A MINIATURIZED MULTI-MODALITY IMAGING SYSTEM FOR DIELECTRIC

MATERIALS EVALUATION

By

Paul Probst

A THESIS

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Electrical and Computer Engineering – Master of Science

2021

ABSTRACT

A MINIATURIZED MULTI-MODALITY IMAGING SYSTEM FOR DIELECTRIC

MATERIALS EVALUATION

By

Paul Probst

With a rapidly decaying infrastructure, the need for improved and more efficient structural health
monitoring and nondestructive evaluation (NDE) only grows with each passing day. Without
comprehensive evaluation of structural issues, repairs tend to be temporary and haphazard. The
quality of repairs utilizing composite materials cannot be adequately inspected, as there exists
a gap within the material coverage in NDE methods caused by the unique anisotropic dielectric
characteristics of composite materials. Due to the inherent relationship of dielectric permittivity
to electromagnetic theory, and the lack of compact and accurate imaging systems that are not
prohibitively expensive for the frequency range (10 KHz to 9 GHz), low frequency capacitive
imaging and high frequency microwave imaging have been chosen as the modalities for the system
presented in this thesis.
First, the governing theory behind low frequency imaging is detailed, with an emphasis on the
capacitive imaging for dielectric characterization of composites. A miniaturized low frequency
capacitive imaging system is designed to operate at a frequency range from 10 kHz to 200 MHz
with a dynamic range of 113 dB. Additionally, high-Q capacitive resonance probes are designed
to be used with the miniaturized imaging system and the results are compared with the preexisting
table-top system. Second, general electromagnetic theory is explained and a miniaturized near-field
microwave imaging system is designed for dielectric evaluation. The microwave system operates
from 1 GHz to 9 GHz and has the ability to work with various probes. Third, both low and high
frequency systems are integrated together to provide a wideband multi-modality imaging system.
The system demonstrates comparatively accurate results for a fraction of the price of existing
systems. The compact and practical nature of this system makes it an optimal tool that can be
utilized not only in the lab, but implemented in field conditions as well.

ACKNOWLEDGMENTS

I would first like to acknowledge my advisor, Dr Yiming Deng for his support, guidance and the
level of freedom he provided me throughout my graduate studies. I would also like to thank Dr
Prem Chahal and Dr Lalita Udpa from my committee for being available for technical guidance. In
particular, Dr Prem Chahal allowed me to use his lab equipment when our lab equipment was out
for repairs, and gave invaluable technical input when I was designing the probes. I am grateful for
their guidance and support, and appreciate the opportunity to work with all three professors.
I would like to thank my coworkers at the Nondestructive Evaluation Laboratory (NDEL), for
their guidance and input, and would especially like to thank Deepak Kumar for his mentorship
throughout my graduate studies. I would like to thank all my professors and classmates over the
years, for the opportunity to work with them, and for their help in shaping my understanding of
numerous topics in electrical engineering. In particular I would like to thank Dr Tim Hogan, Dr
Sunil Chakrapani, and Dr Edward Rothwell for their interesting classes, and Dr Chakrapani for
allowing me to be a teaching assistant for him for a semester. I would like to thank the supporting
staff at the Department of Electrical and Computer Engineering, especially Meagan and Laurene,
for their assistance with anything and everything related to administrative requests. I would also
like to thank Brian Wright for unquestioningly processing and fabricating every antenna, sensor,
and 3D printing design I sent him, no matter how many times.
I would also like to thank the
funding sponsor for this project, Consolidated Edison, for allowing me the opportunity to work on
it. Finally I would like to thank my parents, Pete and Vilija for their unwavering support and love.
I would like to thank my grandfather, Viktor and brother Peter for the inspiration to be an engineer
and continue my education. I guess my brother Jon paid for a vacation that one time and never lets
me hear the end of it so. Thanks bud. Also, my girlfriend Maddy, who provided an untold amount
of support and was always there for me at the end of a long day. Lastly I would like to thank anyone
and everyone who isn’t here to see the end of the journey, particularly Robby, rest in peace.

iii

TABLE OF CONTENTS

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CHAPTER 1

LIST OF TABLES .
LIST OF FIGURES .

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INTRODUCTION .
1.1 Traditional NDE methods .
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1.2 System overview .
1.3 Thesis Objectives .
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vi
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CHAPTER 2 LOW FREQUENCY CAPACITIVE IMAGING TECHNIQUE . . . . . . . .
2.1 Low Frequency Physics Overview . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Capacitive Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2 Dielectric Evaluation .

1
1
2
2

4
4
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5

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2.2.1 Review of State-of-the-art low frequency methods for dielectric proper-

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ties measurement
Probes
2.2.1.1
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2.2.1.2
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1.4

7
7
7
8
9
CHAPTER 3 HIGH FREQUENCY MICROWAVE IMAGING TECHNIQUE . . . . . . . 10
3.1 Microwave Imaging Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.1 Near Field Microwave Theory . . . . . . . . . . . . . . . . . . . . . . . . 12
Far Field Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2
Scattering Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.3
3.1.3.1
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Scattering Parameters and Power

3.2 Dielectric Evaluation .

3.2.1 Review of State-of-the-art high frequency methods for dielectric proper-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
ties measurement
Probes
3.2.1.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1.2
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1.4
CHAPTER 4 MINIATURIZED CAPACITIVE IMAGING SYSTEM . . . . . . . . . . . . 30
4.1 Probe Design and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 System Design and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 System Comparison, Lock-in and Mini . . . . . . . . . . . . . . . . . . . . . . .
. 37
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4.5 Conclusion .
CHAPTER 5 MINIATURIZED MICROWAVE IMAGING SYSTEM . . . . . . . . . . . . 57

Importance of High Q .

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5.1.1 Wave-guide .
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5.1.2 Coaxial probe . .

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5.2 System Comparison, Table top and Mini
5.3 Conclusion . .
5.4

5.1 Probe and System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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. 57
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Importance of System Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4.1 Edge detection vs penetration . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Low and High frequency system comparisons . . . . . . . . . . . . . . . . . . . . 72
CHAPTER 6 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . 73
Importance of System Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1.1 Edge detection vs penetration . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1.2 Low and High frequency system comparisons . . . . . . . . . . . . . . . . 74
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DIELECTRIC LENS . . . . . . . . . . . . . . . . . . . . . . . . . 77
MICROWAVE NETWORK ANALYSIS . . . . . . . . . . . . . . . 81
. 95
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6.2 Conclusions .
6.3 Future work .
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APPENDICES .
APPENDIX A
APPENDIX B
BIBLIOGRAPHY .

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6.1

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LIST OF TABLES

Table 2.1: Dielectric properties of various materials [25] [26]

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Table 2.2: Common low frequency system relative comparison . . . . . . . . . . . . . . .

6

9

Table 3.1: Common High frequency system comparison . . . . . . . . . . . . . . . . . . . 29

Table 4.1: Capacitive probe dimensions

. . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Table 4.2: Frequency Shift of probes, at varying permittivities . . . . . . . . . . . . . . . . 35

Table 4.3: Plate 1 size and scan dimension table

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Table 4.4: Plate 2 size, scan, and defect dimension table . . . . . . . . . . . . . . . . . . . 44

Table 4.5: Plate 4 size, scan, and defect dimension table . . . . . . . . . . . . . . . . . . . 46

Table 4.6: Plate 5 size, scan, and defect dimension table . . . . . . . . . . . . . . . . . . . 48

Table 4.7: Plate 6 size, scan, and defect dimension table . . . . . . . . . . . . . . . . . . . 50

Table 4.8: Plate 7 size, scan, and defect dimension table . . . . . . . . . . . . . . . . . . . 52

Table B.1: Split ring resonator dimension table . . . . . . . . . . . . . . . . . . . . . . . . 94

vi

LIST OF FIGURES

Figure 2.1: Capacitive probe imaging setup used for imaging within this thesis . . . . . . .

6

Figure 4.1: LC tank resonance probe design . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Figure 4.2: Low Frequency Probes designed for the system . . . . . . . . . . . . . . . . .

. 31

Figure 4.3: Measured reflection coefficient for Probe 2 (1 mm Plate separation) . . . . . .

. 32

Figure 4.4: Measured reflection coefficient for Probe 4 (1.5 mm Plate separation) . . . . .

. 32

Figure 4.5: Measured reflection coefficient for Probe 6 (2 mm Plate separation) . . . . . .

. 33

Figure 4.6: Measured reflection coefficient for Probe 1 (1 mm Plate separation) . . . . . .

. 33

Figure 4.7: Measured reflection coefficient for Probe 3 (1.5 mm Plate separation) . . . . .

. 34

Figure 4.8: Measured reflection coefficient for Probe 5 (2 mm Plate separation) . . . . . .

. 34

Figure 4.9: Frequency shift comparison between probes 1 through 6 . . . . . . . . . . . . . 35

Figure 4.10: Block Diagram of Low Frequency system . . . . . . . . . . . . . . . . . . . . 37

Figure 4.11: Aluminum calibration sample results comparison (a) table top system, (b)

mini system .

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. .

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Figure 4.13: GFRP impact damage sample results comparison (a) table top system, (b)

. .

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mini system .

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Figure 4.15: PE sample subsurface defect results comparison (a) table top system, (b) mini

system .

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Figure 4.17: Plate 1 Defect Map .

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Figure 4.18: Plate 1 results comparison (a) table top system, (b) mini system . . . . . . . . . 43

Figure 4.20: Plate 2 Defect Map .

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Figure 4.21: Plate 2 results comparison (a) table top system, (b) mini system . . . . . . . . . 45

Figure 4.23: Plate 4 Defect Map .

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vii

Figure 4.24: Plate 4 results comparison (a) table top system, (b) mini system . . . . . . . . . 47

Figure 4.26: Plate 5 Defect Map .

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Figure 4.27: Plate 5 results comparison (a) table top system, (b) mini system . . . . . . . . . 49

Figure 4.29: Plate 6 Defect Map .

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Figure 4.30: Plate 6 results comparison (a) table top system, (b) mini system . . . . . . . . . 51

Figure 4.32: Plate 7 Defect Map .

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Figure 4.33: Plate 7 results comparison (a) table top system, (b) mini system . . . . . . . . . 53

Figure 4.35: PE Sample Probe setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 54

Figure 4.36: Probe equivalent circuit

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Figure 5.1: Aluminum Calibration sample, table top system (Coax)

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

Figure 5.2: Aluminum Calibration sample, mini system scan (Coax) . . . . . . . . . . . . . 60

Figure 5.3: Aluminum Calibration sample, table top system (Wave-guide) . . . . . . . . .

. 61

Figure 5.4: Aluminum Calibration sample, first square, mini system (Coax) . . . . . . . .

. 62

Figure 5.5: Aluminum Calibration sample, first square, mini system (Wave-guide)

. . . . . 63

Figure 5.6: Polyethylene, table top system (Wave-guide)

. . . . . . . . . . . . . . . . . .

. 64

Figure 5.7: CFRP Plate 1, Wave-guide, mini system . . . . . . . . . . . . . . . . . . . . . 65

Figure 5.8: CFRP Plate 2, Wave-guide, mini system . . . . . . . . . . . . . . . . . . . . . 66

Figure 5.9: CFRP Plate 4, Wave-guide, mini system . . . . . . . . . . . . . . . . . . . . . 67

Figure 5.10: CFRP Plate 5, Wave-guide, mini system . . . . . . . . . . . . . . . . . . . . . 68

Figure 5.11: CFRP Plate 6, Wave-guide, mini system . . . . . . . . . . . . . . . . . . . . . 69

Figure 5.12: CFRP Plate 7, Wave-guide, mini system . . . . . . . . . . . . . . . . . . . . . 70

Figure A.1: Focusing of RF waves radiated from patch antenna using dielectric lens . . . . . 78

Figure A.2: Designed and fabricated lens with the dimensions . . . . . . . . . . . . . . . . 78

viii

Figure A.3: Simulated and Measured gain of the patch antenna at 5.3 GHz without lens . . . 79

Figure A.4: Simulated and Measured gain of the patch antenna at 5.3 GHz with lens

. . . . 80

Figure B.1: Simulated split ring resonator with dimension parameters . . . . . . . . . . . . 93

Figure B.2: Simulated split ring resonator response . . . . . . . . . . . . . . . . . . . . .

