Novel Techniques for the Design and Characterization of
Electromagnetic Devices with Application to Multilayer Structures
and Waveguide Filters
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
Daniel Lee Faircloth
Certificate of Approval:
Sadasiva M. Rao
Professor
Electrical and Computer Engineering
Michael E. Baginski, Chair
Associate Professor
Electrical and Computer Engineering
Lloyd S. Riggs
Professor
Electrical and Computer Engineering
SooYoung Lee
Professor
Electrical and Computer Engineering
Stephen L. McFarland
Dean, Graduate School
Novel Techniques for the Design and Characterization of
Electromagnetic Devices with Application to Multilayer Structures
and Waveguide Filters
Daniel Lee Faircloth
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
May 11, 2006
Novel Techniques for the Design and Characterization of
Electromagnetic Devices with Application to Multilayer Structures
and Waveguide Filters
Daniel Lee Faircloth
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon the request of individuals or institutions and at their expense. The author reserves
all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Daniel Lee Faircloth, son of Sam A. Faircloth and Carlene K. Wade, was born
August 13, 1981 in Opelika, Alabama. He graduated from the Alabama School of Fine
Arts in 1999. Beginning in the Fall of 1999, he attended Auburn University on a number
of scholarships and graduated in August 2002 with a Bachelors of Electrical Engineering
degree. In the same month, he entered graduate school at Auburn University after
accepting a NASA Graduate Student Researchers Program Fellowship supported through
the Electromagnetics Research Branch of Langley Research Center in Hampton, Virginia.
In December 2003, he received the Master of Science degree in Electrical Engineering.
He is married to Lisa P. Faircloth with whom he has a son Joseph A. Faircloth.
iv
Dissertation Abstract
Novel Techniques for the Design and Characterization of
Electromagnetic Devices with Application to Multilayer Structures
and Waveguide Filters
Daniel Lee Faircloth
Doctor of Philosophy, May 11, 2006
(M.S., Auburn University, 2003)
(B.S., Auburn University, 2002)
158 Typed Pages
Directed by Michael E. Baginksi
In this work, optimization methods are presented for the analysis of inverse and
design problems in electromagnetics. The current and future demands placed on design
engineers necessitate the hybridization of forward solvers (e.g., computational and empir
ical methods) with novel optimization algorithms to yield engineering development and
analysis tools which are largely autonomous and not restricted to the familiar principles
employed in traditional design work. Two specific problems are addressed to demonstrate
the capabilities of the proposed methods.
First, several optimization algorithms (i.e., Sequential Quadratic Programming, the
Genetic Algorithm, and Particle Swarm Optimization) are presented for the estimation
of complex constitutive parameters of multilayered materials. Using X band waveguide
Sparameter measurements, the complex constitutive parameters of each individual layer
are extracted. The results are compared to measurements as well as those of single layer
techniques which estimate the constitutive parameters of individual materials.
v
The second problem addressed is the automated design of waveguide filters. Initially,
a study is conducted into whether dielectric slabs randomly doped with conducting in
clusions such as short, thin wires or thin patches could yield useful frequency dependent
reflection and transmission behaviors when placed inside a waveguide. Results obtained
by placing conducting patches on the slab?s surface were found promising. Therefore, op
timization techniques were then employed to find the appropriate arrangement of patches
of the dielectric?s surface so that the resulting transmission response closely matched the
response specified by the user. Results of this study were verified by fabrication and
measurement for X band filters and, in all cases, found to be in excellent agreement.
vi
Acknowledgments
The author would like to express his sincere gratitude to Dr. Michael E. Baginski
and Dr. Manohar D. Deshpande for supporting him throughout his research. He would
also like to extend his thanks to Dr. Sadasiva M. Rao, Dr. Lloyd S. Riggs, Dr. Stuart M.
Wentworth, and Dr. SooYoung Lee for their friendship and guidance. To his parents,
he thanks them for always pushing him in the right direction and for all the years of
encouragement. He thanks his wife Lisa and son Joey for always understanding the late
nights and giving him the constant love and support he needed. Finally, to the Lord
Jesus Christ, the author thanks Him for the blessings and gifts he has been given.
vii
Style manual or journal used IEEE Transactions on Microwave Theory and
Techniques (together with the style known as ?auphd?).
Computer software used The document preparation package TeXnicCenter and
LATEX with the stylefile auphd.sty. Illustrations prepared using Canvas X.
viii
Table of Contents
List of Figures xi
List of Tables xiv
1 Introduction 1
1.1 Electromagnetic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Overview and Historical Perspective . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Constitutive Parameter Extraction . . . . . . . . . . . . . . . . . . 11
1.3.3 Waveguide Filter Optimization . . . . . . . . . . . . . . . . . . . . 13
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Complex Permittivity Extraction 15
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Error Function Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Sequential Quadratic Programming . . . . . . . . . . . . . . . . . . 20
2.4.2 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.3 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . 27
2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Measurements Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.1 Single Layer Measurements . . . . . . . . . . . . . . . . . . . . . . 36
2.6.2 Two Layer Measurements . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.3 Three Layer Measurements . . . . . . . . . . . . . . . . . . . . . . 48
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Complex Constitutive Parameter Extraction 52
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Extraction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 WolfsonWentworth Method . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 CP Extraction Method Modifications . . . . . . . . . . . . . . . . 53
3.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 Single Layer Measurements . . . . . . . . . . . . . . . . . . . . . . 57
3.4.2 Two Layer Measurements . . . . . . . . . . . . . . . . . . . . . . . 65
ix
3.4.3 Three Layer Measurements . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Waveguide Filters Using Conductor Doping of Dielectrics 71
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Numerical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 Absorbing Boundary Condition . . . . . . . . . . . . . . . . . . . . 79
4.3.2 ModeMatching Domain Truncation . . . . . . . . . . . . . . . . . 82
4.3.3 Scattering parameters . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.4 Asymptotic Waveform Evaluation . . . . . . . . . . . . . . . . . . 88
4.4 Method Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5 Waveguide Filter Optimization Using Surface Patches 106
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.2 Optimization Technique . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.3 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.4 Forward Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3.1 Notch Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.2 Bandpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.3 Low Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3.4 High Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6 Conclusions 126
6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1.1 Constitutive Parameter Extraction . . . . . . . . . . . . . . . . . . 128
6.1.2 Waveguide Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.1.3 Extension To Antenna Design . . . . . . . . . . . . . . . . . . . . . 129
Bibliography 131
x
List of Figures
1.1 Rastrigrin?s function which has many local minima and a global minimum
located at (0,0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Error surface wellsuited to local optimization problems. . . . . . . . . . . 8
2.1 Rectangular waveguide loaded with nlayer sample. . . . . . . . . . . . . . 17
2.2 Flowchart illustrating the Genetic Algorithm procedure. . . . . . . . . . . 23
2.3 Crossover between two parents using binary string encoding. . . . . . . . 25
2.4 Comparison of convergence for the traditional GA and GA?s using three
redundancy removal methods. The convergence rate of PSO is also shown. 28
2.5 Illustration of the neighborhood structure implemented in PSO. Each par
ticle is connected to its two neighbors to the left and right. . . . . . . . . 30
2.6 Flowchart illustrating the PSO procedure. . . . . . . . . . . . . . . . . . . 33
2.7 Magnitude of S11 and S21 for single layer Bakelite sample. The calculated
Sparameters are generated using the GA with error function (2.9). . . . . 37
2.8 Magnitude of S11 and S21 for single layer Garlock Rubber sample. The cal
culated Sparameters are generated using the GA with error function (2.9). 38
3.1 Comparison of MPSQP and WolfsonWentworth method extracted com
plex permittivity values for F125 sample. . . . . . . . . . . . . . . . . . . 61
3.2 Comparison of MPSQP and WolfsonWentworth method extracted com
plex permeability values for F125 sample. . . . . . . . . . . . . . . . . . . 62
3.3 Measured and generated Sparameter magnitudes for the F125 sample. . . 63
3.4 Measured and generated Sparameter phases for the F125 sample. . . . . 64
3.5 Measured and generated Sparameter magnitudes for the F125/Teflon
sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
xi
3.6 Measured and generated Sparameter phases for the F125/Teflon sample. 68
4.1 Loaded waveguide geometry with doped slab. Boundary notations for
FEM formulation are indicated. . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Discretized slab illustrating wires lying along tetrahedral edges. Notice
some edges join to form long, erratically shaped wires. Wires may also be
located on interior edges but are not shown here. . . . . . . . . . . . . . . 75
4.3 Tetrahedral element showing edge and node numbering. . . . . . . . . . . 78
4.4 Waveguide mesh used for method verification. . . . . . . . . . . . . . . . . 92
4.5 Comparison of FEM and theoretical S11 for the loaded waveguide. . . . . 93
4.6 Comparison of FEM and theoretical S21 for the loaded waveguide. . . . . 94
4.7 Comparison of magnitude of S21 for triangular patch doping using present
method and HFSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.8 A wide variation of S21 responses is shown for the cases of randomly
embedded wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.9 Illustration of the wide range of notch behaviors observed even when wire
connections within the dielectric are not allowed. . . . . . . . . . . . . . . 100
4.10 Magnitude of S11 and S21 for typical case of a dielectric slab doped with
50 2 mm wires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.11 Comparison of magnitude of S21 for typical cases of a dielectric slab doped
with 10, 20, and 35 3 mm wires. . . . . . . . . . . . . . . . . . . . . . . . 103
4.12 Comparison of magnitude of S21 for cases of a dielectric slab doped with
20, 30, and 75 triangular patches. . . . . . . . . . . . . . . . . . . . . . . . 104
5.1 Rectangular waveguide with optimized filter. . . . . . . . . . . . . . . . . 107
5.2 Gridded substrate face showing metallization for fabrication and FEM
simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 Flowchart illustrating the GA procedure with geometry refinement, re
dundancy removal, and high mutation procedures. . . . . . . . . . . . . . 111
xii
5.4 The lower left quarter of the waveguide filter is shown illustrating the
rules governing the geometry refinement process. The inset illustrates the
problem with cornerconnected patches. . . . . . . . . . . . . . . . . . . . 114
5.5 Comparison of ideal data, MM/FEM, HFSS, and measurements. The
MM/FEM matches well with the ideal (requested) response. However,
the geometry generated during the process yields a different response in
actuality as given by HFSS and measurements. . . . . . . . . . . . . . . . 116
5.6 HFSS model used for optimization . . . . . . . . . . . . . . . . . . . . . . 118
5.7 Fabricated samples produced using the patterns generated by the opti
mization procedure. A dime is shown for size reference. . . . . . . . . . . 119
5.8 Comparison of S21 responses of ideal, simulated, and measured notch filter.120
5.9 Comparison of S21 responses of ideal, simulated, and measured bandpass
filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.10 Comparison of S21 responses of ideal, simulated, and measured low pass
filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.11 Comparison of S21 responses of ideal, simulated, and measured high pass
filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
xiii
List of Tables
2.1 Complex Permittivities and Thicknesses for the 1, 2, and 3Layer Com
puter Generated Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Results using ideal Sparameters . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Extracted Permittivity & Error for Bakelite . . . . . . . . . . . . . . . . . 40
2.4 Extracted Permittivity & Error for Ceramic . . . . . . . . . . . . . . . . . 41
2.5 Extracted Permittivity & Error for Garlock Rubber . . . . . . . . . . . . 42
2.6 Extracted Permittivity & Error for Nano Material . . . . . . . . . . . . . 43
2.7 Extracted Permittivity & Error for Garlock/Bakelite . . . . . . . . . . . . 45
2.8 Extracted Permittivity & Error for Garlock/Ceramic . . . . . . . . . . . . 46
2.9 Extracted Permittivity & Error for Garlock/Nano Material . . . . . . . . 47
2.10 Extracted Permittivity & Error for Nano Material/Garlock/Garlock . . . 49
3.1 Single Layer Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 MPSQP Error for Single Layer Sample . . . . . . . . . . . . . . . . . . . . 60
3.3 Performance Comparison of GA and MPSQP . . . . . . . . . . . . . . . . 60
3.4 Results for Two Layer Samples . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5 Results for F125/F40/Teflon Sample . . . . . . . . . . . . . . . . . . . . . 69
xiv
Chapter 1
Introduction
Over the last several years, both consumer and military demands for better per
forming yet smaller and cheaper electromagnetic (EM) devices have overcome design
engineers? ability to rely solely on traditional design techniques [1]. Specifically, per
formance requirements for EM devices have created the need for new techniques and
technologies to be developed in order to meet the everincreasing demand. Prior to the
early 1980?s, much of the design work was conducted via trial and error using empirical
or semiempirical formulations. However, with the significant gains in computational
power and simultaneous development of accurate numerical electromagnetic solution
techniques in the 1980?s, designers could now reliably simulate many designs before ever
moving to the fabrication and measurement stage. Although this development was a
significant milestone in the evolution of EM design, the actual design process still relied
heavily on trial and error since engineers continued to use their intuition for design mod
ifications. Of late, computational analysis tools and optimization methods have become
hybridized which has allowed designers to realize novel devices that would otherwise
have been impossible to conceive through traditional methods. This union of analysis
and optimization tools has also opened possibilities for reverse engineering processes
(inverse problems) such as characterization of complex manufactured devices and novel
multilayered materials to be used in Monolithic Microwave Integrated Circuits (MMIC).
