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A wideband two-layer microwave measurement method for the electrical characterization of thin materials

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A Wideband Two-Layer Microwave Measurement Method for the
Electrical Characterization o f Thin Materials
by
Trevor Cameron Williams
B.Eng., University of Victoria, 2000
A Thesis Submitted in Partial Fulfillment o f the
Requirements for the Degree of
MASTER OF APPLIED SCIENCE
in the Department o f
Electrical and Computer Engineering
We accept this thesis as conforming
to the required standard
Dr. M A. StjJ/chly, Supervisor (Dept, of Electrical and Computer Engineering)
/—a Ll ^
Dr. J. Bomemanp, Department Member (Dept, o f Electrical and Computer Engineering)
Dr. J. Provan, Outside Member (Dept. o< f Mechanical Engineering)
Dr. G. Beer, External Examiner (Dept, of Physics)
© Trevor Cameron Williams, 2002
UNIVERSITY OF VICTORIA
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by any means, without the permission o f the author
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Supervisor: M.A. Stuchly
Abstract
A novel measurement method was developed to characterize the electrical properties of
specific materials within strict requirements. The main measurement parameters needed
include the ability to measure thin, lossy materials over a broad frequency range in a
relatively expedient manner. The method developed uses a two-layer structure, consisting
of a layer o f thin, flexible unknown material supported by a thicker, rigid known material
that enables accurate placement inside a waveguide for measurement. A non-linear leastsquares optimization algorithm is used to converge on the complex permittivity and
permeability o f the material. An uncertainty analysis is performed to investigate optimal
thicknesses o f both the sample layer and the supporting layer. Simulations in FDTD were
constructed to explore the effects o f sample layers that are not o f exact dimensions.
Results from many different materials show the repeatability, and accuracy of the data set
convergences, as well as several limitations o f the procedure.
Examiners
z
/lu.c. Cwk
-J
Dr. M.A./Stuchly, Supervisor (Dept, of Electrical and Computer Engineering)
Dr. J. Bomemann, Department Member (Dept, o f Electrical and Computer Engineering)
Dr. J. Provah, Outside Member
Membt (Dept, o f Mechanical Engineering)
Dr. G. Beer, External Examiner (Dept, of Physics)
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Table of Contents
1
Introduction.............................................................................................................1
Background................................................................................................................... 1
Research Objectives..................................................................................................... 2
Thesis Contributions....................................................................................................2
Thesis Outline...............................................................................................................3
2
Definitions and Background Literature Review................................................. 5
2.1 Definitions.....................................................................................................................5
2.2 Measurement Methods o f Electrical Properties.........................................................5
2.2.1 Time-Domain and Frequency-Domain Measurements...................................... 5
2.2.2 Resonant and Wideband Measurements............................................................. 6
2.2.3 Measurement Method Selected............................................................................ 7
2.3 Guided Wave Measurement Techniques.................................................................... 8
2.3.1 Classical Nicholson-Ross-Weir (NRW) Method [4][6].................................... 8
2.3.2 Multi Sample / Multi Sample-Position Methods...............................................16
2.3.3 Short-Circuited Waveguide Method...................................................................17
2.3.4 Non-Linear Optimization.................................................................................... 18
2.4 Conclusions................................................................................................................. 20
3
Method o f Analysis and Measurements............................................................. 22
3.1 Two-layer Measurements Method........................................................................... 22
3.1.1 Governing Equations in a Waveguide............................................................... 22
3.1.2 Advantages and Disadvantages......................................................................... 25
3.1.3 Sources o f Uncertainty........................................................................................26
3.2 Iterative Solution........................................................................................................ 27
3.2.1 Single Value Optimization in Narrow Frequency Bands.................................27
3.2.2 Polynomial Whole-Band Optimization.............................................................29
3.2.3 Causality of Solution..........................................................................................30
3.3 Main Sources o f Uncertainty..................................................................................... 31
3.3.1 Sample Thickness............................................................................................... 31
3.3.2 Uncertainty in Vector Network Analyzer D ata................................................ 32
3.3.3 Small Gaps in Unknown Sample.......................................................................36
3.4 Modeling of Sample Imperfections.......................................................................... 37
3.4.1 VNA Simulation..................................................................................................43
3.4.2 FDTD Evaluation o f Gaps and Facial Inconsistencies.....................................47
4
Results o f Uncertainty Analysis and Measurements........................................48
4.1 Uncertainty Analysis................................................................................................. 48
4.1.1 Material 1, Dielectric Constant = 80, Loss Factor = 10.................................. 51
4.1.2 Material 2, Dielectric Constant = 20, Loss Factor = 40.................................. 58
4.1.3 Material 3, Dielectric Constant = 100, Loss Factor = 200.............................. 61
4.1.4 Material A, Dielectric Constant = 3, Loss Factor = 0.0001.............................64
4.1.5 Summary Uncertainty Plots............................................................................... 67
4.2 Measurement R esults................................................................................................74
4.2.1 Sample Thickness Optimization........................................................................ 78
1.1
1.2
1.3
1.4
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5
Modeling Results and their Evaluation..............................................................84
5.1 Comparison o f FDTD Results with Measurements...............................................84
5.2 Waveguide gap effect............................................................................................... 88
5.2.1 Full Gaps.............................................................................................................. 88
5.2.2 Dents in Sample...................................................................................................94
5.3 Summary o f FDTD Facial Inconsistency Simulations......................................... 100
6
Conclusions........................................................................................................ 101
6 .1 Measurement Method............................................................................................. 101
6.2 Non-Linear Optimization....................................................................................... 102
6.3 Equipment Uncertainty Analysis...........................................................................102
6.4 FDTD Modeling of Sample Facial Inconsistencies..............................................103
6.5 Results..................................................................................................................... 103
6 .6 Future W ork............................................................................................................104
Bibliography..........................................................................................................................B1
7
Appendix............................................................................................................. A1
7.1 Definitions................................................................................................................ A1
7.2 Summary Uncertainty Plots (Acrylic thickness of 3mm)..................................... A4
7.3 VNA HP8720 Uncertainty Specification Sheet.....................................................A7
V ita........................................................................................................................................ VI
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V
List o f Figures
Figure 2.1 Instability problems of the classic NRW explicit solutions [8]
9
Figure 2.2 Sample in rectangular waveguide with sample holder. Represented as three two-port devices
10
Figure 2.3 Sample in rectangular waveguide with different length sampleholder
11
Figure 2.4 Sample in rectangular waveguide with no sample holder
12
Figure 2.S Simulated Sn parameter for a lossless non-magnetic
14
Figure 2.6 Simulated S21 parameter for a lossless non-magnetic
15
Figure 2.7 A.) er and \i, derived from S-Parameters of simulated sample.
16
Figure 2.8 Reproduced from [NIST] to portray smooth curve fit o f least squares solution in contrast to explicitly
solved single-frequency point solutions.
19
Figure 2.9 Known substance with unknown coating on its front face that make
21
Figure 3.1 Cross section of waveguide with sample and acrylic inserted.
22
Figure 3.2 The very narrow frequency sections o f constant complex permittivity approximate the ideal solution
well while each section remains unrelated.
29
Figure 3.3 A two-layer structure with the sample not perfectly matching the backing acrylic (the waveguide
height).
36
Figure 3.4 A two-layer structure with a small dent in the sample.
37
Figure 3.S Graded mesh computational domain constructed for simulations of waveguide.
39
Figure 3.6 Five sofr sources used to excite the fundamental mode, TE|0. in the waveguide. Amplitudes have a
sinusoidal amplitude from 0 to 1.
40
Figure 3.7 A cross section of the waveguide at z = 30mm throughout the entire simulation. The fundamental
mode is clearly visible, as well as the Gaussian Excitation.
40
Figure 3.8 Plot of three fundamental fields (Ey, Hz, Hx) in relation to non-fundamental fields (Ex, Ez, Hy)
collected at z = 15mm, which is 6mm away from excitation and corresponds to the Su collection
point.
41
Figure 3.9 Plot of all three fundamental fields (Ey, Hz, Hx) in relation to non-fundamental fields (Ex, Ez, Hy)
collected at z = 480mm, which is 6mm away from excitation and corresponds to the S2 1 collection
point.
42
Figure 3.10 Plot A shows the Cosine waveform that is multiplied to Plot B, which is the pure Gaussian pulse.
Plot C, shows complete time domain Cosine Gaussian Pulse generated over 4000 time steps, Plot D
displays the narrow band frequency pulse created with 2 18 points (mostly zero-padded) which places
8.191 - 12.4 GHz between discrete points 480 and 727 witha 17.1 MHz frequency step (df).
43
Figure 3.11 Facial inconsistencies modeled; Gaps are on the left, dents on the right.
47
Figure 4.1 Uncertainty o f the dielectric constant (£’r) due to errors in measurements of S u and S2 1
51
Figure 4.2 Uncertainty of the loss factor (£’’r) due to errors in measurements o f S 11 and S2l
52
Figure 4.3 Uncertainty of the dielectric constant (£’r) due to errors in measurements of Sn and S21 - Acrylic =
1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
52
Figure 4.4 Uncertainty of the loss factor (e” ) due to errors in measurements o f Si 1 and S2 1 . Acrylic = I mm (top
left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
53
Figure 4.5 Uncertainty o f the dielectric constant (£’r) due to errors in measurements o f S2t
53
Figure 4.6 Uncertainty o f the loss factor (£” r) due to errors in measurements o f S2i
54
Figure 4.7 Uncertainty o f the dielectric constant (£’r) due to errors in measurements o f S2 1 - Acrylic = 1mm (top
left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
54
Figure4.8 Uncertainty o f the loss factor (£” r) due to errors in measurements o f S: i- Acrylic= lmm (top left),
2mm (top right), 4mm (bottom left), 5mm (bottom right).
55
Figure 4.9 S-parameter plot of two-layer structure at the single frequency o f 10GHz. Sample thickness varies
from 0.01 mm to 3mm
57
Figure 4.10 The magnitude o f both S-parameters of the two-layer structure at 10GHz, with sample thickness
varying from 0.0lmm to 3mm.
57
Figure 4.11 Uncertainty o f the dielectric constant (E’r) due to errors in measurements of S21
59
Figure 4.12 Uncertainty o f the loss factor (£” r) due to errors in measurements o f S2 1
59
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Figure 4.13 Uncertainty o f the dielectric constant (£’r) due to errors in measurements of S2| . Acrylic = 1mm (top
left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
60
Figure 4.14 Uncertainty of the loss factor (E”r) due to errors in measurements o f S2|. Acrylic = lmm (top left),
2mm (top right), 4mm (bottom left), 5mm (bottom right).
60
Figure 4.15 Uncertainty of the dielectric constant (£’r) due to errors in measurements of S2,
62
Figure 4.16 Uncertainty o f the loss factor (£”r) due to errors in measurements o f S2|
62
Figure 4.17 Uncertainty of the dielectric constant (£’r) due to errors in measurements of S2|.Acrylic = lmm (top
left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
63
Figure 4.18 Uncertainty of the loss factor (E"r) due to errors in measurements o f S2i. Acrylic = lmm (top left),
2mm (top right), 4mm (bottom left), 5mm (bottom right).
63
Figure 4.19 Uncertainty of the dielectric constant (£’r) due to errors in measurements of S21
64
Figure 4.20 Uncertainty o f the loss factor (£”r) due to errors in measurements o f S2i
65
Figure 4.21 Uncertainty of the dielectric constant (£’r) due to errors in measurements of S2|. Acrylic = 1mm (top
left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
65
Figure 4.22 Uncertainty of the loss factor (£”r) due to errors in measurements o f S2|. Acrylic = lmm (top left),
2mm (top right), 4mm (bottom left), 5mm (bottom right).
66
Figure 4.23 Uncertainty of the dielectric constant due to errors in measurements o f S2|. Five different ratios o f
material at 10GHz, a sample thickness of 0.1 mm and an acrylic thickness of 1mm.
68
Figure 4.24 Uncertainty of the imaginary permittivity due to errors in measurements of S2). Five different ratios
o f material at 10GHz, a sample thickness o f 0.1mm and an acrylic thickness o f lmm. The erp= 5*erpp
has uncertainty too high to be recorded in this plot.
68
Figure 4.25 Uncertainty of the dielectric constant due to errors in measurements o f S2i. Five different ratios o f
material at 10GHz, a sample thickness of lmm and an acrylic thickness o f lmm.
69
Figure 4.26 Uncertainty of the imaginary permittivity due to errors in measurements o f S2|. Five different ratios
o f material at 10GHz, a sample thickness o f 1mm and an acrylic thickness of 1mm.
69
Figure 4.27 Uncertainty o f the dielectric constant due to errors in measurements of S2i. Five different ratios o f
material at 10GHz, a sample thickness of 2mm and an acrylic thickness of 1mm.
70
Figure 4.28 Uncertainty o f the imaginary permittivity due to errors in measurements of S2|. Five different ratios
of material at 10GHz, a sample thickness o f 2mm and an acrylic thickness of lmm.
71
Figure 4.29 Uncertainty o f both the real and imaginary components of permittivity at a dielectric constant o f 100
and a frequency of 10GHz. Acrylic thickness for all plots is lmm. e,p = £’r.e,pp = E”r
72
Figure 4.30 An extreme close-up of both S-Parameters and how data smoothing occurs with gating.
75
Figure 4.31 Results from Batch#5 Sample#4, six separate measurements. Arrows indicate increasing time.
Sample thickness = 0.15mm, Acrylic thickness = 1 .1 8mm
76
Figure 4.32 Results from Batch#7 Sample#4, four separate measurements. Arrows indicate increasing time.
Sample thickness = 0.15mm, Acrylic thickness = 1 .1 8mm
77
Figure 4.33 Results from Batch#9 Sample#5, four separate measurements. Sample thickness = 0.15mm, Acrylic
thickness = 1.18mm
78
Figure 4.34 Entire Batch #4. Thickness constraint envelopes were recorded. Sample numbers correspond to
Table 4.6
79
Figure 4.35 The polynomial solution and the small-section solution. Sample thickness = 0.14mm, Acrylic
thickness = 1.18mm
81
Figure 4.36 Very low-loss sample with very high uncertainty. Also shown is one-layer convergence o f acrylic
only. Sample thickness = 0.16mm, Acrylic thickness = 2.07mm
82
Figure 5.1 S-Parameters showing both VNA-collected results and FDTD-simuiated results for sample B21S2 85
Figure 5.2 Sample B21S2, (Left) Dashed line indicates measurements taken with VNA, solid shows FDTD
simulated results. (Middle, Right) Zoomed in results showing 10GHz comparison. Blues lines show
£r’ , red lines show e ,”
86
Figure 5.3 Comparison o f measured and simulated results for B26S4. Focus is 10GHz results. Green uncertainty
enveloped centered on VNA-measured values.
87
Figure 5.4 The five full gap configurations simulated.
89
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Figure 5.5 FDTD simulated Su parameter for gaps 1-5 for sample B21S2. Blue left: perfect, Green: gap #1,
Magenta: gap #2, Cyan: gap #3, Yellow: gap #4, Blue right: gap #5. Red tips indicate low frequency
data points of S-parameters.
89
Figure 5.6 FDTD simulated S2| parameter for gaps 1-5 for sample B21S2. Blue left: perfect. Green: gap #1,
Magenta: gap #2, Cyan: gap #3, Yellow: gap #4, Blue right: gap #5. Red tips indicate low frequency
data points of S-parameters.
90
Figure 5.7 FDTD simulated results for sample B 21S2 or dielectric constants and loss factors for one perfectly
homogeneous layer and five samples with gap inconsistencies.
90
Figure 5.8 FDTD simulated S| t and S2| parameters for gaps 2-4 for sample B26S4. Blue: perfect. Magenta: gap
#2, Cyan: gap #3, Yellow: gap #4,. Red tips indicate low frequency data points of S-parameters. 92
Figure 5.9 FDTD simulated results for sample B26S4 of dielectric constants and loss factors for one perfectly
homogeneous layer and three samples with gap inconsistencies.
93
Figure 5.10 The five dent configurations simulated.
95
Figure 5.11 FDTD simulated Sn and S2t parameters for sample B21S2, gaps 1-5. Blue left: perfect. Green: dent
#1, Magenta: dent #2, Cyan: dent #3, Yellow: dent #4, Blue right: dent #5. Red tips indicate the low
frequency data points o f the S-parameters.
96
Figure 5.12 FDTD simulated results of dielectric constants and loss factors for one perfectly homogeneous layer
and five samples with dent inconsistencies. Sample B21S2. On enlarged region plot, E 'r on left, £”r
on right. Dent #5 omitted from £’r 10GHz enlarged region plot.
97
Figure 5.13 FDTD simulated Sn and S2| parameters for sample B26S4, dents 2-4. Blue left: perfect. Magenta:
dent #2, Cyan: dent #3, Yellow: dent #4. Red tips indicate low frequency data points o f Sparameters.
98
Figure 5.14 Results of dielectric constants and loss factors for one perfectly homogeneous layer and three
samples with dent inconsistencies. Sample B26S4.
99
Figure 7.1 Uncertainty of the dielectric constant due to errors in measurements o f S2|. Five different ratios of
material at 10GHz, a sample thickness o f 0.1 mm and an acrylic thickness of 3mm.
A4
Figure 7.2 Uncertainty of the loss factor due to errors in measurements of S2|. Five different ratios o f material at
10GHz, a sample thickness o f 0.1 mm and an acrylic thickness of 3mm.
A4
Figure 7.3 Uncertainty of the dielectric constant due to errors in measurements o f S2|. Five different ratios of
material at 10GHz, a sample thickness o f lmm and an acrylic thickness of 3mm.
A5
Figure 7.4 Uncertainty of the loss factor due to errors in measurements of S2i. Five different ratios o f material at
10GHz, a sample thickness o f lmm and an acrylic thickness of 3mm.
AS
Figure 7.5 Uncertainty of the dielectric constant due to errors in measurements o f S2I. Five different ratios of
material at 10GHz, a sample thickness o f 2mm and an acrylic thickness of 3mm.
A6
Figure 7.6 Uncertainty of the loss factor due to errors in measurements of S2|. Five different ratios o f material at
10GHz, a sample thickness of 0. lmm and an acrylic thickness of 3mm.
A6
Figure 7.7 HP8720 Equipment Uncertainty Specification Sheet
A7
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List of Tables
Table 4.1 The dielectric constant and loss factor of the four materials analyzed.................................................49
Table 4.2 Displaying cut-off frequency, and guided half-wavelength, and guided quarter-wavelength distances
for the X-band............................................................................................................................................58
Table 4.3 Sample thickness = 0.1mm, acrylic thickness = lmm, frequency = 10GHz...................................... 73
Table 4.4 Sample thickness = lmm, acrylic thickness = lmm, frequency = 10GHz..........................................73
Table 4.5 Sample thickness = 2mm, acrylic thickness = lmm, frequency = 10GHz..........................................73
Table 4.6 Data corresponding to Figure 4.34 showing variable thickness with S% constraint........................... 80
Table 4.7 Data Set displaying difficulties with optimization convergence. All optimized solutions of thickness
lay on the constraint edge o f the envelope, except for the two shown in Red Text.............................81
Table S. 1 Electrical characterization of two representative samples at 10GHz................................................... 84
Table 5.2 Comparison o f measured and simulated complex permittivity values at 10GHz................................ 87
Table 5.3 Summary table showing differences in simulated dielectric constant and loss factor for different
configurations o f facial inconsistencies. All values at 10GHz...............................................................93
Table 5.4 Summary table showing differences in simulated dielectric constant and loss factor with different
configurations o f facial inconsistencies. All values at 10GHz.............................................................100
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Acknowledgments
Thank you Dr. Stuchly for your patience and understanding to let me accomplish many
other goals simultaneously with this Masters degree. I have bettered m yself in many
aspects during the time spent on this research and I feel I owe a great portion o f it to you.