. 94

ix

CHAPTER 1

INTRODUCTION

1.1 Traditional NDE methods

The market currently contains standalone systems for eddy current probes, but as of the writing
of this, there are no off the shelf systems that explicitly offer capacitive imaging capability, nor
are there any systems for microwave imaging. Multi-modality systems exist, but none of them
incorporate both microwave and capacitive imaging systems. The systems that exist in the market
are also prohibitively expensive, with the price of a standalone eddy current system reaching over
fifty-thousand USD, whereas the system that will be presented in this paper was designed, built,
and implemented for a fraction of the price. Traditional, off the shelf NDE methods for composite
materials include but are not limited to ultrasound, [1,2] X-ray/CT, [3] and eddy-current systems. [4]
Within research though, high frequency microwave and low frequency capacitive modalities have
been tested numerous times [5–8] and proven to characterize defects as reliably as the traditional
methods, with the added caveat of being much cheaper to produce and upkeep. Microwave
systems utilizing RF waves have been used to image and monitor infrastructure like oil and gas
lines [9], and specifically used for composite scanning [10] as well as been used experimentally
to excite thermoacoustic waves to detect defects acoustically [11]. Higher frequency scanning
of plastics alone have been proven to reliably image defects utilizing RFID tags. [12] Microwave
scans of composite CF-metal samples have shown to accurately image steel corrosion. [13] Systems
similar to this wherein NDE has been performed utilizing portable systems have been shown to
reliably image carbon fiber samples as well, [14] besides the use of more traditional research
systems used for imaging carbon-fiber shown to reliably image defects utilizing ion beams [15] and
using magnetic induction tomography [16]. CT scans have been utilized for composite CF-metal
defect detection. [17] Other reliable results have been obtained for electromagnetically imaging
glass-fiber reinforced composites [18] and through ultrasonic scanning via nonlinear harmonic

1

generation [19]; far-field microwave scanning, which isn’t strictly utilized but will be briefly
discussed later, has proven reliable for imaging GFRP-composite materials. [20] Outside of NDE,
microwave analysis has been reliably utilized to perform quality control on packaged food [21] with
system improvements occurring regularly [22] [23] or to perform quality control on increased or
decreased acidity when the situation calls for it. [24]

1.2 System overview

The system presented in this thesis was built for specific composite scanning and will be utilized
in the field, though it is not limited to this application specifically, and can be improved further as
will be detailed below. Typical scanning systems will include hardware and software elements, but
due to the possibility of untrained personnel utilizing the system in the field, the hardware side has
been emphasized for this system so that all the debugging can be done before any field operations
take place. The software created for this system is minimal, and mostly serves the purpose of
visualization of the data collected. The system has been designed so that results can be acquired
quickly and economically.

1.3 Thesis Objectives

The scope of this thesis is to demonstrate a compact, standalone, economical multi-modality
NDE system, while also establishing an explanation and justification of the utilized methods. The
thesis organization is as follows: Chapter 2 gives an overview of low frequency imaging theory,
and details the design process and compares cost, sensitivity, and size of commonly used low
frequency systems. In chapter 3, a detailed overview of microwave theory is presented within the
scope of NDE with the majority of the details serving as explanation for how the DAQ method for
microwave was utilized within the system, how the probes work physically, why certain theories
are so important, and goes on to perform a similar analysis of relative size, cost, and sensitivity of
commonly used microwave imaging systems. Chapter 4 details the low frequency probes designed
for the system, and shows a direct comparison of the results obtained from a table top system

2

utilizing a lock-in amplifier to the results acquired from the designed low frequency mini system.
In chapter 5, the probes used with the minituarized microwave system are briefly discussed, along
with a comparison of results obtained from the microwave system that validate results acquired
from the capacitive imaging system. In chapter 6, the capability of integration of both systems for
the purpose of a multi modality imaging system is discussed, and the strengths and weaknesses of
both imaging systems is covered. The ultimate conclusion of the thesis is also presented, and a
brief road map for improvements to the system moving forward is proposed. In appendix A, work
performed fabricating a dielectric lens for implementation within the same frequency band as the
high frequency system presented in this thesis is discussed, though that work was not explicitly
done for this thesis. In appendix B, a more detailed background of microwave network analysis is
covered, and a simulated split ring resonator design, its dimensions, and response, is shown.

3

CHAPTER 2

LOW FREQUENCY CAPACITIVE IMAGING TECHNIQUE

2.1 Low Frequency Physics Overview

2.1.1 Capacitive Theory

Shortwave capacitive probes were chosen due to their lower frequency allowing deeper penetration
of the carbon fiber layers. This allows for defects present deeper than the microwave penetration
depth to be identified. The design of shortwave capacitive probes follows the same logic as
microwave probes, as the prevailing design theory is relatively similar: the capacitive probe detects
a change in dielectric coefficient that allows for contrast images to be developed, leading to defect
identification. Many of the equations, however can be simplified due to lower frequencies involved.
Shortwave probes have been proven to reliably image defects in composites due to the innate ability
to sense changes in dielectric permittivity with one of the most frequently utilized probes being a
parallel plate capacitor in some degree, where the governing equations of these probes is defined as

where C, the capacitance, is defined as

Zcap =

1
jωC

C = 
 A
d

(2.1)

(2.2)

Where A is the area of the probe, and d is the distance between the excitation and ground. For the
probe utilized in the system an LC tank architecture is used due to the ability for much higher Q
factor, and thus, a much more sensitive probe, or much larger change to the resultant impedance
imposed from a dielectric permittivity change. The impedance is defined as

(cid:113)

Z =

R2 + (XL − XC)2

while the Q factor for an LC tank is defined as

Q =

1
R

4

(cid:114) L

C

(2.3)

(2.4)

and can be simplified to

and even further simplified to

where

(cid:115)

L2
LC

Q = R

Q = ω0L
R

ω0 =

1√
LC

Due to this, the resonant frequency of the probe can be inferred as

f0 =

1
√

2π

LC

(2.5)

(2.6)

(2.7)

(2.8)

The probe is read in reflection mode, as at the resonant frequency, when the Q is highest, the
reflective coefficient of the probe is near maximum, allowing for total reflection. The introduction
of materials will shift the resonant frequency much more sharply based on the Q value, and will
have specific responses for common defect dielectric values, such as air. The sharp change in the
resultant values of the reflection coefficient that arise from the introduction of defects in contrast to
the consistent values acquired when the uniform material is scanned allow for defects to be viewed
when the data is input to an imaging software like MATLAB.

2.2 Dielectric Evaluation

The resonance frequency of a capacitive probe is inherently related to the capacitance and
inductance values, as seen in equation 2.8, with the equation for capacitance shown in equation
2.2. By using a fixed inductance value and a capacitor utilizing parallel plates or concentric circles
that contain a fixed area, and a fixed distance between the two, the only variable in the equation
becomes 
, the relative permitvitty. By printing a capacitive plate onto a board with a known
dielectric constant, the only way to induce a change in relative permitvitty is by interacting with the
capacitive fields by bringing a sample within the field range. This interaction allows for imaging
and characterization of samples with defects in them, due to the nature of defects frequently having
specific relative permittivities associated with them. A crudely drawn example of this can be seen

5

in the figure below. Dielectric defects in samples can be easily scanned when reading a resonance

Figure 2.1: Capacitive probe imaging setup used for imaging within this thesis

probe in reflection mode due to the frequency shift that will be induced by a sudden change in
permittivity. The reflected power and phase can be back-calculated from values read out in the
form of in phase (I) and quadrature phase (Q). The approximate location, and size of the defect can
be inferred and a sample can be characterized. Various dielectric characteristics of defects present
in samples scanned are shown in the table below, with results shown later in the paper.

Material
Air
Water
Sand
Clay (soil)
Humid Soil
Polyethylene
GFRP
Mold (fungus)
Resins


r

≈ 1.0006

80
3-5
5-40
30
2.2-2.4
≈ 2-4

1.15-1.6 [27]

2.4-4.1

Table 2.1: Dielectric properties of various materials [25] [26]

6

2.2.1 Review of State-of-the-art low frequency methods for dielectric properties measure-

ment

2.2.1.1 Probes

The most common capacitve probe is any variation of a capacitive plate where the signal is fed
into one plate, with the other plate being grounded for reflection mode, or with a signal being
fed into one plate and read on the other plate in transmission mode. The probe can be viewed as
a 1 or 2 port network depending on the read method utilized. Probes that do not utilize an LC
resonance architecture have a relatively low Q value, leading to a loss in sensitivity. The probe
itself is important as plates with wider separation will have greater penetration depth, though they
will be less sensitive to dielectric change, ie the resonance frequency will not shift as much. Probes
with smaller separation between the plates will be extremely sensitive, or the resonance frequency
will shift by much more, but the field penetration depth and overall distance of field output will be
much lesser.

2.2.1.2 Systems

Current common methods of capacitive imaging include the use of a lock-in amplifier, a direct
read-out of change in capacitance in farads using a microcontroller and voltage excitation, using an
impedance analyzer to detect a change in capacitance. Ectane is an off the shelf product capable
of preforming capacitive scans do to its relatively low frequency excitation. These methods can
provide accurate data, and a SR844 lock-in amplifier was used for calibration and as a benchmark
when debugging and prototyping the created mini system, but a lock-in amplifier is a bulky piece
of equipment, and can cost several thousand dollars1; moreover a separate signal generator to excite
the probe is required, adding another few thousand to the total price tag. Direct readout utilizing
a micro controller is very compact, and is very cheap, but the level of accuracy is low due to the
inability for this method to read resonance probes efficiently. Impedance analyzers are arguably
as accurate as lock-in amplifiers, though again, impedance analyzers are relatively bulky pieces of

1SR844, thinksrs.com

7

equipment, and can run tens of thousands of dollars in price2. From this there seems to be a trade
off between accuracy, size, and price. Ectane is relatively compact, as far as standalone systems
go, though it is roughly the same volume as the lock-in amplifier and signal generator, and can
produce accurate scans. This option seems optimal until price is taken into account, as Ectane’s
sticker price is over eight thousand dollars a month to rent3. To buy used, the system can cost thirty
thousand dollars 4 while the price for a new system can easily run over fifty thousand dollars 5. So
is a compact, accurate, and relatively cheap system an impossibility? Not at all. The low frequency
system presented in this paper performs at least as well as a lock-in amplifier, was prototyped and
built for less than a thousand dollars, and can take up a volume of 252 inches3.

2.2.1.3 Applications

Capacitive imaging systems can be utilized to scan dielectric materials for defects, and in a modern
world with more complex materials, more and more infrastructure components and industries in
general are using composite materials that have dielectric properties. Composite plastics currently
are widely used in pipelines, and in industries including aerospace, defense, and energy. The ability
for complex materials to fail catastrophically means there needs to be some sort of quality control
and capacitive imaging is broadly used for this purpose. [8, 28] An additional application specific
problem unrelated to price, size, but tangentially related to sensitivity, is the ability for a system
to be used in field conditions, and not solely research lab conditions. The lock-in amplifier, signal
generator, and impedance analyzer are too delicate and too bulky for field conditions, and are not
designed for such ventures. The direct read system can be adapted for field use, but it needs heavy
modification to achieve this capability. The Ectane system is specifically built for field use, but
again, at the price of ≈fifty times the price of the designed mini system.

2Keysight E4990A
3kwipped.com (Ectane 2)
4eBay
5bullseye calibration

8

2.2.1.4 Limitations

A comparison of the aforementioned systems and the designed mini system in regards to sensitivity,
price, overall size, and field use capability summarizing the above section can be seen in the table
below.

System
Lock-in Amplifier
Impedance Analyzer
Direct read
Ectane
Mini system

Relative Sensitivity

Medium-High

High

Low
High
High

Price

$10,000+
$25,000+

Relative Size

Bulky
Bulky

$30+
Very Compact
$50,000+
≈$1,000 Very Compact

Compact

Field use?

X
X
(cid:88)
(cid:88)
(cid:88)

Table 2.2: Common low frequency system relative comparison

9

CHAPTER 3

HIGH FREQUENCY MICROWAVE IMAGING TECHNIQUE

The most important general form equations utilized within the following sections are modifications
and variations of the generalized three dimensional wave equation, which takes the form

∂2(cid:174)q
∂t2 = ∂2(cid:174)q

∂z2 + ∂2(cid:174)q

∂ y2 + ∂2(cid:174)q

∂z2

1
v2

(3.1)

where q((cid:174)r, t) is the displacement and is a function of the medium at a point (cid:174)r = (x, y, z) at a time t.
A more useful form of the equation is utilized in the form,
c2 )Ψ((cid:174)r, ω) = 0
(cid:90)∞
which incorporates the Fourier-transform of the wave equation
−∞ Ψ((cid:174)r, ω)e−jωtdω

(∇ × + ω2

Ψ((cid:174)r, t) =

(3.3)

(3.2)

for evaluation of the wave equation in the frequency domain. Expansion of the above equations
will be touched on in the proceeding sections.

