■S3J
'■a
0.6
0.4
0.2
0
2
3
x (m m )
4
5
6
Figure 4.14: The normalized power patterns in the x-direction at 24 GHz inside dielectric
materials with constant loss factor.
Table 4.3: The half-power widths for materials with dif 'erent permittivities at 24 GHz.
dielectric constant
3-jO.l
5-jO.l
7-jO.l
9-jO.l
half-power width (mm)
5.16
4.81
4.8
4.81
115
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3-jO.l
— - 5-jO.l
- - - 7-jO.l
9-jO.l
S ' 0.8
*55
c
&
S
0.6
£
1
0.4
0
10
20
z (m m )
30
40
50
Figure 4.15: The normalized power patterns in the z-direction at 24 GHz in side dielectric
materials with constant loss factor.
Table 4.4: The distances at which the power density drops to 32.8% of its original value
_______________ for materials with different permittivities at 24 GHz._______________
dielectric constant
3-j0.1
5-j0.1
9-jO.l
7-j0.1
distances (mm)
5.2
6.6
8.1
9.1
IV .3 .2 .2 Influence o f Loss Factor
The imaginary part of a dielectric material is called the loss factor and it describes
the ability of the material to absorb microwave energy. All the calculations in this section
were performed at a frequency of 24 GHz in the K-band. Figure 4.16 shows the power
patterns in the x-direction calculated for materials with dielectric constants of (5-j0.1, 5j0.5, 5-jl and 5-j2). The figure indicates that as the loss factor increases, the half-power
width increases slightly.
Table 4.5 shows the half-power widths obtained for each
dielectric.
116
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The patterns in the z-direction were also calculated at the same frequency for the
same dielectric materials. Figure 4.17 shows the patterns corresponding to the different
calculations. The figure indicates that as the loss factor increases, the distances at which
the power density drops to 32.8% of its original value decreases.
Table 4.6 shows the
distances obtained for each dielectric material.
5-jO.l
— - S-j0.5
- - - 5-jl
S-J2
0
2
3
x (m m )
4
5
6
Figure 4.16: The normalized power patterns in the x-direction at 24 GHz inside dielectric
materials with constant permittivity.
Table 4.5: The half-power widths for materials with different loss factors at 24 GHz.
dielectric constant
5-jO.l
5-j2
5-j0.5
5-jl
half-power width (mm)
4.81
4.88
4.96
5.04
Table 4.6: The distances at which the power density drops to 32.8% of its original value
for materials with different loss factors at 24 GHz.
dielectric constant
5-jO.l
5-j0.5
5-jl
5-j2
distances (mm)
6.6
5.1
3.85
2.6
117
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5-jO.l
— - S-j0.5
—- - 5-jl
— S-j2
0.8
0 .6
-I*
a.
0.4
0
10
20
z (m m )
30
40
50
Figure 4.17: The normalized power patterns in the z-direction at 24 GHz inside dielectric
materials with constant permittivity.
Finally, for a changing permittivity and loss factor, while maintaining a constant
loss tangent, the patterns are calculated. Figure 4.18 shows the power patterns in the xdirection calculated for materials with dielectric constants of (5-j0.1, 10-j0.2 and 15-jO.3).
The figure indicates that the half-power width increases as the values of both the
permittivity and the loss factor increase which means that the influence of the imaginary
part dominates at this point in the x-direction.
The patterns in the z-direction were also calculated at the same frequency for the
same dielectric materials. Figure 4.19 shows the panems corresponding to the different
calculations. The figure indicates that the distances at which the power density drops to
32.8% of its original value increases at the beginning (i.e. for 10-j0.2) and then it begins to
decrease (for 15-j0.3) which means that the permittivity dominates at the beginning and
after the loss factor exceeds a certain value it takes over the properties of the material.
118
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>.
e
&
0
5-jO.l
10-J02
0.8
- - - 15-jOJ
0.6
£
1
0.4
T3
§
2
0.2
0
2
4
3
5
6
x (m m )
Figure 4.18: The normalized power patterns in the x-direction at 24 GHz inside dielectric
materials with constant loss tangent.
5-jO.I
— - 10-j0.2
C/J
a
- - - 15-j0.3
a
u
O
*
£
•o
os
a
O 0.2
0
10
20
z (m m )
30
40
50
Figure 4.19: The normalized power patterns in the z-direction at 24 GHz inside dielectric
materials with constant loss tangent.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IV.4 Summary and Remarks
Near-field microwave imaging is a very powerful NDT tool for inspecting the
integrity of dielectric composite materials. Experimental and theoretical results have shown
that very high resolutions are obtainable in all directions. To interpret the information in a
near-field microwave image, the effect of all the factors influencing the formation of an
image must be understood. One of the prime factors influencing an image is the near-field
properties of the radiator (i.e. the interaction of the fields with the material). The fields in
the near-field region of the radiator were formulated and used to understand the properties
of the open-ended rectangular waveguide probe. The effects and locations of sidelobes are
now understood and can be used to obtain information about the shape and orientation of a
defect.
The formulations are general and can be used in obtaining the fields in a multi­
layered structures backed by either an infinite half-space of material or by a conducting
sheet. The properties of the fields in the near-field region were investigated as a function of
the frequency of operation, waveguide dimensions and dielectric properties of the material
under inspection. The waveguide dimensions influence the spatial resolution drastically, as
the frequency band increases (i.e. the waveguide dimensions decrease) the spatial
resolution increases as well. On the other hand, lower frequency bands showed that higher
distances at which the power density drops to 32.8% of its original value are obtained (i.e.
larger waveguide dimensions).
So, depending on the application the frequency of
operation and the frequency band (waveguide dimensions) can be determined. So, given a
certain frequency, if the goal is to have maximum penetration in to material, the lowest
frequency band that contains the frequency is the best band to operate within. On the other
hand, if the goal is to obtain maximum resolution, the highest frequency band that contains
the frequency is the best to operate within.