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1
Chapter 1
1
Introduction
A measurement method was required for the electrical characterization of thin, flexible
materials intended to perform as radar absorbers at microwave frequencies. After a
review o f many measurement techniques previously developed, it was determined that an
extension to the available techniques was needed for this application. A two-layer
waveguide measurement technique has been selected and is described in this thesis. To
evaluate the technique, an uncertainty analysis is carried out, several measurement results
are analyzed, and a Finite Difference Time Domain (FDTD) simulation is performed that
evaluates sample imperfections.
1.1
Background
The motivation for this research was provided by an interest and a need to evaluate novel
materials for radar cross section (RCS) reduction. Researchers at the Canadian Forces
Base (CFB) Esquimalt, Canada, have been developing radar absorbing materials (RAM)
based on the specific organic polymers, Polypyrrole (PPy) and Polyaniline (PAni). These
materials exhibit unique microwave behaviour if manufactured properly. RAM needs to
have relatively high dc and ac-conductivities and a relatively low dielectric constant.
Both real and imaginary parts of the permittivity have to be controlled reliably in material
manufacture. Design of structures reducing RCS depends on available suitable materials.
New materials with unique, controllable microwave properties would dramatically
increase the degree of freedom of design. In the process of developing new RAM,
measurements of its permittivity and permeability are critical. Since initially, the
available samples were made relatively small, it was decided that measurements in the Xband (8.2-12.4 GHz) would suffice. Also, initially only very few materials were expected
to have relative permeability greater than 1. Furthermore, it proved difficult to create
homogeneously thick sample coatings o f PPy and PAni on the known substrate.
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2
1.2
Research Objectives
The objectives o f this research and thesis are:
• To establish a specific measurement method for the microwave
characterization of thin flexible lossy materials.
• To develop iterative (optimization) procedures for obtaining complex
permittivity (er) and permeability (jir) from measured complex scattering
parameters (Su and S21).
• To ensure that measurements are repeatable and relatively fast.
• To examine the uncertainty associated with the apparatus used during the
measurements.
• To model the system in FDTD in order to explore how small facial
inconsistencies in the sample affect the measurement uncertainty and
optimization convergence when solving for complex permittivity.
1.3
Thesis Contributions
The main contribution of this thesis is the development of the specific measurement
configuration used to characterize the electrical properties of thin flexible electricallyunknown samples. The measurement technique was developed under the constraints of
several factors, all of which were met. The measurement process had to be a relatively
inexpensive and fast way o f measuring a large number o f samples. A detailed uncertainty
analysis was performed for this novel measurement technique for many different types of
samples, but the main focus was on lossy non-magnetic materials on the order o f 0 .2 lmm thick. To converge on a solution (permittivity and permeability o f a material) a non­
linear least-squares optimization procedure was developed. Using the curve-fitting
concept devised at NIST [1], a refined program was developed to process the input from
a Vector Network Analyzer (VNA), and to calculate the complex permittivity and
permeability o f a material. Uncertainty was then calculated based on the VNA accuracy
specifications in collecting S-Parameters (magnitude and phase). Although complex
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3
permeability is not explored in this thesis, the convergence program maintains the ability
to accommodate magnetic materials, though no testing or uncertainty analysis has been
performed. The procedures described for the permittivity are directly applicable for
permeability. The measurement technique developed is transferable to free-space
measurements and will be used with broadband horn antennas once the ability to deposit
these new materials over large surfaces is attained. FDTD simulations were performed to
illustrate the effect that slight imperfections in the sample layer had on data collection
and subsequent permittivity determination.
1.4
Thesis Outline
Chapter 2 contains a review o f previously developed measurement methods. Several
methods are discussed in detail and their drawbacks are identified with respect to the
measurements, which is the goal in this thesis. Relevant waveguide theory is discussed,
as well as the electrical characterization o f materials at microwave frequencies. Reasons
for selecting the two-layer approach are given.
Chapter 3 describes the selected method (the two-layer technique) and procedures needed
to obtain electrical characterization of materials. The optimization procedures developed
for this work are covered in detail and reasons are given why one method may be better
at converging on a solution than another for a specific set o f collected data. Sources o f
uncertainty are enumerated. Uncertainty due to the VNA is derived. Uncertainties due to
small gaps or dents existing in the sample are noted and FDTD simulation is introduced
as a way o f investigating this practical problem. The FDTD simulation of the entire
waveguide system is presented along with safeguards used to ensure an accurate solution
for the data collected. The procedure to convert data from a simulated waveguide system
to a useable set o f values mimicking an actual waveguide set up and calibration is
presented.
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4
Chapter 4 gives the results of the uncertainty analysis developed in Chapter 3 as well as
several measurement results of samples delivered to the lab. Four general materials are
introduced and their electrical characterizations presented. Each material is subsequently
analyzed in terms o f its uncertainty in relation to its thickness as well as the thickness of
the supporting structure. Useful uncertainty plots portraying materials with various
sample and acrylic thicknesses are then presented, and trends and recommended
configurations are explored. Measurement results o f seven materials are then presented
with the optimization convergence characteristics analyzed. Limitations of the two-layer
optimization approach are shown for several of the presented materials.
In Chapter 5, the FDTD simulation results and evaluations are shown. A comparison
between actual sample measurements and FDTD simulated measurements is given to
validate the computational domain setup. Two lossy materials are used in this
comparison. Facial inconsistencies in the form o f gaps and dents in each o f these two
samples are then simulated. Five gap and dent configurations are explored for one
material, three configurations for the other. The gap and dent results are compared to the
results for a perfectly homogeneous sample layer.
Chapter 6 contains conclusions comparing the quality o f the solution to the initial
problem. General recommendations regarding functionality and limitations of the
optimization procedure are given. Directions for future work on this measurement
technique are given and possible avenues of further research suggested.
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5
Chapter 2
2
Definitions and Background Literature Review
2.1
Definitions
Conventional definitions and notations used in this thesis are given in Appendix 7.1.
They include the permittivity, permeability, characteristic impedance, reflection and
transmission coefficients, and scattering parameter definitions.
2.2
Measurement Methods of Electrical Properties
Electrical properties o f materials have been characterized successfully by many different
methods. A comprehensive literature review was performed to determine a well-suited
measurement method for the project. After consideration o f the project requirements, a
specific measurement method and sample configuration was selected. A outline o f the
different approaches toward material measurement is given, a more comprehensive
review is given elsewhere [2 ]
2.2.1
Time-Domain and Frequency-Domain Measurements
Time-Domain (TD) measurements use a material response to the spectral content o f a fast
rise-time (subnanosecond) pulse [3]. The scattering parameters are determined by taking
the Fourier transform of this response [4]. This type of measurement has only become
possible in the early 1970’s due to the development of fast sampling oscilloscopes and
tunnel-diode step generators with picosecond rise times [2]. Advantages o f TD
measurements include a potentially lower equipment cost and the ability to take
measurements without the need of a calibration procedure [2]. However, resolution and
accuracy are often sacrificed in TD techniques, if broadband measurements are required
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6
at high frequencies. This, however, is becoming less o f a problem with advancing
technology [2 ].
Frequency-Domain (FD) measurements require the use o f an Automatic (or Vector)
Network Analyzer (VNA) that collects the scattering parameters (reflection and
transmission coefficients) directly. Other methods have been used in the past, but they are
cumbersome and less accurate [5]. After a calibration is performed and a measurement
plane is established, the scattering parameters can be read at a frequency resolution
dependant on the particular analyzer [6 ]. The disadvantages of this method are the cost
o f the VNA, and a time-consuming calibration procedure that must be performed before
every measurement if accurate results are desired.
2.2.2
Resonant and Wideband Measurements
Resonant measurement methods can be performed in either a closed-cavity or an open
environment. These methods rely on the change o f the Q-factor and the resonant
frequency of a resonator after the test sample has been inserted. In many cases, very
specific geometries and exact dimensions o f the test sample are required. The most
popular is the cylindrical TMoio cavity, which requires a rod-shaped solid sample, or a
liquid in a tube [2]. Other cavity methods can be used that have similar restrictions on
sample shape. Open resonators are more convenient than cavity methods, but at relatively
low frequencies large homogeneous samples are required. Both types of resonant
methods can only measure the properties in a very narrow frequency band requiring a
different measuring device or sample size at each frequency range [7]. Mathematical
expressions required to solve for the electrical characteristics in one cavity do not convert
to different cavity geometries. Only for the perturbation method does the analysis remain
unchanged [5].
Wideband measurements can be performed in a closed environment using TEM, TE or
TM waves, or in free-space using plane waves. Coaxial lines or waveguides can be
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7
loaded with a test sample provided the correct sample geometry is achievable. A very
wide frequency range can be attained, particularly for coaxial line measurements. Usually
the fundamental mode is utilized as the only propagating wave. This limits the
complexity of mathematical expressions. Again, exact geometries of the test samples are
required to limit the uncertainty o f the measurements. Free space measurements require
larger samples, as the directivity o f the antenna is the limiting factor on sample size.
However, as long as a sufficiently large homogeneously thick sample can be prepared,
the width and height of the sample does not have to have exact dimensions. The
mathematical expressions required for the determination of the electrical properties o f the
test sample do not need to be altered significantly when transferring between
fundamental-mode waveguide and free-space measurements.
2.2.3
Measurement Method Selected
Because of the requirement to obtain data in a relatively wide frequency range, and the
availability of a VNA (HP 8720), further focus has been restricted to frequency domain
wideband measurements.
Wideband measurements of small samples are possible using coaxial lines, but the test
sample geometry is difficult to machine accurately. Neither the manufacturing process
nor the facilities to mill samples suitable for coaxial measurement were available at the
start o f this research. Rectangular cross-section samples are much easier to manufacture
with high tolerances. Because only small thin sheets of sample material were readily
available, X-band waveguide was chosen for the measurements.
The same method and technique required to process the scattering parameters to
determine the electrical properties of the test sample with minimum uncertainty can be
directly carried over to other frequency ranges o f the rectangular waveguide, as well as to
free-space. Additionally, since only thin flexible samples were available, the proper
measurement technique was needed to accommodate this limitation.
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8
2.3
Guided Wave Measurement Techniques
A brief outline o f the equations required to solve for the complex permittivity and
permeability using the scattering parameters will be given in the following chapters. This
method was established in 1970 and is still predominately used today for measurements
o f relatively low to medium loss materials. Limitations of this method will be explained
and several different solutions shown.
2.3.1
Classical Nicholson-Ross-Weir (NRW) Method [4][6]
Nicholson and Ross derived an explicit solution for the permittivity and permeability
from the measured scattering parameters. The solution was based on transmission line
theory. These explicit solutions were derived before the invention of a practical
ANA/VNA, therefore their work was performed in the time domain. After the advent o f
the ANA/VNA, Weir [6 ] further developed the technique in the frequency domain. The
method is often referred to as the NRW method.
Limitations in the NRW method encouraged further research in this area. At certain
frequencies close to and including guided half-wavelength (Xg/ 2 ) multiples in the sample,
the NRW equations completely breakdown [8 ]. Figure 2.1 illustrates the common
problems associated with a sample thickness that is A,g/2 or a multiple at some frequency
for a low-loss material [8 ]. This limits the sample to thicknesses less than Xg/2 o f the
highest frequency. However, this often results in the scattering coefficient values that are
in the range where VNA exhibits large uncertainties [8 ], [1]. An in depth uncertainty
analysis o f data collected from the VNA using thin samples relates to this research and is
presented later in this chapter.
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9
2.5
10
0
12
Fr«quMcy(GHz)
10
12
Prequwef(QH<
Figure 2.1 Instability problems o f the classic NRW explicit solutions [8 ]
Although the NRW method is explicit, it requires a relatively good initial guess o f the
electrical properties measured. This can be a limiting factor when the substance is
completely unknown. With a poor initial guess, the equations may produce wrong values
with no warning o f failure. The importance of the initial guess arises from an infinite
number o f roots existing for equation (2.11). An initial guess is required to determine the
correct root coefficient for equation (2 . 11 ) and is accomplished by solving for the group
delay [6 ].
A brief outline is now given detailing the NRW method o f explicit determination o f the
electrical properties of materials. Parameters Er and |ir o f a homogeneous sample in a
waveguide (or, in general any transmission line) can be derived by modeling the sample
under test as three two-port devices. These are two planes at which reflections between
the two materials are considered (Di and D3), and a section of the waveguide uniformly
filled with the material (Dj). This is shown in Figure 2.2.
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10
Port 2
Plano 03
Plane D1
I • Length of Sample
and Sample Holder
Port 1
Figure 2.2 Sample in rectangular waveguide with sample holder. Represented as three
two-port devices
The matrices defining these sections are:
r
z>,=
' 0
i - r '
D, =
t + r
- r
-T
T
D,
‘ - r
=
0
i-r
i + r
(2. 1)
r
where T and T are complex as defined in the Appendix 7.1.
When cascaded, the three matrices form:
D 4 =
Dxo D 2 ° Di
T r ( i - T 2)‘
’ T ( i - r 2 )'
i - r :T 2
i - t 2t 2
[ T ( i - r : )‘
’ r ( i - T 2 )'
■^11
=
Al
S\2
(2.2)
S 22
[ l - r 2T 2
where, for isotropic, reciprocal (e.g. non-ferrite) materials:
S„
=
r(i-T2)'
t - r 2T 2
Sl2
= 5 ,, =
T (i-r2)
i - r 2T2
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(2.3)
11
Modifications must be made to these equations when the sample holder is not o f the exact
thickness of the sample. In this case, a linear transformation translates the S-Parameter
matrix to the calibration planes, shown in Figure 2.3.
Port 2
Plan* 03
12 - Langth batwean Plana 2
and Sampla
I - Length of Sampla
and Sampla Holder
Plana D1
- Length between Plana 1
and Sampla
Port 1
Figure 2.3 Sample in rectangular waveguide with different length sample holder
Translation can be performed as:
dJ
(2.4)
=0£>ae
where:
-r»L,
©=
0
(2.5)
Therefore:
5,, —e
' r ( i - T 2)'
|_i—r 2x 2J
5,, = e -2i^ [' ri (- i r- T2T2)‘2 _
s ,, = S p =e-r>^*L'-' * T ( i - r 2)'
[_ 1—r 2x 2J
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(2.6)
12
In the experimental procedure used, the sample was placed flush with the calibration
plane, as shown in Figure 2.4.
Port 2
Sample 02
Plane 01
I - Length of Sample
Port 1
Figure 2.4 Sample in rectangular waveguide with no sample holder
For the setup o f Figure 2.4, Li = 0, and L2 = -L. The equations for the S-parameters
reduce to:
r(i-T2)'
[ i - r 2T2J
’r(i-T2)'
S„ = e 2r°L
|_i-r2x2J
S,, = S r
'T(i-r2)'
[ t - r 2T2
= e r°L
To solve for the permittivity o f permeability explicitly, only two S-Parameters are
needed. Therefore, the system (three complex equations) is over-determined.
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13
Using S21, T is solved for and replaced in the equation for S| 1. Solving for T then results
in:
r=K±J{tc2-i)
=
" 2
( 2 .8 )
2s„
and:
T =
s n +s2i- r
i - ( s lt+5:,)r
(2.9)
Using previous and appendix definitions for T, T, and y, explicit equations can be
obtained for complex p.r and Er. These are:
ro+iD
vA , , 0 - r )
r
f 1)
\
'is !'
e. =
(2. 10)
fl.
1
VC /
With Xq and Xc being the free-space wavelength and waveguide cut-off wavelength
respectively and:
-U fl'
( 2 . 11)
T
A2
The expression ln(l/T) contains multiple roots. Solving for the correct root requires a
guess o f the permittivity, or an analysis o f the group delay [6 ].
For a low loss, non-magnetic sample, when L = Xg/2, Si 1 -> 0, and S21
C
2
K =
_C
1, it leads to
- 4.1
(2. 12)
25,
as the magnitude o f T < 1 :
0
c
o
T=
^
tl(-» 0 )
+C
^
( " ^ I I I —. 0 )
+
I)
5
_r
* ( — 0)
l ( — I ) ^ '( - . 0 ]
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(2.13)
14
As T
1:
(2.14)
The equations dictate that either |ir or er is equal to zero. This is not a physical solution,
therefore |ir and Er are no longer separable.
Theoretically, the equations are only non-separable at one frequency. However, since
computers have a finite numerical accuracy, the result is a small frequency band within
the frequency range of interest that is affected. To illustrate the problem, a sample with er
= 2.6+j0.0001 and |ir = I is evaluated, and the results are shown in Figure 2.5 and Figure
2 .6 .
S11 - P olar plot without error of non-magnetic sa m p le with ei= 2 .6 * f0 001
270
Figure 2.5 Simulated Si i parameter for a lossless non-magnetic
sample of £r = 2.6 and thickness 2.256cm
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15
S21 - P o la r plot w ithout error o f n o n - m a g n e tic s a m p le w ith er=2.6-*j*0.001
120
0.8
0.6
150
02
180
210
330
240
300
270
Figure 2.6 Simulated S21 parameter for a lossless non-magnetic
sample of er = 2.6 and length 2.256cm
It can be noted that on one side of the frequency range the Sn parameter approaches zero
and the S21 parameter approaches one. In both cases, the errors in measurements tend to
infinity.
The material properties derived from the S-Parameters using equation (2.12) - (2.14) are
shown in Figure 2.7. Figure 2.7a illustrates the half-wavelength problem, Figure 2.7b
displays the narrow band where results cannot be explicitly derived. The half-wavelength
multiple in the frequency range of the X-band waveguide for this sample occurs at 9 194
720 620 Hz. The frequency resolution for these graphs has been increased to highlight the
half-wavelength problem, and does not represent true VNA measurements. It would
however, if the half-wavelength multiple occurred near a standard VNA 1.05 MHz step.
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16
R oil Com ponents of Permiltnaty tntf Perm eoB44r
1
3
't
25
B««l Com ponents o f Pofm4tiwtf
<5 25
2
1
2
i
P«>mo«feibtY
*
a
i
15
H
3
1
15
*
1 05
!■ *
2
a
0
O SS
09
0 95
i
1 05
'<
1 15
F’o o u o n c f. x-8«nd W noGuiO* Bongo
9 194/
12
( , g 14
9 194/
9 t9 4 /
9 194/
9 194/
Froquoncv. X-B*n4 Wi»«Ogt00 P tn p o
A
9 194/
( , q*
B
Figure 2.7 A.) er and |ir derived from S-Parameters o f simulated sample.
B.) Narrow frequency band illustrating breakdown o f explicit equations.
Apart from the problems related to the half-wavelength ‘transparency of the sample’, and
non-separability of er and |ir, uncertainty in broadband measurements of lossless non­
magnetic materials results from the VNA having the largest error in phase when the Sn
parameter approaches zero, and in magnitude when the S| i parameter approaches 1. This,
in practical measurements, means that a wide frequency band around the half-wavelength
multiple is unusable due to an unacceptably large error.
To overcome these limitations of the NRW approach, other methods have been proposed.
Namely, multi sample/multi sample-position methods, and short-circuited waveguide
methods.