3.1 Microwave Imaging Theory

Low-power high Q factor microwave probes are commonly chosen as an imaging method due
to their ability to detect and characterize defects within dielectric materials. The probes needed to
be designed using the following characteristic equations due to their governing of the response of
the probes. When powered, the probes radiate high frequency electromagnetic waves that penetrate
the carbon fiber layers to, at most, the steel. The waves are reflected back to the probe, and the
defects are characterized utilizing changes in the reflection coefficient. The reflection coefficient
from a sample would remain consistent with a consistent height of probe, frequency, and power, so
steps were made in the system design to ensure that both frequency and power delivered remained
as stable as possible. Without defects, the reflection coefficient would return one value, while with
defects, the value would be different enough that a contrast image could be obtained using the

10

magnitude and phase of the reflection coefficient. Depending on the nature of the defect and its
size, certain characterizations can be made about the signal response, though if defect types have a
cross over in dielectric constant, this characterization cannot be guaranteed to be entirely accurate,
E.g. if a an epoxy delamination and a soil contamination dielectric coefficient range from 
 = 3−15
and 
 = 8 − 20 then a measured response indicating a dielectric coefficient of 
 = 8 − 12 would
not be able to differentiate between the two, unless other information is reliably known, if either
has a consistent size. The returned results would indicate that a defect was present though, and
repairs would have to be made regardless. The design of microwave probes for this project took
place utilizing ANSYS HFSS, as well as MATLAB, to solve the following equations. The most
important relations of electromagnetic theory are as follows: The electromagnetic wave equation
describing the electric and magnetic fields is defined as

for the electric field and

for the magnetic field, where

ph∇2 − ∂2
(v2

∂t2 ) (cid:174)E = 0

ph∇2 − ∂2
(v2

∂t2 ) (cid:174)B = 0

vph =

1√

µ


(3.4)

(3.5)

(3.6)

is the speed of light. Following this, the most important relations, and arguably some of the most
important mathematical equations derived in all of human history, is the collection of equations
known as Maxwell’s equations, presented here in differential equation form, and take the forms

∇ · (cid:174)D = ρ
∇ · (cid:174)B = 0
−∂ (cid:174)B
∇ × (cid:174)E =
∂t
∇ × (cid:174)H = (cid:174)J + ∂ (cid:174)D

∂t

(3.7a)

(3.7b)

(3.7c)

(3.7d)

where 3.7a is Gauss’s law, relating the electric field to charge and dielectric constant and fundamen-
tally proves the existence of electric dipoles. Conversely, 3.7b is Gauss’s law for magnetism which

11

shows the divergence of a magnetic field is zero, or that a magnetic monopole does not exist. 3.7c
is the Maxwell-Faraday equation or Faraday’s law of induction, and predicts how a magnetic field
will interact with an electric circuit to produce electromotive force. This equation was touched on
earlier for an in depth explanation of eddy current system performance. Finally, 3.7d is known as
Ampère’s circuital law, which relates the electric field and induced current charge on a closed loop
to the intrinsic magnetic field of the circuit.

3.1.1 Near Field Microwave Theory

There are different regions within microwave fields dependent on distance, the near field, and the
far field. In far field analysis, the waves can be treated like planar waves, and the power dissipates
extremely rapidly after exiting the near field region. The waves within the near field act very
differently, and the relationship between the electric and magnetic fields is very complicated, as
will be explained shortly. The electromagnetic fields are much more powerful in the near field, and
allows for imaging due to the high intensity of the fields. The boundary between near and far field
can be found using the Fraunhofer distance equation

d =

2D2
λ

(3.8)

where D is the largest dimension of the device that is radiating (eg. a loop antenna’s diameter) and
λ is the wavelength of the signal. The near field is split into the non-radiative or reactive near field,
the nearest part of the near field, and the radiative near field or Fresnal region, the farthest part of
the near field. The reactive near field covers up to the distance λ
2π while the Fresnal region covers
between this distance and the Fraunhofer distance. The reactive region can also be characterized by
any location less than or equal to 0.62
λ while the Fresnal region covers between this distance
and d from the above equation. Near field scanning modalities utilize oscillating fields that do not
allow for the propagation of EM waves by scattering them due to the geometry of the probe. [29]
The oscillation of the fields allows for a net energy flow of zero, due to the Poynting vector averaged
over an oscillation equaling zero. [29] These fields are referred to as evanescent and create waves

(cid:113) D3

12

with imaginary wavenumbers that store reactive energy. This reactive energy can be either electric,
or magnetic, or both. A sample interacting with the near field of the probe will change the energy,
altering the EM response of the probe. The creation of an evanescent field in a sample relies on the
size of the probe tip D and must be small enough to satisfy |kd|D << 1, where kd is the complex
wave number of the device under test given by:
√

kd = ω


0
d µ0 µd

(3.9)

The evanescent fields from the probe interact with the DUT and allow for characterization through
the change in EM waves. The size of the field(s) is roughly equivalent to the probe size D, which
dictates the resolution of the scan more so than any properties of the DUT. This type of imaging
is called NF microwave microscopy and is a very exciting topic in the imaging world due to the
use of microwave frequencies to image samples with resolutions of orders of magnitude smaller
than a wavelength. The energy contained in the reactive field goes through a type of feedback loop
and can load the system ’more’ than it already is, allowing for the energy that is ’returning’ from
the DUT/scanned sample be isolated from the input signal, and evidently this allows for accurate
imaging to take place due to perturbations in the signal signifying defects/characterizing features
in the sample. To effectively create a NF microwave probe, one must be able to calculate the
impedance and reflection coefficients. And to make it worthwhile, we would want to do it using
transmission lines because lumped components are extremely passé. The impedance of a probe tip
can be found using

(cid:90)

V

(wm − we)d3x

Zp = Rp + j Xp

where a NF tip reactance can be found using
4ω
|Ii|2
wm = BH∗

where wm is magnetic energy density,

Xp =

4

and we is electric energy density,

we = E D∗

4

13

(3.10)

(3.11)

(3.12)

(3.13)

(cid:90)

and Ii is the harmonic input current at the tip terminal. [30] Rp can be found using

Rp = ω
|Ii|2

|E|2 + 
0
”|E|2 + µ0 µ”|H|2)d3x

V ( σ
ω

(3.14)

The impedance of an electric tip can be expressed in relation to the coupling capacitance between
the probe tip and sample where

1
Cc

+

1
Cc2

=

1
Cc1

and Cstr represents the stray capcitance from the probe tip, as

1
ZtE

= (

1

jωCc

+ Zs)−1 + jωCstr

(3.15)

(3.16)

(3.17)

where Zs is the NF sample impedance. [30] The impedance of magnetic tip, when |kd|D >> 1 can
be found using

ZtM = jωL0 + ω2M2
Zs + jωL1

where L0 is the loop self inductance, L1 is the sample inductance, and M is the probe tip-sample
inductance. [30] The impedance of an electric tip should satisfy |ZtE| >> Z0 while a magnetic tip
should satisfy |ZtM| << Z0 thus the complex reflection coefficient from the tips can be found using

ΓE = exp− 2Z0RpE
R2
pE + X2
pE

exp j

2Z0XpE
R2
pE + X2
pE

where ΓE can also be expressed as

and

can also be expressed as

ΓE =|ΓE| exp jθE =

ΓM =|ΓM| exp jθM =

ZpE − Z0
ZpE + Z0

ZpM − Z0
ZpM + Z0

ΓM = exp−2RpM
Z0

exp−j

2XpM
Z0

(3.18)

(3.19)

(3.20)

(3.21)

These equations, however, deal primarily with phase, and when the phase measurement/calculation
is not accurate enough resonant frequency and Q factor for the probe tip would also need to be

14

calculated. The equations for resonant frequency and (un-loaded) Q factor of an electric field probe
are as follows

F =

2πL√

πn

0 µ0
e f f

Q = ω(cid:48)

2ω(cid:48)(cid:48) =

Z0XpE
pE + X2
pE)

[1 +

πn(R2
(πn + Z0XpE
R2
pE+X2
pE
2Z0RpE
2h(cid:48)(cid:48)L +
R2
pE+X2
pE

)

]

(3.22)

(3.23)

(3.24)

where L is the length of the probe and

h = h(cid:48) − jh(cid:48)(cid:48) = ω

(cid:113)

(
0
e f f µ0) − jh(cid:48)(cid:48)

(3.25)

(3.26)

is the complex propagation constant of the transmission line. [30] If instead of a probe, the design
was to be a cavity (eg. a waveguide) there is cavity perturbation theory which shows that for a low
loss dielectric film of unknown (complex) permittivity the frequency associated can be found using

(cid:90)

∆F
F

∼= −(


(cid:48)
f ilm − 1) 
0
4W

E1E2dV

Vs

along with associated Q factor

∆

1
Q

= −tanδ

2
(cid:48)

(cid:48)
f ilm − 1

f ilm

∆F
F

with the desired solutions being the electric fields E1 and E2. [30] Near field power applications
can be utilized using an equation for power received, PRx vs power transmitted, PT x for antennas
that are either both magnetic or electric, or mis-matched which are respectively as follows

PRx
PT x

= GT xGRx

4

(

PRx
PT x

= GT xGRx

(

4

1

1

1
(kr)6 )
(kr)4
(kr)2
1
1
(kr)4 )

(kr)2

(3.27)

(3.28)

with GT x and GRx representing the gain of the transmitting and receiving antennas respectively,
while r is the range between the antennas and k = 2π

λ . [31]

15

3.1.2 Far Field Analysis

Analysis of microwave networks in the far field differs from near-field analysis in that many of the
equations become simplified. The main point of interest for far field analysis is arguably the Friis
transmission equation,

PR = PT + GT + GR + 20log10( λ
4πd

)

(3.29)

where PR is the power received, PT is the power transmitted, GT is the gain of the transmitting
antenna, GR is the gain of the receiving antenna, all in dBm; λ is the wavelength in meters, and d is
the distance between the antennas in meters. In terms of far field scanning of samples, the sample
would be viewed as the receiving antenna and the gain would be minimum, or very low, dependent
on composition of the sample. The other important relation is

λ =

c
√

f


r

where c is the speed of light in m
s , f is the frequency in Hz, and 
r is the relative permittivity. Since
the near entirety of the sample scans were performed within the near field, this section will remain
brief.

3.1.3 Scattering Parameters

When designing a microwave system, it must be treated like a network to adequately simulate the
possible results. When analyzing the network, the important components to monitor include the
propagation of the system, or the power of the electromagnetic fields that will be interacting with the
samples being imaged. The propagation is also important for knowing the response of the probe by
knowing the locations of the near and far field. Another important component is the transmission
lines involved in the system. Transmission lines dictate the intrinsic impedance of the system at a
specific frequency and the effectiveness of the designed probe and system responses. Transmission
lines are a very fundamental part of microwave network analysis and electromagnetic theory as a
whole, and are further explored in B.1.1. One of the most important, if not the most important,
components of microwave network analysis and system design are the scattering parameters. These

16

parameters are intrinsically linked with the frequency of the system. They give information about the
reflection and transmission coefficients, and the overall responses of the reflection and transmission
of the system as a whole. The resultant reflection and transmission coefficients in a well designed
system are inversely related: when transmission is maximum, reflection is minimum, and vice
versa. This is very important for dictating how data will be acquired, and which read mode to use,
either reflection on one port, or reflection on two ports. With single port reflection the probe injects
electromagnetic waves into the sample and "listens" for the reflected wave. The time difference in
sent and received signals can dictate the depth of a defect, and the power reflected will indicate
the composition of the defect. The scattering parameter response of the both the microwave and
capacitive probes were used to tune the operating frequency for both. A brief explanation of
scattering parameters is included below: From [32] the scattering matrix is defined as:



V−
1
V−
2
...
V−
N





=

S11
S21
...
SN1

S12

. . . S1N
...

...

. . .

SNN





V+
1
V+
2
...
V+
N



(cid:12)(cid:12)(cid:12)(cid:12)V+

17

N is defined as the forward voltage, and V−

Where V+
N is the reverse voltage. The [S] matrix is
comprised of scattering parameters, and is defined in relation to the incident and reflected voltage
waves:

[V−] = [S][V+]

(3.30)

(3.31)

where a specific matrix element is defined as
V−
i
V+
j

Si j =

k =0 for k(cid:54)=j
j and measuring the reflected amplitude V−

So by driving port j with an incident voltage V+
i on
port i the element Si j can be found. These measurements can be made using a Virtual Network
Analyzer (VNA) to determine very important properties. When measuring, the incident waves
on all ports except the jth port are terminated with matched loads to avoid reflections. The
reflection coefficient on any port i is determined by matching all ports and defined as Sii whereas

the transmission coefficient from port j to port i, when all other ports are matched, is Si j. These
properties are intrinsically related to the impedance and admittance matrices, which are defined as

or

[V] = [Z][I]

(3.32)

where Zi j is the impedance seen when driving port j with current Ij and open circuiting all other
ports. The current and voltage are related to the impedance by the equation

The admittance matrix [Y] is the inverse of the [Z] matrix, that is

(3.33)

(3.34)

(3.35)

(3.36)



V1
V2
...
VN





=

Z11 Z12
Z21
...
ZN1

. . .

. . . Z1N
...

...

ZNN





I1
I2
...
IN



(cid:12)(cid:12)(cid:12)(cid:12)Ik=0 for k(cid:54)=j

Zi j = Vi
Ij

[Y] = [Z]−1





V1
V2
...
VN



Y11 Y12
Y21
...
YN1

. . .

Yi j = Ii
Vj

. . . Y1N
...

...