1
1.1 Electromagnetic Optimization
Historically, optimization has been applied to electromagnetics in two ways: viz.
inverse problems [2] and design problems [3]. An inverse problem is the task of using
limited information such as scattered fields or Sparameters to obtain the physical char
acteristics of the scatterer. The parameters being solved for often include geometric
shape and dimensions, constitutive parameters (CP), position, and orientation. Opti
mization algorithms typically require some initial estimate for the unknown parameters
and iterate until values are found which yield within the tolerance limits the same quan
tity as that which was measured. The solution of an inverse problem is typically quite
difficult since limited information is available and, therefore, not sufficient for satisfying
uniqueness. Hence, there is the possibility that many solutions may return nearly iden
tical information. To help curtail this problem, constraints are usually placed on the
physical characteristics of the object thereby reducing the chance of encountering mul
tiple solutions. As an example, consider trying to determine the physical characteristics
of a scatterer using only a small sample of the radiated field. Without any assumptions
made, this problem lacks sufficient information to yield a meaningful solution. However,
if the scatterer shape is assumed known, then the CP may be calculated to a given degree
of accuracy.
Design optimization problems closely resemble their inverse counterparts with the
exception that the information provided to the optimizer is different. Inverse solvers
typically utilize measurements, whereas design optimizers are fed design goals as specified
by the design engineer. For example, a certain bandwidth characteristic may be desired
2
for a patch antenna, and an optimization can be performed to determine the shape of
the patch that will yield the required results [4,5].
1.2 Problem Statement
This research focuses on the development of novel optimization tools to automate
the design and characterization process for a wide range of EM engineering problems.
Toward this end, a series of specific problems are addressed even though the methods
presented may easily be adapted to a variety of similar tasks. Initially, the inverse prob
lem technique is presented in which the complex permittivity of a single layer structure or
each individual layer of a multilayer structure are extracted using Sparameter data from
a loaded waveguide. Using this method, the complex permittivity of each layer in the
structure is extracted using a variety of optimization techniques. One of the fundamental
challenges associated with optimization techniques is the proper balance of convergence
rate and robustness (i.e., their ability to adequately search the entire solution space).
Therefore, the initial focus is placed on methods that improve the convergence character
istics of the algorithms implemented here without sacrificing robustness. In addition to
considering different optimization techniques, a thorough investigation is conducted into
the effects of different error functions on the convergence characteristics and accuracy of
the optimization algorithms. By varying the mathematical form of the error functions
as well as the quality and quantity of Sparameter information included, conclusions
may be drawn about the maximum number of layers for which a specific error function
is effective. Some consideration is also given to the calibration effort required to make
3
accurate Sparameter phase and magnitude measurements since the availability and ac
curacy of this information will have a dramatic effect on each algorithm?s performance.
After this analysis, the method is further modified to include the extraction of complex
permeability. Additional discussion of the optimization algorithms is given as this higher
dimensionality problem requires a more robust procedure.
The second primary area of focus pertains to the use of the aforementioned op
timization algorithms to automate the design of EM devices. Specifically, this work
addresses the feasibility of obtaining (optimizing) reflection and transmission properties
using dielectric slabs as waveguide filters. The filters are analyzed using a hybrid Mode
Matching/Finite Element Method (MM/FEM) which acts as the forward solver within
the context of the optimization algorithm and has been shown to be an efficient algo
rithm for loaded waveguide problems [6,7]. Two analyses are considered with regard to
the geometry of the filters. In the first, an attempt is made to characterize the feasibil
ity of obtaining useful filter responses by embedding randomly oriented thin conducting
wires and other conducting objects (e.g., thin patches and small volumetric inclusions)
within the dielectric slab. Such a material would be quite novel since it would possess
designable filter characteristics and yet maintain similar visual and structural proper
ties. For the practical applicability of this type of filter, the responses of the randomly
doped slabs would be required to maintain a certain level of consistency for given doping
levels (densities) and inclusion sizes. Consequently, a thorough analysis is conducted
into whether or not this method is both computationally and physically practical. In
the second part, desired waveguide filter responses are found by optimizing the shape
4
of conducting patch on the surface of the dielectric slab. This method has practical
advantages since fabrication of such a filter is relatively simple.
1.3 Overview and Historical Perspective
1.3.1 Optimization Techniques
Optimization refers to the solution of a problem that can be cast in the form
f (vectorx) = 0 where vectorx is an ndimensional vector of parameters, and f is referred to as
the error, objective, or fitness function. Depending on the problem to be solved, each of
the parameters xn may be continuous or discrete and may be bounded or unbounded.
The problem may also be subject to equality constraints of the form ci (vectorx) = 0 and/or
inequality constraints of the form gj (vectorx) ? 0. Within the field of optimization, there are
generally two classes of methods: local and global (the classes are also referred to as
deterministic and stochastic, respectively).
Local Optimization Techniques
Local optimization methods utilize information about an error function in the vicin
ity of the current iteration?s parameter values. Chronologically, local methods were the
first to be developed and utilized for engineering development since they can usually
locate the optimal solution with a minimal number of error function evaluations depend
ing on the initial guess and complexity of the algorithm. Generally, these techniques are
subdivided into two categories: direct search methods and gradientbased methods [8].
Direct search methods were first developed in the 1950?s [9] and received much
attention due to the fact that they do not explicitly utilize information about the gradient
5
of the error function. At that time, this attribute was particularly beneficial since the
numerical computation of gradients was difficult due to computational limitations. These
were also the only methods available for optimization of discontinuous search spaces.
However, they were all but abandoned by the optimization community beginning in the
early 1970?s for a number of reasons outlined by Swann including the fact that there
were no mathematical proofs that such methods would ever converge to a minimum [10].
Within the last 10 years, however, direct search methods have regained popularity as
certain proofs are now available guaranteeing convergence [11?14].
Gradientbased optimization, as the name implies, relies on the gradient of the er
ror function to formulate a search direction from which the function minimum can be
found. The Steepest Descent Methods [15] and others like them, in their simplest forms,
require the search space be continuous in its first derivative. Higher order methods such
as Newton?s Method [15] require that the Hessian (secondorder derivative) be available.
Other methods include QuasiNewton methods, LevenbergMarquardt (LM) [16], Se
quential Quadratic Programming (SQP) [15,17?21], etc. These algorithms often obtain
the minimizing parameters with high accuracy and low computational cost provided that
the gradient and Hessian are easily and accurately calculable. When these methods are
applied to problems where the derivatives are not analytically available and must be cal
culated by numerical means or where the search space is discontinuous, their accuracy
generally suffers or, in the worst case, may fail altogether. A similar problem occurs
when the search space has many minima such as in the case of Rastrigin?s function, a
6
function often used to benchmark optimization algorithms (see Fig. 1.1) [22]. An ex
ample error surface for which local methods are particularly wellsuited (surfaces having
only one minimum) is shown in Fig. 1.2.
X
Y
?1 ?0.5 0 0.5 1?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
1
Figure 1.1: Rastrigrin?s function which has many local minima and a global minimum
located at (0,0).
Of the gradientbased methods, SQP is currently considered to be one of the most
robust nonlinear programming (NP) methods available for optimization problems [20].
NP refers to the class of optimization problems in which the error functions are non
linear in their parameters and constraints. Although SQP is considered to be a local
optimization method, it utilizes features which enhance its ability to avoid local min
ima. By taking a quadratic expansion of the nonlinear error function, the Hessian and
Lagrangian operators can be efficiently approximated and used to create a Quadratic
7
X
Y
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 1.2: Error surface wellsuited to local optimization problems.
8
Programming subproblem. The solution of the subproblem results in the selection of
the line search direction. Depending on the accuracy of the initial guess, SQP has been
shown to exhibit second order convergence toward a local minimum. Therefore, when
local methods are needed in the problems presented in this dissertation, SQP is utilized.
For further discussion of SQP, the interested reader is referred to [15,17?19,21].
Global Optimization Techniques
Global optimization refers to the task of finding the absolute best set of parameters,
usually over a very large search space, which will minimize the error function. A large
number of methods have been developed to handle these tasks which include techniques
wellsuited for combinatorial problems, discontinuous problems, constrained problems,
etc. Over the years, the methods have been grouped according to their approach to
the optimization. These groups include Branch and Bound Methods [23], Clustering
Methods [24], Evolutionary Algorithms [25], Statistical Methods [26], and miscellaneous
and hybrid techniques [27]. Of late, Evolutionary Algorithms including the Genetic Al
gorithm (GA) [28], Particle Swarm Optimization (PSO) [29], and Differential Evolution
(DE) [30] have received considerable attention. These methods generally rely on stochas
tic procedures to prevent the local minimum trapping that may hinder local methods.