2.3.2
Multi Sample / Multi Sample-Position Methods
The main goal of these methods focuses on taking multiple measurements o f one sample
in different positions inside the transmission line, or one measurement o f samples o f
varying thickness. This method is mostly used for low-loss materials, where the NRW
equations exhibit high uncertainty.
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17
These methods rely on the sample to be either:
accurately positioned within the transmission line in two or more
locations [9], [10]
symmetric and of the same width, length, and thickness [9], [10], [11],
[ 12]
manufactured at specified thicknesses, e.g. double the thickness of
another sample [ 11 ]
In all cases, the sample has to be self-supported within a waveguide, or to be machined to
fit inside a coaxial line. This was not possible with the samples of interest in this work.
The samples could not be positioned within the waveguide without collapsing and
accuracy o f placement was very poor. Also, different thicknesses of sample could not be
produced without changing the substrate on which the sample was grown, therefore,
changing its electrical properties. These methods, thus, were not suitable.
2.3.3
Short-Circuited Waveguide Method
Two main methods have dominated this measurement technique, von Hippel’s use o f a
fixed short-circuit to determine the electrical characteristics in a relatively narrow
frequency band [13], and the sliding-short method [14]. One difficulty with utilizing a
short-circuit, or a sliding-short, is the reliance on the Si i parameter alone, which typically
have the higher uncertainty associated with the measurements. In the case of the fixed
short-circuit method [I], an explicit solution can be obtained for the permittivity. On the
other hand, Maze [14], using 48 or more positions in an actuator-controlled sliding-short
had to solve for permittivity and permeability iteratively.
To use Maze’s method, the sample must be self supporting and placed very accurately
while measurements are taken; this was not possible with the samples of interest. Also, a
sliding short was not available. Because o f the desire to take wideband measurements
with maximum accuracy, von Hippel’s method did not seem practical. Very thin samples
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18
do not work well when placed flush with a short circuit, as the electric field is
approximately zero at this location.
2.3.4
Non-Linear Optimization
All methods in chapters 2.3.1 - 2.3.3, solve for the permittivity and, if applicable, the
permeability explicitly (with exception o f [14]). A non-linear least squares optimization
approach instead of relying on the NRW equations or the modified explicit solutions of
the several other methods was proposed by James Baker-Jarvis et al. at the National
Institute o f Science and Technology (NIST) [1], [15], [16].
The NIST approach uses least-squares to solve for the coefficients o f polynomials that
describe the complex permittivity (and permeability) over the entire frequency range of
interest. An explanation of nonlinear optimization techniques is covered in Chapter 4 and
is not reviewed here with a complete derivation of the non-linear optimization technique,
the Levenberg-Marquardt Method, in [17].
The advantage of the optimization technique stems from the fact that no explicit
equations are necessary for the material properties. Therefore, no initial guess is required
as the multiple solutions arising from equation (2.11) are not an issue. Also, sections in
the frequency band that are known to be within high equipment uncertainty can be
weighted differently from the rest of the collected data. Because optimization is used,
convergence on only select parameters o f the collected data is an option, e.g. only
converging on S21 magnitude and phase data.
Using the optimization technique, a theoretical relationship between the dielectric
constant and the loss factor, as frequency is varied, can be represented. This relationship
is known as the Kramers-Kronig (KK) relation. The KK relationship can be used as an
optimization constraint in combination with the selection of the measured data, to further
facilitate the optimization procedure. For the data where the Su parameter is close to 1 (a
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19
high uncertainty area for the VNA), more weight in the optimization algorithm can be
placed on the KK relationship. Where data is in the low uncertainty area, the KK
relationship can be set up to only play a role in detecting intermittent erroneous data.
As least-squares is a curve-fitting routine, a very smooth curve is generated that is not
drastically affected by single-frequency erroneous points, unlike explicit solutions. Figure
2.8 [1] shows a result o f the permittivity obtained in a very wide frequency band. The
large variations in the non-optimized solution come from data collection that corresponds
to points of very high uncertainty in the VNA.
Cor a teadad glaaa over 0.043-11 <JHs tor the
) aad poirt-by-potat Hrh «iipi > ( ------- )
C
Figure 2.8 Reproduced from [ 1] to portray smooth curve fit o f least squares solution in
contrast to explicitly solved single-frequency point solutions.
With this optimization approach, it is possible to converge to a solution when
determining both permittivity and permeability even when using the short-circuit
approach. This, however, requires at least two positions or two sample thicknesses.
A very robust optimization algorithm must be used for characterization o f high-dielectric
constant materials, since several local minima may be converged to before finding the
global minimum. If the algorithm does not recognize local minima, the results will not be
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20
correct. An approximate initial guess could be used to speed up convergence, but is not
needed as critically as for explicit solutions, as long as computing time is not a factor.
The optimization technique seems to be the most promising solution method, as it has
several advantages. The use of the KK relation is important and is a definite advantage.
However, even without this relationship, the optimization technique seems superior, due
to the curve-fitting nature o f least squares. In the single point-by-point explicit methods,
many points have very high accuracy while many points have unacceptable accuracy.
With the curve fit, the high accuracy o f some points may be sacrificed to a very small
degree but the low accuracy points are brought within error tolerances. This makes the
overall data set acceptable rather than only select points. A more detailed description of
this method is given in Chapter 3.2 as this method is used for permittivity determination
in this thesis.
2.4
Conclusions
With the specific restraints placed on measurement arrangements, none of the methods
previously described were directly suitable for the test samples o f interest. Therefore an
alternative solution was required. Support for the material sample and accurate placement
within the sample holder were needed. A two-layer approach with a material o f known
electrical characteristics as a supporting structure was chosen and investigated. Figure
2.9 shows the sample configuration, which is to be placed inside a waveguide. This
relatively large (to the unknown sample) and rigid structure has the ability to be placed
very accurately before and during data collection. This setup can also aid in thickness
measurement of the unknown sample, as the known sample can be measured very
accurately before and after the unknown sample has been set on the structure, and the
subtraction o f these two values equals the thickness o f the unknown sample.
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21
Knowi Substance
Unknown Substance
Figure 2.9
Known substance with unknown coating on its front face that make
up the sample to be measured.
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22
Chapter 3
3
Method of Analysis and Measurements
To successfully measure materials o f interest and satisfy several constraints imposed, a
modification to the measurement arrangement from previous work was required. The
main challenge is the small thickness and flexibility of the samples, coupled with the
requirement to measure the electrical characteristics in a broader frequency range than
cavity methods allow for. Additional conditions are imposed by the need for
measurements of large numbers o f different materials over a relatively brief time-span.
The two-layer approach has been selected as the direction to focus the measurement
effort.
The equations governing the two-layer method are presented in this chapter, followed by
an outline of advantages and limitations of this method. An iterative solution o f the
equations is discussed next, and an analysis of sources of uncertainty is also presented.
An FDTD program has been used for determination of errors that cannot be evaluated
analytically, and the FDTD modeling of the problem is briefly described.
3.1
3.1.1
Two-layer Measurements Method
Governing Equations in a Waveguide
The two-layer configuration is shown in Figure 3.1
Zo
2
Zo
Figure 3.1 Cross section o f waveguide with sample and acrylic inserted.
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23
where Zo is the characteristic impedance o f the empty waveguide, and Z*j and Z *2 are the
characteristic impedances o f the respective material inside the waveguide. A (*) indicates
a complex number. Reflection coefficients,
r\, T*2,
and
T*3, are
at each respective
medium interface, as shown in Figure 3.1.
The characteristic impedance are defined in the three different media as:
where eo is the permittivity in free space (8.85 x 10‘12), po is the permeability in free
space (4tc
x
10‘7),
e \i
and e *r2 are the complex relative permittivities o f the respective
materials shown in Figure 3.1. The relative permittivities are defined as e \i = e*i / eo and
e *r2 = e *2 / eo, fc is the cut-off frequency o f the waveguide and is equal to 6.65GHz for Xband waveguide used in this work, f is the frequency (typically 401 equally spaced data
points between 8.2GHz — 12.4GHz are collected). Impedance, Z%, in Equation (3.1) is a
general representation for any material, although in this thesis it is a real number, as the
loss factor o f acrylic is negligibly small. Parameters e *r2 and p *r2 from now on will be
denoted as er2 and p r2 for simplicity, although they are complex.
The reflection coefficients at the three interfaces, indicated by arrows, are:
p _ ^ 2 ~ Zo
1 z 2+z0
p* _
~ ^2
2 z; +z,
p. _ ZQ—Z\
3 z0+z;
(3.2)
Rearranging equations (3.2) and expressing T*2 as a combination o f H and T*3 gives:
z;= -z 04 —!1 °r;+ i
z ,= -z 0H±!
2 °r,-i
r: = -(r 3+r,)
- r/^+ i
(33)
Transmission coefficients within the two materials are defined as:
T* =
t ' =e~jrzdl
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(3-4)
24
where the propagation constants are:
r Iwt \ z
co
Y\
= ■
c
CO
Yi =
I-------------
(3.5)
— V ^ r2 « r2
C
Ka J
where ‘a’ is the width of the waveguide (2.286cm for the X-band).
The reflection and transmission coefficients are used to calculate S*u and S*2i at the
outside boundary of the interfaces of the two-layer structure i.e. S*n on the material 1
side, S*2i on the material 2 side. For simplicity, stars are suppressed in all expressions
that follow.
5 =^
v,
S 2, =
u2
(3.6)
V,
where:
", = r, (l -
t ,!
)•(r,: - t,: )+ r, (1- r/x ,3)•(1- t,’)
=t,t,(i—r,’ )•(i—r32)
v, = v :
=r,rJ(i-T!:)(i-T 13)+(i-rJ!T!1)-(i-r,!Tli)
(3.7)
(3.8)
(3.9)
Equation (3.6) for S21 does not accurately reflect the S21 collected from the VNA. For
Equation (3.6) to correctly represent actual collected data, a sample holder of the exact
thickness o f the two layer structure would have to be inserted between the two calibration
planes established during calibration. This is not practical for a large number of samples
that have varying thicknesses. To correct for this, the calibration plane must be moved
toward the load the distance o f the combined thicknesses o f the two materials. This is
done in Equation (3.10), and the corrected S21 is labeled S210 The scattering parameters
measured by the VNA are transferred to the plane of reference, thus:
(3.10)
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25
where, for a non-magnetic medium:
V
7
with fc being the cut-off frequency, denoted:
with d| and d2 is the thickness o f the sample and the acrylic backing, respectively.
Equations (3.6) and (3.10) now relate Sn and Sjic to the variables er), £ri, (iri, pr2, d|, and
d 2 for a given waveguide operating in the fundamental mode. If material two has known
dielectric constant and thickness, then Si i and S2ic are functions o f only e*ri, p.\i, and dj.
3.1.2
Advantages and Disadvantages
The primary advantage of the two-layer method is that it facilitates measurements o f a
thin flexible material in the waveguide. Without the second layer, the unknown material
cannot be placed accurately. Because o f the material flexibility, even if one edge of the
material were placed accurately, there would be no guarantee that the thin sheet was not
concave in the center or on a slant. Another serious concern is related to the manipulation
o f very thin, flexible and fragile samples, as their integrity may be compromised. Without
a supporting structure, the unknown material can easily be damaged during the first
measurement preventing repeated measurements, which are often required for
comparison.
One o f the major problems with the one-layer approach, as discussed in Chapter 2, was
the ‘breakdown’ o f the explicit solution o f the NRW equations for guided half­
wavelength multiples. Furthermore, when approaching the equation breakdown, the VNA
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26
measurements have extremely high uncertainty, as the |Su| parameter approaches zero
and IS21I approaches one. As discussed in chapter 2, and documented in the literature [I],
a way to combat this is to use sufficiently thin samples of unknown material. However,
this leads to a very small phase change in S21, and consequently to high uncertainty. By
using the two-layer method, it is possible to use different backing materials (material 2 )
or backing material with different thickness to bypass conditions o f high uncertainty (|S| i|
close to 0 ).
Disadvantages of having the supporting block include added uncertainty due to the
uncertainty in material thickness, and reliance on consistent material characteristics o f the
known support dielectric. A minor disadvantage includes the added time of sample
preparation. However, this can be alleviated with the mass production of material 2
blocks to very high tolerances, all from the same batch o f acrylic. As will be explored in
further detail, the uncertainty in the solution for thin lossy samples is slightly increased
by the presence of material 2 , which is obviously a disadvantage.
3.1.3
Sources of Uncertainty
Many sources of uncertainty related to the mechanics of the measurement process have
been reduced over the course of this project. These include: movement of the coaxial
cables, repeated disconnection of the waveguide, and jostling o f the coax-coax and coaxwaveguide junctions, during and afrer calibration and measurements. These sources have
been reduced by two vice-grips secured to the sample table holding the waveguide on
each side o f the calibration plane. This way, very minimal movement o f the whole system
is achieved during and afrer calibration. During the calibration, matched-load and short
circuit attachments, as well as subsequent sample material insertions, only a slight twist
of one waveguide is required. This drastically reduces movements o f the coaxial lines
and torque on the junctions.
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27
More important sources of uncertainty include thickness measurement o f both the
unknown sample and the known material, uncertainty in the collected data from the
VNA, and uncertainty due to small gaps in the unknown and known materials. The latter
will be discussed in Chapter 3.3.
3.2
Iterative Solution
Since the equations associated with the two-layer method are much more complicated
than for the single layer solution, an explicit solution for e \ and ji*r is not feasible. Thus,
optimization is used. Several optimization algorithms can be used, as many excellent
routines for solving non-linear equations exist in commercial mathematical programs.
Two ways o f solving for e*r and |i*r were explored, both for specific reasons. Ideally,
causality should be used in the optimization, as discussed in chapters 3.2.1 and 3.2.2.
However for reasons outlined in chapter 3.2.3 the two optimization methods used are
non-causal.
3.2.1
Single Value Optimization in Narrow Frequency Bands
This routine was created on the assumption that within a very narrow frequency range,
on the order of 2-10MHz, e \ and fi\ are constants. The X-band is divided into a userdetermined number o f small sections (typically ~40) that are each treated completely
separately from the others. Each section contains at least 6 data points collected from the
VNA. Because the data points collected are complex, this gives 12 known convergence
points in the system. Typically, only e \ is solved for, as magnetic materials have not been
explored in as much detail, but when both e \ and |i*r are needed, this corresponds to 4
unknowns to be converged to. Optimization does not require an over-determined system
or even an equally determined system, but the more points provided, the more accurate
the convergence for the narrow frequency band.
The optimization routine uses the
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28
MatLab ‘Isqnonlin’ function, which uses either the Gauss-Newton or LevenburgMarquart methods (user-determinable).
The splitting of the frequency band into small sections was done initially because of an
advantage over solving for one e \ and | i \ for the entire frequency range. This allowed
both the permittivity and permeability to vary over the 8.2-12.4GHz frequency band.
Although the X-band is not at all considered to be a wide-frequency band, it is known
that the permittivity can vary inside this range. To dictate that the permittivity and/or the
permeability must only be a single value would incur large errors. Figure 3.2 displays the
ideal parameters of salt water, and the converged upon values for the narrow band
optimization routine solution. It is clearly seen that while the narrow band single value
optimization is not error free, it produces incomparably more accurate results than a
complete X-band single value solution. Figure 3.2 only displays a portion o f the X-band
(9.5 - 10.5GHz) frequency range to better show the piecewise (staircased) convergence
of the algorithm. Better convergence would be attained with more sub-bands. A total of
30 (for entire X-band) are used in Figure 3.2, while 50 is the maximum. In Figure 3.2 and
in subsequent figures throughout, e’r is equal to ep and e,p, and e”r is equal to epP and e^p.
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29
Narrow FrcQjcncy Optimization Solution for an
Ideal Salt Water Samnle.
Frequency (GHz)
Figure 3.2 The very narrow frequency sections o f constant complex permittivity
approximate the ideal solution well while each section remains unrelated.
Another advantage of this method pertains to erroneous data readings. Any completely
erroneous S-parameter readings are contained inside a small portion o f results and are
quite apparent. If a single value for permittivity were converged upon with several
erroneous data points in the set, the value would be skewed in the direction of the error.
Results on these look ‘staircased’. It may appear tempting to fit a straight line into this
piecewise solution; however, this is not likely to be a correct solution for most samples
that are measured.
3.2.2
Polynomial Whole-Band Optimization
While the method o f Chapter 3.2.1 is quite robust, a more accurate solution can be
attained by optimizing the coefficients o f a polynomial to represent each part o f the
complex permittivity. Using a 4th order polynomial, this routine converges to the
coefficients required to minimize the squared error (distance) from each individually
solved frequency point. A ten-coefficient polynomial was found to be a good balance
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30
between processing time required and reliability of the algorithm. If more coefficients
were used, the algorithm becomes less reliable in terms o f convergence.
The whole-band optimization uses the piecewise solution of chapter 3.2.1 to provide an
initial guess. Once these coefficients are supplied, the optimization refines the solution.
This method works well and gives smooth full X-band results.
It is the method
predominately employed to obtain the results o f chapter 4. Under some circumstances,
this polynomial method has difficulty converging to a reasonable result. In this case, the
method of chapter 3.2.1 is used. An example of the erroneous solution is shown in
chapter 4.2.1.
3.2.3
Causality of Solution
The polynomial convergence requires introducing constraints regarding how e’r and e” r
are inter-related. The method outlined in chapter 3.2.2 discussed polynomial convergence
without this inter-relation.
Using constraining equations developed by Debye, two
separate polynomials for epsilon prime and double prime should be solved for in the ideal
situation. Unfortunately, when these constraints were applied, it was found that the
convergence highly depended upon the initial guess. According to the equations (3.13),
the variables to converge to include £«, £r», and a time constant, x [18].
(3.13)
For a single relaxation process, £ „ is the real part of the complex permittivity at very low
frequency, £ ^ is the real part o f the complex permittivity at a high frequency limit, and x
is the relaxation time constant [18]. Therefore, if e re, £r», and x are known, then the
complex permittivity at any frequency is determined.
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31
With these constraints, the optimization never converged to a consistent solution. The
problem is due to the too narrow frequency range o f the X-band to give a unique
relationship between e’r and e” r. Also, a material may exhibit more than one Debye
relaxation [18]. Multiple Debye relaxations, represented by Equation (3.14), can be set up
in the optimization algorithm, but this adds more variables and still does not guarantee
that a unique, or even close to accurate solution, can be attained. With the multiple Debye
relaxations, once again solutions were highly dependant upon the initial guess. Therefore
it was decided not to use this convergence method until measurements in a much broader
frequency range were available. However, multi-waveguide, or free-space, measurements
are beyond the scope of this thesis.
A material exhibiting more than one Debye term can be described as in equation (3.14).
e ' M = e „ + e" e~ - + £" - £~ +...
(3.14)
l +(awj2
3.3
l + (awr i )‘
Main Sources of Uncertainty
The sources o f error listed in Chapter 3.1 constitute only a part o f the uncertainty
associated with measuring materials in a waveguide. With due care, these uncertainties
can be minimized. Three main sources o f uncertainty are: sample thickness
measurements, uncertainty in the collected data from the VNA, and small gaps or facial
inconsistencies in the sample material.
3.3.1 Sample Thickness
While the acrylic backing layer is very rigid and milled to very high tolerances, the
unknown sample is often either grown directly on the acrylic, or prepared separately and
placed on the backing acrylic. When the sample is grown directly on the acrylic, the total
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32
material thickness is measured and the acrylic thickness subtracted to give the unknown
sample thickness. When the sample is placed on the acrylic, it can be measured only
when it is not so thin and flexible that the measurement process would damage it. In
many cases, even when the sample is grown without the acrylic, it is first only measured
once on the protective backing. In addition to being flexible, the samples often have a
pliable, rubber-like consistency, which makes measuring the material with calipers quite
challenging, as no pressure must be exerted on the sample. In many cases, two people
measuring the same sample arrive at slightly different results, differing up to 0 . 1mm.