YNN

(cid:12)(cid:12)(cid:12)(cid:12)Vk=0 for k(cid:54)=j
(cid:73)

¯Ea × ¯Hb =

¯Eb × ¯Ha · d ¯s

S

(cid:73)

S

and has the form

where Yi j can be found as



I1
I2
...
IN



=



The [Z] and [Y] matrices have special properties when carried through the reciprocity theorem,

¯Ha,

¯Eb and ¯Hb are the electric and magnetic fields anywhere in the network from
where ¯Ea,
independent sources a and b, and S in this case is the closed surface following the boundaries of
the network. The fields due to the sources can be evaluated as

18

¯E1a = V1a ¯e1, ¯H1a = I1a ¯h1
¯E1b = V1b ¯e1, ¯H1b = I1b ¯h1
¯E2a = V2a ¯e2, ¯H2a = I2a ¯h2
¯E2b = V2b ¯e2, ¯H2b = I2b ¯h2

(3.37)

where the transverse modal fields of ports 1 and 2 are ¯e1, ¯h1 and ¯e2, ¯h2 respectively, and ¯E1a is the
transverse elctric field at plane t1 of port 1 due to source a. Upon substitution of 3.37 into 3.36 the
resultant equation is

(V1aI1b − V1bI1a)

¯e1 × ¯h1 · d ¯s + (V2aI2b − V2bI2a)

S1

¯e2 × ¯h2 · d ¯s = 0

S2

(3.38)

where S1 and S2 are cross sectional areas at the planes of ports 1 and 2. This can be rewritten when
taking into account the power formula

(cid:90)

(cid:90)

(cid:90)

P = V I*
2

(cid:90)

(3.39)

(3.40)

(3.41)

where I* is the conjugate [I] matrix and implies that C1 = C2 = 1 so that Eqn. 3.38 can be rewritten
as

¯e1 × ¯h1 · d ¯s =

S1

¯e2 × ¯h2 · d ¯s = 1

S2

and reduces Eqn. 3.38 to

V1aI1b − V1bI1a + V2aI2b − V2bI2a = 0

When applying the above to a 2 port network and respective 2 × 2 admittance matrix to eliminate
the currents gives

I1 = Y11V1 + Y12V2
I2 = Y21V1 + Y22V2

which can be substituted into Eqn. 3.41 to give

(V1aV2b − V1bV2a)(Y12 − Y21) = 0

(3.42)

19

where the independent sources a and b allow the voltages V1a , V1b , V2a, and V2b to take on
arbitrary values, and thus to satisfy 3.42 for any and all sources, Yi j = Yji. Because the admittance
and impedance matrices are inverses, if [Y] is symmetric, then so too is [Z].

The admittance and impedance matrices are often filled with complex values, which makes their
analysis arduous and complex: this is why the scattering matrix as the measure can be expressed
in terms of input and output power. The terms within the impedance and admittance matrices can
be written as

Vn = V+
In = I+

n + V−
n − I−

n
n = V+

n − V−

n

so the definition from Eqn. 3.32 can be used with the above equations to give the relations

[Z][I] = [Z][V+] − [Z][V−] = [V] = [V+] + [V−]

or

([Z] + [U])[V−] = ([Z] − [U])[V+])

where [U] is the unitary or identity matrix and is defined as

(3.43)

(3.44)

(3.45)

(3.46)

(3.47)



[U] =

1 0 . . . 0
...
0 1
...
0

...
. . . 1



comparing equations 3.30 and 3.44 leads to the equivalency

or in terms of [S]

[S] = ([Z] + [U])−1([Z] − [U])

[Z] = ([U] + [S])([U] − [S])−1

and results in writing the scattering matrix in terms of values of the impedance matrix, and for a
single port network this can be reduced to

S11 = z11 − 1
z11 + 1

20

(3.48)

The scattering parameters thus relate the magnitude and phase of incident and reflected waves
traveling on and from a microwave network respectively, and change based on their reference
points. The [S] matrix can be written in terms of the incident and reflected voltages,

[V−] = [S][V+]
[V(cid:48)−] = [S(cid:48)][V(cid:48)+]

(3.49)

where the first equation in Eqn. 3.49 refers to the original planes, and the second equation refer to
new planes. With

where Z0 is the characteristic impedance, the new wave amplitudes can be related to the original
amplitudes by

0 e−j βz + V−
V(z) = V+
0 e j βz
e−j βz − V−
V+
0
0
e j βz
Z0
Z0

I(z) =

V(cid:48)+
n = V+
V(cid:48)−
n = V−

n e jθn
n e−jθn

(3.50)

(3.51)

[V(cid:48)−] = [S]

[V(cid:48)+]

(3.52)

where the subscript is in reference to the new reference plane located at zn = ln with the original
plane arbitrarily placed at zn = 0. and

θn = βnln

is the electric length of the reference plane shift of port n, or in matrix form for an N port network



e jθ1

. . .

0



0
...

. . .
e jθ2
...

. . .

e jθN



Multiplying the RHS by the inverse matrix on the LHS results in

[V(cid:48)−] =

e−jθ1

. . .

0

. . .
e−jθ2
...

. . .

0
...

e−jθN

e−jθ1

. . .

0

. . .
e−jθ2
...

. . .

0
...

e−jθN



[V(cid:48)+]



. . .
e−jθ2
...

. . .

0
...

e−jθN

e−jθ1

. . .



0





[S]

21

which when compared to Eqn. 3.49 shows

e−jθ1

[S(cid:48)] =


 =

0

. . .

. . .

[S]

0
...

e−jθN

. . .
e−jθ2
...



(Z11−Z0)(Z22+Z0)−Z12Z21

2Z21Z0

∆Z

∆Z

S11 S12

S21 S22

e−jθ1

. . .

0

. . .
e−jθ2
...

. . .

0
...

e−jθN



(Z11−Z0)(Z22−Z0)−Z12Z21

2Z12Z0

∆Z

∆Z



nn = e−2jθnSnn meaning the phase is shifted by twice the electrical length, because the
Where S(cid:48)
wave travels the length twice, upon incident and reflection. The mathematical relationship between
the S, and Z matrices is as follows

where

∆Z = (Z11 + Z0)(Z22 + Z0) − Z12Z21

And is important for converting Impedance matrices to Scattering parameter matrices.

3.1.3.1 Scattering Parameters and Power

In regards to Eqn. 3.43, the total voltage and current seen in network analysis can be rewritten as

0

0 + V−
V = V+
0 − V−
V+
Z0

I =

0

(3.56)

To find the incident and reflected wave amplitudes in terms of total voltage and current, Eqn. 3.56
must be inverted resulting in

(3.53)

(3.54)

(3.55)

(3.57)

0 = V + Z0I
V+
0 = V − Z0I
V−

2

2

22

so that the average load power can be written as

Re(V I*)

PL =

=

2
0 V−*
0 |2−V+
Re(|V+
0 |2−|V−
(|V+
0 |2)

0 + V+*

0 V−

0 − |V−
0 |2

1
2Z0
1
=
2Z0
0 V−
as the quantities V+
0 are purely imaginary. This result though, is only valid when the
characteristic impedance is real and does not apply when Z0 is complex. The reflection coefficient
Γp for the power can be defined as

0 V−*

and V+*

(3.58)

0

Γp =

V − Z∗
RI
V + ZRI

=

ZL − Z∗
ZL + Z∗

R

R

(3.59)

where ZL and ZR are the load and reference impedance respectively, and when ZR = Z0 and is real,
reduces to

Γ = ZL − Z0
ZL + Z0

(3.60)

For a basic circuit system consisting of an input voltage V0 from a generator with impedance Zg
through a load of impedance ZL along with the voltage and current equations from 3.50 the voltage,
current, and power equations can be written as

V = V0

ZL

ZL + Zg
V0

I =

PL =

ZL + Zg
V2
0
2

RL

|ZL + Zg|2

(3.61)

where

ZL = RL + j XL

These equations are very important, as when designing a microwave probe, the power delivered
to the load needs to be maximum as the load in this case is the probe itself. The power is
maximum when Zg = Z∗
L, and when maximum power is delivered to the probe, the reflection from
a sample ΓDUT must be maximum to ensure maximum power is detected from from a material

23

being scanned. For probes read in reflection mode, a maximum ΓL allows for an extremely sharp Q
and very sensitive changes in Z0 allowing for defects to be detected in the sample. When probes are
in read mode, the S11 parameter is equivalent to ΓDUT, and this is why the scattering parameters are
so important when designing microwave probes. This proof can be found by utilizing the ABCD-
parameters, also called the transmission matrix. These parameters are used when the networks
being analyzed are cascaded, or there is more than one network being analyzed so that the networks
can be combined. This is the case in this system as the probe, DAQ and excitation hardware are
independent network, but can be cascaded to achieve the overall output via the ABCD-parameters.
For two two-port networks, the ABCD-matrix is defined as

or,

When cascaded, the equations are

V1 = AV2 + BI2
I1 = CV2 + DI2

I1

I1

I2

C D

I2

C1 D1

V2

V1
 =
A B

V1
V2

A1 B1
 =
V2
V3

A2 B2
 =
V1
 =
A1 B1
A2 B2

S11 S12
 =
 A+B/Z0−CZ0−D

A+B/Z0+CZ+D
A+B/Z0+CZ0+D

S21 S22

C1 D1

C2 D2

I3

C2 D2

I2





I1

V3

I3





2(AD−BC)

A+B/Z0+CZ0+D
−A+B/Z0−CZ0+D
A+B/Z0+CZ0+D

2

24

So that when Eqn. 3.64b is substituted into Eqn. 3.64a the resultant equation becomes

The relation between the S-matrix and ABCD- matrix is as follows,

(3.62)

(3.63)

(3.64a)

(3.64b)

(3.65)

(3.66)

To see how the reflection coefficient, Γ is related to the S11 parameter, the equation B.21 is used
for γ and the identities

cosh(x) = ex + e−x
sinh(x) = ex − e−x

2

2

(3.67a)

(3.67b)

are used when regarding the ABCD and Z parameters vs the S parameters. When the limits of
a line of length l, with characteristic impedance Z0 and input impedance ZL are taken to ∞, the
relation is

|S11| = lim
l→∞

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

L)sinh(γl)
L + Z2

0)sinh(γl)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

0 − Z2
(Z2

2ZL Z0cosh(γl) + (Z2
0 − Z2
(Z2
2ZL Z0eγl + (Z2

L)eγl
L + Z2

0)eγl

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(Z0 − ZL)(ZL + Z0)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z0 − ZL

2ZL Z0 + Z2

L + Z2
0

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

≈

=

=

Z0 + ZL

= |Γ|

(3.68a)

(3.68b)

(3.68c)

(3.68d)

(3.68e)

with Eqn. 3.68d equivalent to Eqn. 3.60 This mathematical proof and the preceding relations are
what allow the microwave system to operate in read mode. The power is put through port 1, and
the reflected power is read on port 1, and due to the amplitude and phase of the reflected signal,
a defect is able to be identified by the precise loss of power seen in the amplitude, and the area is
able to be generalized by the phase loss.

3.2 Dielectric Evaluation

As shown in the theory section, dielectric permittvity 
 is an extremely important value for the
response of a microwave probe. The method of action, simplified, is relatively the same as the
low frequency capacitive method. A probe with fixed values will react a known way in air, with a
resonant frequency F0 and when a material with a different permittvity is introduced, the response

25

will change. Microwave probes are read in either reflection mode, treating the probe as a single
port network, with the S11 value equivalent to reflection coefficient read out using the voltage value
to back calculate the power and phase of the reflected signal. Probes are also commonly read in
two port network configuration, exciting one port and reading on the other. In this configuration,
the S21 or S12 value is used, dependent on the port of excitation and reading, as the transmission
coefficient to perform the same power and phase calculations. Due to the higher frequency utilized
in microwave scanning, the resonance shift seen from varying permittvity is much greater than
capacitive scanning. This greater sensitivity, however comes with a loss of penetration depth.
Another drawback to microwave dielectric evaluation is the inability to utilize lumped components,
which are easier to debug and spontaneously alter for a different response.

3.2.1 Review of State-of-the-art high frequency methods for dielectric properties measure-

ment

3.2.1.1 Probes

Commonly utilized probes for microwave scanning include waveguides, coaxial cables, split ring
resonators, and patch antennas. Waveguides have high penetration depth, but can suffer from
decreased sensitivity. Coaxial cables are inverse to this, and can be highly sensitive, but suffer from
low penetration depth. Split ring resonators can have relatively good sensitivity and penetration
depth, but can perform inadequately on thicker samples. Patch antennas can be utilized for
microwave imaging of surface defects, but may not be able to penetrate very far, or if they can,
may not be sensitive to small defects. The quality factor associated with split ring resonators make
them an attractive choice for microwave imaging, but another downside to their use is the inability
to spontaneously alter their response if the fabrication process did not create a resonator matching
simulation results, or if the simulation itself was flawed making the results inconsistent. For this
reason, a coaxial probe and waveguide were used to calibrate the mini microwave system due to
their reliability for the samples imaged.