In addition, they are particularly wellsuited for problems having search spaces which
are discontinuous, constrained, multidimensional, noisy, and multiextremal. Unfor
tunately, this robustness results in a greater computational expense since many error
function evaluations are typically required to reach a good solution. Also, the methods
9
are considered inexact because their convergence usually suffers once a solution is found
very close to the global minimum.
Of GA?s, PSO, and DE, GA?s are chronologically the first to have been devel
oped [31?33]. GA?s are based on the concepts of evolutionary biology in which a ?sur
vival of the fittest? paradigm is adopted. The GA begins with selection of an initial
population of parameter values and selectively evolves the population toward the global
minimum of the search space. This method has been extensively researched and applied
to many problems found within the EM community since the error functions encoun
tered here typically possess many of the qualities of difficult NP problems previously
mentioned in addition to the fact that the error functions are computationally expensive
to evaluate [34]. In fact, GA?s have been applied successfully to nearly all areas of elec
tromagnetics including antenna design [35], phased arrays [36], and radar absorbers [37].
Thorough discussions regarding applications of GA?s to electrical engineering problems
can be found in [34,38,39].
PSO is similar to the GA in that it is a populationbased technique which utilizes
information from the population to move toward a globally optimal solution. PSO,
developed by Eberhart and Kennedy in 1995, is based on social theories related to bird
flocking, insect swarms, and fish schools [29]. PSO begins with an initial population of
?particles? each of which has a particular location and velocity within the search space.
Utilizing the information of their ?neighbors?, each particle can adjust its flight direction
and speed until all particles arrive at the global minima of the search space. Advanced
algorithms employ many features such as fully connected neighborhoods, particle inertia,
etc. which can accelerate convergence for particular applications [40,41]. Recently,
10
PSO has been applied to several EM problems including antenna design [42] and array
synthesis [43].
DE is also a populationbased technique and was developed by Price and Storn to
solve the Chebychev Polynomial fitting problem [30]. It relies on simple vector algebra
operations on the population members to quickly generate new population members
which are closer to the global minimum. This technique has been shown in certain
instances to converge faster than other evolutionary algorithms [44] and is also wellsuited
for parallelization. Although DE has yet to receive much attention in EM optimization,
some research has been conducted on antenna arrays [45] and inverse scattering [46].
Of the evolutionary algorithms discussed, the GA and PSO are utilized in this
work with the GA receiving a more extensive consideration. Details of the specific
implementations and modifications of the algorithms are given in later sections.
1.3.2 Constitutive Parameter Extraction
Multilayer substrate materials are currently used for many practical applications
that include Microwave Integrated Circuits (MIC), MMIC [47], radomes, spatial filters
for antenna beam shaping [48], and Frequency Selective Surfaces (FSS) [49].
By choosing the appropriate thicknesses and material parameters for the layers,
it is possible to synthesize composite structures with novel electromagnetic properties
otherwise not found in a single material [50]. Recently, attention has focused on non
destructive methods of determining the constitutive parameters of each individual layer
of the substrate [51,52].
11
There are a large number of methods for determining the permittivity or perme
ability of a single homogeneous sample or the effective ?bulk? properties of a layered
material. These methods include splitcylinder resonators, cavity resonators, TE10 split
post dielectric and magnetic resonators, whisperinggallery resonators, transmissionline
and waveguide techniques, etc. [53?55]. For measurements of the complex permittivity
and permeability over broad frequency bands (e.g., X or L bands), transmission line
or waveguide techniques are generally preferred even though the achievable accuracy is
reduced due to unavoidable measurement errors [50,56].
Some recent methods used to accurately estimate the complex permittivity of in
dividual layers of a multilayered or inhomogeneous structure are given by Sanadiki and
Mostafavi [57], Zwick et al. [52], and Deshpande and Dudley [50]. Sanadiki and Mostafavi
provide a method of solving the inverse scattering problem using a least squares error
approach. This method is only tested against computer generated data and may be
sensitive to errors associated with measurements. Zwick et al., utilize a GA to find the
complex constitutive parameters for a multilayered sample by an evolutionary process.
Their method requires measurements be obtained over a frequency range or as a func
tion of incidence angle for a given frequency and does not require phase information for
the transmission or reflection coefficients. Deshpande and Dudley?s algorithm employs
SQP and utilizes both magnitude and phase information of the measured Sparameters
which, in turn, requires very accurate system calibration. To the author?s knowledge,
no published work has address the issue of extracting both complex permittivity and
permeability from individual layers of a multilayered structure.
12
1.3.3 Waveguide Filter Optimization
In recent years, there has been a steady increase in the demand for fast and effi
cient passive waveguide filter design tools applicable to Eplane filters [58,59], Hplane
filters [60], dualband filters [61], etc. Thus, many researchers have begun exploiting
optimization techniques to alleviate the burden of traditional analytical design meth
ods [62?67]. Although the majority of earlier work in filter design is focused on the de
velopment of longitudinal filters (i.e., structures which are oriented longitudinally within
the waveguide) [58,59,61?67], of late, some work has been conducted in the design and
optimization of transverse filters (i.e., structures which have their prominent dimension
oriented parallel to the waveguide face) [68?70]. Transverse filters have the advantage of
being lightweight, low profile, and easy to manufacture. Additionally, transverse filters
do not require modification of the supporting waveguide structure as do some longitudi
nal filters [67].
Lockyer and Vardaxoglou [68] and Seager et al. [69] have investigated transverse
filter performance using twolayer aperture arrays within the waveguide [68,69] and were
able to realize narrow bandpass filters using structures ?1 mm thick. They have shown
that, for narrowband applications, a transverse filter can be constructed quite easily.
Alternatively, Monorchio et al. [70] developed an optimization technique to design a
wideband transverse filter for square waveguides based on a GA and Method of Moments
(MoM) solver [71]. Their approach optimized a FSS screen which transmitted K and Ka
bands while reflecting X and Ku bands.
Since transverse waveguide filters are basically bounded FSS, a brief history of FSS
optimization is also warranted. This topic has received significant attention due the
13
complexity of the methods involved in creating these structures [71?78]. Manara et
al. [71] utilized a GA optimization procedure to refine the geometry of a multiband,
single layer FSS structure which could efficiently transmit L and S bands while reflecting
the Ka band. Parker et al. [73] also employed a GA to reduce the angular dependence
of the reflection band for a FSS. Bozzi et al. [75] and Li et al. [78] applied fast solvers to
the optimization problem thereby making the analysis of realistically complex structures
feasible. Chakravarty and Mittra [74,76,77] demonstrated the MicroGA?s ability to
optimize multilayer spatial filters efficiently when several design parameters were allowed
to vary over realistic ranges. Finally, Ohira et al. [72] utilized the GA to optimize mask
shape for FSS elements using printed circuit technology.
1.4 Thesis Outline
The remainder of this dissertation is organized in the following manner. Chap
ters 2 & 3 present the formulation, optimization algorithm, and results for the complex
constitutive parameter extraction technique. Chapter 4 presents the numerical technique
and results pertaining to the embedded conductor filters. In Chapter 5, modifications
of the optimization algorithms presented thus far are discussed as well as results for the
waveguide filters generated using surface patches. Finally, Chapter 6 presents conclusions
and suggestions for future work.
14
Chapter 2
Complex Permittivity Extraction
2.1 Overview
In this chapter, the performance of several optimization techniques is evaluated
when applied to the multilayer complex permittivity extraction problem. Sparameter
measurements obtained from a loaded waveguide serve as an input to the inversion algo
rithms. The methods to be considered are Sequential Quadratic Programming (SQP),
Genetic Algorithm (GA), and Particle Swarm Optimization (PSO). Their sensitivity
and performance relative to the choice of error function is also investigated. Simulations
are performed using computer generated Sparameters to quantify the performance of
each algorithm under ideal conditions. In order to determine the robustness of these
methods, the extracted permittivities determined by the algorithms are used to generate
Sparameter data sets for comparison to the measured Sparameters.
Sparameter X band waveguide measurements, provided by NASA Langley Re
search Center [50], were made using the following dielectric materials: Bakelite, Ceramic,
GarlockRubber, and Nano Material. All of the materials are lowloss, nonmagnetic,
and were found to have complex permittivities that remained nearly constant over the X
band [50]. They were used to create planar, single layer samples of various thicknesses,
with all samples having X band waveguide crosssectional dimensions. The multilayer di
electric structures were fabricated by placing single layer samples adjacent to one another
(see Fig. 2.1). All measurements were obtained using an HP8510C Vector Network An
alyzer for frequencies of 8.2  12.4 GHz (X band). Additionally, the complex permittivity
15
of each single layer sample was determined using the Agilent 85071 Materials Measure
ment Software [79,80] and compared to the values obtained from the algorithms.
2.2 Theory
Assuming that only the dominate TE10 mode propagates in the loaded waveguide,
as shown in Fig. 2.1, the formulation of the Sparameters can be expressed in terms of
each layer?s thickness (dn) and unknown permittivity (??n) using ABCDparameters as
shown below [81,82]:
?
??An Bn
Cn Dn
?
??=
?
?? cosh(?ndn) Zn sinh(?ndn)
sinh(?ndn)/Zn cosh(?ndn)
?
?? (2.1)
?
??A B
C D
?
??= nproductdisplay
i=1
?
??Ai Bi
Ci Di
?
?? (2.2)
where ?n and Zn are the propagation constant and wave impedance of the nth layer,
respectively, and can be expressed as
Zn = j???
n
(2.3)
and
?n =
radicalBig
(?/a)2 ??2???c (2.4)
where a is the longer dimension of the waveguide.
16
Figure 2.1: Rectangular waveguide loaded with nlayer sample.
17
The ABCDparameters can be directly converted to Sparameters using the following
equations
S11 = (A+B/Z0 ?CZ0 ?D)/X
S12 = 2(AD?CB)/X
S21 = 2/X
S22 = (?A+B/Z0 ?CZ0 +D)/X
X = A+B/Z0 +CZ0 +D
(2.5)
where Z0 is the empty waveguide impedance.
2.3 Error Function Selection
The error functions used in the investigation are of the basic form given by
Err =
radicaltpradicalvertex
radicalvertexradicalbt 1
N
Nsummationdisplay
i=1
bracketleftBig
f
parenleftBig
[S]f,i
parenrightBig
?f
parenleftBig
[S]m,i
parenrightBigbracketrightBig2
(2.6)
whereN isthetotalnumberoffrequencypoints, [S]f,i aretheformulatedSparameters(2.5)
evaluated at frequency point i, and [S]m,i are the measured Sparameters at frequency
point i. Error functions can generally be grouped into two categories: (1) error func
tions used to minimize differences in both phase and magnitude of the measured and
formulated scattering parameters, and (2) error functions that minimize only magnitude
variations of the measured and formulated scattering parameters. For the purposes of
this study, three error function definitions were used and found to accurately obtain the
permittivity of single and multilayer structures:
18
Err1 =
radicaltpradicalvertex
radicalvertexradicalvertex
radicalvertexradicalvertex
radicalbt
1
N
Nsummationdisplay
i=1
?