With the high tolerance o f the backing acrylic, slight force was needed to place the block
inside the waveguide. Also, because of the waveguide arrangement, the structure was
placed in the waveguide acrylic side first. The two-layer structure was then slid even with
the calibration plane. This slight compression of the unknown material was distributed
uniformly by using a waveguide short-circuit to apply pressure to the structure until it
was precisely even with the calibration plane.
Both factors in sample handling give rise to an uncertainty in its thickness. To rectify this
situation, the sample thickness can be added as a variable in the optimization algorithm
with its initially measured thickness as the initial guess. A constraint is placed on the
envelope of possible thicknesses the optimization program may converge to; often set at
5 or 10%. Comparing the originally measured thickness with the converged-to thickness
is a good indication o f how well the optimization program performed. This, will be
discussed briefly in chapter S
3.3.2
Uncertainty in Vector Network Analyzer Data
Equations (3.15) and (3.16) represent general expressions for the uncertainty o f both real
and imaginary permittivity and permeability introduced by the measured scattering
parameters [I]. The brief derivation that follows evaluates complex permittivity
dependent on Sit. The real and imaginary uncertainty components can be evaluated
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33
separately by separating the complex term o f permittivity into its real and imaginary part.
To evaluate the uncertainty o f complex permittivity depending on both Sn and S21, the
square root o f the addition of the squared terms is taken. The same is applied to the
uncertainty o f the permeability.
v
rI _
**
(3.15)
AIS.
d0„
-r 1
_ 1
M' ri
V
ty'rl
' P
» * .
(3.16)
Mrt
where a = 11 or 21, superscript * stands for prime (‘) or double prime (“), A0 is the
uncertainty in the phase o f the scattering parameters, A|Sa| is the uncertainty of the
magnitude o f the scattering parameter. Both A6 and A|Sa| are provided in the System
Performance Specifications supplied with the VNA 8720C used Figure 7.7.
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34
Uncertainty of complex permittivity depending on S2 1 -
A£r, _
1
a*;, AS,,
a|s2,|
d 02l
(3.17)
w
V
1
a*;,
as,. as,,
as,, _ as,, ar, as,, azj
a^'“"af'ai^+"a^"ai^
1 2
3
(3.18)
4
Equation (3.18) has 4 unrelated components that have been labeled 1-4 and will be
evaluated individually; only the significant derivation steps will be shown.
(3.19)
^ = -2 r,r,r 2(i-r32)
ar,
|^ - =r 3(i- r,2)-(i- r22)- 2r,r,2(1- r32r22)
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(3.20)
35
2r,r,r3(1- r,2)- i r rr, (1- r 32r,2)
(3 .2 2 )
where m, U2,vi, and v2 can be found in equations (3.7), (3.8), and (3.9), respectively. Su
and S2i are found in equation (3.6). Ti, r 2, H , T| and T2 are stated in equations (3.2) and
(3.4).
where Ti and Z| are stated in equations (3.2) and (3.3)
Therefore:
» L -----------------(z, +Z 0)2TJo2Mr
d£r‘,
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(3.23)
where Ti and yi are given by equations (3.4) and (3.5)
Therefore:
(3.24)
3.3.3
Small Gaps in Unknown Sample
On occasion, test samples have been delivered with small facial inconsistencies, either
appearing as dents on the face of the sample or gaps straight through to the backing
acrylic. Depending on the material, preparation of a homogeneously thick sample may
not have been possible. Also, when the sample is made separately and placed on the
acrylic, it may not have exact dimensions. The sample may be up to 1mm too short in
height (see Figure 3.3), but more typically is about 0.1mm shorter than the acrylic.
1mm
Figure 3.3 A two-layer structure with the sample not perfectly matching the backing
acrylic (the waveguide height).
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37
Figure 3.3 illustrates the ‘worst-case* sample imperfection. Typically, as mentioned
previously, if any inconsistencies are present, they occur as small gaps between the
sample and the waveguide or dents located on the external face. Figure 3.4 shows a
typical dent configuration.
0.5mm
Figure 3.4 A two-layer structure with a small dent in the sample.
It is impossible to analytically determine the uncertainty produced by these
unsymmetrical gaps and dents in the sample surface. A Finite Difference Time Domain
(FDTD) simulation package has been used to model these inconsistencies in test samples
and the results are shown in chapter S. A complete list of simulated sample facial defects
is found in Chapter 3.4.2
3.4
Modeling of Sample Imperfections
The FDTD modeling method was chosen to perform the simulations for two main
reasons. The time domain approach enabled the gathering o f information required at all
frequencies of interest in one simulation. This was a significant advantage, as the
measurements entailed collecting data over the entire X-band. The second reason was the
availability of tested FDTD software that had been developed at the UVic
Electromagnetics Lab. As the structures simulated (waveguide, rectangular blocks o f
dielectric material) and the use o f Perfectly Matched Layers (PMLs) were previously
tested components o f this FDTD package, no additional test procedures were needed to
complete the modeling.
The FDTD program enables two ways of constructing the finite computational domain.
These include using a standard cubic / parallelepiped Yee-cell grid [19], [20] of constant
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38
volumes, or the graded mesh [19], [20]. The graded mesh allows for higher resolution
inside the computational domain in specific areas and lower resolution in the areas of
lesser interest. This enables the computing time of simulations, e.g. those performed for
this thesis, to remain reasonable. A two-dimensional FDTD approach was not feasible in
this case as transient and fundamental mode analysis is needed to ensure the propagating
waves interacted with the sample and acrylic in the same manner as in a real X-band
waveguide. Also, asymmetrical gap and facial inconsistencies are introduced in chapter S.
To simulate the waveguide, termination o f four of the six computational domain walls
was done with infinitely conductive electric walls. These four walls simulated the
waveguide walls along the x and y axis. To terminate the positive and negative z-axis, 9layer Berenger's PML with approximately -60dB reflection for normal incidence were
used. This amount was adequate as the VNA noise floor is approximately at -40dB, and
there was no need to sacrifice computational time to further reduce the reflection. Graded
mesh was used in two directions, namely the y and z-axis. It was needed in the y-axis to
increase the resolution near one edge o f the waveguide to simulate thin gaps. The cell
size for this direction varied from 0.5mm to 0.1mm with a maximum side-by-side cell
difference o f 11.56%. The graded mesh in the z-axis was needed as the samples were in
the range o f 0 . 1mm thick, and this resolution was not needed in the rest of the free-space
waveguide. The maximum side-by-side cell difference in the z direction was 2.5%. The
x-axis (modeled at 23mm in width) and the y-axis (modeled at 10mm in height)
corresponded well to the actual X-band waveguide dimensions of 22.86mm and
10.16mm, respectively.
The computational domain in the z direction needed to be relatively long for several
reasons. The first reason was higher order non-propagating evanescent waves generated
by the excitation. The distance between the excitation and the material had to be
sufficient to ensure that only the TEoi was present. The second reason was related to data
collection. From the excitation plane, the wave travels along the positive z-axis o f the
waveguide, it first encounters the Su data collection point that is recording the Ey and Hx
fields at every time step. The full excitation pulse must completely travel through this
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39
data collection point before reaching the sample and reflecting back in the negative z
direction. Both incident and reflected waves must be completely distinguishable to enable
splitting the time sequence o f the simulation, in order to determine the Sn parameter.
500 mm
968 voxels
23 mm
23 voxels
Figure 3.5 Graded mesh computational domain constructed for simulations of
waveguide.
Smallest voxel size = (1 x 0.1 x 0.1) mm
Largest voxel size = (1 x 0.5 x 1) mm
Number of voxels = 756976
The excitation used simulated the Ey field that exists in the waveguide for the TEio mode.
As shown in Figure 3.6, the Ey was excited with a sinusoidal amplitude pattern to
stimulate the TEio mode. The sinusoid was approximated with five equally space voltage
spikes across the waveguide, at 4mm, 8 mm, 12mm, 15mm, and 19mm. The field inside
the waveguide was then sampled at points further along the z axis to ensure that TEio (the
fundamental mode) was the only significant mode. The source was modeled as a ‘Soft
Source’. This is opposed to the more common ‘Hard Source’. Both types of sources
dictate an electric or magnetic field o f a certain amplitude in the computational domain at
a series of time steps. A hard source did not fit the specific criteria for the simulations, as
the wave needed to travel back through the excitation point o f origin in the negative z
direction once the excitation was zero (or turned off). A hard source is and remains a
short circuit, much like a physical antenna, even when turned off. This is not acceptable
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40
in a waveguide where metal wires have a significant impact on the propagation o f the
wave. A soft source, once turned off, becomes free space, as required.
0.608 0.924
1
0.924 0.609
Figure 3.6 Five soft sources used to excite the fundamental mode, TEio. in the
waveguide. Amplitudes have a sinusoidal amplitude from 0 to 1.
E at X . 2mm intervals inside waveguide
Pointe collected at S ,, Data Collection Point
Time Steps
X Component in Waveguide (mn
Figure 3.7 A cross section of the waveguide at z = 30mm throughout the entire
simulation. The fundamental mode is clearly visible, as well as the Gaussian Excitation.
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41
Comparison of fundamental Ey, Hx. and H . wilh non-fundamental Ex, E;, and Hy a t S ,, collection point
'
0
J
'Jl
'
■'
- - T- ■
1T11
I
1
.lif lj 1lU .
- ,Jli ,
jp 1
1
1
'
!
1
uTui'uT
111
1
2000
4000
6000
8000
10000
Time Steps
12000
14000
16000
16000
2000
4000
6000
10000
6000
Time Steps
12000
14000
16000
16000
Figure 3.8 Plot of three fundamental fields (Ey, Hz, Hx) in relation to non-fundamental
fields (Ex, Ez, Hy) collected at z = 15mm, which is 6mm away from excitation and
corresponds to the Su collection point.
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42
Comparison of fundamental Ey Hx, and H, wilh non-fundamental E(. E2, and Hy a t S 2, collection point
2000
4000
5000
8000
10000
Time Steps
12000
14000
16000
18000
.*10
^ 2
u. *
H.
,ii
I
-«*r
o
I-1
1-2
-3
o_
2000
4000
6000
8000
10000
Time Steps
12000
14000
16000
18000
Figure 3.9 Plot o f all three fundamental fields (Ey, Hz, Hx) in relation to non­
fundamental fields (Ex, Ez, Hy) collected at z = 480mm, which is 6mm away from
excitation and corresponds to the S21 collection point.
Excitation is accomplished with a Frequency Shifted Gaussian Pulse (FSGP) centered at
10GHz and approximately 4GHz wide. The pulse is created over 4000 time steps with a
discrete time difference (dt) o f 2.2354*1013 s. Figure 3.10c is the time domain plot of the
pulse, while Figure 3.10d displays the frequency domain plot. It is noted that the time
domain pulse starts at time step = I with an amplitude o f 0.0138, which is a deviation
from zero. This may produce higher frequency transients, however, they are very low in
magnitude. The frequency domain plot of Figure 3.10d shows the narrow band pulse
generated, the bars on the plot indicating the frequency band o f interest (8.2 - 12.4 GHz).
The time step numbers that correspond to the frequency band bars are explained in
chapter 3.4.1
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Figure 3.10 Plot A shows the Cosine waveform that is multiplied to Plot B, which is the
pure Gaussian pulse. Plot C, shows complete time domain Cosine Gaussian Pulse
generated over 4000 time steps, Plot D displays the narrow band frequency pulse created
with 2 18 points (mostly zero-padded) which places 8.191 - 12.4 GHz between discrete
points 480 and 727 with a 17.1 MHz frequency step (df).
A total o f 18000 time steps were used which corresponded to a total simulation time of
40.24ns with a dt o f 2.2354xl0'13 s.
3.4.1
VNA Simulation
In order to manipulate the data to simulate data collection from the VNA, calibration o f
the FDTD S-parameters is essential. Standard calibration equations are used after four
specific FDTD simulations are completed; three simulations are for Sii and one for S21.
After determining where the calibration plane had to be placed inside the modeled
waveguide, the first simulation was a short circuit. The calibration plane was located on
the z-axis value where all the subsequent materials to be measured would be placed. For
all simulations, this value was set at z = 250mm with the total z-axis length o f the
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44
waveguide being 500mm. Also, before calibration, all observation and excitation points
had to be determined and remain unchanged, so as not to change the phase angle of the
collected Su and S21 parameters.
The calibration process for the Su parameter involves three specific simulations [21].
These three simulations are required to create a determined system of three equations
with three unknowns. The three vector unknowns in the system are
E d f, E sf, E r f
which
relate to three specific calibration requirements for a VNA. The Directivity Error, Edf,
compensates for the power that is reflected back to the VNA, by devices such as couplers
and transitions, before it reaches the sample. The Source Match Error,
E sf,
compensates
for the imperfect transitions from the system back to the source. A small portion of the
signal is reflected back into the system and interacts with the sample for repeated times.
The Tracking Error,
E rf,
is caused by variations in magnitude and phase flatness versus
frequency between the test and reference signal paths. Equation (3.25) shows the three
main uncertainty parameters in a system [21], where
Sum
is measured S u , and
Sua
is
actual. Equation (3.27) shows the system equation with three unknowns. Three
circumstances exist where the S ua is known and they are outlined next. The first, a
perfect electric conductor (PEC) is used for calibration plane termination. The
information gathered from this simulation is given the label Susc, where the
Sua
is
known to be -1 (or magnitude 1, phase 180°) and corresponds to equation (3.26)a. The
second simulation is an open circuit. This is not possible in practice, but can be taken
advantage o f while modeling. A perfect magnetic conductor (PMC) is placed at the same
location as the short circuit and maintains an
Sua
of 1 (or magnitude 1, phase 0°)
throughout the entire frequency band. This corresponds to equation (3.26)b. A small
compensation is added to the gathered data, as the PMC specified at a certain interface in
FDTD, actually exists half way between the voxels. Therefore, to compensate for the
slight shift in phase that would occur, 0.05mm o f phase shift is added to the collected Su
data (for the grid size o f 0.1mm). The data collected from this calibration run was labeled
Suoc- The third and final calibration procedure is the matched load. This was
accomplished by placing the PML discussed in chapter 3.4 exactly at the calibration
plane. The third procedure, labeled S u mi, is not very critical in FDTD simulation,
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45
especially with the simple setup being discussed, and the result is very close to zero, this
corresponding to equation (3.26)c.
s ,, u
= e „ + 1 t.SF • 01I/(
<
General Form o f Calibration Equation
S
C
=E
i ( 1) ' e kf
^ l - £ s f (-l)
(a)
_
C
_
—/T
s
_£
+
^
l- £ * .( l)
(b)
3-25>
(3.26)
^
x 0 • E rf
***
1 - £ JF o
(c)
Once the three vectors are established, S ua is solved for to produce equation (3.27). Any
measured Si i is now applied to this equation before analysis.
S UA = ----E s f C^i t Af
(3.27)
^
df
) "F E
rf
Calibration o f S21 only involves one simulation, namely a full waveguide with no sample
inside and no discontinuities at the calibration plane. Once data for calibration and data
for an unknown material have been collected, relevant data are selected from the
observation record. After the truncation is completed, a Fast Fourier Transform (FFT) is
performed on both data arrays. Zero Padding is utilized to increase frequency resolution
(df decreases). The equally sized arrays are then divided. This gives the magnitude and
phase difference data, as both arrays are complex. An Inverse Fast Fourier Transform
(IFFT) is applied to the array to obtain time domain data.
FFT(Sl m ) N
S Ma = IFFT FFT(Sl m ) /
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(3.28)
46
Since the purpose o f simulating the measurement procedures in the FDTD is to mimic the
actual collected data from the VNA, the data collected from the FDTD will have to be
manipulated to be as similar as possible to the VNA output. Two programs were written
to condition the modeled data, one for Su and one for S21. The FDTD output produces a
spectral data set o f 4096 magnitude and phase output points, while the VNA produces a
set of 401 data points. Upon inspection, approximately 3600 points of the data set from
the FDTD program were within the X-Band frequency. For Su after calibration, out o f
these -3600 points, every ninth was recorded into a text file in the same format as the HP
VNA output. This is a three-column set o f data containing the frequency (from 8.212.4GHz), magnitude and phase arrays.
To condition the S21 output, a more complicated procedure had to be constructed. As the
current state o f the data contained the frequency response from 0 to 2236.8GHz (a result
from a 218 zero padding and a dt = 2.23535034e-l3), only a very small portion of this
array was needed. Based on the highest frequency of 2236.8GHz and the array size, this
corresponded to a discrete frequency step of df = 17.1 MHz. It was determined that array
steps 480 and 727 were the lower and upper points of the X-Band frequency range of
interest. Once a frequency array was established that contained the S21 data in (727480=) 247 points, and the associated calibrated magnitude and phase S21 data were
linked, a polynomial is fit to this data. A very high degree polynomial (20) was used, as
computation time or memory limitations were not an issue. Two high degree polynomials
now existed, both in the form y = C + C|X + C2X2 + ... one o f S21 magnitude the other of
phase, both with frequency corresponding to x. From the polynomial, the SI 1 frequency
array was systematically inserted as the values for x in both polynomials, and the
magnitude and phase of S21 recorded.
Two files with exact frequency arrays and corresponding S-Parameters now exist,
simulating the VNA. Once all other relevant data is inserted into the files, the data is
ready to be processed with the same procedure as standard VNA data.
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47
3.4.2
FDTD Evaluation of Gaps and Facial Inconsistencies
Simulations were conducted with the l st-layer (the unknown material) having symmetric
and asymmetric inconsistencies. The configurations are illustrated in Figure 3.11. Results
o f simulations are in Chapter S.3.
0.5mm
0.1mm
9.9mm
□
O
0.5mm
□
.9 mm
O
0.5mm
□
Figure 3.11 Facial inconsistencies modeled; Gaps are on the left, dents on the right.
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48
Chapter 4
4
Results of Uncertainty Analysis and Measurements
The electrical properties o f the test samples varied with e r ranging from close to 1 to
several hundred, while e r ranged from ~0 to several hundred. With materials having also
different electrical characteristics within each batch, an intelligent initial guess for the
optimization algorithm was not possible.
Not all results from the measurements are presented, for several reasons. Many o f the
samples received were too thin and lossless, too inhomogeneous, or o f a thickness
resulting in high uncertainty. Results shown are for samples that have acceptable
‘perceived’ uncertainty. ‘Perceived’ uncertainty is determined based on consultation of
uncertainty plots of thickness o f both layers versus complex permittivity, perceived
thickness measurement accuracy and acceptable deviation in terms of sample thickness
and facial inconsistencies (dents and gaps). Static conductivities have been measured for
several o f the materials at their point of manufacture and is plotted on several graphs (as
converted to the DC conductivity component of the imaginary permittivity). Equation
(4.1)[18] portrays this relationship and its definition is explained in detail in chapter 7.1.
€'
^d c _ + ^ac_
(4J)
coe0 (oeQ
4.1
Uncertainty Analysis
The thicknesses of the sample and backing acrylic both have a profound effect on the
uncertainty of the measurements. The value o f permittivity also has a large effect on the
uncertainty. While the thickness is determined before any measurements are made,
optimizing the thickness for the particular complex permittivity of an uncharacterized
material is impossible. Therefore, minimizing the potential uncertainty is the goal when
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49
choosing a sample and acrylic thickness for the initial round o f measurements of a new
sample.