Results indicated that within the same
120
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frequency band, operating at higher frequencies improves the spatial resolution and the
distances at which the power density drops to 32.8% of its original value slightly. As the
permittivity increases, the spatial resolution changes minimally and the distances at which
the power density drops to 32.8% of its original value improves slightly. As the loss factor
increases, the spatial resolution decreases slightly while the distances at which the power
density drops to 32.8% of its original value decreases significantly. These results have a
very important practical ramifications. For instance in a hose structure, the testing goal is
to inspect the wall of the hose independent of the material that is filling the hose (water, oil,
gas, etc.). For that the distances at which the power density drops to 32.8% of its original
value must be limited to close to the inner side of the wall.
To remedy this problem a
dielectric slab can be placed in front of the waveguide to obtain the required distance.
Up to this point the field properties in the near-field region of an open ended
rectangular waveguide and their interaction with a dielectric material have been investigated.
In chapter 5 a study of the mechanism by which the fields interact with an inclusion will be
presented. An effective dielectric constant formula will be used to model the reflection
properties o f dielectric structures. The influence of the non-uniformity associated with the
electric field distribution at the aperture of the waveguide will be investigated and
incorporated in calculating the effective dielectric constant.
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CHAPTER V
Influences o f The E ffective D ielectric Constant a n d N on-L inear Probe
A perture F ield D istribution on N ear-F ield M icrow ave Im ages
In this part of the investigation, the interaction of the fields, radiating from an openended rectangular waveguide, with a defect is investigated to further understand the
mechanism of near-field microwave image formation.
An effective dielectric constant
formula is used to model the effect of two and three dielectric half-spaces situated side-byside, covered and not covered by layers of dielectric materials, when scanned by an openended rectangular waveguide probe. The influence of the non-uniformity associated with
the electric field distribution at the waveguide aperture is investigated as well. A linear and
a non-linear volume fraction calculation methods are incorporated to calculate the effective
dielectric constant of a specimen made of two or three dielectric half-spaces. Subsequently,
the phase and magnitude of the reflection coefficient at the waveguide aperture arc
calculated. Theoretical and experimental results are presented and compared.
As mentioned earlier, a near-field microwave image is the result of several factors
such as the probe type (e.g. open-ended rectangular and circular waveguides, coaxial lines,
etc.), antenna pattern (i.e. main lobe, sidelobes and half-power beamwidth), and
geometrical and physical properties (e.g. dielectric constant) of both the inclusion (defect)
and the host (background) dielectric material under inspection [Qad.95].
The field
properties of an open-ended rectangular waveguide radiator and their interaction with
several background materials were discussed in Chapter 4. The field/defect interaction will
be investigated in this chapter.
122
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V .l
The Effective Dielectric Constant o f A Medium
To understand the information conveyed by a microwave image, the basics of
wave/inclusion interactions must be understood. Dielectric materials are characterized by
their relative (to free-space) complex dielectric constant, er = e ’r - j£”r. The reflection
coefficient measured at the waveguide aperture is a function of the effective dielectric
constant of the specimen under inspection. Thus, while the sensor scans over a dielectric
specimen, as long as the effective dielectric constant of the specimen in front of the
waveguide sensor does not change (i.e. absence of an inclusion), the effective reflection
coefficient remains unchanged, and consequently a constant signal (voltage) is recorded.
However, once the effective dielectric constant of the specimen changes (i.e. presence of an
inclusion or scanning over one dielectric while moving into another), the effective reflection
coefficient changes, and consequently the recorded signal changes indicating the presence
of the inclusion. The properties of this recorded signal then provide information about the
type and geometry o f the inclusion [Chapter 3].
The production o f near-field microwave images can be described by noting that the
waveguide aperture, as it scans a medium with an inclusion, continually “sees” an effective
dielectric material whose dielectric properties are a combination of the host and the
inclusion. The effective dielectric constant of a material is defined as the ratio between the
average electric flux displacement D and the average electric field E
D = e tffE
where,
(5.1)
is the effective dielectric constant of the medium. This equation applies to
isotropic media (most dielectrics are isotropic).
The effective dielectric constant must
123
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satisfy the relation el < 6 ^ < eu, where, £, is the lower dielectric constant (of either the
host or the inclusion) in the mixture and eu is the higher dielectric constant in the mixture.
The displacement is given by
D = e E +P
(5.2)
where, £ is the dielectric constant of the material and P is the polarization in the material.
The poloraization depends on the polarizability o f the material.
V.2
E ffective D ielectric Constant F orm ulae
To model the effective dielectric constant of a specimen with an inclusion some type
o f an effective dielectric constant model or formula may be used as long as it reflects the
nature of the overall dielectric properties of the medium.
V.2.1
Far- F ie ld vs. Near-Field
In the far-field region the effective dielectric constant of a medium is influenced by
the physical and geometrical properties of both the host and the inclusion materials. In the
far-field the incident plane wave uniformly irradiates an object. The factor that influences
the effective dielectric constant the most is the volume fraction of each constituent. On the
other hand, in the near-field region one of the dominant factors influencing the effective
dielectric constant of a medium is the field distribution of the probing sensor. The volume
fraction of each constituent in the medium is therefore influenced (weighted) by the field
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distribution of the probe. Figure 5.1 shows the dominant mode electric field distribution at
a waveguide aperture. Figure 5.1a illustrates the fact that the electric field is maximum in
the center of the broad dimension of the waveguide and sinusoidally diminishes to zero at
the edges. Figure 5.1b shows the uniformity of this electric field distribution for a fixed
location on the broad dimension and as a function of different locations on the narrow
dimension of the aperture.
y
b
x
a
(a)
y
A
L .,.1
i
> x
a
(b)
Figure 5.1: Dominant mode electric field distribution at a waveguide aperture.