??
?
parenleftBig
Re
parenleftBig
[S]f,i
parenrightBig
?Re
parenleftBig
[S]m,i
parenrightBigparenrightBig2
+
parenleftBig
Im
parenleftBig
[S]f,i
parenrightBig
?Im
parenleftBig
[S]m,i
parenrightBigparenrightBig2
?
??
? (2.7)
Err2 =
radicaltpradicalvertex
radicalvertexradicalbt 1
N
Nsummationdisplay
i=1
parenleftBigvextendsinglevextendsingle
vextendsingle[S]f,i
vextendsinglevextendsingle
vextendsingle?
vextendsinglevextendsingle
vextendsingle[S]m,i
vextendsinglevextendsingle
vextendsingle
parenrightBig2
(2.8)
Err3 =
radicaltpradicalvertex
radicalvertexradicalbt 1
N
Nsummationdisplay
i=1
Mi +P1,i +P2,i (2.9)
where
Mi =
parenleftBigvextendsinglevextendsingle
vextendsingle[S]f,i
vextendsinglevextendsingle
vextendsingle?
vextendsinglevextendsingle
vextendsingle[S]m,i
vextendsinglevextendsingle
vextendsingle
parenrightBig2
Pf1,i = 1?S112f,i ?S212f,i
Pf2,i = 1?S222f,i ?S122f,i
Pm1,i = 1?S112m,i ?S212m,i
Pm2,i = 1?S222m,i ?S122m,i
P1,i = (Pf1,i ?Pm1,i)2
P2,i = (Pf2,i ?Pm2,i)2
Err1, given in (2.7), includes both phase and magnitude information. It is a slightly
modified form of Deshpande and Dudley?s error function [50] and is applicable to materi
als where the permittivity remains approximately constant over the measured frequency
range. The inclusion of measurement information over the entire frequency range tends
to minimize the effects of instrumentation error in the calculations. Err2, given in (2.8),
requires only magnitude information of the scattering parameters and is representative
19
of the fitness (error) function used by Zwick et al. [52] and Queffelec and Gelin [54] .
The third error function Err3, given in (2.9), is unique in that it uses only magnitude
information but includes terms accounting for dissipated power. It was initially believed
that the inclusion of the power terms would increase the algorithms? ability to accurately
determine the imaginary part of the permittivity. It should be noted that Eqns. (2.8)
and (2.9) are calculated in decibels which was found to decrease the time required to
find the global minimum in the solution space [52].
2.4 Optimization Algorithms
2.4.1 Sequential Quadratic Programming
Since publication of a series of expository papers in the 1970?s, SQP has become the
most popular method for solving Nonlinear Programming (NP) problems [83?85]. SQP,
itself, is not a specific NP algorithm but a framework under which many implementations
have evolved. A thorough treatment of the mathematical details of SQP and its sub
algorithms is beyond the scope of this dissertation, and interested readers are referred
to the foundational papers on SQP [18,86,87]. Instead, a brief and general overview
is favored since specific implementations are readily available and deserve individual
treatment [88]. As mentioned in Section 1.3.1, the goal of NP is to solve the problem
f (vectorx) = 0 (2.10)
subject to the constraints
20
ci (vectorx) = 0
gj (vectorx) ? 0
(2.11)
where vectorx is the vector of optimization parameters, ci (vectorx) are the set of equality constraints
on vectorx, and gj (vectorx) ? 0 are the set of inequality constraints on vectorx. SQP?s novelty lies in its
ability to solve problems efficiently which are nonlinearly constrained. This is possible
through the quadratic approximation of the Lagrangian function. The Langrangian
function, in its unmodified form, is given by
L = f(vectorx)+vectoruTvectorc(vectorx)+vectorvTvectorg(vectorx) (2.12)
where vectoru and vectorv are the Lagrangian multipliers. The reasoning behind forming the
quadratic subproblem is that a number of techniques are available for solving error
functions which are quadratic and subject to linear constraints. Therefore, the approx
imation is formed as a quadratic function around the current iterate?s location, and
the constraints are linearized. Once the quadratic approximation is formed, this can
be solved by a number of Quadratic Programming (QP) methods [15,89,90] to form a
line search direction. This eventually leads to the selection of the next evaluation point
which will converge to the minimum of f.
The basic SQP outline is given as follows:
1. Obtain initial parameters (i.e., initial guess).
2. Solve the QP problem at this point to obtain the line search direction.
3. Choose a step length such that the error is reduced sufficiently.
21
4. Update the current iterate and Lagrangian multipliers for the next iteration.
5. Check stopping criteria (convergence).
6. Proceed to next iteration.
2.4.2 Genetic Algorithm
The GA, the foundations of which are found in evolutionary biology [25,31?33], is
based on the concept of evolution. Therefore, much of the terminology used to explain
the algorithm is biological in nature. GA?s, like the other global optimization algo
rithms used in this work, are somewhat immune to local minimum trapping since they
employ stochastic processes. GA?s find application in a wide variety of fields such as bio
science [91], neural networks [92], economics [93], and many more [94]. The remainder
of the section outlines the GA implemented in this work.
The GA begins with the creation of a population of parameter values randomly dis
tributed throughout the solution space (see Fig. 2.2). Each member of the population is
referred to as a ?chromosome? and contains one ?gene? storing a value for each parame
ter in the function to be minimized. In traditional GA?s, each gene is encoded as an nbit
binary string which is used for creation of the next population. Upon creation of the
initial population, the socalled fitness of each chromosome is evaluated by determining
the error function value associated with that chromosome?s list of parameters.
After fitness evaluation, a new population (generation) is created. Construction
of the new population begins by selecting a subset population representing the ?best?
chromosomes (i.e., those chromosomes having the lowest error function value). These
22
X59X65X73
X4EX6F
X4EX6F
X59X65X73
X44X65X74X65X72X6DX69X6EX65X20
X45X6CX69X74X65X20
X43X68X69X6CX64X72X65X6E
X49X6EX69X74X69X61X6CX69X7AX65X20
X50X6FX70X75X6CX61X74X69X6FX6E
X45X76X61X6CX75X61X74X65X20
X46X69X74X6EX65X73X73
X53X65X6CX65X63X74X69X6FX6E
X43X72X6FX73X73X6FX76X65X72
X4DX75X74X61X74X69X6FX6E
X50X6FX70X75X6CX61X74X69X6FX6EX20
X52X65X64X75X6EX64X61X6EX63X69X65X73X3F
X52X65X6DX6FX76X65X20
X43X6FX70X69X65X73
X52X65X69X6EX69X74X69X61X6CX69X7AX65X20
X54X6FX20X46X69X6CX6CX20
X50X6FX70X75X6CX61X74X69X6FX6E
X45X78X69X74X20X43X6FX6EX64X69X74X69X6FX6EX3F
X45X78X69X74
Figure 2.2: Flowchart illustrating the Genetic Algorithm procedure.
23
?elite? chromosomes are inserted into the new population ensuring that these solutions
will not be eliminated from the new generation.
The crossover stage is the defining phase of the GA. First, two ?parent? chromo
somes are selected from the current generation using the binary tournament selection
method [38]. In binary tournament selection, two chromosomes are picked at random.
The chromosome with the best fitness is chosen as the first parent. This process is re
peated for the second parent. Following selection of the parent chromosomes, the GA
generates a random number p ? [0,1]. If p > pcrossover, the crossover rate, the two
parents are copied directly into the new population. Otherwise, a random position in
the parents? binary string of encoded information is selected (see Fig. 2.3). The first
?child? is formed using the string of information to the left of the crossover point in the
first parent and the information to the right of the crossover point in the second parent.
The second child is formed using the remaining genetic information from each parent.
This process of selection and crossover is repeated until the new population is completed.
Using this concept, chromosomes with poor fitness that possibly possess ?good? genetic
information have the opportunity to pass their traits on to future generations.
The next major phase of the GA is the mutation stage. For each child in the
new population (elite children excluded), a random number m ? [0,1] is generated. If
m > pmutation, the mutation rate, no change is made to the chromosome. However, if
m < pmutation, one or two random bits of the chromosome are transposed (i.e., 1 ? 0).
This reduces the likelihood of local minima trapping.
The GA will discontinue if a chromosome is found to have a fitness value below 10?5
or if the maximum number of generations has been reached. Depending on the number
24
1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1
0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0
Crossover Point
1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0
0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1
Parent 1
Parent 2
Child 1
Child 2
Figure 2.3: Crossover between two parents using binary string encoding.
25
of layers being characterized, the maximum number of generations is allowed to vary.
The specific details of this parameter?s selection will be discussed in Section 2.5.
GA Redundancy Removal
At each generation, during the creation of the new population, some chromosomes
may be duplicated. Several novel redundancy removal schemes were developed and their
effect on convergence rate was analyzed. The first scheme developed is referred to as
the Complete Elite Redundancy Removal GA (CERRGA). After a significant number
of generations, the GA converges to a population consisting of identical chromosomes
representing the best obtained solution to the fitness function (with the exception of
the occasional mutation). At this point, a traditional GA exits since simple mutation
alone is a very inefficient way of searching for new genetic information that will more
efficiently minimize the error function. In the CERRGA method, the GA checks for this
redundancy in the population for each new generation. When this situation occurs, one
copy of this redundant chromosome is kept as well as any mutations. The rest of the
population slots are filled with random chromosomes in the same manner in which the
initial population was generated, and the algorithm is restarted at the current iteration.
The next redundancy removal scheme employed was termed Incomplete Elite Re
dundancy Removal GA (IERRGA). This method is very similar to CERRGA in that
elite chromosome redundancy is eliminated. However, IERRGA allows a factor to be set
that controls how much of the population is filled with elite chromosomes before the re
dundancies are eliminated. In the results presented shortly, a 90% IERRGA scheme was
used. Therefore, the algorithm monitored the number of elite chromosome redundancies
26
present in the population. If that number exceeded 90%, only one copy was kept and
the remaining population slots were filled with random chromosomes. This differs from
CERRGA in that the new random chromosomes have a greater chance of reproducing
with one of the elite copies. Therefore, in theory, there may be a greater number of
?good? chromosomes in the population at any one time than with CERRGA.
The last redundancy removal method monitors the population at each generation
and removes all redundancies. Therefore, only one copy of any chromosome is present
in the population. All emptied population slots are, as always, filled with random chro
mosomes. This scheme is termed the Total Redundancy Removal GA (TRRGA).
These methods were tested using a computer generated Sparameter data set for one
layer (see Section 2.5). Fig. 2.4 shows the average fitness value of the best chromosome
at each generation from 20 simulations for all of the methods presented as well as the
unaltered GA (i.e., no redundancy removal). From the figure, it is obvious that the
TRRGA method provides significantly faster convergence than the traditional GA and
CERRGA and is better than IERRGA by approximately 80 generations. Therefore, the
results presented in the remaining sections correspond to those obtained by the TRRGA.