In the following sections, only realistic (practical) sample thickness and complex
permittivity ranges are analyzed. Typical sample thickness has been in the order of 0.1 1.0mm, although several successful manufacturing attempts at a thicker sample have
been made. The thickest sample analyzed for uncertainty is 3.0mm. Acrylic thickness is
treated similarly. The minimum thickness available is 1.2mm, while acrylic thicknesses
over 5.2mm become too difficult (awkward) to insert into the waveguide.
All results of the uncertainty analysis presented in chapters 4.1.1 - 4.1.3 assume ideal
samples with no facial inconsistencies. Four combinations of property characteristics
have been chosen as a representative of the samples tested. The characteristics of these
four materials are summarized in Table 4.1.
Table 4.1 The dielectric constant and loss factor of the four materials analyzed.
Material
1
2
3
A
Real Permittivity
Imaginary Permittivity
( E ’r)
( E ” r)
80
20
100
3
-10
-40
-200
-0.0001
Data in Table 4.1 do not show separately static (DC) and AC conductivities, only the
imaginary permittivity (loss factor). The static conductivity is measured directly after the
sample is manufactured and can be subtracted from the effective conductivity if desired,
in post-processing. All of these samples and subsequent plots assume non-magnetic
materials (|i’r = 1, M-’V = 0).
Uncertainty o f real and imaginary parts of the permittivity determined from both S| i and
S21 will be displayed for Material 1. After the analysis, it became apparent that for non­
magnetic materials is was unnecessary to include the uncertainty plots due to the
combined uncertainty of the S-Parameters.
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50
Chapters 4.1.1. - 4.1.4 analyze the materials 1,2,3, and A, listed in Table 4.1,
respectively. Three different uncertainty plots are presented, the unique format o f which
are beneficial for the analysis. All contain the uncertainty of either e’ or e” due to Si i and
S21 (or due solely to S21) on the vertical axis. The plots are as follows.
Plot Type I: Three-dimensional plots with sample thickness and acrylic thickness as
variables on the opposing axes (e.g. Figure 4.1, Figure 4.2). These plots are for a specific
material (e.g. £’ = 80, e” = 10) at a single frequency (e.g. f = 10GHz). The thickness of
the sample material varies from 0.1mm-3mm, while the acrylic varies from 1mm to 5mm.
Although the plots include sample thicknesses greater than 1mm, these are very rarely
manufactured in practice. Measurement uncertainties of many materials are reduced
when their thickness is increased beyond 1mm (as will be shown in Chapter 4.1.1. 4.1.4.), therefore including these sample thicknesses that may possibly be attainable in
the future is useful.
Plot Type 2: Pseudo three-dimensional plots of uncertainty vs. sample thickness with
view oriented directly down the frequency axis (8.2-12.4GHz) (e.g. Figure 4.3, Figure
4.4). These are given in a series of four plots of increasing acrylic thickness (1mm, 2mm,
4mm, 5mm) These plots are very useful as the uncertainty for many materials varies
substantially over the X-band. Each plot shows the entire envelope of uncertainty over
the frequency range of interest.
Plot Type 3: Two-dimensional plots containing five X = e’/ e” ratios. In these plots, e’r
increases from 1 - 1000, while frequency remains at 10GHz, and sample and acrylic
thickness remain constant. Values for X were chosen to be 0.2, 0.5, 1, 2, and 5, as these
values best represent many measured values.
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51
4.1.1
Material 1, Dielectric Constant = 80, Loss Factor = 10
eP • #0 ePO
„ • 10. sample
thickness 0.1-3mm. mstensi thickness
^
freq. 10GHz
0.9 .
0.4.
0.1
1
Thickness of matenal (mm)
Thickness of Acetic (mm)
Figure 4.1 Uncertainty o f the dielectric constant (e\) due to errors in measurements o f
Sii and S21
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52
Cp — 8 0 6pp — 10,
sample thickness 0.1-3mm. material thickness 1-Smm. freq. 10GHz
Thickness of matenal (mm)
Thiekness of Acrylic (mm)
Figure 4.2 Uncertainty of the loss factor (e” r) due to errors in measurements of Su and
S21
crQ• 00 er„Q• 10 Freq • 9 - 12 GHz Sample thickness • 0.1 - 3mm
0.3 r
2
1
Thickness of Sample
xIO J
Thiekness of Somple
Thickness of Sample
x 10 '
Thickness of Sample
2
1
0
x
0
s iq °
Figure 4.3 Uncertainty of the dielectric constant (e’r) due to errors in measurements o f
Su and S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom
right).
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53
e r • ?? er
• ?? Freq > 9 - 1 2 GHz Sample thickness • 0.5 - 3mm
Thickness of Sample
110~3
Thickness of Sample
-J
3
Thickness of Sample
x
2
1
Thiekness of Sample
0
* ro 3
Figure 4.4 Uncertainty o f the loss factor (e” ) due to errors in measurements o f Sn and
S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
ep • 90 c9Q • 10. sample thickness 0.1-3mm. material thickness 1-5mrn. freq. IOGHi
0.1
1
Thickness of material (mm)
Thickness of Acrylic (mm)
Figure 4.5 Uncertainty o f the dielectric constant (e’r) due to errors in measurements of
S21
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54
eQ■ 80
• 10. sample thickness 0.1-3m m material thickness 1-5mm. freq. 10GHz
Thickness of matenal (mm|
Thickness of Acrylic (mm)
Figure 4.6 Uncertainty of the loss factor (e” r) due to errors in measurements of S21
erp • 60 er_Qa 10 Freq • 9 - 12 GHz Sample thickness ■ 0.1 - 3mm
0-1 r
2
Thickness of Sample
%
1
Thickness of Sample
0
* io*'J
0.1 r
€ 0.05 r
! 0.0S
2
Thickness of Sample
x 1Q-3
1
Thickness of Sample
0
x iq ' 3
Figure 4.7 Uncertainty o f the dielectric constant (e’r) due to errors in measurements o f
S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
erp ■ BOCTpp • 10 Freq • 9 - 12 GHz Sample thickness • 0.1 - 3mm
■
2
0.9 r
m
1
Thickness of Sample
0
0
IM
3
Thickness of Sample
, io '3
3
TNckness of Sample
Thickness of Sample
Figure 4.8 Uncertainty of the loss factor (e” r) due to errors in measurements o f S 21.
Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
Both Figure 4.1 and Figure 4.2 show relative uncertainties due to errors in measurement
of Su and S21 - Although the uncertainty decreases down to between 5% and 15% when
the thickness of the sample is 1.6mm, for other thicknesses, it is unacceptably large. For
this sample, a very thin sample coupled with a thin acrylic is beneficial, as the uncertainty
rapidly decreases for real and imaginary permittivities near the smallest values.
The uncertainty is unacceptably high mostly because of the Si 1 uncertainty term. After
analysis o f several other plots of various materials, it was determined that the inclusion o f
Su uncertainty into the optimization algorithm contributed more uncertainty to the
convergence than was beneficial from the extra data points. The Su uncertainty cannot be
eliminated when the material is expected to have magnetic properties, but for materials
with |i.’r = 1, only Su or S21 data suffices. Uncertainties are presented for material 1 that
include Su and S21, but for the remaining materials, only the uncertainty o f S21 is given.
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56
Although acceptable uncertainty can be achieved using the parameters from Figure 4.1
and Figure 4.2 because prior knowledge o f the material properties is not available, very
high uncertainty would likely result.
Figure 4.3 and Figure 4.4 again show the uncertainties o f both Sn and S21. If the main
priority for this sample were to determine only the dielectric constant, then maintaining a
sample thickness greater than 1.6mm ensures an uncertainty o f less than 17%.
Unfortunately, the uncertainty for the imaginary permittivity is very large in this case for
a large sample thickness. There is an abrupt drop in uncertainty when the sample is
approximately 1.6mm but the slopes on either side o f this value are very steep and
achieve a value o f 100% very quickly.
Figure 4.5 and Figure 4.6 show the uncertainty of the same material when only S21 data
are used. The uncertainty is much lower when compared to Figure 4.1 and Figure 4.2,
respectively. Several combinations of thicknesses o f the sample or acrylic are acceptable
for the dielectric constant. The maximum and minimum errors occur for various sample
and acrylic thicknesses. Their locations, however, cannot be easily explained by the
relative (to the guided wavelength) sample and acrylic thickness. The relevant
thicknesses are given in Table 4.2. The wave undergoes multiple reflections at the three
interfaces, thus a quarter and half-wave transformer analogy cannot be used. However,
the minima and maxima o f the uncertainty can be easily explained on the polar plots
shown in Figure 4.9. With reference to Figure 7.7 in the appendix, it is clear that the Si 1
uncertainty is very large with approaching a magnitude value of 1. Both Figure 4.9 and
Figure 4.10 show Su approaching 1 for specific thicknesses of sample.
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57
S „ and Sg, at 10GHz, for a matarial with a rp = 80 and a rpp = 10
120
0.8
0.6
\3 0
150
Blue
Green
Purple
Cyan
n
■»' acryUc - 1mm
- acrylic - 2mm
• acrylic * 4mm
- acryliq * 5mm
180
330
210
240
270
Figure 4.9 S-parameter plot o f two-layer structure at the single frequency o f 10GHz.
Sample thickness varies from 0.01mm to 3mm
Magnitude of both S11 and S21 at 10GHz
for a matarial with a r_ = 80 an d e rnn = 10
osj0.8
is 0.7 i
Blue
Green
Purple
Cyan
-
acrylic * 1mm
acrylic*2m m
acrylic*4m m
acrylic • 5mm
0.3
0.1
I
1.5
2
Thickness ot A crylic in mm
2.5
Figure 4.10 The magnitude o f both S-parameters o f the two-layer structure at 10GHz,
with sample thickness varying from 0.01mm to 3mm.
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58
Table 4.2 Displaying cut-off frequency, and guided half-wavelength, and guided
quarter-wavelength distances for the X-band.
Free-Space (Inside Waveguide)
Material 1 (e’r = 80)
(Inside Waveguide)
Acrylic (e’r = 2.65)
(Inside Waveguide)
fc (GHz)
6.56
X,. (mm)
40
0.5A* (mm)
20
0.25A* (nun)
10
0.734
3.4
1.7
0.85
4.03
20
10
5
Figure 4.7 and Figure 4.8 show the uncertainties due to only S21. It can be seen that the
majority o f the values are much lower than when Su is included, as noted previously.
The high uncertainty region is now visible and exists (Figure 4.7) for a sample thickness
o f between 1 - 1.4mm. However, the uncertainty does not exceed 8% for any sample or
acrylic thickness at any frequency. Very low uncertainties are obtained for samples
thicker than about 1.5mm. Also, as desired for this work, low uncertainty is obtained for
thicknesses of 0.1 -1 mm with 1-2mm acrylic.
4.1.2
Material 2, Dielectric Constant = 20, Loss Factor = 40
Figure 4.11 to Figure 4.14 present the results o f the uncertainty analysis based on S21 data
only.
Material 2 is very consistent in terms o f trends in uncertainty values for both the
dielectric constant and the loss factor. This makes the optimum thickness obvious. As
seen in Figure 4.11, the uncertainty o f the dielectric constant steadily decreases with
increasing sample thickness, and increases slightly with acrylic thickness.
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59
ep • 20
• 40. sample thickness 0.1-3mm, material thickness 1-5mm, freq. 10GHz
TNckness of material (mm)
TNckness of Acrylic (mm)
Figure 4.11 Uncertainty of the dielectric constant (e’r) due to errors in measurements o f
S21
e • 20 c
D
0.1
3C
• 40. sample thickness 0.1-3mm, matenal thickness 1-5mm» freq 10GHz
j
I
i
Thickness at material (mmt
Thickness at Acrylic (mm)
Figure 4.12 Uncertainty of the loss factor (e’\ ) due to errors in measurements of S21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
er„ • 20 er^ • 40 Freq * 9 - 1 2 GHz Sample thickness ■ 0.1 - 3mm
0.1
O.OS
I 0.05
3
Thiekness of Sample
x io ’3
TNckness of Sample
* iq ' j
TNckness of Sample
x io*j
0.1
I.0.0S
0.09
Thiekness of Sample
Figure 4.13 Uncertainty of the dielectric constant (e’r) due to errors in measurements of
S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
erp ■ 20 erCO <• 40 Freq • 9 -12 GHz Sample thehneas • 0.1 - 3mm
f
fgO.OS
3
| 0.05
0
Thickness of Sample
TNckness of Sample
x io '2
* iq -
0.1
£
| 0.05
I 0.05
3K
0
TNckness of Sample
%
TNckness of Sample
Figure 4.14 Uncertainty of the loss factor (e”r) due to errors in measurements o f S21.
Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
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61
The envelope o f uncertainty over the entire frequency range (Figure 4.13 and Figure
4.14) remains nearly constant with changing acrylic thickness. If only very thin samples
are considered, a thin acrylic thickness is preferred. Thus, an acrylic thickness of
approximately 1.Omm is optimal for samples thinner than 0.5mm. This advantage is very
small. For material 2, measurements benefit from a sample thickness in excess of 0.7mm,
and placed on the thinnest acrylic available, e.g. 3mm or 1.2mm.
Samples o f the permittivity similar to Material 2 were quite common and usually had
thicknesses o f approximately 0.2 - 0.5mm. A concerted effort was made to increase this
thickness once these plots were analyzed.
4.1.3
Material 3, Dielectric Constant = 100, Loss Factor = 200
This material is similar to the material of chapter 4.1.2 and was also very common. Once
again, a thick sample with a thin acrylic is the optimal configuration, as apparent from
Figure 4.17 and Figure 4.18.
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62
ep *100
■ 200. sample thickness 0.1-3mm. matenal thickness 1-Smm. freq. 10GHz
TNckness ol matenal (mm)
TNckness of Acrylic (mm)
Figure 4.15 Uncertainty o f the dielectric constant (e’r) due to errors in measurements o f
S21
e„C> 100 e 9
_C• 200. sample thickness 0.1-3mm. material thickness 1-5mm, freq 10GHz
0.1
-
I 0.0S 4
3
TNckness of material (mm)
TNckness of Acrylic (mm)
Figure 4.16 Uncertainty o f the loss factor (e”r) due to errors in measurements o f S 21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
erp • 100 er^ • 200 Freq • 9 -1 2 GHz Sample tfvckness • 0.1 - 3mm
0.1
1 0.05
I 0.05
Thidinejs of Sample
x 10
0
Thiekness of Sample
x 1fl--
Thickness of Sample
t
0.1
£c■
1 O.OS
a
£
| 0.05
3K
TNckness of Sample
x
Figure 4.17 Uncertainty of the dielectric constant (e’r) due to errors in measurements of
S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
erp • 100 erx • 200 Freq ■ 9 - 12 GHz Sample thickness « 0.1 - 3mm
0.1
9
I 0.05
O.OS
Thiekness of Sample
TNckness of Sample
x i q°
Thickness of Sample
x lfl-i
*c
i 0.05
1 0.05
TNckness of Sample
xIO'
Figure 4.18 Uncertainty o f the loss factor (e”r) due to errors in measurements of S21.
Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
A thick sample of such a high-dielectric constant material may produce smaller
measurement uncertainty. These uncertainty equations do not consider over-moding
within the sample. Other modes that may be excited upon transition between propagation
mediums and which are evanescent everywhere [18], but inside the sample, may
contribute additional convergence error.
4.1.4
Material A, Dielectric Constant = 3, Loss Factor = 0.0001
Figure 4.19 to Figure 4.22 present the results o f the uncertainty analysis for this material.
The uncertainty of the loss factor of this material is very large unless the sample is
sufficiently thick. This is not unexpected, as any deviation from 0.0001 (considering an
accuracy o f 4 decimal places collected from the VNA) will result in over 100%.
Therefore, for this material, plots o f the loss factor can be used as a guide and cannot be
considered accurate.
e* • 3
• 0. sample Vtiekness 0.1-3mm. material thickness l-Smm. frcq. 10GHz
0.6 .
0.2.
2.5
1.5
0.5
Thickness of material (mm)
0.1
Thickness of Acrylic (mm)
Figure 4.19 Uncertainty o f the dielectric constant (e’r) due to errors in measurements o f
S21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
e • 3 e
• 0. sample OiicKness 0.1-3mm, material thickness 1-Smm, freq. 10GHz
Thickness of material (mm)
Thickness of Acrylic (mm)
Figure 4.20 Uncertainty of the loss factor (e”r) due to errors in measurements o f S21
er9 • 3 ercc • 0 Freq > 9 - 1 2 GHz Sample thickness • 0.1 - 3mm
0.1
i
! 0.0s
loos
2
1
Thickness of Sample
x10
Thickness of Sample
Thickness of Sample
x
Thickness of Sample
•3
0
0.1
f°«
x
Figure 4.21 Uncertainty of the dielectric constant (e’r) due to errors in measurements of
S21. Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
erp • 3 er^ ■ 0 Freq * 9 - 1 2 GHz Sample thickness • 0.5 - 3mm
0.5
0.5
0.2
0.2
0.1
0.T
Thickness of Sample
2
x
1
Thickness of Sample
Thickness of Sample
0
x
2
1
Thickness of Sample
*io*3
x
0
Figure 4.22 Uncertainty o f the loss factor (e”r) due to errors in measurements o f S21.
Acrylic = 1mm (top left), 2mm (top right), 4mm (bottom left), 5mm (bottom right).
Samples o f this material with thickness greater than 1mm can be measured with
uncertainty o f 5% or less, if acrylic thickness is approximately 1mm.
Figure 4.21 and Figure 4.22 clearly show increasing uncertainty of measurement with
decreasing sample thickness. In both cases, the envelope also ‘thickens’ with increased
acrylic thickness. For this material, a sample o f over 1.5mm backed by the thinnest
available acrylic (1.2mm) would be optimal and produce a measurement with uncertainty
o f less than 5%. Unfortunately, most very low-loss samples measured had a sample
thickness o f less than 0.3mm resulting in very large uncertainty.
The uncertainty analysis presented so far indicates that for the non-magnetic materials
having the dielectric constant above 10 and loss factor typically greater than 20, the
optimal acrylic thickness is roughly 1mm. Therefore, the analysis that follows is limited
to that thickness. Data for another acrylic thickness is given in Appendix 0. It should be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
noted, that most o f the measurements o f actual materials were made with acrylic
thickness equal to 1.2mm. Only some measurements made before the uncertainty analysis
had been completed were made with thicker acrylic.
4.1.5
Summary Uncertainty Plots
The plots of this section are intended to give an overall picture of a large collection o f
sample measurement uncertainties depending on thickness and acrylic thickness. These
plots proved very useful to quickly ascertain an approximate uncertainty associated with
any material measured. Each plot shows single frequency data, with a constant sample
and acrylic thickness, with varying dielectric constant, which is related by a specific
constant ratio to the loss factor. All figures contained in this section are at a frequency o f
10GHz, and an acrylic thickness o f 1mm (data for an acrylic thickness o f 3mm can be
found in Appendix 0). The frequency o f 10GHz was chosen, as it is approximately
central in the X-band frequency range, and gives a good approximation for the whole
range. Sample thicknesses are 0.1mm, 1mm, and 2mm.
Figure 4.23 and Figure 4.24 show the uncertainty for a very thin (0.1mm) sample. It is
seen clearly, that for any dielectric constant of less than 20, the uncertainty is very large.