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V.2.2
Volume Fraction Calculation
To account for the non-uniformity of the electric field distribution at the waveguide
aperture a non-linear volume fraction calculation approach may be incorporated in
accordance with the non-linear integrating influence of the aperture field distribution (i.e.
modification to volume fraction used in calculating the effective dielectric constant based on
this non-linearity). Figure 5.1 indicates that the distribution o f the electric field is not linear
along the broad dimension o f the waveguide sensor. Thus, depending on the position of a
certain physical volume of a constituent with respect to the aperture field distribution the
influence of that constituent in the overall effective dielectric constant varies. So, one must
consider the fact that the weighting influence of the electric field distribution at the
waveguide aperture must also be taken into account since the contribution of each material
in the specimen is effectively integrated over the aperture in a non-linear fashion (i.e.
sinusoidally). Therefore, the non-uniformity of the field distribution at the waveguide
aperture suggests that a non-linear volume fraction calculation must be used to calculate the
volume fraction of each dielectric material as a function of its position with respect to the
waveguide aperture. Consequently, Equation 5.3 which provides for a volume fraction
weighted by the aperture field distribution is used in conjunction with effective dielectric
constant formulae to obtain a more relevant effective dielectric constant values as a function
of the scanning distance (i.e. changing volume fractions as the waveguide scans the
structure). Hence,
(5.3)
“
initia l
where V could be either V, (volume fraction of material 1, host,)or V2 (volume fraction of
126
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material 2, inclusion,) (while V, + V2 = 1) and the integration is carried along the broad
dimension of the waveguide (i.e. the x-axis). To obtain a volume fraction that can be used
in conjunction with the effective dielectric constant equation (0 < V, and V2 < 1) for each
dielectric material, the portion of the aperture that is covered by the dielectric is normalized
with respect to the broad dimension o f the waveguide aperture, a.
The initial and final
values in Equation (5.3) correspond to the normalized position of each material. Since the
host and the inclusion materials are assumed to be infinite in the z- and y-directions, the
normalized volume fraction is a function only of the different locations on the broad
dimension of the waveguide (i.e. the x-axis).
To illustrate the importance of using Equation 5.3 to calculate the volume fraction of
a material in the near-field of an open-ended rectangular waveguide, three slab like
inclusions with physical sizes corresponding to 10%, 30% and 50% the area of the aperture
are considered next. The volume fraction of the background material corresponding to each
of the three slabs was calculated using the linear (the weight of a material is dependent only
on the aperture area covered by that material) and the non-linear approaches. Figure 5.2
shows the volume fractions of background material calculated using the both approaches as
a slab with a thickness of 10% moves through the aperture.
The volume fraction is
calculated as the slab moves in 1% steps through the broad dimension of the waveguide.
The linear approach values are those physically enclosed by the aperture, while the non­
linear approach values are weighted by the field distribution according to Equation 5.3.
Figures 5.3 and 5.4 show the volume fractions calculated using the linear and nonlinear
apoproaches for the 30% and 50% inclusions, respectively. In Figures 5.2-5.4 the linear
model shows that the volume fraction changes linearly as the slab enters and leaves the
aperture of the waveguide, while the non-linear model shows that due to the field
distribution the volume fraction of the background material does not vary as much once the
slab is entering or leaving the aperture. This is because the weight o f the material at these
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positions is small since the field at these locations is weak.
Once the slab is within the
aperture the linear model yields a constant volume fraction, while the non-linear model
yields varying values depending on the position within the aperture. At the middle of the
aperture, the field is very strong, the linear model does not vary. However, the non-linear
model indicates that due to the strength of the field at the middle of the aperture, the
influence of the 10% slab, when it is in the middle of the slab, is more than the physical
size and is equivalent to a 16% slab in a linear model. Figures 5.3 and 5.4 demonstrate a
similar behavior for wider slabs. Figure 5.4 shows that for the 50% slab, the two models
intersect when the slab is in front of half the waveguide.
i
N'on-Linear
0.95
— — - Linear
0.9
0.85
0.8
0
20
40
60
S can P o sitio n
80
100
Figure 5.2: The volume fraction as a function of scan position as a slab of 10 % the area of
the aperture is being seen by the aperture.
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—
Non-Linear
— - Linear
0.9
e
o
•3
8
&
o
0.8
0.7
o
>
0.6
0
20
40
100
60
80
S can P o sitio n
120
Figure 5.3: The volume fraction as a function of scan position as a slab of 30 % the area of
the aperture is being seen by the aperture.
0.9
-S
Non-Linear
— * Linear
£
o
E
-o
>
0.6
0.5
0.4
0.3
0.2
0
50
100
150
S can P o sitio n
Figure 5.4: The volume fraction as a function of scan position as a slab of 50 % the area of
the aperture is being seen by the aperture.
129
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V.2.3
Effective D ielecrtic Constant Form ulae
The ability of treating a random dielectric medium (a medium with more than one
dielectric) with an effective dielectric property that contains the information on how the
inhomogenities of the material affect the reflection properties is essential to researchers in
many areas such as, microwave industrial and medical applications, remote sensing and
material science. The literature is full of different effective dielectric constant formulae
developed for many structures o f heterogeneous materials. For this study, three formulae
are considered and compared to each other. To simulate the experimentally obtained data
analytically, the formula that was based on a more rigorous investigation is used, as
discussed later.
V.2.3.1
Sim ple A verage E ffective D ielectric C onstant Formula
In this case, the calculation of the effective dielectric constant is based on averaging
the dielectric constants of each material “seen” by the aperture weighted directly by its
volume fraction. The formula is given by
= VfrX + V2^2
(5-4)
where e r, erl and er2 are the specimen’s effective relative dielectric constant, the relative
dielectric constant of material 1 and the relative dielectric constant of material 2,
respectively. Likewise, V, and V2 are the volume fractions ( 0 < {V ,, V2} < 1 and V, + V2
= 1) of material 1 and material 2, respectively.
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V.2.3.2
Rayleigh Effective Dielectric Constant Formula
This effective dielectric constant formula is among the most commonly used.