26

3.2.1.2 Systems

A common method of near and far field microwave imaging is by using a Vector Network Analyzer
(VNA), or Phasor Network Analyzer (PNA) as the excitation and read component of a scanning
system, with a computer connection for interfacing. [33] [34] [35] This is because the VNA/PNA
has the capability of showing the S-parameter network response of a microwave probe over a
wide frequency band, and allows for the recording of several different resonance frequency shifts,
allowing for accurate amplitude and phase readings, a crucial element to microwave imaging. [36]
The main drawback is again, the relative size and price of the system, as VNA/PNA equipment size
is larger than any system mentioned in the low frequency section, and a price tag for a relatively
low frequency VNA system (26.5GHz) is well over a hundred thousand dollars1. The microwave
frequencies utilized by the mini systems were limited well below this, with the coax probe operating
around ≈ 4GHz, and the waveguide operating around ≈ 7.8GHz. If this limit is met with VNA
choice, the price is still in excess of eighty three thousand dollars2. This price obviously comes
with high sensitivity, but the VNA/PNA itself is a general use equipment and is not strictly limited
to imaging. The whole point of using it is for port matching and network evaluation. Besides that,
the time it takes to perform a scan using it is highly excessive, with scans taking as long as eighteen
hours when a scan using the same dimension can be run using a different table top system or the
mini system taking as little as thirty minutes. Specific miniaturized microwave systems are common
within literature [37] but these systems still utilize a table top RF source, that still is as bulky as
a lock-in amplifier volumetrically, and can run over eight thousand dollars refurbished. With the
above price tags associated with microwave imaging, an off the shelf system on par with the Ectane
system would be astronomically expensive, if it even existed; consumer market microwave imaging
systems have yet to be mass produced or sold due to the prohibitively expensive cost of general
use equipment like the VNA. The microwave system designed for this paper utilizes relatively the
same miniaturized back-end read circuit as is referenced in [37] but is excited by a source the size

1Keysight website, N5222B PNA
2Keysight N5221B

27

of a cellular phone, and in addition implements an equally small I/Q demodulator allowing for the
acquisition of I and Q data simultaneously, thus allowing for direct calculation of power and phase
data at half the scan time.

3.2.1.3 Applications

Microwave imaging systems, like capacitive imaging systems have been, and still are frequently
utilized for characterization and imaging of defects in composite materials. [13] [38] [37] With
long standing reliability and consistency of composite results obtained using microwave scanning
systems, recent research has even tested the capabilities on carbon-fibre-reinforced polymer com-
posites (CFRP) to promising results. [14] This imaging can easily be undertaken in a research
setting, but again, one of largest drawbacks is the inability to test probes and the system as a whole
in field conditions. VNA and PNA systems are well known for their relative delicacy, and their
need for repeated calibration in the event that the VNA is compromised in any way, i.e. heavy
breathing in some cases; a calibration kit for one of the quoted systems earlier starts at three
thousand. This is without mentioning that the cables which are interconnected between the VNA
and probe are astoundingly delicate, and should under no circumstance be bent too much or moved
excessively as a single cable can cost more than two thousand dollars. Even when using previously
mentioned miniaturized reading circuits, the system source is still only capable of operating under
lab conditions.

3.2.1.4 Limitations

A comparison of the aforementioned systems and the designed mini system in regards to sensitivity,
price, overall size, and field use capability summarizing the above section can be seen in the table
below.

28

System
VNA/PNA
High Frequency Source
Off the Shelf System
Mini system

Relative Sensitivity

Very High

High
(cid:154)
High

Price

$80,000+
$10,000+
≈$600

(cid:154)

Relative Size
Very Bulky

Bulky

(cid:154)

Very Compact

Field use?

X
X
(cid:154)
(cid:88)

Table 3.1: Common High frequency system comparison

29

CHAPTER 4

MINIATURIZED CAPACITIVE IMAGING SYSTEM

4.1 Probe Design and Characterization

Inductor-capacitor circuits forming resonant tanks one of the commonly used dielectric probes,
wherein one of the component variables in equation 2.8, either inductance or capacitance, are fixed
allowing for a shift in resonance frequency to rely solely on the unfixed variable. Based on the
capacitive theory equations shown in 2.1.1, specifically, equations 2.2 and 2.8, resonance probes
were designed with the idea of fixing the inductance value, allowing for any change in center
frequency to come from a change in the relative dielectric permittivity. A diagram of the resonant
probes is included below in figure 4.1. Six unique high-Q resonance probes were designed for use

Figure 4.1: LC tank resonance probe design

in combination with the system and are shown below in figure 4.2. The capacitors on for C1 and
C2 are identical, though the inductors within the LC tank architecture differ. The same pattern
follows for C3 and C4, as well as C5 and C6. Probes 1, 3, and 5, are designed for approximately

30

Figure 4.2: Low Frequency Probes designed for the system

Inner ring diameter Ring separation Total diameter

Probe Outer ring diameter
1
2
3
4
5
6

2 mm
2 mm
1.5 mm
1.5 mm
1 mm
1 mm

8 mm
8 mm
8 mm
8 mm
8 mm
8 mm

1 mm
1 mm
1.5 mm
1.5 mm
2 mm
2 mm

14 mm
14 mm
14 mm
14 mm
14 mm
14 mm

Table 4.1: Capacitive probe dimensions

12 MHz and utilize a fixed inductance value of 56 µH, while probes 2, 4, and 6 are designed for
approximately 5 MHz and utilize a fixed inductance value of 75 µH. The probes were measured
using a Virtual network analyzer, and their sensitivities were initially tested using samples with
known dielectric permittvities, such that a comparison in the frequency shift could be obtained
upon full measurement of all probes. Initial measurements were obtained in air (
r ≈ 1) and Rogers
boards with 
r = 3.66, 6, and 10. The resultant measurements for the probes are as follows from
low to high frequency, and least to greatest plate separation:

The resultant frequency shift can be seen as a comparison between all six probes following the

probe responses, in figure 4.9.

31

Figure 4.3: Measured reflection coefficient for Probe 2 (1 mm Plate separation)

Figure 4.4: Measured reflection coefficient for Probe 4 (1.5 mm Plate separation)

32

Figure 4.5: Measured reflection coefficient for Probe 6 (2 mm Plate separation)

Figure 4.6: Measured reflection coefficient for Probe 1 (1 mm Plate separation)

33

Figure 4.7: Measured reflection coefficient for Probe 3 (1.5 mm Plate separation)

Figure 4.8: Measured reflection coefficient for Probe 5 (2 mm Plate separation)

34

Figure 4.9: Frequency shift comparison between probes 1 through 6

Probe number
1
2
3
4
5
6


r = 3.66
789 KHz
50 KHz
600 KHz
77 KHz
580 KHz
98 KHz


r = 6

1284 KHz
80 KHz
1061 KHz
133 KHz
932 KHz
188 KHz


r = 10
1787 KHz
138 KHz
1455KHz
200 KHz
1316 KHz
264KHz

Table 4.2: Frequency Shift of probes, at varying permittivities

4.2 Importance of High Q

The quality factor of a probe is inherently related to its sensitivity. The frequency shift seen by
the introduction of a dielectric directly correlates to a change in capacitance seen by the concentric
circle capacitor plate, and the reflected power measured. The higher the quality factor, the smaller
change in dielectric permittvity can be detected, correlating to a smaller change in capacitance.
Knowing the relation between specific permittivties, the frequency shift, and the resultant reflected
power can allow for characterization of detected defects based upon the response at a specific input
frequency and power.

35

4.3 System Design and Characterization

The low frequency miniaturized system has been designed with the idea of replacing a table top
signal generator, lock-in amplifier, and data acquisition system. The system utilizes the AD9958/59
evaluation board as a multi-channel frequency synthesizer; two channels are used, one for the probe
excitation, and one for the I/Q demodulator local oscillator (LO). The demodulation is processed
utilizing an AD8333 I/Q demodulator and phase shifter. A separate board has been designed
utilizing Altium PCB software that allows for a single power connection to power both evaluation
boards. The designed PCB also contains an amplifier for the LO signal, as the I/Q LO power has to
be within an acceptable range, and the power level is extremely important in regards to the quality
of the acquired signal that is to be demodulated. Low pass filters, in the form of an integrator,
have also been designed and implemented within the power/amplifier PCB to allow for filtration
of any higher frequency noise on the acquired signal, as the output value is read in a DC voltage.
Without filtration, the acquired signal is extremely noisy and would require software to clearly see
the results, thus the LPFs act as hardware noise reduction. The results acquired, shown further
below in section 4.3, are figures composed of the raw acquired I or Q data, and the back-calculated
phase or power, depending on graph axis. The full results have been simplified to allow for easy
viewing, thus only one or the other is shown in detail.

A block diagram of the designed system can be seen below. Where the Micro-processing unit
included will be a raspberry pi, and communicates with an low frequency RF source through an
SPI Bus. The signal will run through a bi-directional coupler and the output will be amplified and
run through another coupler so that the main input signal can be fed to the coil, while the reflected
signal and base reference signal will be fed to an I/Q demodulator to parse the amplitude and phase
of the recorded data. This will be run through a low pass filter to remove any higher order harmonic
signals and noise, before running through an ADC and back to the micro-processing unit through
an SPI bus for data acquisition and review of results. Tests run without the LPF proved to be
difficult to reliably image any types of defects without software filtering. The hardware LPF was
designed specifically for this application and has been included to alleviate computational strain

36

Figure 4.10: Block Diagram of Low Frequency system

placed on data manipulation utilizing the micro-processing unit. Increased noise can be attributed
to the inclusion of the amplifiers, though they are necessary to acquire any usable data from usage
of the system. The I/Q demodulator demodulates to the in-phase and quadrature phase sinusoidal
signals that are 90deg out of phase from each other. The acquired in phase and quadrature phase
signals are fed into an imaging code and processed into the reflected power and phase using

(4.1a)

(4.1b)

(4.1c)

(4.1d)

(cid:112)

π

R =

X2 + Y2
RdBm = 10 ∗ log10(R)
φrad = tan−1(Y
)
X
φdeg = φrad ∗ 180

where X is the in-phase component, Y is the quadrature phase component, RdBm is the reflected
power in dBm, and φdeg is the reflected phase in degrees.
4.4 System Comparison, Lock-in and Mini

The results acquired are shown below. Scans were run utilizing parameters as identical as could
be achieved, in regards to lift off distance, probe speed, and sampling rate. Samples include an
aluminum sample, with square, triangle, and circle cut outs. The smallest feature size is the 0.5

37

mm radius circle. A polyethylene sample with sub-surface triangle cut outs was also scanned to
test the probe penetration depth with both systems. A GFRP sample with surface impact damage
was scanned to test the capability of dielectric evaluation utilizing the designed probe and system.
Finally, with all aforementioned results, carbon fiber wrapped steel plate samples provided by
Consolidated Edison were scanned. The defect maps provided are included before each plate
sample result to show the system results in comparison to the apparent defects contained within the
plate samples. The objective for this result comparison is to show that the designed mini system
can perform at least as well, if not better, in relative comparison to results acquired using the table
top lock-in system.

38

(a)

(b)

Figure 4.11: Aluminum calibration sample results comparison (a) table top system, (b) mini system

39

(a)

(b)

Figure 4.13: GFRP impact damage sample results comparison (a) table top system, (b) mini system

40

(a)

(b)

Figure 4.15: PE sample subsurface defect results comparison (a) table top system, (b) mini system

41

Figure 4.17: Plate 1 Defect Map

Parameter

dimension

A
B
C
D

300mm
300mm
240mm
240mm

Table 4.3: Plate 1 size and scan dimension table

42

(a)

(b)

Figure 4.18: Plate 1 results comparison (a) table top system, (b) mini system

43

Figure 4.20: Plate 2 Defect Map

Parameter Dimension Defect Detection

A
B
C
D
E
F
G

300mm
300mm
240mm
240mm
25mm dia Mold
50mm dia Mold
25mm dia Mold

X
X
X

Table 4.4: Plate 2 size, scan, and defect dimension table

44

(a)

(b)

Figure 4.21: Plate 2 results comparison (a) table top system, (b) mini system

45

Figure 4.23: Plate 4 Defect Map

Parameter Dimension Defect Detection

A
B
C
D
E
F
G

300mm
300mm
240mm
240mm
25mm dia Air gap
25mm dia Air gap
25mm dia Air gap

(cid:88)

(cid:88)

(cid:88)

Table 4.5: Plate 4 size, scan, and defect dimension table

46

(a)

(b)

Figure 4.24: Plate 4 results comparison (a) table top system, (b) mini system

47

Figure 4.26: Plate 5 Defect Map

Parameter Dimension

Defect

Detection

A
B
C
D
E
F
G

300mm
300mm
200mm
240mm

12.5mm dia Soil contamination
Soil contamination
50mm dia
25mm dia
Soil contamination

(cid:88)

(cid:88)

(cid:88)

Table 4.6: Plate 5 size, scan, and defect dimension table

48

(a)

(b)

Figure 4.27: Plate 5 results comparison (a) table top system, (b) mini system

49

Figure 4.29: Plate 6 Defect Map

Parameter

Dimension

Defect

Detection

A
B
C
D
E
F

300mm
300mm
240mm
240mm

12.5mm square Lack of Epoxy
Lack of Epoxy
50mm square

partial

(cid:88)

Table 4.7: Plate 6 size, scan, and defect dimension table

50

(a)

(b)

Figure 4.30: Plate 6 results comparison (a) table top system, (b) mini system

51

Figure 4.32: Plate 7 Defect Map

Parameter

Dimension

Defect

Detection

A
B
C
D
E
F
G
H
I

300mm
300mm
240mm
240mm
25mm dia
12.5mm dia
25mm dia

Mold
Air

Soil contamination

X
(cid:88)

(cid:88)

25mm square

Lack of epoxy

partial

50mm dia

Air

(cid:88)

Table 4.8: Plate 7 size, scan, and defect dimension table

52

(a)

(b)

Figure 4.33: Plate 7 results comparison (a) table top system, (b) mini system

53

4.5 Conclusion

The compared results above show that the mini system preforms almost exactly the same as the
table top system in all cases, except with slightly increased noise visible in the PE sample results.
This could be due to the way that the specific scan was run, and the variation in flatness of the
sample itself. The PE scan setup was somewhat unorthodox as can be seen below in figure 4.35.