The uncertainty has an inverse relationship to the ratio e’r / e”r. For e’ = 0.2 x e” the
uncertainty in e ’r is large, while it is small for e”r. This trend is consistent throughout the
results, such that when a large difference exists between e’ and e”, and e’ is between I
and ~50, then there are large differences in the corresponding uncertainties.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
frequency - lOOfz. sample ttvekneas • 0.1mm. material thickness ■ 1mm
0.1
er0 • 2crBC
er • 5*er
0.0S
100
1000
e."o
Figure 4.23 Uncertainty o f the dielectric constant due to errors in measurements o f S 21.
Five different ratios of material at 10GHz, a sample thickness of 0 .1mm and an acrylic
thickness of 1mm.
frequency - 10GHz. sample tK k n ess ■ 0 1mm. m atenal thickness • 1mm
• 2erM
er • 5*er
f
I
IV aosi-I
I
!
100
1000
"3
Figure 4.24 Uncertainty o f the imaginary permittivity due to errors in measurements of
S21. Five different ratios o f material at 10GHz, a sample thickness o f 0.1mm and an
acrylic thickness o f 1mm. The erp= 5 x erpp has uncertainty too high to be recorded in this
plot.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
0.1
frequency - 10GHz. am ple thickness • 1mm, metenal thickness ■ 1mm
*
2Cf«
0
90
er • 5*er
f
0.06
3
100
1000
e.s
Figure 4.25 Uncertainty of the dielectric constant due to errors in measurements of S21.
Five different ratios of material at 10GHz, a sample thickness o f 1mm and an acrylic
thickness o f 1mm.
frequency - 10GHz. sample thickness • 1mm. materiel thickness • 1mm
0
er . 2er
..80
er m5*er
f
i
0.05 (-
c.
’3
Figure 4.26 Uncertainty o f the imaginary permittivity due to errors in measurements of
S21. Five different ratios of material at 10GHz, a sample thickness o f 1mm and an acrylic
thickness o f 1mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
With an increasing sample thickness, materials o f lower dielectric constant can be
measured with a greater degree o f certainty. Figure 4.25 and Figure 4.26 both show that
an uncertainty of less than 5% is realizable for some materials even with a dielectric
constant as low as 10. The red line on both plots indicates a material with the same real
and imaginary permittivity components at 10GHz. With a dielectric constant and loss
factor o f 7, this material measurement uncertainty is an acceptable 5%.
frequency - 10GHz. sample thickness ■ 2mm. matents! thickness ■ 1mm
era • -2cr
.s o
er.. fier
o.osr
100
e.3
Figure 4.27 Uncertainty o f the dielectric constant due to errors in measurements of S21.
Five different ratios o f material at 10GHz, a sample thickness of 2mm and an acrylic
thickness of 1mm.
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71
frequency - 10GHz. sample thickness • 2mm. material thickness ■ 1mm
0.1
er • 2er
er mO'er
0L
1
100
10
e3
Figure 4.28 Uncertainty o f the imaginary permittivity due to errors in measurements o f
S21. Five different ratios of material at 10GHz, a sample thickness of 2mm and an acrylic
thickness o f I mm.
When the sample thickness is 2mm, as in Figure 4.27 and Figure 4.28, the ratios of E’r =
0.2 x e”r and e \ = 0.5 x e”r start to show a similar trend in both plots, once the dielectric
constant reaches 100 (the imaginary permittivity is 500 and 200, respectively). The high
uncertainty is now due to too much wave attenuation by the sample for the VNA data to
be accurate. This is in contrast to the results in Figure 4.23 and Figure 4.24 where the
attenuation was not large enough to alter the S21 magnitude or phase data from an empty
‘through’ connection. For the cases of Figure 4.23 and Figure 4.24, the reason the
uncertainty is kept below 100% when a sample is very thin and lossless is the backing
acrylic, which provides additional attenuation.
Figure 4.29 shows close-up views of the results of Figure 4.23 through Figure 4.28 for
the dielectric constant = 100. Here, a sample thickness o f 1mm is a good compromise for
any loss factor except e”r = 20 (as it is out o f range for the e”r uncertainty plot for this
thickness). For very thin samples of 0.1mm, uncertainties below 5% for both parts o f the
permittivity are only obtained for e’r = e”r. All thin samples (0.1mm) however, can be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
72
measured with uncertainty of less than 10% for 50 < e”r < 200 (likely 300). Thicker
samples (2mm) show very low uncertainty for all e”r.
Uncertainly of
0.1mm 1mm 2mm
Uncertainty of #rpp
0.1mm 1mm 2mm
•rp = 100
0.05
•rpp =500
•rpp = 200
•rpp = 100
•rpp = 50
erpp = 20
\
0.05
I
\\
X
ou
J
/
\
..
i1
100
. t
.
100
100
Figure 4.29 Uncertainty of both the real and imaginary components o f permittivity at a
dielectric constant o f 100 and a frequency of 10GHz. Acrylic thickness for all plots is
1m m . e,p = e ’r, e^ p = e ” r
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73
Summary o f Material Uncertainty
Table 4.3 Sample thickness = 0.1mm, acrylic thickness = 1mm, frequency = 10GHz
Ratio Coeff.
X
(e’r = Xe”r)
0.2
0.5
1
2
5
Uncertainty o f e”r
Uncertainty o f e ’r
< 5%
N/A
e’r >200
e’r >54
e ’r >38
e ’r >32
< 10%
N/A
e’’r >400
e”r >54
e”r >19
e”r>6.4
N/A
e ’r >22
e ’r >29
e ’r >28
e ’r >28
N/A
e”r >44
e”r >29
e” >14
e”r>5.6
< 10%
< 5%
e’r >9.0
e’r >22
e’r >54
e’r >300
N/A
e”r >45
e”r >44
e”r >54
e”r >150
N/A
e ’r >4.0
e ’r >8.5
e ’, >29
e ’r >51
N/A
e"r >20
e”r >17
e”r >29
e”r >26
N/A
Table 4.4 Sample thickness = 1mm, acrylic thickness = I mm, frequency = 10GHz
Ratio Coeff.
X
(e’r = Xe”r)
0.2
0.5
1
2
5
Uncertainty o f e”r
Uncertainty o f e ’r
<5%
e ’r >52
e 'r >18
e ’r >6.3
e’r >5
e ’r >4.4
<5%
< 10%
e”r >260
e”r >36
e”r >6.3
e”r >2.5
e”, >.88
e ’r >10
e ’r >3.4
e ’r >3
e ’r >2.1
e ’r >1.4
e”r >50
e”r >6.8
e”r >3
e”r > l.l
e”r >.08
e’r >l
e’r >3.4
e ’r >6.3
e’r >120
e’r >190
< 10%
e”r >5
e”r >6.8
e”r >6.3
e”r >60
e”r >38
e ’r> l
e ’r >l
e ’r >3
e ’r >4.4
e ’r >180
e”r >5
e’’r >2
e”r >3
e”r >2.2
e”r >36
Table 4.5 Sample thickness = 2mm, acrylic thickness = 1mm, frequency = 10GHz
Uncertainty o f e’V
Uncertainty o f e ’r
<5%
e ’r >12
e ’r >5.9
e ’r >3.2
e’r >1.8
eV >1.7
e”r >60
e”r >10
e”r >3.2
e”r>0.9
e”r >.34
< 10%
e ’r >3.1
e ’r > l.l
e ’r >1
e ’r >1
e ’r >1
e”r>15
e”r>2.2
e”r> l
e”r >0.5
e”r>.2
<5%
e’r >1
e’r>l
e’r >3.2
e’r >22
eV >49
< 10%
e”r >5
e”r >2
e’V>3.2
e”r> U
e”r>10
e ’r> l
e ’r >l
e ’r > l
e ’r >2
e ’r >35
e’V>5
e”r >2
e”r> l
e”r> l
A
k.
w
Ratio Coeff.
X
(e’r = Xe”r)
0.2
0.5
1
2
5
Summary data are given in Table 4.3 - Table 4.5 as the sample thickness increases from
0.1mm to 1mm to 2mm, while the acrylic thickness and frequency remain unchanged.
The ideal pair o f values, e’r > 1 and e”r > 0, would mean that any material would be
below the specific uncertainty. It is clearly seen that as the thickness o f the sample
increases, the required pairs o f values for £’r and e”r decreases, with the best result for a
sample thickness of 2mm and ratio coefficient of 1.
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74
4.2
Measurement Results
The first set of plots (Figure 4.31 through Figure 4.34) shows measurement results for
several materials for which consistent results have been attained. Throughout the entire
set of figures in this section, a dark blue line indicates the dielectric constant (e’r), while
the dotted red line shows the imaginary permittivity (e”r). Several measurements o f each
material were taken to show repeatability and to observe trends in the characteristics
(expected decrease o f conductivity) o f the sample. For many materials degradation
occurred with time resulting in a decrease of effective conductivity. A green uncertainty
envelope is shown for the ‘Original’ measurement but is left off subsequent
measurements. All results are close (in uncertainty) to the original, so there is no need to
show uncertainty limits for all measurements, as this would needlessly clutter the plots.
Dates are given for each measurement, as this is important for the developers o f these
materials.
At the time of writing this thesis, batches of materials were still periodically delivered to
the lab for characterization. To categorize results, a batch number was assigned to each
delivery. A total o f 33 batches o f 3 to 30 samples each had been delivered. This
collection o f results shows representative material batches, as well as substrates on which
the ppy, or similar conducting polymer, was deposited. Among the different substrates
used were silicon rubber, cotton fabric, hardening liquid (epoxy, glue), carbon sheeting,
and plastic brills. Depending on the substrate used for the polymer, different effective
permittivities were expected, even if the concentration o f polymer was constant. This is
understandable,
as
different
substrates
have
differing
thicknesses,
electrical
characteristics, and densities. The chemist preparing the materials decided which
substrate the polymer would be bonded too, and therefore only the overall electrical
characterization is in the scope o f this thesis. This information was added to give the
reader an idea that differing substrates were used to either increase the thickness of
samples, or to enable the increase in concentration o f a specific polymer.
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75
Most o f the repeatedly measured samples were originally measured with gating turned o ff
on the VNA. The measurements were then taken again after another calibration was
performed (labeled ‘redo’ on the figures). For the third measurement, a gating filter was
placed on the S-Parameters (labeled ‘gated’ on the figures). Gating essentially tries to
filter out additional noise and reflections that were not calibrated out, or were introduced
after calibration (e.g. movement o f coaxial cables). A bandpass filter is imposed on the SParameters before they are displayed on the VNA and downloaded for data processing.
This bandpass filter is user definable and is set to bracket the main pulse o f energy
reflected off (or transmitted through) the two-layer sample. Although gating makes a
smoother curve, it also appears that there is a phase shift. An example o f non-gated and
gated Sii and S21 parameters is shown in Figure 4.30. This is the raw data used to
determine e’r and e”r shown in Figure 4.31. The bandpass filter was set sufficiently broad
for all measurements, as the main focus was to measure these samples in bulk and not be
required to alter the filter characteristics for each sample, only for each batch.
Gated and Non-Gated S-Parameters,
Blue is Nan-Gated
Red is Gated
Figure 4.30 An extreme close-up of both S-Parameters and how data smoothing occurs
with gating.
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76
Batch #5 sample 4, original, redo, gated, redo gated
Gated Aug09 results, and Gated Aug22
120
100
80
H 60
at
a
2
40
OngmeJ measurement taken July 24
Redo measurement taken Juty 30
Gated measurement taken July 31
Redo Gated measurement taken Aug 02
Gated measuremenb taken Aug09
Gated measurements taken Aug22
Static Conductivity ■ 0.960 S/on
20
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
Frequency (GHz)
Figure 4.31 Results from Batch#5 Sample#4, six separate measurements. Arrows
indicate increasing time. Sample thickness = 0.1 Smm, Acrylic thickness = 1.18mm
The complex permittivities of the sample in Figure 4.31 have properties opposite that of
‘Material 2’ analyzed with the uncertainty plots of Chapter 4.1.1. Six separate
measurements with six individual calibrations were performed for this sample over a one
month time period. It is seen that while the dielectric constant changes only within the
uncertainty limits, the imaginary permittivity decreases quite significantly over the time
period. The dielectric constant is seen to have four overlapping results from
measurements taken on July 31st, August 2nd, 9th, and 22nd.
Taking the approximate properties of this material at 10GHz to be e’r = 10 and e”r = 80, it
is possible to examine the uncertainty plots and determine whether a different sample
thickness would give lower uncertainty. Referring to Figure 7.1 through Figure 7.6, from
uncertainty plots 3a in Appendix 0, with a sample ratio of e’r = 0.125 x e”r, the closest
analyzed is a ratio equal to 0.5. The upper left subplot of each figure is the closest match
to this material. In this case, an increase in sample thickness would decrease the
uncertainty from ~30% for e’r to below 5%, if a sample thickness o f at least 1.5mm was
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77
available. The uncertainty difference would not be that drastic in the case of e”r, as it
already has an acceptable uncertainty of 3.3%, but it would also decrease.
Batch #7 (PS240701), Sample *4, Original, Redo, Gated,
Redo pated-fMutta-
200 r
180
160
9
'e
S’
2
140
Original meostffemcnt taken Aug06
Redo measurement taken Aug 09
Gated measurement taken Aug 13
Redo Gated measurement taken Aug 16
120
Static Conductwity • 1.40 S/cm
*
10080
60-
♦
40h
201
8.5
9
9.5
10
10.5
11
Frequency (GHz)
11.5
12
12.5
Figure 4.32 Results from Batch#7 Sample#4, four separate measurements. Arrows
indicate increasing time. Sample thickness = 0.15mm, Acrylic thickness = 1.18mm
The measurement results in Figure 4.32 closely resemble the permittivity o f ‘Material 4’
considered during the uncertainty analysis. At 10GHz, this sample shows e’r = 78 and e”r
= 178. Once again it can be seen that the imaginary permittivity gets smaller with time,
while the dielectric constant is within the uncertainty envelope and only changes slightly
with the changing conductivity. The results o f the uncertainty analysis o f Figure 4.15 and
Figure 4.16 clearly show that the uncertainty for this material is acceptable, being 5.6%
and 2.2%, respectively for 8’r and e”r. It would be possible to decrease this further if the
sample thickness were increased. From Figure 4.15 and Figure 4.16 is can be noted that
the sample thickness of 0.15mm in one o f the largest uncertainty regions plotted. By
increasing the thickness to I mm, the uncertainty would drop to 3% and 1.7%,
respectively.
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78
B atch 19 (PSQ10801), S am ple #5, O riginal, Redo, G a te d , a n d
i-teqo aatea neauita
...
%
8.5
9
9.5
10
10.5
11
F re q u e n cy (GHz)
11.5
12
12.5
Figure 4.33 Results from Batch#9 Sample#5, four separate measurements. Sample
thickness = 0.15mm, Acrylic thickness = 1 .1 8mm
The low loss sample o f Figure 4.33 shows an uncertainty o f 32% and 8.4% respectively
for e’r and e”r. All measurements are contained within the uncertainty limits except for
the degrading of conductivity, which happened over a period o f two weeks. In the case of
these low loss samples, a dramatic decrease in uncertainty occurs with even slight
increases in sample thickness. This is very apparent from Figure 4.23 and Figure 4.24 for
this sample, namely e’r = 0.3 le’Vat 10GHz.
4.2.1
Sample Thickness Optimization
As mentioned in Chapter 3, the thickness of the sample was often included in the
optimization. The caliper measurement was given as an initial guess and a userdeterminable constraint was placed on this parameter, usually 5-10%. Examination o f the
convergence of the thickness gave an indication o f how well the optimization program
fared with the data. If the converged thickness o f the sample was on the edges of the
constraint envelope, it showed the algorithm needed to compensate somewhat for poor
data. If the optimized thickness changed by only a fraction o f a percent, then the results
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79
were usually quite reliable and repeatable. Figure 4.34 shows the results for all samples
from the same batch, where the thickness o f the sample was questionable because of the
substrate the polymer was deposited in. Original thickness measurements and optimized
thickness were recorded, as summarized in Table 4.6. The substrate used for this batch
was a plastic brill (resembling a plastic SOS pad) with the polymer grown around every
fiber. It was expected that the difference in sample thickness, between measured and
optimized, would be uncharacteristically large because o f the nature of the material, but
this was not the case.
Batch #4, TH3 White and Green, TH5, TH6, TH7, TH8, July 17 and 18
2i--------------*----------------------------------- 1------------- ;
Sample 9
Sample 10 -1 2
Sample 1 - 8
Sample 7. 8 .sam ple H
........— Sample 9
'
,\y„
P*8 5 ^ S a m p l e 3. 4 "
8.5
9
9.S
'' -•:
10
10.S
11
Frequency (GHz)
V
Sample 10.12
-
11.5
.....
12
■
12.5
Figure 4.34 Entire Batch #4. Thickness constraint envelopes were recorded. Sample
numbers correspond to Table 4.6
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80
Table 4.6 Data corresponding to Figure 4.34 showing variable thickness with 5%
constraint.
1
2
Optimized
Thickness
(mm)
2.55
2.54
3.26
3.27
TH6
TH7
TH7
2.57
2.57
3.27
3.27
3.35
3.35
3.62
3.62
2.47
2.47
3
4
5
6
7
8
9
10
3.32
3.34
3.61
3.62
2.4
2.47
0.90
0.30
0.28
0
2.92
0
2.59 <— > 2.35
TH8
TH8
2.77
2.77
2.76
2.77
0.36
0
2.90 <— > 12.63
TH3 White
TH3 White
TH3 Green
TH3 Green
TH5
TH5
TH6
11
12
Percent
Difference
(%)
0.78
1.18
0.31
0
5% Thickness
Constraint
(mm)
Caliper
Measurement
(mm)
Sample
2.70 <— > 2.44
3.43 <— >3.11
3.52 <— >3.18
3.80 <— > 3.44
As shown in Table 4.6, the optimization program did not change the thickness of the
sample significantly (although within the constraint imposed, it was able to change it as
much as 5%).
Figure 4.35 shows a data set o f a sample (from Batch #7) for which difficulties were
experienced with polynomial convergence.
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81
Batch *7 (PS240701), Sample #1, Original, Redo, Gated,
and Redo, Gated remits
,------------’Original an d R edo
Original a n d R edo
F requency (GHz)
Figure 4.35 The polynomial solution and the small-section solution. Sample thickness =
0.14mm, Acrylic thickness = 1 .1 8mm
Table 4.7 Data Set displaying difficulties with optimization convergence. All optimized
solutions o f thickness lay on the constraint edge of the envelope, except for the two
shown in Red Text.
Measurement
Original
Redo
Gated
Redo Gated
Sample Thickness o f 0.14mm
5% Thickness Constraint = 0.147m m «— »0.133mm
10% Thickness Constraint = 0.154mm <— >0.126mm
20% Thickness Constraint = 0.168mm <— >0.112mm
10% Solution
20% Solution
5% Solution
0.154
0.147
0.153
0.154
0.147
0.156
0.154
0.168
0.147
0.154
0.168
0.147
The Gated and Redo Gated results o f Figure 4.35 in polynomial form are obviously
erroneous. This material is a perfect example o f the polynomial convergence algorithm
having difficulty as previously mentioned. This type o f erroneous result happened very
rarely, (approximately 1 time out of 30) and shows how the ‘small-section’ method can
be used as a back up. If the sample was again delivered on a different thickness o f
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82
acrylic, or itself was a different thickness, this erroneous convergence difficulty would
most likely not be repeated.