It
was originally developed to simulate spherical inclusions, and it is applied to many
heterogeneous structures. The formula is given by
(5.5)
where e r, er, and er2 are the specimen’s effective relative dielectric constant, the relative
dielectric constant of material 1 and the relative dielectric constant of material 2,
respectively. Likewise, V, and V2 are the volume fractions ( 0 < {V , , V2} < 1 and V, + V2
= 1) of material 1 and material 2, respectively.
V.2.3.3
K .W a k in ’s E ffective D ielectric C onstant F orm ula
This effective dielectric constant formula was obtained through rigorous and
numerous simulations using finite element methods in which the effect of dielectric
polarization and the infringing electric K.Wakin distribution at the boundary regions
between the host and the inclusion were taken into account [Wak.93].
The formula is
given by
(5.6)
131
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where £r, Erl and er2 are the specimen’s effective relative dielectric constant, the relative
dielectric constant o f cement paste and the relative dielectric constant of, respectively.
Likewise, V, and V2 are the volume fractions ( 0 < [V, , V2) < 1 and V, + V2 = 1 ) of
material 1 and material 2, respectively [Wak.93].
V.2.3.4
Com parison
The three formulae were used to calculate the effective dielectric constant of a
mixture composed o f two dielectric materials. The linear and non-linear volume fraction
calculation methods were used in conjunction with each formula. A mixture of two infinite
half-spaces of materials arranged side-by-side as shown in Figure 5.5, in this particular
case, cement paste with water-to-cement (w/c) ratio of 0.55 resulting in a measured er =
4.3 - j0.07 and ethylene propylene diene or EPDM which is the primary constituent of
rubber with a measured er = 1.88 - j0.02, was modeled using all three effective dielectric
constant formulae. The three formulae were used with both the linear and the non-linear
models to calculate the effective dielectric constant of the structure described in Figure 5.5.
Figure 5.6 shows the values of the real part of the effective dielectric constant obtained
using K.Wakin’s effective dielectric constant formula as a function of location in front of
the aperture. The figure indicates that the values obtained using both approaches (linear
and non-linear) match at the beginning and at the end (i.e. when one of the two materials is
occupying a small area in front of the aperture). The two approaches match as well when
50% of the aperture is covered by each material. The maximum percentage difference in
the real part of the dielectric constant is around 12%. Figures 5.7 and 5.8 show the same
when using the Rayleigh and the simple average formulae, respectively.
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For these
formulae, similar percentage differences between the linear and the non-linear approaches
are present. Figures 5.9-5.11 show the results obtained for the imaginary pan of the
effective dielectric constant using the three formulae with the linear and the non-linear
approaches. A maximum difference of around 16 % is observed in the results obtained
using K.Wakin’s formula. Similar differences are observed for the other two formulae.
Figure 5.12 shows the real part of the effective dielectric constant obtained using the three
formulae with the non-linear approach as a waveguide scans over the structure shown in
Figure 5.5. The figure indicates that on both ends of the graph (i.e. when one of the two
constituents dominates the region seen by the sensor) the three formulae results match. A
maximum difference o f around 14% is obtained between K.Wakin’s and the simple
average formulae. Figure 5.13 shows the results obtained for the imaginary part. The
results presented above indicate that the effective dielectric constant is influenced by the
volume fraction calculation method and the formula used.
So, by using the non-linear
approach and the proper effective dielectric constant formula is important in obtaining
correct effective dielectric constant values. In the next sections the K.Wakin effective
dielectric constant formula will be used since it was obtained through rigorous and
numerous simulation using finite element methods in which the effect of dielectric
polarization and the infringing electric flux distribution at the boundary regions between the
host and the inclusion were taken into account.
133
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Waveguide
Sensor
------ ^
Scan Direction
Cement Paste
EPDM
e r = 4 .3 -j0 .0 7
£r = 1.88 - j0.02
Figure 5.5: Two infinite half-spaces of cement paste and EPDM arranged side-by-side to
model a large defect.
—
Non-Linear
— - Linear
- \
3.5
1.5
0
0.2
0.6
0.4
S ca n P o sitio n
0.8
Figure 5.6: Real part of the effective dielectric constant obtained using the K.Wakin lines
formula.
134
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43
— Non-Linear
— * Linear
4
3
2.5
0
0.2
0.4
0.6
S can P o sitio n
0.8
Figure 5.7: Real part of the effective dielectric constant obtained using the Rayleigh
formula.
Non-Linear
— - Average Linear
4
3.5
V
3
\
2.5
2
1.5
0
0.2
0.4
0.6
S can P o sitio n
0.8
Figure 5.8: Real part of the effective dielectric constant obtained using the average
formula.
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-0.01
-
0.02
-0.03
-0.04
-0.05
-0.06
- - — Non-Linear
— - Linear
-0.07
-0.08
0
0.2
0.6
0.4
S can P o s itio n
0.8
1
Figure 5.9: Imaginary part of the effective dielectric constant obtained using the K.Wakin
lines formula.
-o.oi
-
0.02
-0.03
-0.04
u
-0.05
-0.06
Non-Linear
— - Linear
-0.07
-0.08
0
0.2
0.4
0.6
S can P o sitio n
0.8
Figure 5.10: Imaginary part of the effective dielectric constant obtained using the Rayleigh
formula.
136
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-0.01
-
0.02
-0.03
-0.04
u
-0.05
—
-0.06
Non-Linear
— - Linear
-0.07
-0.08
0
0.2
0.4
0.6
S can P o sitio n
0.8
Figure 5.11: Imaginary part of the effective dielectric constant obtained using the average
formula.
4.5
N- .
4
—
Flux Line
— - Rayleigh
• - - Average
3
2.5
1.5
0
0.2
0.4
0.6
S can P o sitio n
0.8
Figure 5.12: Real part of the effective dielectric constant obtained using the three formulae
with the non-linear approach.