2.4.3 Particle Swarm Optimization
PSO was developed by Eberhart and Kennedy [29] as an optimization procedure
based on models of social behaviors such as bird flocking. In many ways, it bears a strong
resemblance to the GA except that there are no genetic operators such as crossover and
mutation. PSO handles the generation of a ?new population? in a much different way as
will be explained. Similar to the GA, PSO has received much attention in a variety of
27
0 50 100 150 200
10?6
10?4
10?2
100
Generation
Fitness
Traditional GA
All redundancies
Elite redundancies
90% Elite redundancies
PSO
Figure 2.4: Comparison of convergence for the traditional GA and GA?s using three
redundancy removal methods. The convergence rate of PSO is also shown.
28
fields including EM as previously mentioned. For a thorough review of previous research
regarding PSO, the interested reader is referred to Hu [95].
The PSO algorithm utilizes a population of ?particles? which are given a position
and velocity within the search space. Each of these particles, depending on the complex
ity of the algorithm, has a set of neighbors, socalled individuality and sociality factors
which determine how much they are influenced by their neighbors, and an inertia factor
which includes the effects of previous iterations. Each particle usually is also given a
?memory? of the best solution it has encountered which can also be made to affect the
particle?s propagation through the solution space. The following paragraphs detail the
PSO algorithm implemented in this work.
As with most Evolutionary Algorithms, PSO begins by initializing a population of
particles with a random location and velocity (see Fig. 2.6). In this implementation,
individuality and sociality factors and neighbors are also initialized. A common choice
of individuality and sociality factors is 2 [29]. A fiveparticle neighborhood is established
so that each particle is influenced by the two particles to its left and right (i.e., particle
3 has neighbors 1, 2, 4, and 5 while particle 4 has neighbors 2, 3, 5, and 6) as shown
in Fig. 2.5. The inertia factor is initialized to 1 such that the particle is also influenced
heavily by the previous iteration?s information.
Each particle?s fitness is then tested by evaluating the fitness function (forward
solution). If this fitness is better than the best previous fitness of that particle, the
current location is stored as the best. Also, if any fitness is encountered which is below
a fitness threshold, then the algorithm ends because a sufficient solution has been found.
29
X31 X32
X33
X34
X35
X36
X37
X38
X39X31X30X31X31
X31X32
X31X33
X31X34
X31X35
X31X36
X31X37
X31X38
Figure 2.5: Illustration of the neighborhood structure implemented in PSO. Each particle
is connected to its two neighbors to the left and right.
30
After every particle?s fitness has been evaluated, the velocity and, hence, next lo
cation can be calculated. Below, a pseudocode is presented to explain this calculation.
This process is repeated for each particle.
%Return the neighbor who has had the best fitness
n = GetNeighborWithBestFitnessSoFar()
%Return the best location of that neighbor
n_Best = GetBestLocationOfParticle(n)
%Multiply the sociality factor by a random number [0,1]
Reduced_S = S_Factor*rand1
%Multiply reduced sociality factor by difference between
%neighbor?s best location and particle?s current location
n_Change = Reduced_S*(n_Best  p_Current)
%Multiply individuality factor by a random number [0,1]
Reduced_I = I_Factor*rand2
%Multiply reduced individuality factor by difference between
%particle?s best location and current location
p_Change = Reduced_I*(p_Best  p_Current)
%Get change in location for the particle by summing up all
%change contributions including the inertia factor multiplied
%by the current velocity
Change = inertia*p_Velocity + n_Change + p_Change
%Check to see if the change is greater than what is allowed.
%This helps limit changes that are too small or too large
%so that the particles better cover the search space
p_Velocity = CheckChange(Change, Max_Change, Min_Change)
%Update next location by adding current location to the
%calculated change in location
p_Next = p_Current + p_Velocity
Once this process is complete, the particles? locations are updated for the next
iteration, and the procedure of fitness evaluation begins again. At each iteration, the
inertia factor is reduced by a uniform amount until it reaches 0.2. By reducing the
inertia at each iteration, the particles are slowly forced to converge. It was found that
this method resulted in a faster convergence rate than simply leaving the factor set to 1.
The PSO algorithm presented here was tested using the same one layer computer
generated data as presented in the previous section. Again, 20 runs were performed
31
and the average convergence rate calculated. Fig. 2.4 shows the convergence curve for
the PSO as compared to the GA?s presented. While the PSO returns equally accurate
answers as compared to the GA, it requires many more iterations to do so than the
TRRGA method. Therefore, a complete analysis of the cases considered in the next
sections was not conducted using PSO. For brevity, the cases that were analyzed using
PSO are not presented since their results are the same as what was found using the GA.
2.5 Numerical Results
Computer generated Sparameter data sets for 1, 2, and 3layer cases over X
band frequencies were initially used to test the accuracy of each extraction scheme.
The respective complex permittivities and thicknesses are shown in Table 2.1 (single
layer permittivity extraction used layer 1; 2layer extraction used layers 1 and 2; 3layer
extraction used all layers ordered accordingly). The initial complex permittivity guess
for all layers was set to ??initial = 5?j0.4, while the search space for both algorithms was
limited to 1 < Re(??c) < 10 and 0 < Im(??c) < 0.8.
Table 2.1: Complex Permittivities and Thicknesses for the 1, 2, and 3Layer Computer
Generated Data Sets
Layer ??n dn (mm)
1 7?j0.01 1
2 3?j0.02 10
3 2?j0.1 2
The SQP algorithm terminated for error function or directional derivative values
? 10?16 or after 5000 iterations. The GA?s population size was set to a value of 100
and terminated after 200, 1000, and 5000 generations for the 1, 2, and 3layer cases,
32
X49X6EX69X74X69X61X6CX69X7AX65X20
X50X6FX70X75X6CX61X74X69X6FX6E
X45X76X61X6CX75X61X74X65X20
X46X69X74X6EX65X73X73
X43X61X6CX63X75X6CX61X74X65X20X4EX65X77X20
X4CX6FX63X61X74X69X6FX6EX73
X55X70X64X61X74X65X20
X50X61X72X74X69X63X6CX65X73
X52X65X64X75X63X65X20
X49X6EX65X72X74X69X61
X45X78X69X74X20
X43X6FX6EX64X69X74X69X6FX6EX3F X45X78X69X74
X4EX6F
X59X65X73
X49X6EX64X69X76X69X64X75X61X6CX69X74X79
X53X6FX63X69X61X6CX69X74X79
X4EX65X69X67X68X62X6FX72X73
X49X6EX65X72X74X69X61
Figure 2.6: Flowchart illustrating the PSO procedure.
33
respectively. The GA utilized an 80% crossover rate and 10% mutation rate (these values
are within the range of nominal values given by [38]).
In Table 2.2, the results for the 1, 2, and 3layer cases using each algorithm (single
or multiobjective) and error function are shown. An ?X? in the table indicates that the
algorithm extracted incorrect complex permittivity values for the indicated sample (i.e.,
the algorithm became trapped in a local minimum). For the multiobjective cases, each
term in the error functions (2.7)(2.9) becomes an element of an error function vector.
For instance, in a multiobjective format, Eqn. (2.8) would then be given by
???Err =
?
??
??
??
??
??
??
??
?
radicalBigg
1
N
Nsummationtext
i=1
(S11,f,i?S11,m,i)2
radicalBigg
1
N
Nsummationtext
i=1
(S21,f,i?S21,m,i)2
radicalBigg
1
N
Nsummationtext
i=1
(S12,f,i?S12,m,i)2
radicalBigg
1
N
Nsummationtext
i=1
(S22,f,i?S22,m,i)2
?
??
??
??
??
??
??
??
?
. (2.13)
The vectorization of the error functions was expected to increase the sensitivity of the
SQP algorithm.
The results shown in the table indicate that all methods performed extremely well
for the single and double layer complex permittivity extraction. However, permittivity
extraction for the 3layer sample was unsuccessful using SQP with error functions (2.8)
and (2.9). This is due to local minimum trapping indicating a poor initial estimate
(guess) of the complex permittivity. Alternatively, the GA was able to successfully
extract complex permittivity values using all three error functions independent of the
number of layers. It should also be noted that the error for both the real and imaginary
34
Table 2.2: Results using ideal Sparameters
Error One Layer Two Layers Three LayersAlgorithm Objective
Function Error Time (s) Error Time (s) Error Time (s)
(2.7) 1.837e9 59.1 1.729e8 4.3 2.431e8 6.8
Single (2.8) 2.702e8 1.9 1.780e7 65.6 X X
(2.9) 1.959e11 50.1 1.396e7 65.6 X X
(2.7) 1.712e13 2.1 2.012e9 72.7 6.245e9 83.3
Multiple (2.8) 6.397e9 54.4 8.538e8 4.4 X X
SQP
(2.9) 2.183e9 54.2 1.656e7 3.1 X X
(2.7) 3.561e7 68.8 2.636e4 969 2.079e4 6467.2
GA Single (2.8) 3.106e6 103.1 2.003e3 1136.4 1.151e2 9683.3
(2.9) 3.147e6 124.7 7.266e3 2059 1.058e2 11715
35
parts of the complex permittivity is less than O(10?7) for the SQP using (2.7) and less
than O(10?3) for the GA using any of (2.7)(2.9) for the 1, 2, and 3layer cases.
2.6 Measurements Results
2.6.1 Single Layer Measurements
For measurements of single layer materials, samples of Bakelite (d = 3.277 mm), Ce
ramic (d = 2.845 mm), Garlock Rubber (d = 1.702 mm), and Nano Material (d = 3.099
mm) were used. Sparameter data sets were generated using the extracted permittivity
value from each algorithm and then compared to the measured Sparameter values. Ad
ditionally, the Agilent 85071 Materials Measurement Software was used to extract the
single layer permittivities for comparison to the values returned by SQP and the GA.
Fig. 2.7 shows a comparison of S11 and S21 for the measured Sparameters of the
Bakelite sample and those generated from the extracted permittivities returned by the
GA using error function (2.9). Fig. 2.8 shows similar results for the Garlock sample.
The excellent agreement shown in both figures is also observed for all other materials?
Sparameter comparisons using all algorithm/error function combinations.
36
8.5 9 9.5 10 10.5 11 11.5 120.5
0.55
0.6
0.65
0.7
0.75
0.8
Frequency (GHz)
S
S11,f
S11,m
S21,m S
21,f
Figure 2.7: Magnitude of S11 and S21 for single layer Bakelite sample. The calculated
Sparameters are generated using the GA with error function (2.9).
37
8.5 9 9.5 10 10.5 11 11.5 120.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
S
S11,f
S11,m
S21,m
S21,f
Figure 2.8: Magnitude of S11 and S21 for single layer Garlock Rubber sample. The
calculated Sparameters are generated using the GA with error function (2.9).