Even though realistic and repeatable results were obtained for the sample of Figure 4.35,
because o f the constraint-edge convergence o f the thickness, the results were not
considered to be accurately determined by the uncertainty limits. For this particular
sample, facial inconsistencies were believed responsible for the convergence errors. Only
when the thickness constraints were relaxed to 20% did the optimization converge to a
central location inside the envelope, and only for two samples (shown in red text on
Table 4.7). This is not considered acceptable.
Batch #10, TH280601, Sample #6 Original, Redo,
5 Gated Results and Gated Blank Sapiple with High Uncart;
Original
4.5 (-
3.5 r
8 3‘
3
S2.5F
?
2 2
Blank
Ongmai meaa*ement taken Sep 17
Redo measurement taken Sep 19
Gated measurement taken Sep 10
S t a l e C o n d u c tiv ity ■ 0 .0 0 0 3 0 3 S t a n
1.51-
0.5
Blank
8.5
9.5
10
10.5
11
Frequency (GHz)
11.5
12
12.5
Figure 4.36 Very low-loss sample with very high uncertainty. Also shown is one-layer
convergence o f acrylic only. Sample thickness = 0 .16mm, Acrylic thickness = 2.07mm
Figure 4.36 is included to explicitly illustrate one o f the limitations of this measurement
method, namely very thin, lossless materials. From Figure 4.19, the dielectric constant
uncertainty for this material is approximately 55% at 10GHz. Considering the Redo
measurement, this would place the uncertainty limits at 6.2 and 2.8%, which the other
two measurements fall within. Lossless thick samples, e.g. pure acrylic indicated with
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83
‘Blank’ on the figure, show low uncertainty in measurements. At 10GHz, the Blank
dielectric constant solution is 2.71, while many text references have this material
classified as having an e \ between 2.50 and 2.75, depending on the manufacturer [13].
Since e”r = 0, the measurement data (mean e”r = 0.0044) clearly show the small
erroneous results.
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84
Chapter 5
5
Modeling Results and their Evaluation
Two central ideas are explored in this chapter concerning the FDTD simulations. The
first, a comparison of the simulated results obtained with results from measurements
taken, and the second, the analysis of specific facial inconsistencies that occur in the
sample layer in practice.
5.1
Comparison of FDTD Results with Measurements
The FDTD-collected data was compared to the VNA-collected data as part o f the FDTD
analysis, before facial inconsistencies were explored. Two samples were chosen from the
unknown materials supplied by the DND, namely, Batch #21 Sample 2 (B21S2), and
Batch #26 Sample 4 (B26S4). Both samples were 0.14mm thick and adhered to a
1.18mm block of acrylic with er’ = 2.65. Electrical characteristics o f B21S2 and B26S4
are given in Table 5.1.
Table 5.1 Electrical characterization o f two representative samples at 10GHz
Sample
VNA measured e.
Static Conductivity (<u,)
B21S2
7.18 - i 3 1.8
17.7
B26S4
156.5- i 174
96.9
The procedure is as follows: The VNA is calibrated and S-Parameter data collected after
the insertion of B21S2. The optimization routine is used on the VNA collected SParameters and an array o f frequency dependant complex permittivities is converged
upon. The complex permittivity from the VNA at 10GHz is used in the material
characterization in the FDTD code. The FDTD simulation is performed and a second
array o f S-Parameters is collected. The optimization routine is used on the FDTDcollected S-Parameters and a second array o f complex permittivity values is converged
upon. There now exists an array of VNA S-Parameters and complex permittivities, and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
another array of FDTD S-Parameters and complex permittivities. These two sets of two
arrays are now compared.
It should be noted that the FDTD data could only legitimately be compared to the VNA
data at 10GHz and not in the whole X-band frequency range. On the other hand, the blue
curves in Figure 5.1 show the FDTD data over this entire range, compared to the red
curves showing the VNA S-Parameters. It is possible for a material to be set up to be
frequency dependant using the FDTD software. However, the FDTD algorithm accounts
for the frequency dispersion represented by the Debye equation. Development o f a
polynomially frequency-dependent dispersion in the FDTD code was beyond the scope o f
this thesis.
S-Parameters. Sj j and S 2 1 of VNA Results and FDTD Results
130
90
oa
06
tso
0.4
02
180 1
330
300
270
Red - VNA
Blue - FDTD
Figure 5.1 S-Parameters showing both VNA-collected results and FDTD-simulated
results for sample B21S2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
Comparison batwean Batch #21 Sample 2 and FDTD Simulation
40
3f
10
35
33.5
9.5
1
9
33
8.5
32.5
8
c
I
32
7 .5 1
[
31.5
7j
31 j
6.5 j
6|
6.51
8.5
9.5
10.5
Frequency (GHz)
12.5
30.5 i
10
30 ^
10
Frequency (GHz)
Figure 5.2 Sample B21S2, (Left) Dashed line indicates measurements taken with VNA,
solid shows FDTD simulated results. (Middle, Right) Zoomed in results showing 10GHz
comparison. Blues lines show er\ red lines show er”
Figure 5.2 shows measured and simulated data for the complex permittivities for B21S2.
Data at 10GHz are highlighted and measurement error uncertainty envelope is placed
around VNA-measured data. Firstly, it can be seen that the simulated and measured
parameters are both well within the uncertainty limits. Secondly, the values obtained
from the simulation of the dielectric constant and loss factor at 10GHz corresponding to
the measured S21 are er’ = 7.81 and er” = 31.6, respectively. Thirdly, the difference in er”
values is in this case very small, less than 1%.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
B atch 26 S am ple #4. C om parison of M e a su re d a n d Sim ulated R e s u lts
190
180
Met
LfttsFacfar
170
160
Mm
150
140
9.75
10.25
10.5
F requency (GHz)
Figure 5.3 Comparison o f measured and simulated results for B26S4. Focus is 10GHz
results. Green uncertainty enveloped centered on VNA-measured values.
Uncertainties in measurements of er’ for sample B26S4 are much lower than for B21S2
and approximately equal for er”, as illustrated in Figure 5.3. Table 5.2 summarizes the
results of the comparison.
Table 5.2 Comparison of measured and simulated complex permittivity values at 10GHz
Sample
B21S2
B26S4
Measured
7.18 ±1.59
156.5 ±6.50
Er’
Simulation
Results
7.81
155.9
%
Difference
8.8
0.38
Measured
31.8 ±1.55
174.2 ±6.00
Er”
Simulation
Results
31.6
175.7
%
Difference
0.63
0.86
The small differences between measured and simulated results (primarily in sr’ of B21S2)
are at an acceptable level, assuming that it has been sufficiently well documented that the
measurement results presented are within the computed limits o f uncertainty o f the VNA
use. For the FDTD simulations performed, specific care has been taken to limit to
negligible errors due to: reflections from the graded mesh (maximum voxel dimension
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88
difference = 11%), evanescent modes (sufficient distance between probes for Su and S 21
and sample interfaces), and PMLs. The stability factor was at the conservative level of
0.95, and the source excitation was given sufficient amount of time steps for the Gaussian
Pulse to start and end very close to zero (0.0138 with a normalized maximum of 1).
A very small permutation of the S-Parameters between the VNA and FDTD collected
values gives rise to differing permittivity values, in the case of B21S2, as large as 8.8%.
It would be a very worthwhile endeavor to explore the effect on converged permittivity
from slight variations in the S-Parameters from facial inconsistencies routinely seen in
practice. To elucidate errors caused by gaps in the sample and other imperfections,
selected representative cases are simulated with the FDTD and analyzed.
5.2
Waveguide gap effect
Well defined facial inconsistencies (gaps and dents) in the test sample have been placed
in various spots and the FDTD convergence results analyzed. This simulates the problem
o f completely covering the acrylic supporting structure with a perfectly homogeneous
layer free o f small irregularities. Simulations of two types of the facial inconsistency
were performed, namely full gaps in the unknown samples, and partial gaps (dents).
Select configurations have been drawn in Chapter 3, while a full collection of simulated
models can be found in Figure 5.4 and Figure 5.10. Gap configuration #2 in both o f these
figures is the most probable facial inconsistency, with Gap configuration #4 being the
next most likely.
5.2.1
Full Gaps
Gaps were placed in areas close to the edge o f the sample where they were most likely to
occur, not where they would have the biggest theoretical impact on the TE 10 electric field.
Gap #5 is an extreme inconsistency but is not completely unfounded for simulation, as
similar samples have been delivered for measurement.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
o.Mmm
18 mm
j t.ism m
0.1mm -
l ^
^
9.9mm-
I i
D
Gap *2
Gap *1
0 Mmm
I 1 18 mm
l i
9mm
1mm9mm-
- O
j 1ismm
I I
□
Gap *4
Gap *3
o u14mm
mr
18mm
V
Gap *5
Figure 5.4 The five full gap configurations simulated.
S-Parameter Sf j of Five Gapped Samples, a
Perfect Sample, and no sample
Figure 5.5 FDTD simulated Su parameter for gaps 1-5 for sample B21S2. Blue left:
perfect, Green: gap #1, Magenta: gap #2, Cyan: gap #3, Yellow: gap #4, Blue right: gap
#5. Red tips indicate low frequency data points of S-parameters.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
S-Parameter Sj) of Five Gapped Samples, a
Perfect Sample, and no sample
Figure 5.6 FDTD simulated S21 parameter for gaps 1-5 for sample B21S2. Blue left:
perfect, Green: gap #1, Magenta: gap #2, Cyan: gap #3, Yellow: gap #4, Blue right: gap
#5. Red tips indicate low frequency data points of S-parameters.
FDTD Simulation Results Comparison
lap #1
'erfect
Gap #3
Gap *2
iap #4
Gap #5
G a p *1
lap #2 /G a p #3/G a p §4 /Perfect
/
8.5
10.5
Frequency (6Hz)
9.5
—
11.5
Figure 5.7 FDTD simulated results for sample B21S2 or dielectric constants and loss
factors for one perfectly homogeneous layer and five samples with gap inconsistencies.
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91
O f the five configurations of gaps and dents, #2 and #4 were the most common. An
inconsistency centering on the middle of the x-axis like #3 and #5 is expected to make a
larger impact than an inconsistency on the edges o f the x-axis like #4. This is because of
the half-period sinusoidal magnitude configuration o f the TEio mode in the waveguide.
In Figure 5.5 and Figure 5.6, one set of S-Parameters is displayed of a simulation run
without any sample at all, only acrylic. The acrylic placement was still 0.14mm recessed
from the calibration plane to be consistent with the other simulation runs. Displaying this
result alongside a result from a very large inconsistency like Gap #5 shows that even
when the entire 4mm center (x-axis) of the sample is missing, the S-Parameters, and
finally the permittivity, are still reasonable close to the homogeneous layer.
The sample with higher conductivity (B26S4) was also simulated. Gap #1 and Gap #5
configurations are not shown as the S-Parameters and consequent permittivities were too
far removed from the ‘Perfect’ layer (Figure 5.8 and Figure 5.9).
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92
S-Param eters o f Three Gapped Samples, a Perfect Sample, and
no sample.
, < ii\ \
120
0.6
150
0.4
330
210
300
240
270
Figure 5.8 FDTD simulated S| i and S21 parameters for gaps 2-4 for sample B26S4.
Blue: perfect, Magenta: gap #2, Cyan: gap #3, Yellow: gap #4,. Red tips indicate low
frequency data points o f S-parameters.
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93
FDTD Simulation Results Comparison
200
Perfect«'
180
G ap #3
ip#4 e"
8)
■o
a
c
O
)
CD
Z
160
P e rfe c t £'
lap #2 e'
140
Gap #2 e'
120
8.5
9.5
10
10.5
11.5
Frequency (GHz)
Figure 5.9 FDTD simulated results for sample B26S4 of dielectric constants and loss
factors for one perfectly homogeneous layer and three samples with gap inconsistencies.
As shown in Figure 5.8 and Figure 5.9, certain gap inconsistencies in this sample have a
much larger effect on the data collected and consequently the permittivity convergence.
Table 5.3 displays a comparison and summary o f permittivity results at 10GHz.
Table 5.3 Summary table showing differences in simulated dielectric constant and loss
factor for different configurations of facial inconsistencies. All values at 10GHz.
Sample
B21S2
B26S4
B21S2
B21S2
B26S4
B21S2
B26S4
B21S2
B26S4
B21S2
Condition
Perfect
Gap #1
Gap #2
Gap #3
Gap #4
Gap #5
e ’r
7.81
155.94
9.63
9.01
144.4
7.99
140
7.84
153.6
5.22
AE’r / e ’r
0.19
0.13
0.08
0 .0 2
0.11
0.004
0.015
0.50
tf ” r
32.0
175.72
26.63
30.76
116.59
30.60
171.21
31.76
175.34
24.08
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Ae’V /e’V
0 .2 0
0.04
0.50
0.05
0.03
0.008
0 .0 0 2
0.33
94
The values in bold in Table 5.3 display the solution o f the perfect (no facial
inconsistency) sample, to which all other samples are compared. O f particular interest are
values contained in the heavy outline, namely the Ae’V / e”r values o f 0.04 and 0.50, for
B21S2 and B26S4, respectively. The increase in error associated with this particular gap
configuration with an increase in conductivity indicates that even a relatively small and
often encountered facial inconsistency can significantly alter the convergence solution.
Again, referring to Table 5.3, it is apparent that each gap configuration must be
considered separately in terms o f expected solution deviation from a perfect sample. This
is illustrated by loss factor deviation of gap configuration #3 and #4, which is smaller
with greater conductivity while gap #2 deviation is significantly larger. The opposite is
true for the comparison o f the dielectric constant.
5.2.2
Dents in Sample
In the effort to model even more realistic defects, the following simulations were carried
out. Many samples delivered to the lab had similar problems to those described in this
chapter. The chemists experienced difficulties getting completely homogeneously thick
samples on the acrylic backing. It would have been unfeasible to measure most o f the
materials without a solid supporting structure.
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95
0.06 mm
006
8 rr
mm
18 mm
006m m
0oarr
8mm
UT
n
_______
*IlfBmm
4
9.9m m -* M
Dent *i
0 06 mm
0 0 8 mm
1.18mm
1m m —♦
9mm
1
4
□
Dent *2
I
0 06mm
0.08 m m
M 8 mm
U4
L
9mm—*
□
Dent *4
Dent #3
0 06 mm
loosmm
1
1 18mm
1. 4
p = D
Dent *5
Figure 5.10 The five dent configurations simulated.
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96
S-P aram eters of Five Dented Sam ples, a
Perfect Sam ple, and no sam ple
120
0.8
0.6
ISO
0.4
0.2
180
No Sam ple
'N o S am ple
330
300
Figure 5.11 FDTD simulated Su and S21 parameters for sample B21S2, gaps 1-5. Blue
left: perfect, Green: dent #1, Magenta: dent #2, Cyan: dent #3, Yellow: dent #4, Blue
right: dent #5. Red tips indicate the low frequency data points o f the S-parameters.
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97
FDTD Simulation Results Comparison
40
30
E 20
8.5
9.5
10.5
Frequency (GHz)
11.5
10GHz Region Enlarged
7.85- Dent (1
7.84
32 /Perfect
< « « L /D e n t9 4
31.5 —
31 ' D e n t 8 3 ^ ^
7.83-
Perfect
30.5
jU ^ D e n tJII^ ^
7.82- Dent 84
29.5
7.81
Dent 12
7.8 Dent 83
29
24-5 J e n t 85
28
7.79-
—■
10
10
Figure S. 12 FDTD simulated results o f dielectric constants and loss factors for one
perfectly homogeneous layer and five samples with dent inconsistencies. Sample B21S2.
On enlarged region plot, e’r on left, e”r on right. Dent #5 omitted from e’r 10GHz
enlarged region plot.
Simulation results of the dents resemble the results for the gaps although with less
dramatic changes. Figure 5.12 shows that small facial inconsistencies (e.g. Dent #2, #3,
and #4) in some areas only slightly affect the results o f the loss factor and dielectric
constant.
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98
Sample B26S4 was simulated in the same way, and, once again only configurations #2,
#3, and #4 were included (Figure 5.13 and Figure 5.14).
S-Parameters o f Three Dented Samples, a Perfect Sample,
and no sample i—ra-----------------120
0.9
0.6
150
N« Sm
fir*
330
!10l
300
240
270
Figure 5.13 FDTD simulated Si i and S21 parameters for sample B26S4, dents 2-4. Blue
left: perfect, Magenta: dent #2, Cyan: dent #3, Yellow: dent #4. Red tips indicate low
frequency data points o f S-parameters.
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99
FDTD Simulation Results Comparison
200
180
160
O)
140
120
8.5
9.5
10.5
11.5
Frequency (GHz)
10GHz Region Enlarged
Perfect s'
174 ^^^erfect i*
Dent f 4 c'
15s[
Dent >4 e*N
173
Dent #2 s'
V^Dent # 2 e*
172
I54[
\ D e n t f 3 e*
Dent #3 c'
1
152:
171
170t
10
10
Figure 5.14 Results of dielectric constants and loss factors for one perfectly
homogeneous layer and three samples with dent inconsistencies. Sample B26S4.
Table 5.4 shows a comparison o f errors between each sample and the respective ‘Perfect’
layer. It also gives a comparison between the same dents on low and high conductivity
samples.
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100
Table 5.4 Summary table showing differences in simulated dielectric constant and loss
factor with different configurations of facial inconsistencies. All values at 10GHz.
Sample
B21S2
B26S4
B21S2
B21S2
B26S4
B21S2
B26S4
B21S2
B26S4
B21S2
Condition
Perfect
Dent #1
Dent #2
Dent #3
Dent #4
Dent #5
E’r
7.81
155.94
7.85
7.80
154.4
7.80
153.0
7.81
155.4
6.95
Ae’r / e ’r
0.005
0.001
0.010
0.001
0.019
0
0.004
0.12
e’V
32.0
175.72
30.93
31.75
174.59
31.56
173.0
31.92
175.31
28.53
Ae”r / e”r
0.035
0.008
0.007
0.014
0.016
0.002
0.002
0.12
The values in bold in Table 5.4 display the solution o f the perfect (no facial
inconsistency) sample, to which all other samples are compared. Contrary to the data
shown in Table 5.3, there are no significant differences in errors between similar gaps.
Many o f the errors incurred with the dents are insignificant.
5.3
Summary of FDTD Facial Inconsistency Simulations
A comparison between VNA and FDTD generated data was performed on two samples
o f differing dielectric and conductive properties. The differences between measured and
simulated dielectric constant and loss factor were very small with 8.8% being the largest
difference between two results. FDTD simulation o f small facial inconsistencies on the
two samples were generated and compared to the ‘perfect’ solution o f the completely
homogeneous sample. Very realistic gaps and dents proved to permute the final solution
by as much as 50% and 2% respectively.
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101
Chapter 6
6
Conclusions
A novel microwave measurement method was introduced for the characterization o f
permittivity and permeability o f materials. The method was developed to conform to a set
o f specific criteria. The measurement method utilized a supporting structure of known
electrical properties inserted with the unknown material into the waveguide. An iterative
algorithm was used to converge on the material characteristics. An explicit equation was
not possible given the increased complexity of dependence on sought parameters on
measured data compared to the more traditional one-layer measurement methods. An
equipment uncertainty analysis was performed to find the optimum thickness o f the
sample and supporting acrylic second layer as a function o f the properties of the sample.
General thickness guidelines were developed for both layers. The FDTD method was
used to quantify the effect o f small gaps and dents on the surface of the sample on the
final solution. Several results were shown, displaying many properties o f the convergence
o f the algorithm. Limitations o f the method were also illustrated by the results that were
obviously erroneous.