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-0.01
-
0.02
-0.03
-0.04
CJ
-0.05
-0.06
— — - Average
— — • Flux Line
• * • • • Rayleigh
-0.07
-0.08
0
0.2
0.4
0.6
S c a n P o sitio n
0.8
1
Figure 5.13: Imaginary part of the effective dielectric constant obtained using the three
formulae with the non-linear approach.
V.3
Results
Thus far, the effective dielectric constant, using three different methods, has been
calculated. Consequently, the phase and the magnitude of the reflection coefficient at the
waveguide aperture may be analytically calculated, using the electromagnetic code
developed for describing the interaction of the fields radiated from an open-ended
rectangular waveguide into a multi-layered composite structure [Bak.94], [Chapter 2], and
compared with measurements. As a first step towards modeling a finite sized inclusion
such as those discussed in Chapter 3, simpler cases are analyzed first and more complicated
ones are analyzed later on using the approach outlined above.
To illustrate the influence of the field distribution at the waveguide aperture in the
effective dielectric constant, several sets of specimens with known properties were
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prepared. In the following sections, descriptions of the specimens and the obtained results
are presented. An HP8510 vector network analyzer was used to record the actual phase
(<J>°) and magnitude (ID) of the reflection coefficient ( r = Hie1**) at the waveguide aperture
as a function of position.
V.3.1
Two In fin ite Half-Spaces
This specimen was prepared to model a simple transition from one infinite half­
space to another simulating the presence of a large inclusion (as a first step in the modeling
process). The specimen was produced by arranging, side-by-side, a large cube of cement
paste with a water to cement (w/c) ratio o f 0.55 resulting in a measured er = 4.3 - j0.07 and
ethylene propylene diene or EPDM which is the primary constituent of rubber with a
measured e r = 1.88 - j0.02, as shown in Figure 5.5.
The dielectric constant values
reported in this chapter were all measured at 10 GHz which is the frequency used to
conduct this scan (aperture dimensions are a=22.86 mm by 6=10.16 mm). These two
materials were used because they are relatively homogeneous, they were available in the
laboratory and their dielectric properties had previously been measured for other purposes
[Boi.97], [Gan.94]. Each cube of material was about 8” x 8” x 8” (21 cm x 21 cm x 21
cm) to simulate an infinite half-space situation.
The specimen was scanned starting with the waveguide totally on the cement paste
and ending with it totally on the EPDM, as shown in Figure 5.5.
When the interface
between these two materials is within the waveguide aperture, each material is partially
contributing to the measured reflection coefficient.
Therefore, the measured reflection
coefficient is due to the effective dielectric constant of these two materials exposed to the
139
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aperture.
To implement the scanning results theoretically, the K.Wakin line effective
dielectric constant formula was used to calculate the effective dielectric constant of an
effective medium exposed to the waveguide aperture.
To validate the experimental results theoretically, at first the effective dielectric
constant formula is used with a linear volume fraction calculation method. Figures 5 .14a-b
show the measured and calculated phase and magnitude of the reflection coefficient as a
function of the scanning distance, respectively. Figure 5.14a shows a maximum phase
difference of about 10 degrees between the measured and the calculated results, while
Figure 5.14b shows that the magnitudes differ by more than 0.05 (0 < ID <1). The trend
of the calculated results do not agree with their measured counterparts very well.
Consequently, the linear volume fraction assumption is shown not to be valid.
Figures 5.15a-b show the phase and magnitude of the reflection coefficient
calculated incorporating the non-linear volume fraction calculation. Clearly, in this case the
calculated results agree much better with the measured results when compared to the results
of the linear volume fraction calculation shown in Figures 5.14a-b.
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-140
Measured
Calculated
S can n in g D is ta n c e (m m )
0.5
Measured
Calculated
0.45
[_ 0.4
035
0.3
0
9
18
27
S can n in g D is ta n c e (m m )
Figure 5.14: a) The phase and b) magnitude of the reflection coefficient at the aperture of
the waveguide using the linear model for the specimen shown in Figure 5.5.
141
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-140
Measured
Calculated
S c a n n in g D is ta n c e (m m )
0.5
Measured
Calculated
0.45
.4
035
0.3
0
9
18
27
S c a n n in g D is ta n c e (m m )
Figure 5.15: a) The phase and b) magnitude of the reflection coefficient at the aperture of
the waveguide using the non-linear model for the specimen shown in Figure 5.5.
142
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V.3.2
Surface Slab in An Infinite Half-Space
The second set of specimens was prepared to model an inclusion with a finite
dimension as the second step in the modeling process. This set includes two specimens
with slightly different properties. The first specimen was produced by placing a slab of
Plexiglas with er = 2.56 - j0.02 and a thickness of 4 mm in between two large and identical
cement paste blocks with a water to cement (w/c) ratio of 0.45 having a measured er = 5 .8
- j0.2, as shown in Figure 5.16.
To illustrate the effect of the width of the inclusion
(Plexiglas), a second specimen similar to the one shown in Figure 5.16 was produced with
a Plexiglas slab whose thickness is 5.8 mm. The Plexiglas inclusions were large in length
and height and are thus assumed infinite in the y- and z- directions and are only finite in the
x-direction.
As the sensor, operating at 10 GHz, scanned over each specimen, as shown in
Figure 5.16, the magnitude of the reflection coefficient ITI was recorded.
First the
specimen with the 4 mm thick Plexiglas slab was scanned. The scan is 32 mm long and it
starts and ends with the aperture approximately 3 mm away from the cement/Plexiglas
interface on each side. Figure 5.17 shows the experimental and calculated results obtained
using the linear and non-linear volume fraction models. The linear model shows that once
the Plexiglas inclusion is within the aperture, the measured ITI remains fairly constant
which is in disagreement with the measurements. However, the results o f the non-linear
volume fraction model agree well with the measured results.
Figure 5.18 shows the
measured and calculated in for the 5.8 mm-thick Plexiglas slab. The scan is 36 mm long
and it starts and ends with the aperture approximately 3 mm away from the interface on
each side. Again, the results of the non-linear volume fraction model agrees well with the
measured results. Furthermore, the width associated with Figures 5.17 and 5.18, as a
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function of scanning distance, is larger for Figure 5.18 than for Figure 5.17, as expected.