38
Tables 2.32.6 show the results of the extracted complex permittivities for each
material using all optimization algorithms and error functions. Since error functions (2.8)
and (2.9) are calculated in decibels, a direct comparison of the error function values is not
necessarily indicative of the accuracy of the solution. Rather, summations of the RMS
errors between the magnitudes and phases of the formulated and measured Sparameters
(see Columns 67 in Tables 2.32.6) are better qualifiers of the accuracy of each solution
and expressed by
summationtext
ij
radicalBig
Sij,f2 ?Sij,m2
summationtext
ij
radicalBig
(negationslash Sij,f)2 ?(negationslash Sij,m)2
. (2.14)
The RMS errors between the measured Sparameters and those generated using the
Agilent 85071 Materials Measurement Software are also listed for comparison. It can
be concluded that all of the optimization techniques were effective in minimizing the
RMS errors of the Sparameters (both phase and magnitude). However, the GA using
either (2.8) or (2.9) was found to consistently produce the lowest magnitude error of the
Sparameters (column VI) and maintained a phase error generally no worse than the
other extraction methods.
39
Table 2.3: Extracted Permittivity & Error for Bakelite
ErrorAlgorithm Objective
Function ?
? ??? summationtextSij
err
summationtextnegationslash S
ijerr
(2.7) 3.7244 0.2237 0.02875 0.19899
Single (2.8) 3.7907 0.2528 0.01339 0.20948
(2.9) 3.7909 0.2530 0.01339 0.20952
(2.7) 3.5124 0.3464 0.09824 0.29814
Multiple (2.8) 3.7896 0.2656 0.01434 0.20685
SQP
(2.9) 3.8018 0.2577 0.01364 0.21206
(2.7) 3.7244 0.2238 0.02876 0.19898
GA Single (2.8) 3.7907 0.2528 0.01339 0.20948
(2.9) 3.7910 0.2530 0.01339 0.20954
85071 X X 3.6032 0.2347 0.06947 0.24004
40
Table 2.4: Extracted Permittivity & Error for Ceramic
ErrorAlgorithm Objective
Function ?
? ??? summationtextSij
err
summationtextnegationslash S
ijerr
(2.7) 1.1422 0.0000 0.03263 0.15330
Single (2.8) 1.1819 0.0004 0.00415 0.19111
(2.9) 1.1819 0.0006 0.00415 0.19151
(2.7) 1.1344 0.0000 0.03888 0.14662
Multiple (2.8) 1.1782 0.0518 0.04234 0.70040
SQP
(2.9) 1.1823 0.0050 0.00670 0.21894
(2.7) 1.1423 0.0000 0.03261 0.15333
GA Single (2.8) 1.1818 0.0001 0.00417 0.18989
(2.9) 1.1820 0.0010 0.00422 0.19331
85071 X X 1.1484 0.0043 0.02802 0.18255
41
Table 2.5: Extracted Permittivity & Error for Garlock Rubber
ErrorAlgorithm Objective
Function ?
? ??? summationtextSij
err
summationtextnegationslash S
ijerr
(2.7) 7.7850 0.0409 0.04200 0.13297
Single (2.8) 7.9837 0.1642 0.01669 0.17187
(2.9) 8.1358 0.1304 0.02075 0.20980
(2.7) 7.6184 0.0869 0.06706 0.08846
Multiple (2.8) 8.0340 0.1604 0.01425 0.18396
SQP
(2.9) 8.0347 0.1602 0.01424 0.18415
(2.7) 7.6184 0.0869 0.06707 0.08846
GA Single (2.8) 8.0340 0.1604 0.01425 0.18396
(2.9) 8.0348 0.1602 0.01424 0.18417
85071 X X 7.5320 0.1610 0.08267 0.06454
42
Table 2.6: Extracted Permittivity & Error for Nano Material
ErrorAlgorithm Objective
Function ?
? ??? summationtextSij
err
summationtextnegationslash S
ijerr
(2.7) 2.5165 0.0000 0.05783 0.08785
Single (2.8) 2.5989 0.0119 0.01058 0.18065
(2.9) 2.5989 0.0119 0.01058 0.18061
(2.7) 2.4916 0.0000 0.07457 0.06210
Multiple (2.8) 2.5842 0.0000 0.01671 0.15915
SQP
(2.9) 2.5837 0.0064 0.01530 0.16201
(2.7) 2.5165 0.0000 0.05779 0.08791
GA Single (2.8) 2.5989 0.0119 0.01057 0.18069
(2.9) 2.5989 0.0119 0.01057 0.18069
85071 X X 2.6872 0.0260 0.05989 0.27670
43
2.6.2 Two Layer Measurements
The 2layer structures used for the Sparameter measurements were Garlock Rub
ber/Bakelite, Garlock Rubber/Ceramic, and Garlock Rubber/Nano Material with Gar
lock Rubber used as the first layer for all cases. As discussed in Section 2.6.1, the three
error functions (2.7) (2.9) were minimized using SQP and the GA. No change was made
to any paramater of the SQP algorithim, whereas the GA was allowed 1000 generations
before termination. The extracted permittivities were used to generate Sparameters
data sets for RMS error computation and were also contrasted to the Agilent 85071 re
sults of the previous section. It should be mentioned that the Agilent 85071 software can
only determine the permittivity of single materials or provide bulk permittivity estimates
for composite structures.
Tables 2.72.9 show the extracted permittivities and Sparameter RMS errors for
the three 2layer samples. Using 2layer Sparameter data sets generated from the single
layer Agilent 85071 extracted permittivities, the RMS errors between these data sets and
the measured Sparameters were calculated and are also included in the tables. For the
Garlock Rubber/Bakelite sample, all algorithm/error function combinations achieved
RMS magnitude errors O(10?2) and phase errors O(10?2 ? 10?1). The magnitude of
the extracted permittivities for each layer showed excellent agreement with the extracted
single layer values. However, the imaginary part of the extracted permittivity for Garlock
Rubber showed an increased value when compared to the values given in Table 2.5.
For the Garlock Rubber/Ceramic and Garlock Rubber/Nano Material samples (see
Tables 2.8 and 2.9), the SQP algorithm was only effective at estimating the complex
permittivity values using error function (2.7). Using error functions (2.8) and (2.9),
44
Table 2.7: Extracted Permittivity & Error for Garlock/Bakelite
Error Garlock Garlock Layer 2 Layer 2Algorithm Objective
Function ?? ??? ?? ???
summationtextS
ijerr
summationtextnegationslash S
ijerr
(2.7) 7.9050 0.2207 4.0271 0.3274 0.0400 0.0647
Single (2.8) 7.5613 0.3626 3.9900 0.2563 0.0366 0.1997
(2.9) 7.8488 0.3343 3.9881 0.2583 0.0387 0.1036
(2.7) 7.8211 0.3080 4.0086 0.2908 0.0379 0.0991
Multiple (2.8) 7.9793 0.2813 3.9827 0.2980 0.0410 0.0699
SQP
(2.9) 7.9793 0.2813 3.9827 0.2980 0.0410 0.0699
(2.7) 7.9050 0.2207 4.0271 0.3274 0.0400 0.0647
GA Single (2.8) 7.5620 0.3625 3.9900 0.2563 0.0366 0.1994
(2.9) 7.8523 0.3336 3.9883 0.2585 0.0387 0.1027
85071 X X 7.5320 0.1610 3.6032 0.2347 0.1136 0.3289
45
Table 2.8: Extracted Permittivity & Error for Garlock/Ceramic
Error Garlock Garlock Layer 2 Layer 2Algorithm Objective
Function ?? ??? ?? ???
summationtextS
ijerr
summationtextnegationslash S
ijerr
(2.7) 7.9478 0.1169 1.2254 0.0373 0.0268 0.0601
Single (2.8) 1.0000 0.0712 6.1163 0.0963 0.0687 6.3055
(2.9) 1.0000 0.1167 6.1037 0.0656 0.0695 6.3189
(2.7) 7.8983 0.0405 1.2596 0.0649 0.0378 0.0686
Multiple (2.8) 1.0000 0.2616 6.1208 0.0066 0.0732 6.3536
SQP
(2.9) 1.0000 0.2616 6.1208 0.0066 0.0732 6.3536
(2.7) 7.9479 0.1169 1.2254 0.0373 0.0268 0.0601
GA Single (2.8) 8.0688 0.1916 1.1930 0.0000 0.0144 0.0639
(2.9) 8.0866 0.1927 1.2167 0.0000 0.0146 0.0654
85071 X X 7.5320 0.1610 1.1484 0.0043 0.0803 0.1267
46
Table 2.9: Extracted Permittivity & Error for Garlock/Nano Material
Error Garlock Garlock Layer 2 Layer 2Algorithm Objective
Function ?? ??? ?? ???
summationtextS
ijerr
summationtextnegationslash S
ijerr
(2.7) 7.6909 0.1382 2.7840 0.0585 0.0307 0.0490
Single (2.8) 2.5193 0.1860 5.4421 0.0110 0.0178 6.5273
(2.9) 2.5193 0.1978 5.4555 0.0020 0.0178 6.5223
(2.7) 7.6700 0.1435 2.7751 0.0380 0.0308 0.0552
Multiple (2.8) 2.4387 0.1856 5.3484 0.0152 0.0217 6.5841
SQP
(2.9) 2.4729 0.1596 5.4659 0.0147 0.0204 6.5315
(2.7) 7.6909 0.1382 2.7841 0.0585 0.0307 0.0490
GA Single (2.8) 7.7636 0.2344 2.6881 0.0142 0.0132 0.0942
(2.9) 7.7544 0.2510 2.6866 0.0068 0.0136 0.0972
85071 X X 7.5320 0.1610 2.6872 0.0260 0.0294 0.1103
47
SQP was unable to correctly estimate each layer?s complex permittivity as evidenced
by the unacceptable large phase errors. It is apparent from the discrepancies in the
estimated permittivity values and the large Sparameter phase errors that this was a
result of local minimum trapping. Unlike SQP, the GA was successful in extracting the
complex permittivities using all error functions. The real and imaginary parts of the
permittivities showed excellent agreement with the extracted single layer values, and the
GA achieved RMS magnitude errors O(10?2) and phase errors O(10?2 ? 10?1).
2.6.3 Three Layer Measurements
The Sparameters for a 3layer composite structure (Nano Material/Garlock Rub
ber/Garlock Rubber) were measured. In accordance with the previous sections, Ta
ble 2.10 shows the extracted permittivities and Sparameter RMS errors for the compos
ite structure. SQP was effective in determining the complex permittivites of the sample
for all but the multiobjective form of (2.7). As in the previous cases, the GA succesfully
estimated the permittivities using all error functions. SQP and the GA returned RMS
magnitude errors O(10?2) and phase errors O(10?2 ? 10?1), respectively.