Figure 4.35: PE Sample Probe setup

The probe design itself is based on prior work done in [10] wherein a resonance probe is
designed and implemented using LC tank architecture with the equivalent circuit drawing shown
below in figure 4.36 The capacitive plate probes shown in figure 4.2 were soldered to an inductor
connected across the plates using plated vias through the FR4 board, and a read coil was created
by wrapping 20 AWG copper wire around a suitably sized cylinder (ie a pencil) to fit around the
soldered inductors. The read coil was then glued to the soldered inductor when an appropriate
configuration was found for a sharp resonance peak using a VNA to scan the desired frequency
range. The wires connecting the read coil to the SMA connector were intentionally left long, due
to the dielectric nature of glue. Manufacturers will claim that the effect of the glue is low or
immaterial, but experimental results when fabricating the probes proved this to be wrong, as the

54

Figure 4.36: Probe equivalent circuit

first edition probes with shorter wires lost a lot of Q after the glue dried. The next iteration of
probe included longer wires that could be manipulated by reducing or increasing their separation
after the glue was fully dried. This allowed for mitigation of any dielectric effects of the glue. The
disadvantage was that the wires were no longer fixed, and various glues and epoxies had proven to
be too dielectric for this job. After testing with various materials to fix the wires in place, a PE pipe
was purchased and cut into segments, and glued to the FR4 board such that neither the pipe nor the
glue had a noticeable effect on the response of the sensor, to protect the wires and provide a housing
for them. Common household insulation foam was also purchased and used as a material to fill the
housing as it was capable of securing the wires in place and had dielectric properties much closer

55

to air than the glue did. This allowed for the Q of the sensors to remain after introducing dielectric
glue in between the soldered inductor and read coil. The probe setup shown in 4.35 is a result of
the glue holding the housing to the FR4 board disbonding; the wires and coil were stationary, but
the plate itself was not entirely fixed. A screw driver was placed to maintain downward pressure
onto the capacitive plate. This allowed the plate to conform to the contours of the surface of the PE
sample during the table top scan. Super glue was reapplied to the housing to ensure the probe did
not break, as the entire resonance circuit fabrication process took anywhere from twelve to twenty
four hours to finish completely. By the time the mini system was connected and scanning, the glue
had dried fixing the probe and reducing the amount of conforming the probe did to the sample
surface. As absurd as the improvised probe setup looks, the measured performance of the probes
and acquired results validate the probe design and implementation. The results comparison also
highly emphasizes the importance of signal filtration and validates the design and implementation
of the low pass filtering circuit in the form of an integrator circuit.

56

CHAPTER 5

MINIATURIZED MICROWAVE IMAGING SYSTEM

5.1 Probe and System Description

The probes used for microwave scanning include a cavity waveguide and a monopole antenna
acting as a coaxial cable probe. The mini system aimed to replace the table top RF source, allowing
for a portable microwave scanning system. A higher frequency I/Q demodulator was also utilized
to allow for back calculation of power and phase from acquired data.

5.1.1 Wave-guide

The X band rectangular waveguide used had several resonance points within the 7-12 GHz range but
was utilized in the scans shown below at 7.8 GHz. The waveguide utilized had a lower sensitivity,
but much better penetration depth for the resultant scans than the coaxial, which will be explained
further on in chapter 6. This will also become very apparent when reviewing the results comparison.

5.1.2 Coaxial probe

The probe used as a coaxial probe is actually a monopole antenna utilizing a rigid wire slotted
through a drilled hole in a rogers board that had one side of copper peeled off. An SMA connector
was used to connect the probe to the system with the rigid wire soldered to the signal pin of the
connector, and the ground pins soldered to the copper backside of the rogers board. The "coaxial"
probe was utilized at 4.3GHz, and demonstrated high sensitivity, but low penetration depth.

5.2 System Comparison, Table top and Mini

Calibration scans were performed on the same aluminum sample mentioned in chapter 4, with
a higher level of resolution achieved. The mini microwave system did not perform exactly as well
as the table top system, due to the massively increased phase noise seen at higher frequencies.

57

The amount of random frequencies seen on the output of the I/Q demodulator and every single
resonance frequency associated with the random frequencies caused scans run using the NI DAQ
to appear entirely as noise. As a result, the same low pass filters utilized in the low frequency
system were implemented to disastrous results. The importance of hardware noise filtering at
higher frequencies is a clear take away from the implementation of the microwave mini system.
Due to this filter miscalculation, two digital multi meters were utilized to record both the I and
Q components of the acquired data. The increased phase noise at higher frequencies can be seen
most clearly in figures 5.4 and 5.5. The first six calibrations samples were run using a single DMM
to verify that the system was working correctly, the "coaxial" probe had high sensitivity to small
resolution defects, and the waveguide had penetration depth. The figure form as seen in figures
5.4 and 5.5 is ultimately how the data acquired from two DMMS was graphed, with the top two
sub-figures being in-phase and quadrature phase respectively, while the bottom two graphs are the
back-calculated reflected power and phase. The microwave plate scans further down are minimized
for visibility to just the I component and the back-calculated phase, as these two were the most clear
results. The PE sample was re-run with the microwave system to verify the penetration depth of
the waveguide on subsurface defects, and the results from the monopole scan is not included due
to the results being entirely surface profiling.

58

Figure 5.1: Aluminum Calibration sample, table top system (Coax)

59

Figure 5.2: Aluminum Calibration sample, mini system scan (Coax)

60

Figure 5.3: Aluminum Calibration sample, table top system (Wave-guide)

61

Figure 5.4: Aluminum Calibration sample, first square, mini system (Coax)

62

Figure 5.5: Aluminum Calibration sample, first square, mini system (Wave-guide)

63

Figure 5.6: Polyethylene, table top system (Wave-guide)

64

Figure 5.7: CFRP Plate 1, Wave-guide, mini system

65

Figure 5.8: CFRP Plate 2, Wave-guide, mini system

66

Figure 5.9: CFRP Plate 4, Wave-guide, mini system

67

Figure 5.10: CFRP Plate 5, Wave-guide, mini system

68

Figure 5.11: CFRP Plate 6, Wave-guide, mini system

69

Figure 5.12: CFRP Plate 7, Wave-guide, mini system

70

5.3 Conclusion

The results acquired using the mini microwave system, while noisy, still match the results
acquired from the capacitive mini system with both systems only utilizing hardware noise filtration.
Even with the waveguide calibration sample results as they are in figure 5.3, the results from the
plate scans clearly match, with some defects, like those shown in plate 7 (figure 5.12) becoming
more apparent. This consistency validates the results obtained in chapter 4, and demonstrates
the need for multi system integration, as even though both systems rely inherently upon dielectric
sensitivity, the penetration depth of the microwave scans makes up for its lack of resolution. This
will be discussed more in the next chapter.

5.4 Importance of System Integration

Each system has its own strengths and weaknesses. The capacitive imaging system penetration
depth and signal filtration can be clearly seen from the table top and mini system comparisons,
while it falls short a little in resolution size for smaller defects. The microwave system contrasts
this perfectly, where the penetration depth for highly sensitive probes is nearly inconsequential, as
the PE subsurface defect scan performed with the coax cable wasn’t even worth including. The
waveguide is a sort of middle ground that does both jobs poorly, but is at least helpful for validating
the previous scans and verifying the consistency of both systems. The low frequency system’s
weakness is the high frequency system’s strength, and vise versa. Integrating both systems with
interchangeable probe types would allow efficient dielectric evaluation of composites. The systems
complement each other, and thus the integration of them would create, overall, a much better
system, while still retaining the qualities targeted by the multi modality system. Both systems when
integrated would not only be be relatively cheap, in comparison to the available methods, but carry
the capability of field deployment, all while staying compact. Due to the nature of the boards used
within both systems, the entire integrated system could be fixed into a box with output ports for
their respective probes, and input ports for power and connection to a processing unit allowing for
imaging of dielectric materials with both low frequency capacitive, and high frequency microwave

71

methods.

5.4.1 Edge detection vs penetration

The coaxial probe is the top choice for high resolution surface scans, while the waveguide loses the
edge detection and resolution, it has increased penetration depth, though even then the clarity of
some of the defects that were clearly imaged using capacitive imaging was unparalleled. The low
frequency resonance probes have not only the penetration depth, but very good edge detection for
any surface defects larger than sub-mm. This can be improved upon in the future by optimizing the
low pass filters in the capacitive system more, or by re-fabricating the capacitive resonance probes
with smaller feature sizes.

5.5 Low and High frequency system comparisons

The most notable comparison between the low and high frequency system results is the difference
between the resolution in the aluminum calibration sample, with the capacitive results shown in
figures 4.12b and 4.12a with the smallest defect size being barely visible or unable to be seen
entirely. When the smallest defect is detected, the shape is impossible to make out, the only
conclusion without knowing the defect is actually there is that there is a good chance that it is a
defect and not just noise. This is in sharp contrast to the resolution demonstrated by the coaxial
probe shown in figure 5.1 with all three defects in the final row easily visible and the distinct shape
of a triangle clearly shown. The wave guide edge detection is unsurprisingly poor, as can be seen in
figure 5.3. The problem with the sensitivity in higher frequency imaging probes is that the signal
penetration depth is much less than lower frequencies. The resolution of surface detection, however
is much greater.

72

CHAPTER 6

CONCLUSIONS AND FUTURE WORK

6.1 Importance of System Integration

Each system has its own strengths and weaknesses. The capacitive imaging system penetration
depth and signal filtration can be clearly seen from the table top and mini system comparisons,
while it falls short a little in resolution size for smaller defects. The microwave system contrasts
this perfectly, where the penetration depth for highly sensitive probes is nearly inconsequential, as
the PE subsurface defect scan performed with the coax cable wasn’t even worth including. The
waveguide is a sort of middle ground that does both jobs poorly, but is at least helpful for validating
the previous scans and verifying the consistency of both systems. The low frequency system’s
weakness is the high frequency system’s strength, and vise versa. Integrating both systems with
interchangeable probe types would allow efficient dielectric evaluation of composites. The systems
complement each other, and thus the integration of them would create, overall, a much better
system, while still retaining the qualities targeted by the multi modality system. Both systems when
integrated would not only be be relatively cheap, in comparison to the available methods, but carry
the capability of field deployment, all while staying compact. Due to the nature of the PCB and
components used within both systems, all could be stacked to reduce the space they take up, and
the entire integrated system could be fixed into a box with output ports for their respective probes,
and input ports for power and connection to a processing unit allowing for imaging of dielectric
materials with both low frequency capacitive, and high frequency microwave methods.

6.1.1 Edge detection vs penetration

The coaxial probe is the top choice for high resolution surface scans, while the waveguide loses the
edge detection and resolution, it has increased penetration depth, though even then the clarity of
some of the defects that were clearly imaged using capacitive imaging was unparalleled. The low

73

frequency resonance probes have not only the penetration depth, but very good edge detection for
any surface defects larger than sub-mm. This can be improved upon in the future by optimizing the
low pass filters in the capacitive system more, or by re-fabricating the capacitive resonance probes
with smaller feature sizes.

6.1.2 Low and High frequency system comparisons

The most notable comparison between the low and high frequency system results is the difference
between the resolution in the aluminum calibration sample, with the capacitive results shown in
figures 4.12b and 4.12a with the smallest defect size being barely visible or unable to be seen
entirely. When the smallest defect is detected, the shape is impossible to make out, the only
conclusion without knowing the defect is actually there is that there is a good chance that it is a
defect and not just noise. This is in sharp contrast to the resolution demonstrated by the coaxial
probe shown in figure 5.1 with all three defects in the final row easily visible and the distinct shape
of a triangle clearly shown. The wave guide edge detection is unsurprisingly poor, as can be seen in
figure 5.3. The problem with the sensitivity in higher frequency imaging probes is that the signal
penetration depth is much less than lower frequencies. The resolution of surface detection, however
is much greater.