However, these widths do not correspond to the actual thickness of the Plexiglas inclusions
used in the experiments. This fact clearly shows the non-linear integrating effect that the
waveguide aperture has when used for near-field imaging. An inverse routine may be used
to take this non-linear integrating effect out of such measurements and provide for the
actual inclusions size.
a
Waveguide
Sensor
Scan Direction
Cement Paste
Cement Paste
e f = 5.8 - j0.2
er = 5 .8 - j0.2
Plexiglass
£r = 2.56 - j0.02
Figure 5.16: A thin Plexiglas slab inserted in between an infinite half-space of cement
paste to model a one dimensional defect.
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0.6
Measured
Linear
Non-Linear
• ••
U
0.5
0.45
0.4
0
5
10
20
15
25
30
35
40
S c a n n in g D ista n c e (m m )
Figure 5.17: The magnitude of the reflection coefficient for 4 mm-thick Plexiglas in an
infinite half-space of cement paste.
0.6
Measured
Linear
Non-Linear
•••••••
U 0.5
0.45
0.4
0
5
10
15
20
25
30
35
40
S c a n n in g D ista n c e (m m )
Figure 5.18: The magnitude of the reflection coefficient for 5.8 mm-thick Plexiglas in an
infinite half-space of cement paste.
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V.3.3
Deep Slab in An Infinite Half-Space
This set of specimens (three in all), was prepared to model an inclusion with a finite
dimension under a layer of material as a step forward in the modeling process. Figure 5.19
shows a specimen that was produced by inserting a slab of Plexiglas with er = 2.56 - j0.02
and a thickness of 5.8 mm in an infinite half-space of cement paste with a water to cement
(w/c) ratio of 0.5 having a measured er = 6.35 - j 1.5. The Plexiglas slab was inserted at a
depth of 8 mm from the scan surface, as shown in Figure 5.19. To illustrate the effect of
the inclusion (Plexiglas) width and depth two other specimens, similar to the one shown in
Figure 5.19, were produced. In the first, a Plexiglas slab whose thickness is 5.8 mm was
inserted in a similar cement past infinite half-space at a depth of 4.8 mm. In the second, an
8.8 mm thick Plexiglas slab was inserted in a similar half-space of cement paste at a depth
of 4.8 mm. The Plexiglas inclusions were large in length and height and are thus assumed
infinite in the y- and z- directions and are only finite in the x-direction.
Since the Plexiglas was inserted at different depths in the cement, the field pattern in
an infinite half-space of cement paste was calculated to assure that the beam at all depths is
still confined to within the waveguide aperture. Figure 5.20 shows the normalized YZplane field pattern at 7 GHz (the frequency at which the measurement and calculation are
performed, waveguide dimensions are 34.84 mm by 15.8 mm). The figure indicates that
the fields remain confined to within the aperture dimensions up do depths of around 20
mm.
The three specimens were scanned at a frequency of 7 GHz and the magnitude of
the reflection coefficient ITI was recorded as a function of the scan location. Figures 5.215.23 show the results obtained from scanning the three specimens. The scans are 100 mm
146
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long with a step size of 1 mm. Good agreement between the measurement results and the
theoretical results is obtained. Figures 5.21 and 5.22 show that the width of the detected
inclusion (on both figures) is the same in both cases even though the Plexiglas slabs are at
different depths. This again indicates that the resolution is similar at both depths. The
width associated with Figure 5.23 as a function of the scanning distance is larger than those
associated with Figures 5.21 and 5.22. These results indicate that information about the
depth and width of a defect can be obtained.
a
Waveguide
Sensor
Scan Direction
I
Cement Paste
Ef = 6.35 - j 1.5
Cement Paste
ef = 6.35 - j 1.5
Plexiglass
er = 2.56-j0.02
Figure 5.19: A thin Plexiglas slab inserted at depth d in an infinite half-space of cement
paste to model a one dimensional defect under a layer of material.
Figure 5.20: The normalized YZ-plane field pattern at 7 GHz in an infinite half space of
cement.
147
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0.55
Measurement
Theory
a
0
x
20
i
a
40
60
80
100
S c a n n in g D istan ce (m m )
Figure 5.21: The magnitude of the reflection coefficient for a 5.8 mm-thick Plexiglas slab
8 mm deep in an infinite half-space of cement paste.
Measurement
Theory
u
o
20
40
60
80
100
S c a n n in g D istan ce (m m )
Figure 5.22: The magnitude of the reflection coefficient for a 5.8 mm-thick Plexiglas slab
4.8 mm deep in an infinite half-space of cement paste.
148
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0.6
Measurement
Theory
0.55
X
X
0
20
40
60
80
100
S can n in g D is ta n c e (m m )
Figure 5.23: The magnitude of the reflection coefficient for a 8.8 mm-thick Plexiglas slab
4.8 mm deep in an infinite half-space of cement paste.
V.3.4
Deep Slab in A M ulti-Layered Structure
Finally, to model an inclusions in a multi-layer structure, a 3.17 mm thick layer of
synthetic rubber was laid on top of each of the specimens described in the last section as
shown in Figure 5.24. The magnitude of the reflection coefficient ITI was recorded and
calculated as a function of the scan location. Figures 5.25-5.27 show the results obtained
from 100 mm long scans performed at a frequency of 7 GHz. The different depth as well
as the different width influences are indicated again. In Figure 5.27 a dip like feature is
observed on each side of the defect is observed. The calculation results show this feature
as well. This feature is due to the way the fields interact with a dielectric medium to
produce a certain reflection coefficient.