It was already determined that the algorithms and error functions would return
complex permittivity values nearly indistinguishable from the actual values for all com
puter generated cases (SQP using (2.8) and (2.9) excepted). Therefore, the possible
reasons for the errors sited previously warrant an explanation. The most likely sources
of error for the single layer extractions stem from inaccuracies associated with instru
mentation and sample thickness measurements. In addition to these errors, compound
structures may suffer from the presence of small air gaps between the layers as well as
48
Table 2.10: Extracted Permittivity & Error for Nano Material/Garlock/Garlock
Error Layer 1 Layer 1 Layer 2 Layer 2 Layer 3 Layer 3Algorithm Objective
Function ?? ??? ?? ??? ?? ???
summationtextS
ijerr
summationtextnegationslash S
ijerr
(2.7) 2.8234 0.0615 8.3424 0.2413 8.1992 0.2086 .03571 .06789
Single (2.8) 2.9228 0.0064 8.2085 0.3069 7.3469 0.3813 .01132 .28668
(2.9) 2.9255 0.0133 8.1900 0.3014 7.3365 0.3561 .01178 .28771
(2.7) 2.7634 0.0371 8.3823 0.3812 8.3505 0.3224 .05696 .06735
Multiple (2.8) 4.1458 0.0082 4.7201 0.2681 2.2123 0.8000 .23606 3.6167
SQP
(2.9) 4.6797 0.0000 2.4478 0.6366 2.8608 0.0000 .14824 4.5302
(2.7) 2.8238 0.0625 8.3417 0.2402 8.1981 0.2051 .03568 .06775
GA Single (2.8) 2.9159 0.0125 8.1326 0.3000 7.4216 0.3630 .01254 .26070
(2.9) 2.9259 0.0136 8.1716 0.3000 7.3279 0.3552 .01185 .29188
85071 X X 2.6872 0.0260 7.5320 0.1610 7.5320 0.1610 .10697 .63053
49
sample misalignment. Also, results obtained using (2.7) would be adversely affected by
any uncertainty in the phase planes? positions.
2.7 Summary
In this chapter, the performance of complex permittivity extraction methods based
on SQP and the GA was contrasted using Sparameter measurements for 1, 2, and
3layer samples. Three different error function definitions were also used to quantify
the performance of each algorithm in terms of the amount of Sparameter information
(magnitude only or magnitude and phase) available for the inversion process.
Computer generated Sparameter data was initially used to determine the attainable
accuracy of each algorithm/error function combination. The results of this portion of
the study clearly indicated that the extracted permittivity from the single and multilayer
cases was nearly identical to that used to generate the data when the GA was used. This
was also found to be true for SQP in all but two cases possibly due to local minima
trapping. This demonstrates that the GA is extremely accurate and would therefore be
limited only by the precision of the Sparameter measurements, whereas the performance
of SQP would also likely be limited by the accuracy of the initial guess.
The algorithm/error functions were used to extract the complex permittivity from
single layer Sparameter measurements, and it was evident that all of the optimization
techniques were highly effective at minimizing the RMS error(s) of the Sparameters
(both phase and magnitude). However, the GA using either (2.8) or (2.9) was found to
consistently produce the lowest magnitude error of the Sparameters and hence the best
estimate of the complex permittivity.
50
When the same techniques were used for complex permittivity extraction from mul
tilayer composite structures, the GA, for all cases considered, provided approximately
the same level of accuracy as that observed for the single layer cases. SQP, however,
failed to obtain accurate results for several of the cases considered. In summary, the GA
appeared to be the more robust algorithm in terms of its ability to always achieve a low
Sparameter RMS error and accurately obtain each layer?s complex permittivity.
51
Chapter 3
Complex Constitutive Parameter Extraction
3.1 Overview
This chapter extends the procedure presented in Chapter 2 to allow extraction of
both complex permittivity and permeability for each layer in a multilayer sample. In
order to determine the accuracy of the method, the extracted constitutive parameters
(CP) are used to generate Sparameters for comparison with measured Sparameters.
Also, the single layer, complex CP extraction technique developed by Wolfson and Went
worth [96,97] is employed to provide further verification of the correct operation of the
code.
X band waveguide Sparameter measurements of three materials (Teflon, F40, and
F125) were obtained using an HP8510C Vector Network Analyzer for the frequency
range of 8.2  10 GHz. Teflon was used to ensure the method obtained accurate results
for nonmagnetic and low loss materials. Further, two radarabsorbing materials (RAM),
F40 and F125 [98], were used to demonstrate extraction of complex permittivity and
permeability. Multilayer samples were constructed by placing the single layer samples
adjacent to one another in different combinations.
52
3.2 Extraction Techniques
3.2.1 WolfsonWentworth Method
The WolfsonWentworth method consists of placing a sample in a section of rectan
gular waveguide and measuring the S11 parameter with the waveguide terminated by two
offset shorts of differing length [96,97]. The Sparameter measurements are required to
calculate the input impedance of the sample for each case. The input impedances along
with transmission line equations are then used to extract the complex permittivity and
permeability of the sample over the frequency range of interest. This approach requires
the material sample thickness to be less than onehalf wavelength for the TE10 mode
inside the sample in order to avoid exciting higher order modes. However, a low loss
sample must be thick enough to provide significant reflections. If both of these conditions
cannot be met, spurious data for the extracted values of permittivity and permeability
can result at high frequencies. A detailed description of this method is given in [97].
3.2.2 CP Extraction Method Modifications
As discussed previously, the method of Chapter 2 is a 2port technique requiring
a full set of Sparameters to extract complex permittivity for each layer of an nlayer
sample (see Fig. 2.1). Here, however, the method is modified to also account for mag
netic materials. Appropriate modification of the forward solution only requires that the
propagation constant and wave impedance of (2.1) be expressed as
?i =
radicalbigg
(?/a)2 ??2?0?0
parenleftBig
??r,i ?j???r,i
parenrightBigparenleftBig
??r,i ?j???r,i
parenrightBig
(3.1)
53
Zi =
j??0
parenleftBig
??r,i ?j???r,i
parenrightBig
?i (3.2)
where a is the maximum crosssectional dimension of the waveguide. Calculation of the
Sparameters then follows the same procedure outlined in Section 2.2.
Once again, an error function analysis was carried out due to the fact that the
extraction both both ?? and ?? results in a higher dimensional search space. Therefore,
the conclusions previously drawn may be invalid for this situation. Initially, Err2 (2.8)
was considered and determined to be illsuited for accurately estimating the CP since
Sparameter phase information becomes critical to the extraction process?s ability to
obtain low RMS errors. Therefore, two error functions involving both magnitude and
phase information were tested and are given by
Err = 1N
Nsummationdisplay
i=1
parenleftBigvextendsinglevextendsingle
vextendsingle[S]f,i
vextendsinglevextendsingle
vextendsingle?
vextendsinglevextendsingle
vextendsingle[S]m,i
vextendsinglevextendsingle
vextendsingle
parenrightBig2
+ 1N
Nsummationdisplay
i=1
parenleftBig
negationslash [S]f,i ?negationslash [S]m,i
parenrightBig2 (3.3)
Err = 1N
Nsummationdisplay
i=1
parenleftBig
Re
parenleftBig
[S]f,i
parenrightBig
?Re
parenleftBig
[S]m,i
parenrightBigparenrightBig2
+ 1N
Nsummationdisplay
i=1
parenleftBig
Im
parenleftBig
[S]f,i
parenrightBig
?Im
parenleftBig
[S]m,i
parenrightBigparenrightBig2 (3.4)
where [S]f,i and [S]m,i are the formulated and measured Sparameters at frequency point
i, respectively, and N is the number of frequencies. After a number of studies, (3.4) was
found to give a lower overall RMS error for both magnitude and phase of the formulated
54
and measured Sparameters. This may be due to a number of factors such as numer
ical precision of the calculations and unequal weighting of the magnitude and phase
information in (3.3).
The Genetic Algorithm (GA) of Chapter 2 was determined to be an extremely robust
method for determining accurate values for complex permittivities from Sparameter
measurements. Sequential Quadratic Programming (SQP), a local optimization tech
nique, was only able to accurately determine permittivity values in some cases due to
its severe dependence on the algorithm?s initial starting point (??initial). However, SQP
was found to be 50 to 1700 times faster than the GA depending on the number of layers
(higher speedup for larger number of layers). Also, SQP was shown to more accurately
obtain the value of the global minimum than the GA when the initial starting point was
in the vicinity of the minimum. In this section, a modified SQP algorithm is developed to
exploit SQP?s speed and accuracy while eliminating the issue of local minima trapping.
The novel multipoint SQP (MPSQP) presented here relies on the generation of P
randomly1 distributed initial guesses (??initial, ??initial) to reach the global minimum in the
bounded solution space. The SQP algorithm is performed on each initial guess resulting
in P solutions. If a sufficient number of points are taken, the MPSQP will accurately and
quickly determine the global minimum by taking the solution with minimum error of all
the returned solutions. To ensure that the minimum has been accurately determined, the
MPSQP algorithm is repeated but bounded by a reduced search space centered about the
previously determined solution. In all cases considered, this added step returned results
identical to the results of the initial set of iterations. A study comparing the performance
1Uniform probability density function.
55
of the GA and MPSQP for the extraction of single layer material parameters is presented
in Section 3.4.1.
3.3 Measurements
The test ports of an HP8510C Vector Network Analyzer were connected to WR90
(X band) waveguide via coaxtowaveguide adapters. The calibration procedure follows
Wolfson?s outline of recommended instrument settings for improved measurement accu
racy [99]. Higher accuracy was obtained when operating the analyzer in step sweep mode
with 128X averaging. A 2 ms dwell time was used to account for the analyzer settling
time and propagation delay due to coaxial cable length. Finally, the TE10 mode cutoff
frequency is entered as the waveguide delay (6.557 GHz for the WR90 waveguide used
in this work).
For the WolfsonWentworth method, the measurement reference plane is established
at the open end of the waveguide of sufficient length to allow unwanted modes to attenu
ate before reaching the measurement reference plane. The reference plane was defined at
the end of this waveguide using an offset short calibration procedure [100]. The procedure
uses a short at the reference plane and offset shorts of lengths approximately ?/8 and
3?/8, where ? is chosen to give maximum phase separation for the offset shorts across the
band [100]. For WR90 rectangular waveguide the optimum offset short lengths become
.483 cm and 1.455 cm. Following calibration, S11 is measured for the reference plane
terminated by both offset short loads. A short section of waveguide is then attached to
the reference plane, with the material sample placed in the far end of the guide. S11
is then measured for the sample terminated by both offset short loads. These values
56
along with S11 measured for the offset shorts are inserted into the routine described by
Wolfson [97] to extract ?? and ??.
A throughresponse measurement for the method presented here requires two sec
tions of waveguide for attenuation of unwanted modes. The measurement reference
planes are established at the ends of these two waveguides using a full 2port calibra
tion that employs a short, the pair of offset shorts described in the WolfsonWentworth
approach, and a direct connection. Following calibration, the material sample is placed
into a short section of waveguide which is then inserted between the two reference planes.
A complete set of Sparameters are then measured and the results used in the method
presented here.