6.2 Conclusions

In this paper, several methods of defect classification and characterization are discussed from
an NDE perspective. A gap in the frequency band was identified and capitalized on, allowing for
efficient dielectric evaluation of composite materials utilizing high Q LC tank resonance probes
created for use with the system. The governing theory for both low frequency capacitive scanning
and high frequency microwave scanning has been thoroughly detailed to explain the design and
implementation process of the creation of the system. The miniaturized system has been shown to
produce equivalent results to bulky, expensive, table top equipment, while maintaining a low cost,
compact size, and the capability for field deployment.

74

6.3 Future work

Improvements can be made to the system by utilizing the specific integrated circuits from the
evaluation boards for the low and high frequency sources, as well as their associated I/Q demodulator
source boards. This would further miniaturize the system and ultimately lower the overall cost. The
low pass filter integrators can be improved so that the cut-off frequency is even lower, filtering as
much higher frequency phase noise and harmonics as possible, allowing microwave data acquisition
with the NI DAQ or with the ADC, which would decrease the microwave scan time substantially
and allow for both amplitude and phase to be acquired simultaneously. Modifications can be made
to the system housing allowing for a more robust field deployment in the future.

75

APPENDICES

76

APPENDIX A

DIELECTRIC LENS

The high gain antenna requirement for imaging and communication application has historically
evolved the antenna design. Today, the commonly known high gain antennas are vivaldi, yagi-uda,
horn, and wave-guides, all of which have an excellent gain performance but with the trade-off of
having a high profile [39,40]. The only design with a relatively higher gain and a low profile is the
patch antenna. In this paper, the authors have explored a way to increase the gain of patch antenna
using a dielectric RF lens, while keeping a lower profile than any available high gain antenna.

Lenses are classical tools that have been used for centuries in the field of optics. The lenses
are well known for their focusing, collimating, and diffraction properties. The same principle
have been applied for lower frequency waves, in order to decrease the beamwidth and increased
gain. A number of designs operating at range of frequencies with different shapes and applications
have been previously published in the literature. The primary focus of most of the papers are
characteristics for high gain, low profile, and beam steering capabilities, allowing for improvement
in imaging and communication systems [41–44].

While dielectric lenses are a great tool, in the past they were often difficult to implement due
to the response being dictated by how precisely the geometry could be manufactured to fit the
design specifications. Besides that, before quite recently, high fidelity simulation softwares for
high frequency applications did not exist, making the design of dielectric lenses very difficult, due
to the inability to account for changes in the system due to manufacturing errors.

Because of recent breakthroughs in simulation software, and more importantly, manufacturing
techniques like 3D printing, the design, fabrication, and implementation of RF components has
become much more economical, and easier, allowing for planar antenna excitation in the C-band.
This paper shows a design of dielectric lens and patch antenna in ANSYS High Frequency
Structure Simulator (HFSS), fabrication of the lens using additive manufacturing technique, and
measurement of the patch antenna gain with and without the dielectric lens.

77

Dielectric lenses affect electromagnetic waves similarly to how optical lenses affect light, and
are known for their uses as an electromagnetic beam collimater, as they allow for focusing of the
waves propagated from a source, in this case an antenna. [41] This effect is illustrated in Fig.A.1,

Figure A.1: Focusing of RF waves radiated from patch antenna using dielectric lens

[42] shows that a lens can be a hemispherical due to the wave symmetry.
The VeroWhite material with a dielectric permittivity of 2.8 is used to design the lens in ANSYS
HFSS. In order to mount the lens over a patch antenna operating in C band (5.3 GHz) with the
ground plane of size 50 mm x 50 mm, the lens’s diameter is selected to be 60 mm. The designed
lens and patch antenna with dimensions is shown in Fig.A.2.

Figure A.2: Designed and fabricated lens with the dimensions

78

The Stratasys’s CONNEX3 OBJET350 3D printer is used to fabricate the lens and CNC milling
machine is used to fabricate the patch antenna on Rogers’s 4350 substrate (thickness = 1.52 mm,
dielectric = 3.66, loss tangent = 0.003).

The designed patch antenna at 5.3 GHz is fabricated, and its gain is measured using Satimo’s
antenna measurement system. The simulated and measured gain are shown in Fig.A.3. The
measured gain pattern matches the simulated gain pattern, and the highest measured gain in normal
direction is 5.9 dBi. The measured 3 dB beamwidth for the patch is 76o.

Figure A.3: Simulated and Measured gain of the patch antenna at 5.3 GHz without lens

Next, a lens is mounted on top of the patch antenna. A 15 mm separation is given in between
lens and antenna to eliminate the direct dielectric loading issues. The patch gain is measured again
with the dielectric lens and the measured results are shown in Fig.A.4.

The shown simulated and measured results match closely. The highest measured gain in the
normal direction is 9.4 dBi, which has enhanced the normal patch antenna gain by 3.5 dBi. The 3
dB beamwidth for the patch is reduced from 76o to 43o.

Additionally, the dielectric lens is simulated from 4 GHz to 10 GHz in ANSYS HFSS, where
it performs similarly at all frequencies, ideal for C-band operations. This dielectric lens could be
used in tandem with the microwave probe designed for this project allowing for greater detection
capabilities.

79

Figure A.4: Simulated and Measured gain of the patch antenna at 5.3 GHz with lens

80

APPENDIX B

MICROWAVE NETWORK ANALYSIS

When designing Microwave probes, the Helmholtz equation, Sommerfield radiation condition, and
vector Green’s theorem must be considered thoroughly to understand the radiation conditions of
the designed probe. This section will serve as background for the derivation of the conditions that
need to be solved when simulating and fabricating microwave probes. The vector Green’s theorem
is classified as

ˆn · ( (cid:174)A × ∇ × (cid:174)B − (cid:174)B × ∇ × (cid:174)A)dS =

( (cid:174)B · ∇ × ∇ × (cid:174)A − (cid:174)A · ∇ × ∇ × (cid:174)B)dV

(B.1)
4πR and as a notation from the notes let (cid:174)CG = (cid:174)G1 Green’s

where (cid:174)A = (cid:174)H and (cid:174)B = (cid:174)CG = (cid:174)C e−jkR
function must satisfy the Helmholtz equation,

(cid:73)

S

(cid:90)

V

using Maxwell’s equations

and

∇2G + k2G = 0

∇ × (cid:174)E = − (cid:174)Jm − jωµ (cid:174)H

∇ × (cid:174)H = (cid:174)J + jω
 (cid:174)E

To solve for terms used on the right hand side of the vector Green’s theorem resulting in

∇ × ∇ × (cid:174)H = ∇ × (cid:174)J + jω
∇ × (cid:174)E = ∇ × (cid:174)J − jω
 (cid:174)Jm + k2 (cid:174)H

Solving for the other terms on the RHS of the vector Green’s theorem,

∇ × ∇ × (cid:174)G1 = ∇(∇ · (cid:174)G1) + k2 (cid:174)G1

Plugging these into the volume integral in the vector Green’s theorem,

(cid:90)
(cid:90)

RHS =

=

[G (cid:174)C · (∇ × (cid:174)J − jω
 (cid:174)Jm + k2 (cid:174)H) − (cid:174)H · (∇(∇ · (cid:174)G1) + k2 (cid:174)G1)]dV
[G (cid:174)C · (∇ × (cid:174)J − jω
 (cid:174)Jm) − (cid:174)H · ∇(∇ · (cid:174)G1)]dV

V

V

81

(B.2)

(B.3)

(B.4)

(B.5)

(B.6)

(B.7)

where the terms

where

and the term

So

∇ · (cid:174)H = ρm

µ

∇ · (cid:174)G1 = ∇ · ( (cid:174)CG) = G∇ (cid:174)C + (cid:174)C · ∇G

(cid:174)C(∇2G + k2G = ∇2 (cid:174)CG + k2 (cid:174)CG = 0)

where (cid:174)C is a constant vector meaning so the equation becomes
∇ · (cid:174)G1 = ∇ · ( (cid:174)CG) = (cid:174)C · ∇G

∇ · ( (cid:174)H∇ (cid:174)G1) = (∇ · (cid:174)G1)(∇ · (cid:174)H) + (cid:174)H · ∇(∇ · (cid:174)G1)

(B.8)

(B.9)

(B.10)

(B.11)

plugging back into the earlier equation,

RHS =

(cid:90)
using the substitution of(cid:82)
(cid:90)

V

RHS =

[G (cid:174)C · (∇ × (cid:174)J − jω
 (cid:174)Jm) + (cid:174)C · ∇G ρm
mu

− ∇ · (cid:174)H (cid:174)C · ∇G]dV

(B.12)

separating the positive and negative terms in the integral into their own respective integrals, and

V ∇ · xdV =(cid:72) x · ˆndS the RHS of the vector Green’s theorem becomes

[G (cid:174)C · (∇ × (cid:174)J − jω
 (cid:174)Jm) + (cid:174)C · ∇G ρm
mu

]dV −

V

[( (cid:174)H (cid:174)C · ∇G) · ˆn]dS

(B.13)

(cid:73)

and then moving the surface integral to the LHS of the Green’s theorem to combine like terms, the
LHS becomes now (after plugging in the variables defined earlier)

(cid:73)

ˆn · ( (cid:174)H × ∇ × (cid:174)G1 − (cid:174)G1 × ∇ × (cid:174)H + (cid:174)H (cid:174)C · ∇G)dS

S

(B.14)

∇ × (cid:174)G1 = ∇ × (cid:174)CG = G∇ × (cid:174)C − (cid:174)C × ∇G

G∇ × (cid:174)C = 0

82

using B.5 and

where

the LHS can be simplified to

(cid:73)

ˆn · [ (cid:174)H (cid:174)C · ∇G − ( (cid:174)H × (cid:174)C × ∇G) − (cid:174)CG × ( (cid:174)J + jω
 (cid:174)E)]dS

(B.15)

s

to simplify the LHS further by distributing the ˆn the following relations must be used

ˆn · [ (cid:174)H × (cid:174)C∇G] = [ˆn × (cid:174)H] · (cid:174)C × ∇G = (cid:174)C · [∇G × (ˆn × (cid:174)H)]

and

ˆn · [ (cid:174)C × (cid:174)J + jω
 (cid:174)E] = (cid:174)C · [ˆn × ( (cid:174)J + jω
 (cid:174)E)]

Pulling out (cid:174)C so that the entire vector Green’s theorem can be rewritten as

(cid:174)C ·

(ˆn× (cid:174)H)×∇G + ˆn× (cid:174)JG + jω
 ˆn× (cid:174)EG +∇G ˆn· hv]dS = (cid:174)C
[G(∇ × (cid:174)J − jω
 (cid:174)Jm) + ∇G ρm

]dV

(B.16)

Thus, the constant vector (cid:174)C can be divided from both sides removing it from the equation entirely.

[∇G × (ˆn × (cid:174)H) + jω
 ˆn × (cid:174)HG + ∇G(ˆn · (cid:174)H)]dS =

S

[ (cid:174)J × ∇G − jω
 (cid:174)JmG + ρm

µ

V

∇G]dV

(B.17)

µ

(cid:90)

A surface can be characterized by S = S0tS
 + S∞ on S
 where

G ∼=

1
4πR

(cid:73)

and

(cid:73)

·

(cid:90)

s

V

Taking the limit of the surface integral in the function as 
 → 0 and plugging in the above functions,
where ˆn = ˆR in this case when applied using spherical coordinates,

(cid:73)

ˆR
4π
2 + jω
 ˆR × (cid:174)E
[−( ˆR × (cid:174)H) ×
ˆR
[( ˆR · (cid:174)H)
− (cid:174)H( ˆR · ˆR)
4π

1
4π

1
4π

− ( ˆR · (cid:174)H)

ˆR
r π

− ( ˆR · (cid:174)H)

ˆR
4π
2 ]
2 sin θdθdφ

] sin θdθdφ

(B.18)

(cid:73)

lim

 →0

S

= lim

→0
S

= − (cid:174)H((cid:174)r(cid:48))

∇G ∼= − ˆR
4πR2

83

taking the limit on the surface as it goes towards ∞, the relations for G now become

and

G ∼= e−jkr
4πr

∇G ∼= −jk ˆr

e−jkr
4πr

which can be applied as

(cid:73)

S∞

(cid:73)
(cid:73)

S∞

lim
r →∞

= lim
r→∞
= lim
r→∞
= 0

[(ˆn × (cid:174)H) × ∇G + jω
 ˆn × (cid:174)EG + (ˆn · hv)∇G]dS
e−jkr
4πr

[−jk(−ˆr × (cid:174)H) × ˆr
[−jk(ˆr · (cid:174)H) + jk(ˆr · ˆr) (cid:174)H − jω
 ˆr × (cid:174)E + jk ˆr · (cid:174)E]re−jkr sin θdθdφ

− jω
 ˆr × (cid:174)E

+ jk ˆr · (cid:174)H

e−jkr
4πr

e−jkr
r

S∞

]r2 sin θdθdφ

(B.19)

if r(jk (cid:174)H − jω
 ˆr × (cid:174)E) → 0as r → ∞ which when restated becomes the Sommerfeld radiation while
stating that r (cid:174)H must be finite, and that (cid:174)H = ˆr× (cid:174)E
∇G − jω
 (cid:174)JmG]dV +
(cid:174)H((cid:174)r(cid:48)) =

so, the answer can be written as

[ (cid:174)J × ∇G + ρm

(cid:73)

(cid:90)

η

[(ˆn × (cid:174)H) × ∇G + (ˆn · (cid:174)H)∇G + jω
(ˆn × (cid:174)E)G]dS
(B.20)

S0

µ

V

B.1 Propagation

The propagation constant of a voltage wave on a transmission line can be described as:

Where α and β can be defined in the following equations below:

γ =(cid:112)(R + jωL)(G + jωC) = α + j β
LC(cid:112)1 − rg
LC(cid:112)1 − rg

(cid:114)
(cid:114)

(cid:115)
(cid:115)

[−1 +

[+1 +

1
2

√

√

1 + ( r + g
1 − rg

1 + ( r + g
1 − rg

1
2

α = ω

β = ω

(B.21)

(B.22)

(B.23)

)2]

)2]

84

Where:

r = R
ωL

g = G
ωC

First, rearranging equations B.1 and B.1 to solve for R and G:

R = rωL

G = gωC

Replacing R and G within the original Eqn. B.21 to simplify the problem:

γ =(cid:112)(rωL + jωL)(gωC + jωC)
LC(cid:112)(r + j)(g + j)
LC(cid:112)(rg − 1) + j(r + g)

γ = ω

γ = ω

√

√

After simplifying the propagation constant formula, it is possible to obtain two equations for the
two unknowns, α and β, by squaring both sides of the propagation constant formula then comparing
the real and imaginary components.