149
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Waveguide |
Sensor
Synthetic Rubber •
e r = 4.9 -jO.l
Scan Direction
1
Cement Paste
ef = 6.35 - jl.5
Cement Paste
ef = 6.35- j 1.5
Plexiglass
er = 2.56-j0.02
Figure 5.24: A thin Plexiglas slab inserted at depth d in an infinite half-space of cement
paste under a 3.17 mm thick layer of synthetic rubber to model a one dimensional defect in
a multi-layered structure.
0.5
0.495
u
0.49
Measurement
0.485
Theory
0.48
0
20
40
60
80
100
S c a n n in g D ista n ce (m m )
Figure 5.25: The magnitude of the reflection coefficient for a 5.8 mm-thick Plexiglas slab
8 mm deep in an infinite half-space of cement paste.
150
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0.509
Measurement
Theory
t_
0
20
40
60
SO
100
S ca n n in g D ista n c e (m m )
Figure 5.26: The magnitude o f the reflection coefficient for a 5.8 mm-thick Plexiglas slab
4.8 mm deep in an infinite half-space of cement paste.
Measurement
Theory
0
20
40
60
80
100
S c a n n in g D ista n c e (m m )
Figure 5.27: The magnitude o f the reflection coefficient for a 8.8 mm-thick Plexiglas slab
4.8 mm deep in an infinite half-space of cement paste.
151
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V.4
Summary and Remarks
Near-field microwave imaging using open-ended rectangular waveguide sensors is
an effective nondestructive tool for the inspection of materials particularly when
determining the size and shape o f inclusions.
The non-uniformity associated with the
electric field distribution at the waveguide aperture is an important factor that must be
explicitly taken into account when trying to understand and interpret images obtained in the
near-field of such sensors. In this chapter, this issue was discussed using four sets of
dielectric specimens as steps towards modeling a finite sized inclusion in a host material. A
simple formula was used to obtain the effective dielectric constant o f a material made of two
or three dielectric half-spaces arranged side-by-side covered or uncovered by layers of
dielectric material. The non-uniformity of the field distribution was incorporated in the
effective dielectric constant formula by using a non-linear volume fraction calculation
consistent with the aperture field distribution of the rectangular waveguide probe.
The
results were compared to a linear volume fraction formula and it was determined that the
linear volume fraction method does not sufficiently predict the properties of the specimen
under inspection. The results presented indicate that the effective dielectric constant is
influenced by the volume fraction calculation method and the formula used. So, using the
non-linear approach and the proper effective dielectric constant formula is important in
obtaining correct effective dielectric constant values. The foreword problem of imaging
relatively simple structures and the field-defect interaction is now understood.
More
complicated structures can be analyzed in a similar fashion and the inverse problem of
determining defect size and properties can no be solved.
The volume fraction calculation outlined in this chapter assumes that the distribution
associated with the aperture fields is sinusoidal (i.e. that of the dominant mode only).
152
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However, for a more detailed analysis, the influence of the higher-order modes on the nonlinearity associated with the aperture field distribution must be considered. The properties
of the higher-order modes are influenced by the physical and geometrical properties o f the
structure under consideration and the frequency of operation. Another assumption that was
made by using the sinusoidal (i.e. dominant mode) distribution is that wave bending
doesn’t occur as two dielectric materials with different dielectric properties are partially seen
by the aperture. High dielectric materials tend to pull the fields into them. The same
phenomena happens at the aperture of the waveguide when two materials are seen by the
aperture.
The higher dielectric tend to attract more fields into it, thus, changing the
sinusoidal distribution of the field at the aperture. The variation of the field distribution
(i.e. wave bending) is influenced by the difference in the dielectric properties between the
materials and the position of each dielectric material in front of the aperture. The bending is
significant if the difference between the dielectric properties is large and both dielectrics
occupy relatively large portions of the space in front of (or seen by) the aperture. The
influence of wave bending can be mathematically incorporated using higher-order modes.
In this study the effect of wave bending would be minimal because the dielectric properties
of the materials considered were close to each other. However, for a more detailed study
the higher-order modes and the wave bending effects must be considered.
153
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CHAPTER VI
C onclusions
The accumulation of flaws or defects (and damage) in a composite structure is
closely tied to its loaded physical and mechanical properties such as strength, durability,
stiffness, etc. It is imperative to have a good knowledge of the integrity of a composite
structure before and during use.
In composite media, defects may be divided into two
groups (depending on the size with respect to the footprint (sensing area) of the microwave
sensor), namely: large and small defects. A large defect, is a defect whose area is several
times larger than the footprint (e.g. a disbond or a delamination), and it can be considered
as an extra layer in the structure. This layer can be detected and characterized using the
measured reflection coefficient. If the defect is small (i.e. its extent is smaller than the
footprint) it will have a different interaction with the fields, due to the boundaries and
edges, and this interaction will influence the refection coefficient and consequently the
signal measured by the microwave sensor. Characterization of defects (determining their
sizes, locations and properties), after they are detected, is a very important part o f any
nondestructive testing technique.
In Chapter 2 a theoretical study was conducted to expand on and demonstrate the
ability of utilizing an open-ended rectangular waveguide probe to monitor the existence and
to determine the position of a disbond in a layered composite structure.
Theoretical
analysis of radiation from a rectangular waveguide into layered dielectric composite media
was presented in this chapter.
Initially, variational formulation was evoked to come up
154
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with a stationary expression for the terminating aperture admittance of general cylindrical
waveguides with arbitrary cross section. The formulation was then expanded to take into
account general N-layer media terminated into an infinite half-space or a perfectly
conducting sheet. The results of a theoretical study, using an open-ended rectangular
waveguide radiating into a multi-layered structure, for detecting disbonds in thick sandwich
composite structures and determining their depths unambiguously were presented in
Chapter 2. The results indicated that disbond detection at depths is possible at a number of
frequencies and standoff distances.
Ka-band was shown to be the most optimum
frequency band to operate in. K-band also showed promise for not only disbond detection
but also for depth determination. It was shown that several frequencies and/or standoff
distances may be used for unambiguous depth determination. All of these results involved
the calculation of the phase of the reflection coefficient at the waveguide aperture. It turns
out that other related parameters such as the magnitude and/or the real pan of the reflection
coefficient may also be used in conjunction (or individually) with other parameters to
determine disbond depth unambiguously.