6.1
Measurement Method
The two-layer approach was developed primarily to satisfy the following conditions for a
measurement procedure:
broadband
relatively expedient
limited initial startup costs
accommodate very thin material without the ability o f self-support
accommodate materials only available on thin sheets (essentially unmachinable)
achieve minimum measurement uncertainty
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102
The final measurement configuration required the sample to be placed or grown on one
face o f a known supporting structure that conformed to the height and width dimensions
o f the waveguide.
6.2
Non-Linear Optimization
Two optimization approaches were used, both utilizing the same Levenburg-Marquardt
algorithm but formulated differently. The first approach (and was later used as an initial
guess for the second approach) solved for a single complex permittivity (or both
permittivity and permeability) in approximately 40 separate frequency bands inside the
X-band frequency range. The second optimization solved for the coefficients o f a
polynomial that described the permittivity characteristics over the entire X-band
frequency range.
An option was given to include the thickness of the sample in the optimization o f the
second approach. This proved to be a very quick and useful way to determine whether the
optimization procedure had had difficulty converging on a solution. This optimization
routine proved to be very reliable and quite robust, its limitations were explored in the
Results o f Chapter 5.
6.3
Equipment Uncertainty Analysis
The VNA used to obtain the S-parameters introduces uncertainty in the measurement
data. An uncertainty analysis was performed to determine what combinations of sample
and acrylic thicknesses provided the lowest uncertainty for a wide range o f lossy
materials. Several specific plots were generated that illustrated the effects of change in
sample thickness with constant acrylic thickness and frequency, while general plots were
also shown that encompassed a broader range of materials and thicknesses that could be
used for reference.
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103
6.4
FDTD Modeling of Sample Facial Inconsistencies
Samples measured did not always have perfect dimensions. Often small gaps or dents
existed at the edges o f the sample, sometimes completely exposing the acrylic. A
waveguide measurement model was constructed in FDTD and several gap or dent
configurations seen in practice were simulated. The results were then compared to the
‘perfect’ sample layer and differences analyzed. This was done for both a low-loss and a
medium-loss sample, finding that higher loss in a sample did not directly correspond to a
larger percentage difference between the converged solution o f the perfect sample and the
non-homogeneous sample.
6.5
Results
Example plots were given o f differing materials displaying many different characteristics.
Results were chosen to highlight either strengths or weaknesses associated with the
method. Very consistent measurement repeatability was observed, and consistent tracking
o f material properties as a function o f time (displaying decreasing conductivity). The
results were invariably contained within uncertainty limits. Results were also given that
included data where the sample thickness was allowed to vary in the optimization.
Circumstances where this method worked very well, and when it did not, were shown and
discussed. The benefits o f the ability to retain the optimization data of the non­
polynomial solution (the stair-cased solution) were illustrated. Very low-loss thin sample
results were given with very poor convergence, displaying a limitation of the
optimization program. Finally, a one layer solution of the acrylic being used as the
supporting structure was given.
In summary, the method selected together with the uncertainty analysis and the FDTD
simulations proved to be fully satisfactory for measurement o f the permittivity o f lossy,
thin samples. For most o f the samples, the permittivity could be evaluated with
uncertainty better than 5%.
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104
6.6
Future Work
The measurement program in Matlab is currently being ported to LabView. With its GUI
and communication with the VNA, this will upgrade the usability of the program, if not
the function.
Adding uncertainty in the thickness of the sample into the theoretical calculations to
observe how much error is introduced with small measurement discrepancies would be
very beneficial and may prove to warrant more accurate measurements o f thickness. An
investigation o f over-moding o f relatively thick medium-to-high loss samples would be
interesting and could provide the software program with alternatives if an acceptable
solution cannot be found.
Transferring this method to free space, once the samples become available, will offer a
challenge as signal processing may have to become involved to decrease the interference
by outside factors such as wave scatter and diffraction on the sample edges.
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B- 1
Bibliography
[1]
Baker-Jarvis, J., Janezic, M. D., Grosvenor, J. H. Jr., Geyer, R. G.,
‘Transmission/Reflection and Short-Circuit Line Methods for Measuring
Permittivity and Permeability’ NIST Technical Note 1355 (revised), Dec. 1993.
[2] Afsar, M. N., ‘The Measurement of the Properties of Materials’ Proceedings of the
IEEE, Vol. 74, No. 1, pp. 183-199, Jan. 1986.
[3] Courtenay, C. C., ‘Time Domain Measurement o f the Electromagnetic Properties o f
Materials’ IEEE Transactions on Microwave Theory and Techniques, Vol. 46,
No. 5, pp. 517-522, May 1998.
[4] Nicholson, A.M., Ross, G.F. ‘Measurement o f the Intrinsic Properties o f Materials by
Time-Domain Techniques’ IEEE Transactions On Instrument and Measurement,
Vol. IM-19, No. 4, pp. 377-382, Nov. 1970.
[5] Stuchly, M. A., Stuchly, S. S., ‘Industrial, scientific, medical and domestic
applications o f microwaves’ IEE Proceedings, Vol. 130, Pt. A., No. 8, pp. 467503, Nov. 1983.
[6] Weir, W.B. ‘Automatic Measurement o f Complex Dielectric Constant and
Permeability at Microwave’ Proceedings o f the IEEE, Vol. 62, No. 1, pp. 33-36,
Jan. 1974.
[7] Stuchly, S. S., Matuszewski, M., ‘A Combined Total Reflection-Transmission
Method in Application to Dielectric Spectroscopy’ IEEE Transactions on
Instrumentation and Measurement’ Vol. IM-27, No. 3, pp. 285-288, Sept. 1978.
[8] Boughriet, A.H., Legrand, C., Chapoton, A. ‘Noniterative Stable Transmission /
Reflection Method for Low-Loss Material Complex Permittivity Determination’
IEEE Transactions on Microwave Theory and Techniques, Vol. 45, No. 1, pp. 5257, Jan. 1997.
[9] Wan, C., Nauwelaers, B., De Raedt, W., Van Rossum, M. ‘Complex Permittivity
Measurement Method Based on Asymmetry o f Reciprocal Two-Ports’ Electronic
Letters, May 1996.
[10] Back, K.H., Sung, H.Y., Park, W.S. “A 3-Position Transmission/Reflection Method
for Measuring the Permittivity of Low Loss Materials’ IEEE Microwave and
Guided Wave Letters, Vol. 5, No. 1, pp. 3-5, Jan. 1995.
[11] Wan, C., Nauwelars, B., De Raedt, W., Van Rossum, M. ‘Two New Measurement
Methods for Explicit Determination o f Complex Permittivity’ Determination’
IEEE Transactions on Microwave Theory and Techniques, Vol. 46, No. 11, pp.
1614-1619, Nov. 1998.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B-2
[12] Janezic, M.D., Jargon, J.A., ‘Complex Permittivity Determination from Propagation
Constant Measurements’ IEEE Microwave and Guided Wave Letters, Vol. 9, No.
2, pp. 76-78, Feb. 1999.
[13] Von Hippel, A. R., ‘Dielectric Materials and Applications’ The M. I. T. Press,
Mass., 1966.
[14] Maze, G., Bonnefoy, J. L., Kamarei, M., ‘Microwave Measurement of the Dielectric
Constant Using a Sliding Short-Circuited Waveguide Method’ Technical Feature,
Microwave Journal, pp. 77-88, Oct. 1990.
[15] Jarvis, J., Vanzura, E.J., Kissick, W.A. ‘Improved Technique for Determining
Complex Permittivity with the Transmission / Reflection Method’ IEEE
Transactions on Microwave Theory and Techniques, Vol. 38, No. 8, pp. 10961103, Aug. 1990.
[16] Jarvis, J., Geyer, R.G., Domich, P.D., ‘A Nonlinear Least-Squares Solution with
Causality Constraints Applied to Transmission Line Permittivity and
Permeability Determination’ IEEE Transactions on Instrument and Measurement,
Vol. 41, No. 5, pp. 646-652, Oct. 1992.
[17] Williams, T. ELEC540 Project, ‘Use o f a Non-Linear Least-Squares Strategy in the
Evaluation o f Parameters Concerning the Design of a Multi-Layered Radar
Absorber’. University o f Victoria, 2000
[18] Balanis, C. A., ‘Advanced Engineering Electromagnetics’ Wiley, Toronto, 1989.
[19] Taflove, A., Hagness, S. C., ‘Computational Electrodynamics The FDTD Method
Second Edition’ Artech house, Boston, 2000.
[20] Booton, R. C. Sr. ‘Computational Methods for Electromagnetics and Microwaves’
Wiley, Toronto, 1992.
[21] Vector Network Analyzer Calibration,
http://morph.demon.co.uk/electronics/new.htm
[22] Pozar, D.M. ‘Microwave Engineering’ Second Edition, Wiley and Sons, U.S.A.,
1998.
[23] Ligthart, L.P. ‘A Fast Computational Technique for Accurate Permittivity
Determination Using Transmission Line Methods’ IEEE Transactions on
Microwave Theory and Techniques, Vol. MTT-31, No. 3, pp. 249-254, Mar.
1983.
[24] Rzepecka, M. A., Hamid, M. A. K. ‘Automatic Digital Method for Measuring the
Permittivity o f Thin Dielectric Films’ IEEE Transactions on Microwave Theory
and Techniques, Vol. MTT-20, No. 1, pp. 30-37, Jan. 1972.
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B-3
[25] Cheng, D. K., ‘Field and Wave Electromagnetics’ Second Edition, Addison-Wesley,
U. S. A., 1989.
[26] Iskander, M. F., ‘Electromagnetic Fields and Waves’ Prentice Hall, New Jersey,
1992.
[27] Hayt, W. H. Jr., Buck, J. A., ‘Engineering Electromagnetics’ Sixth Edition,
McGraw-Hill, Toronto, 2001.
[28] Knott, E. F., Shaeffer, J. F., Tuley, M. T., ‘Radar Cross Section Second Edition’
Artech House, Boston, 1993.
[29] Sadiku, M. N. O., ‘Numerical Techniques in Electromagnetics’ CRC Press, Ann
Arbor, 1992.
[30] Pasalic, D. ELEC629 ‘Measurements of Complex Permittivity and Permeability of
Various Materials Using Transmission / Reflection Method’, University of
Victoria, 2000
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A- 1
7
7.1
Appendix
Definitions
Complex Permittivity and Permeability
For non-magnetic materials, the complex relative permittivity defines the material
electrical characteristics. The complex permittivity in Ampere’s Law expressed in phasor
form, includes both conduction and displacement current.
=J +
Vx
MrMo
)
(7.1)
&
J -a E
(7-2)
If the current density (7.2) is represented in terms o f the electric field and conductivity,
Ampere’s Law is represented as in (7.3).
V x 77 = J + ja xE = o E + j a x E
(7 3)
Replacing conductivity with expression (7.4),
(T = ere0a)
(7-4)
and substituting into (7.3) results in (7.5),
V x /7 = OJ£"r£0E + jOJ£r£QE = jO £0E(£r - j £ r )
(7-5)
The relative complex permittivity is defined as er = (e r - j e r), and shown in (7.6),
V x H = jeae0£rE
(7-6)
A non-magnetic material has a relative permeability (|ir) equal to 1. Magnetic materials
possess a complex permeability represented by pr = |i r - jp r. The constants eo and |io are
8.854xl0'l2and l/(c2eo), respectively, where c is the speed of light.
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A-2
Characteristic Impedance
In free space, the characteristic impedance for a TEM wave is equal to:
Za = . f &
(7-7)
In the thesis, characteristic impedance has a general form for a TEio propagating wave in
rectangular waveguide, shown in (7.8).
1
Z=
(7.8)
l-
i
J ,
where f is the frequency, fc is the waveguide cut-off frequency, and both er and |ir may be
real or complex depending on the material through which the TEio wave is propagating.
Reflection and Transmission Coefficient
The voltage reflection coefficient, shown in (7.9), represents the amplitude o f the
reflected voltage wave normalized to the amplitude of the incident voltage wave [22] at a
reference plane. The subscripts ‘0 ’ and ‘ 1’ refer to labels o f the specific material the wave
is incident from (Zo) and the material it is reflected from ( Z |) , respectively.
Z -Z
r = =!——
z, +z
The
voltage transmission coefficient, shown in
(7.9)
(7.10), represents
the normalized
transmitted voltage througha specific medium. Here y is the propagation constant, shown
in (7.11), and is complex for lossy materials.
•p-g-yUc)
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( 7 . 10)
A-3
030 J I.l ~
Y = J•~ r—
(7.11)
For the dominant mode TEio and a waveguide filled with the material having er and ^ r,
the cut-off frequency is (7.12),
c
2 a ^er
(7.12)
where ‘a’ is the greater o f the two waveguide cross-sectional dimensions.
Scattering Parameters
The scattering parameters, Sn and S21, as measured by the VNA, represent the
normalized voltage level change reflected by, and transmitted through a two-port,
respectively. The scattering parameters are calibrated in a plane of reference, usually to
the junction immediately preceding the sample holder. This way, not only amplitude but
also phase-change (with the insertion of a post-calibration object) can be measured.
Equation (7.13) shows the full representation of the S-parameters collected by the VNA.
When
= 0, the system is reduced to the equation (7.14). Therefore, Sn represents the
voltage reflected back from the system, normalized by the original transmitted voltage.
S21 represents the voltage transmitted through the system normalized by the original
voltage o f the incident wave [22].
V
y~..
~su
~K
S22..V2.
Sa ~
V
'V
- V 2 .
-V
K
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(7.13)
(7.14)
A-4
7.2
Summary Uncertainty Plots (Acrylic thickness of 3mm)
frequency - 10GHz. sample tidiness • 0.1mm. material tidiness • 3mm
0.1
er0 • 2er00
er • 8"er
100
1000
*3
Figure 7.1 Uncertainty of the dielectric constant due to errors in measurements of S21.
Five different ratios of material at 10GHz, a sample thickness of 0.1mm and an acrylic
thickness of 3mm.
frequency - 10GHz. sample tidiness • 0.1mm material tidiness • 3mm
er3• ..
2erM
er ■ 3*er
1000
e.’3
Figure 7.2 Uncertainty of the loss factor due to errors in measurements o f S 21. Five
different ratios o f material at 10GHz, a sample thickness o f 0.1mm and an acrylic
thickness o f 3mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A-5
frequency - 10GHz. sample tfsckness • Irrnv material h d u c u • 3mm
0.1
er ■ O.S*er
?
Iu
3K
100
1000
e.3
Figure 7.3 Uncertainty of the dielectric constant due to errors in measurements of Si i.
Five different ratios of material at 10GHz, a sample thickness o f I mm and an acrylic
thickness o f 3mm.
frequency - 10GHz. sample thickness ■ 1mm, material thickness ■ 3mm
e r ■ O.Far
e r • 0.5*er
e r • erPG
8
“ 80
c r ! . 8 #eT
100
1000
e.3
Figure 7.4 Uncertainty of the loss factor due to errors in measurements o f Si i. Five
different ratios o f material at 10GHz, a sample thickness o f 1mm and an acrylic thickness
of 3mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A-6
frequency - 10GHz. sample thickness • 2mm. material thickness • 3mm
0.1
0.05
3
100
'3
e.
Figure 7.5 Uncertainty of the dielectric constant due to errors in measurements of S21.
Five different ratios of material at 10GHz, a sample thickness o f 2mm and an acrylic
thickness of 3mm.
frequency - 10GHz. sample ttvdmess • 2mm. material inclines* • 3mm
f
•§ 0 .0 s
3g
100
1000
e
’3
Figure 7.6 Uncertainty of the loss factor due to errors in measurements o f S21. Five
different ratios o f material at 10GHz, a sample thickness o f 0.1mm and an acrylic
thickness o f 3mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A-7
7.3
VNA HP8720 Uncertainty Specification Sheet
M easurement uncertainty
Reflection m easurem ents
1
>
<®tal
• M
•
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•1 5
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a,
Magnitude
t «• «
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5
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Figure 7.7 HP8720 Equipment Uncertainty Specification Sheet
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I la I
.9 * • a
Vita
Surname: Williams
Given Names: Trevor Cameron
Place o f Birth: Cranbrook, British Columbia, Canada
Educational Institutions Attended:
East Kootenay Community College
University o f Victoria
1994 to 1995
1995 to 2002
Degrees Awarded:
B.Eng.
University o f Victoria
2000
Top Electrical Engineering Graduating Grade Point Average of 2000
Honours and Awards:
BCHydro Scholarship
1998
Nordal Scholarship
1999
URSI Student Competition
1999
NSERC Undergraduate Scholarship
1999
APEG BC Highest GPA Electrical Engineering Graduate of 2000
Advanced Systems Institute Graduate Bursary
2000
NSERC Post Graduate Scholarship A
2000 to 2002
President’s Research Scholarship
2000 and 2001
NSERC Post Graduate Scholarship B
2002 to 2004
Publications:
Williams T., Rahman M., Stuchly M.A., “Dual-Band Meander Antenna for Wireless
Telephones”. Microwave and Optical Technology Letters, vol 24 (2), pp. 81-85, Jan 2000
Articles in Preparation to Refereed Journals:
Williams, T. M.A. Stuchly, “Measurements of Permittivity and Permeability of Thin Lossy
Sheets at Microwave Frequencies”.
Other Refereed Contributions:
Williams, T., M.A. Stuchly, and P. Saville, “Measurement of Thin Radar Absorbing Materials”,
2001 USNC/URSI National Radio Science meeting, URSI DIGEST, p. 27, Boston, Mass., July
8-13, 2001
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Bajwa, A., Williams, T., and Stuchly, M.A., “Design o f Broadband Radar Absorbers with
Genetic Algorithms”, 2001 IEEE Antennas and Propagation Society International Symposium,
Vol. IV, pp. 672-675
Neville, S., Williams, T., Bajwa, A., and Stuchly, M.A., “Measurements and Optimization of
Radar Absorbing Materials: Preliminary Considerations”, Department o f National Defense
Cansmart Workshop, Kingston, Ontario, Sept. 2000
Stuchly, M.A., M. Rahman, M. Potter, and T. Williams, “Modeling Antenna Performance in
Complex Environments”, IEEE 2001 Aerospace Conference, Big Sky, Montana, USA, March
18-25
Non-Refereed Contributions:
Williams, T., Bajwa, A., and Stuchly, M.A., Poster presentation, Advanced Systems Institute
(AS I) Exchange Day, April 2001
Williams, T., Stuchly, M.A., 1999 URSI General Assembly, Presentation, “Design of a DualFrequency, Wideband, Dual-Meander Antenna for Cellphone Applications”, Undergraduate
Student Finalist, August, 1999
Potter, M., Fear, E., Rahman, M., Williams, T., Poster presentation, Advanced Systems Institute
(ASI) Exchange Day, April 1999
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
University of Victoria Partial Copyright License
I hereby grant the right to lend my thesis to users of the University of Victoria Library, and to
make single copies only for such users or in response to a request from the Library of any other
university, or similar institution, on it behalf or for one o f its users. I further agree that
permission for extensive copying o f this thesis for scholarly purposes may be granted by me or a
member o f the University designated by me. It is understood that copying or publication o f this
thesis for financial gain by the University of Victoria shall not be allowed without my written
permission.
Title of Thesis:
A Wideband Two-Layer Microwave Measurement Method for the Electrical Characterization o f
Thin Materials
Author
Trevor Cameron Williams
Aug. 30, 2002
Note: This license is separate and distinct from the non-exclusive license for the National Library o f Canada.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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