√

LC(cid:112)(rg − 1) + j(r + g) = α + j β
LC(cid:112)(rg − 1) + j(r + g)]2 = [α + j β]2

γ = ω

√

γ2 = [ω

γ2 = ω2LC[(rg − 1) + j(r + g)] = (α2 − β2) + j(2αβ)

85

Therefore, the real and imaginary components can be compared as:

RE[γ2] = ω2LC(rg − 1) = α2 − β2

I M[γ2] = ω2LC(r + g) = 2αβ

From Eqn. B.25, α can be solved with respect to β:

2αβ = ω2LC(r + g)

α = ω2LC(r + g)

1
2β

(B.24)

(B.25)

(B.26)

This α equation can be placed into the imaginary component equations from Eqn. B.24:

α2 − β2 = ω2LC(rg − 1)

(ω2LC(r + g)

1
2β

)2 − β2 = ω2LC(rg − 1)

(ω2LC)2(r + g)2 1

4β2 − β2 = ω2LC(rg − 1)

β2 can be distributed across the equations:

Rearranging:

(ω2LC)2(r + g)2 1
4

− (β2)2 = β2ω2LC(rg − 1)

(β2)2 + β2ω2LC(rg − 1) − (ω2LC)2(r + g)2 1
4

= 0

The quadratic equation can be used:

−b ±

x =

√

b2 − 4ac
2a

86

Where x = β2, a = 1, b = ω2LC(rg − 1), and

c = (ω2LC)2(r + g)2 1
4

β2 =

Simplifying:

−ω2LC(rg − 1) ±(cid:113)
[ω2LC(rg − 1) ±(cid:113)

β2 = −1
2

[ω2LC(rg − 1)]2 − 4(ω2LC)2(r + g)2 1
4

2

(B.27)

(cid:113)

(cid:115)
(cid:114)
(cid:114)
(cid:114)

87

(ω2LC)2(rg − 1)2 + (ω2LC)2(r + g)2]

β2 = −1
2

[ω2LC(rg − 1) ± (ω2LC)

(rg − 1)2 + (r + g)2]

[(rg − 1) ±(cid:113)

β2 = −ω2LC

1
2

(rg − 1)2 + (r + g)2]

It is also helpful to pull out a (1 − rg). Consider that in this case rg − 1 = −(1 − rg) and

(rg − 1)2 = (1 − rg)2.

β2 = −ω2LC

1
2

[−(1 − rg) ±(cid:113)

(1 − rg)2 + (r + g)2]

β2 = −ω2LC

1
2

[−(1 − rg) ± (1 − rg)

(1 − rg)2 + (r + g)2

(1 − rg)2

]

(B.28)

From this point, β can be written as:

(cid:115)

β2 = ω2LC(1 − rg)

[1 ∓

1
2

1 + ( r + g
1 − rg

)2]

β =

ω2LC(1 − rg)

LC(cid:112)1 − rg

√

β = ω

(cid:115)

[1 ∓

1
2

1 + ( r + g
1 − rg

)2]

[+1 ∓

1
2

1 + ( r + g
1 − rg

)2]

This equation is similar to Eqn. B.23, but has a sign issue that needs to be resolved. The choice
for this sign is dependent on the answer selected for α. α can be resolved in the same way β was
resolved, starting with the imaginary equivalence equation from Eqn. B.25 to solve β with respect
to α:

2αβ = ω2LC(r + g)

β = ω2LC(r + g)

1
2α

From the real equivalence equation from Eqn. B.24:

α2 − β2 = ω2LC(rg − 1)

α2 − [ω2LC(r + g)

1
2α

]2 = ω2LC(rg − 1)

(α2)2 − [ω2LC(r + g)]2 1

4α2 = ω2LC(rg − 1)

(α2)2 − ω2LC(rg − 1) − [ω2LC(r + g)]2 1

4α2 = 0

The quadratic equation can be used, where Where x = α2, a = 1, b = −ω2LC(rg − 1), and

c = (ω2LC)2(r + g)2 1
4

ω2LC(rg − 1) ±(cid:113)

α2 =

[ω2LC(rg − 1)]2 − 4[(ω2LC)2(r + g)2 1
4]

2

From this point, it is recognizable that there is only one difference between the quadratic
equations when solving for α and β, from the above equation and Eqn. B.27. The only different is
that the b component in the α case is negative while it is possible in the β case. Skipping several

88

similar simplification steps explained while solving for β, similar to Eqn. B.28:

α2 = −ω2LC

1
2

[(1 − rg) ± (1 − rg)

(1 + ( r + g
1 − rg

)2]

α2 = −ω2LC(1 − rg)

Negating inside the parenthesis:

α2 = ω2LC(1 − rg)

Solving for α:

LC(cid:112)1 − rg

√

α = ω

(1 + ( r + g
1 − rg

(1 + ( r + g
1 − rg

)2]

)2]

[1 ±

1
2

[−1 ∓

1
2

(cid:115)

[−1 ∓

1
2

(1 + ( r + g
1 − rg

)2]

(cid:114)

(cid:114)
(cid:114)
(cid:114)

(cid:114)
(cid:114)

Again, this is very close to the form provided in Eqn. B.22, but the minus plus symbol must be
resolved. The final step will be to resolve the symbol by assuming that the plus is selected instead
of the minus, and proving it by chugging the equations into Eqn. B.25. Therefore we assume the
following:

(cid:115)
LC(cid:112)1 − rg
(cid:115)
LC(cid:112)1 − rg

√

α = ω

√

β = ω

[−1 +

1
2

(1 + ( r + g
1 − rg

)2]

1
2

[+1 +

1 + ( r + g
1 − rg

)2]

Place these into the imaginary equivalence equation:

ω2LC(r + g) = 2αβ

(cid:113)

Where A = 1

2[−1 +
square roots of A and B:

√
ω2LC(r + g) = 2ω2LC(1 − rg)
1−rg)2] and B = 1

√
A
1 + ( r+g

2[+1 +

(cid:113)

B

(1 + ( r+g

1−rg)2]. Solving the product of the

√

√
A

B =

√

AB

89

(cid:115)

[−1 +

1
4

√

√
A

B =

(cid:114)

(cid:114)

)2][+1 +

1 + ( r + g
1 − rg

)2]

Consider (X − 1)(X + 1) = X2 − 1, where X =

1 + ( r+g

1−rg)2. In other choices for plus or minus

configurations, this will be different and, as proven soon, will be incorrect. Therefore:

(1 + ( r + g
1 − rg

(cid:113)

(cid:115)
(cid:114)1
(cid:114)1

4

(cid:114)
(cid:114)

[

√

√
A

B =

1 + ( r + g
1 − rg

)2]2 − 1

√

√
A

B =

1 + ( r + g
1 − rg

)2 − 1

4

(cid:114)

√

√
A

B =

1
2

( r + g
1 − rg

)2

√

√
A

B =

1
2

( r + g
1 − rg

)

Placing this equation back and solving both sides:

√
ω2LC(r + g) = 2ω2LC(1 − rg)

√
A

B

ω2LC(r + g) = 2ω2LC(1 − rg)

1
2

( r + g
1 − rg

)

ω2LC(r + g) = ω2LC(r + g)

Therefore, the configurations of both ± symbols in order to obtain the final solutions which are

equivalent to equations B.22 and B.23 respectively:

(cid:115)
LC(cid:112)1 − rg
(cid:115)
LC(cid:112)1 − rg

√

α = ω

√

β = ω

(cid:114)
(cid:114)

[−1 +

1
2

(1 + ( r + g
1 − rg

)2]

1 + ( r + g
1 − rg

)2]

1
2

[+1 +

90

B.1.1 Transmission Lines

Transmission lines are vital for simulating and analyzing the response of a microwave system due
to the very important characteristics of attenuation and phase constant. The attenuation constant
predicts how quickly the wave amplitude will dissipate within the system, while the phase constant
predicts the change in phase per length along the path of the wave. The amplitude and phase are
key in detection of location and composition of defects. For low loss lines, it can be assumed that
r << 1 and g << 1. That is, the following approximations must be true:

+ G(cid:112)L/C]

α ≈ 1
2

[ R√
L/C

(B.29)

(B.30)

)2]

√

LC

β ≈ ω

(cid:115)
LC(cid:112)1 − rg
(cid:114)1

√

LC

α ≈ ω

(cid:114)

[−1 +

1
2

(cid:113)

(1 + ( r + g
1 − rg

[−1 +

2

1 + (r + g)2]

B.1.2 Solving for α

√

α = ω

Consider that if r << 1 and g << 1, then rg << 1. If rg << 1, then 1 − rg ≈ 1. Rewriting:

Consider when (r + g)2 = r2 + g2 + rg. Because r << 1 and g << 1, the effects of r2, g2,
and rg are amplified to be all very less than one. Therefore, (r + g)2 << 1. Because of this, the
approximation

2 ≈ 1 can be used, where X = (r + g)2.

(cid:113)

1 + X

[−1 + 1 +

1
2

(r + g)2

2

]

(B.31)

(cid:115)

√

LC

α ≈ ω

(cid:114)1

4

√

LC

α ≈ ω

(r + g)2

√

α ≈ 1
2 ω

LC(r + g)

91

Substituting r and g from equations B.1 and B.1:

√

α ≈ 1
2 ω

LC( R
ωL

+ G
ωC

)

√

α ≈ 1
2

LC( RC
LC

+ GL
LC

)

α ≈ 1
2

√

(

α ≈ 1
2

)

( RC√
LC

+ GL√
LC
√
C)2 + G(
R
√
LC/(
+ G(cid:112)L/C]

√
L)2/
√
C

α ≈ 1
2

[ R√
L/C

L

)

This equation is the same at Eqn. B.29 and thus concludes the proof of the approximation.

B.1.3 Solving for β

√

β = ω

In a similar process to α, β can be simplified to Eqn. B.31:

(cid:114)

1 + ( r + g
1 − rg

)2]

1
2

[+1 +

(cid:115)
LC(cid:112)1 − rg
(cid:115)

√

β ≈ ω

LC

[+1 + 1 +

1
2

(r + g)2

2

]

(cid:115)
(cid:115)

√

√

LC

LC

β ≈ ω

β ≈ ω

1 +

(r + g)2

4

4 + (r + g)2

4

As solved before, (r + g)2 << 1. This means (r + g)2 << 4, which is scaled four times more

than the previous assumption. Therefore, 4 + (r + g)2 ≈ 4 is a reasonable assumption.

92

(cid:114)4

4

√

LC

β ≈ ω

√

LC

β ≈ ω

This is equivalent towards Eqn. B.30.

B.2 Split Ring Resonator

The previous section’s equations were utilized by ANSYS HFSS to simulate the response for a
split ring resonator around 8 GHz, though the fabrication would have been difficult due to the
small feature size of the split ring resonator. The simulated resonator, dimensions, and results are
included below.

Figure B.1: Simulated split ring resonator with dimension parameters

93

Parameter

A
B
C
D
E
F
G

Minimum ring separation
Maximum ring separation

Dimension
3.3 mm
9.5 mm
9.15 mm
0.29 mm
0.22 mm
1.15 mm
0.57 mm
0.07 mm
0.4 mm

Table B.1: Split ring resonator dimension table

Figure B.2: Simulated split ring resonator response

94

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