The disbond thickness used in this study was assumed to be equal to the thickness
of the adhesive layer (0.28 mm). In practice however, disbond thickness may be in excess
of 0.5 mm to a few millimeters. In such cases the phase difference values that were used
for detection will increase, rendering the disbond much easier to detect.
Additionally,
disbond depth determination in such cases will be easier as well. Since disbond thickness
influences the phase of the reflection coefficient, it is very likely to not only be able to
determine its depth but also its thickness (within a given range) as well.
Multiple disbonds may exist in a sandwich composite as well. This microwave
nondestructive testing method is (should be) capable of detecting multiple disbonds as well
(as part of a future investigation).
155
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The formulation used in this general theoretical model is quite complicated.
However, in the future, it may be worth looking into the possibility of developing an
analytical inverse model to determine disbond properties such as existence, depth and
thickness.
Furthermore, in the study conducted here the effect of dielectric property
variations in each layer o f the composite was not investigated.
In a future study the
influence of such variations (including layer thickness variations) should also be taken into
account as well.
In Chapter 3, near-field microwave imaging of dielectric composite structures using
open-ended rectangular waveguides was studied experimentally. Experimental setups were
presented and their operations were discussed.
The utility of applying near-field
microwave techniques to inspect a wide variety of composite structures with different types
of defects was demonstrated, and several experimental results were presented in Chapter 3.
The inspection of thick layered composite materials is essential in ensuring structural
integrity before and during usage. Many types of defects that cannot be visibly observed
can occur in production and in use situations weakening the structural integrity of the
composite and endangering structures employing such materials. Early detection of defects
is necessary to mitigate damage propagation. The ability of microwaves to penetrate inside
dielectric materials makes microwave NDT techniques very suitable for interrogating
structures made of thick dielectric composites. Three experimental setups were presented
for three different types o f near-field microwave imaging. The effects of frequency of
operation and the standoff distance as measurement optimization parameters to enhance the
sensitivity to a defect were studied and presented.
Experimental results obtained from
scanning a variety of composite samples with different types of embedded defects were
presented. Images of these defective samples were created using a measured voltage that is
related to the phase and/or magnitude of the effective reflection coefficient at the aperture of
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the rectangular waveguide sensor. These images presented detailed information about the
structure and integrity of the inspected samples. On all of these images the size of a defect
matches closely its physical size, indicating the high resolution associated with this
technique.
Chapter 4 was devoted to study the field properties in the near-field region of an
open-ended rectangular waveguide and its interaction with a dielectric material. This study
included investigating the influences of frequency and dielectric properties on the radiation
pattern. One of the prime factors influencing an image is the near-field properties of the
radiator (i.e. the interaction of the fields with the material). The fields in the near-field
region of the radiator were formulated and used to understand the properties of the openended rectangular waveguide probe.
The effects and locations of sidelobes are now
understood and can be used to obtain information about the shape and orientation of a
defect.
The formulations, outlined in Chapter 4, are general and can be used in obtaining
the fields in a multi-layered structures backed by either an infinite half-space of material or
by a conducting sheet.
The properties of the fields in the near-field region were
investigated as a function of the frequency of operation, waveguide dimensions and
dielectric properties o f the material under inspection. The waveguide dimensions influence
the spatial resolution drastically, as the frequency band increases (i.e. the waveguide
dimensions decrease) the spatial resolution increases as well. On the other hand, lower
frequency bands showed that higher penetration depths are obtained (i.e. larger waveguide
dimensions).
So, depending on the application the frequency of operation and the
frequency band (waveguide dimensions) can be determined. So, given a certain frequency,
if the goal is to have maximum penetration in to material, the lowest frequency band that
contains the frequency is the best band to operate within. On the other hand, if the goal is
157
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to obtain maximum resolution, the highest frequency band that contains the frequency is the
best to operate within. Results indicated that within the same frequency band, operating at
higher frequencies improves the spatial resolution and the depth of penetration slightly. As
the permittivity increases, the spatial resolution changes minimally and the penetration
depth improves slightly. As the loss factor increases, the spatial resolution decreases
slightly while the penetration depth decreases significantly.
In Chapter 5 a study of the mechanism by which the fields interact with an inclusion
was presented. An effective dielectric constant formula was used to model the reflection
properties o f dielectric structures. The non-uniformity associated with the electric field
distribution at the waveguide aperture is an important factor that must be explicitly taken
into account when trying to understand and interpret images obtained in the near-field of the
sensor. In Chapter 5, this issue was discussed using four sets of dielectric specimens as
steps towards modeling a finite sized inclusion in a host material. A simple formula was
used to obtain the effective dielectric constant of a material made of two or three dielectric
half-spaces arranged side-by-side covered or uncovered by layers of dielectric material.
The non-uniformity of the field distribution was incorporated in the effective dielectric
constant formula by using a non-linear volume fraction calculation consistent with the
aperture field distribution of the rectangular waveguide probe.
The results presented
indicate that the effective dielectric constant is influenced by the volume fraction calculation
method and the formula used. So, using the non-linear approach and the proper effective
dielectric constant formula is important in obtaining correct effective dielectric constant
values. The volume fraction calculation outlined in Chapter 5 assumed that the distribution
associated with the aperture fields is sinusoidal (i.e. that of the dominant mode only).
However, for a more detailed analysis, the influence of the higher-order modes on the nonlinearity associated with the aperture field distribution must be considered.
Also, the
influence of wave bending due to two dielectric materials with different dielectric properties
158
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being partially seen by the aperture should be considered as a part of a future study.
The foreword problem of imaging relatively simple structures and the field-defect
interaction is now understood. More complicated structures can be analyzed in a similar
fashion and the inverse problem of determining defect size and properties can now be
solved.
159
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