3.4 Results
3.4.1 Single Layer Measurements
For the single layer materials, samples of Teflon (d = 4.8 mm), F40 (d = 1.524 mm),
and F125 (d = 3.3 mm) were used. Sparameter data sets were calculated from the values
extracted using the GA with the appropriately modified forward solver, MPSQP, and the
WolfsonWentworth method. These data sets were in turn compared to the measured
Sparameters. In addition to Sparameter comparisons, the extracted CP from each
algorithm were directly compared. It was initially assumed for the GA and MPSQP
that the CP were constant over the frequency band of interest, whereas the Wolfson
Wentworth method makes no such assumptions. The results presented show that this
assumption was valid, and, therefore, it is used throughout this chapter. For materials
having frequency dependent CP, the algorithm can be easily modified.
57
For all single layer optimizations, the GA utilized a population size of 100, crossover
rate of 80%, and mutation rate of 10%. The redundancy removal scheme of the TRRGA
was also employed to accelerate convergence. Since a single layer problem has a four
dimensional search space, the GA was allowed to iterate for 1000 generations which
corresponds to the parameter utilized in Section 2.6.2 for 4D problems. For MPSQP,
the number of initial guesses (P) was set to 100. The real parts of both the relative
permittivity and permeability were restricted to the range of .1 to 25, while the imaginary
parts were restricted from 0 to 8.
For the Teflon sample, the extracted constitutive parameters are nearly identical
for the GA and MPSQP as shown in Table 3.1. Good agreement is also shown between
these two methods and the results of the WolfsonWentworth method. The constitutive
parameter values listed for the WolfsonWentworth method are the average values over
the frequency band. The RMS errors between the MPSQPgenerated and measured
Sparameter magnitudes and phases are listed in Table 3.2 and show excellent agree
ment. The GA results were nearly identical to the MPSQP results and are not shown.
Analogous results were also obtained for the two RAM samples as shown in the tables.
In all cases, MPSQP returned slightly lower RMS Sparameter errors (generated versus
measured) than the WolfsonWentworth method and the GA. Figs. 3.1 and 3.2 show
comparisons of the extracted CP from MPSQP and the WolfsonWentworth method
over the entire frequency range. Figs. 3.3 and 3.4 show excellent agreement between the
MPSQPgenerated Sparameters and the measured data.
Table 3.3 illustrates the improvement in performance when employing MPSQP in
stead of the GA. Overall, the MPSQP showed an average speedup of 21.1 (speedup
58
taken as the ratio of GA runtime to MPSQP runtime). Additionally, the MPSQP was
able, in all cases, to achieve slightly lower error function values which, in turn, directly
corresponded to lower Sparameter RMS errors. It is also worth noting that in Chapter 2
the speedup of SQP over the GA scaled superlinearly with increasing number of layers.
Although not directly presented here, this was also observed with MPSQP and the GA
for the multilayer cases presented in the following sections. Therefore, it appears that
the MPSQP is an improved method for obtaining the material parameter values not only
in terms of its speed but also its accuracy in determining the global minimum.
Table 3.1: Single Layer Results
Method Teflon F40 F125
GA ?? 2.05 12.7 6.97
MPSQP ?? 2.03 12.7 6.98
Wolfson ?? 2.07 12.9 7.95
GA ??? 0 0.187 0
MPSQP ??? 0 0.203 0
Wolfson ??? 0.027 0.673 0.693
GA ?? 0.958 1.55 0.440
MPSQP ?? 0.965 1.55 0.434
Wolfson ?? 0.952 1.63 0.490
GA ??? 4.52e3 1.24 0.516
MPSQP ??? 4.63e3 1.24 0.515
Wolfson ??? 0.017 1.06 0.474
59
Table 3.2: MPSQP Error for Single Layer Sample
Teflon F40 F125
RMS Error S11 1.47e2 6.60e3 1.25e2
RMS Error S12 8.17e3 5.14e3 7.86e3
RMS Error S21 8.20e3 3.72e3 1.43e2
RMS Error S22 1.29e2 4.35e3 7.04e3
RMS Error negationslash S11 2.85e2 2.60e2 2.08e2
RMS Error negationslash S12 1.01e2 2.05e2 4.71e2
RMS Error negationslash S21 1.50e2 1.41e2 3.97e2
RMS Error negationslash S22 4.82e2 2.77e2 2.15e2
Table 3.3: Performance Comparison of GA and MPSQP
Error Time (sec)Material
GA MPSQP GA MPSQP
Teflon 1.53e3 1.47e3 810.2 48.6
F40 8.93e4 8.93e4 782.6 32.1
F125 1.35e3 1.34e3 790.9 35.6
60
8.5 9 9.5 10?2
0
2
4
6
8
10
12
Frequency (GHz)
??,
???
MPSQP
Wolfson Method
MPSQP ??
Wolfson ??
MPSQP ???Wolfson ???
Figure 3.1: Comparison of MPSQP and WolfsonWentworth method extracted complex
permittivity values for F125 sample.
61
8.5 9 9.5 100
0.2
0.4
0.6
0.8
1
Frequency (GHz)
??,
???
MPSQP
Wolfson Method
MPSQP ??
Wolfson ??MPSQP ???
Wolfson ???
Figure 3.2: Comparison of MPSQP and WolfsonWentworth method extracted complex
permeability values for F125 sample.
62
8.5 9 9.5 100
0.2
0.4
0.6
0.8
1
Frequency (GHz)
S
11
, S
21

MPSQP
Wolfson Method
Measurements
S11 S21
Figure 3.3: Measured and generated Sparameter magnitudes for the F125 sample.
63
8.5 9 9.5 10?200
?150
?100
?50
0
50
100
150
Frequency (GHz)
?S
11
, ?
S 21
MPSQP
Wolfson Method
Measurements
?S11
?S21
Figure 3.4: Measured and generated Sparameter phases for the F125 sample.
64
3.4.2 Two Layer Measurements
ThesamplesusedfortwolayerSparametermeasurementswereF125/Teflon, Teflon/F40,
and F125/F40. To ensure an accurate solution, P was set to 500 initial points. The in
crease in P was necessary since the search space for a two layer problem has 8 dimensions.
The upper and lower bounds remained the same as those of the single layer case.
Table 3.4 shows the extracted parameter values as well as the error function value
and RMS errors between the generated and measured Sparameters. Overall, excellent
agreement is shown by the low error values for all material combinations. Additionally,
the extracted parameter values agree well with the values obtained from the single layer
optimization. Figs. 3.5 and 3.6 show the magnitude and phase of the generated and
measured S11 and S21 for the F125/Teflon sample. As with the data from Table 3.4,
the agreement between the extracted and measured data is well within the tolerance of
instrumentation error.
65
Table 3.4: Results for Two Layer Samples
Sample F125/Teflon Teflon/F40 F125/F40
Layer F125 Teflon Teflon F40 F125 F40
?? 6.82 2.05 2.36 11.8 6.2 14.0
??? 0 0 0 0.091 0 0
?? 0.444 0.997 1.08 1.06 0.402 0.492
??? 0.546 0 9.72e3 1.23 1.78 1.46
Error Function Value 2.29e3 3.02e3 2.73e3
RMS Error S11 1.85e2 7.30e3 7.99e3
RMS Error S12 1.39e2 8.21e3 1.04e2
RMS Error S21 1.17e2 1.03e2 1.04e2
RMS Error S22 2.02e2 2.57e2 1.85e2
RMS Error negationslash S11 2.88e2 2.90e2 4.96e2
RMS Error negationslash S12 4.91e2 2.82e2 6.12e2
RMS Error negationslash S21 4.40e2 4.25e2 6.83e2
RMS Error negationslash S22 2.64e2 5.87e2 5.99e2
66
8.5 9 9.5 100
0.2
0.4
0.6
0.8
1
Frequency (GHz)
S
11
, S
21

MPSQP
Measurements
S11
S21
Figure 3.5: Measured and generated Sparameter magnitudes for the F125/Teflon sam
ple.
67
8.5 9 9.5 10?250
?200
?150
?100
?50
0
50
100
150
Frequency (GHz)
?S
11
, ?
S 21
MPSQP
Measurements
?S11
?S21
Figure 3.6: Measured and generated Sparameter phases for the F125/Teflon sample.
68
3.4.3 Three Layer Measurements
A three layer F125/F40/Teflon sample?s Sparameters were measured and the com
plex constitutive parameters extracted using MPSQP. The upper and lower bounds re
mained unchanged while P was increased to 1000 due to the 12 dimensional search space.
Table 3.5 shows the extracted constitutive parameters for each layer as well as the error
function value and RMS errors between the measured and generated Sparameters. As
in the prior cases, the algorithm was able to successfully match the generated and mea
sured Sparameters as evidenced by the very low RMS error values. Inspection of the
extracted parameter values shows that each of the materials was correctly classified and
agreement with the single and two layer parameter values was very good.
Table 3.5: Results for F125/F40/Teflon Sample
F125 F40 Teflon
?? 5.98 12.1 2.04
??? 0 0 0.107
?? 0.411 1.90 1.08
??? 0.495 1.49 0
Error Function Value 7.61e4
RMS Error S11 7.03e3
RMS Error S12 8.77e3
RMS Error S21 8.48e3
RMS Error S22 1.10e2
RMS Error negationslash S11 1.22e2
RMS Error negationslash S12 5.78e2
RMS Error negationslash S21 4.79e2
RMS Error negationslash S22 2.59e2
69
3.5 Summary
In this chapter, a method was presented for accurately extracting the complex per
mittivity and permeability from each individual layer in a multilayer sample using S
parameter waveguide measurements. The forward solution formulation presented in
Chapter 2 was modified to account for magnetic materials. Also, a modified SQP al
gorithm (MPSQP) was presented which utilized a large number initial guess points to
alleviate the possibility of local minima trapping, a problem with gradientbased opti
mization methods. Use of such an algorithm was beneficial since it provides significant
computational gains over traditional global optimization methods such as the GA. Specif
ically, for the single layer cases presented here, the MPSQP showed an average speedup
of 21.1 over the GA as well as improved accuracy.
Sparameter measurements were conducted on three material samples used to con
struct multilayer samples. The MPSQP was used successfully to extract the complex CP
for each layer. These values were then compared with values extracted using the Wolfson
Wentworth method (single layer cases only). Also, Sparameters were generated using
the extracted values and compared with the measured data. In all cases, results were
found to be in excellent overall agreement with both the WolfsonWentworth method
values and measured data. In summary, the MPSQP was found to be a computationally
efficient and robust algorithm for extracting complex CP from multilayer materials.
70
Chapter 4
Waveguide Filters Using Conductor Doping of Dielectrics
4.1 Overview
In the previous chapters, novel optimization algorithms were presented for the so
lution of inverse problems, namely, complex constitutive parameter extraction of mul
tilayered materials. In this and the remaining chapters, the topic of optimal design
is addressed. The specific problem considered is that of optimizing the geometry of
waveguide filters to realize a specified transmission response.
Novel frequencydependent reflectivity and transmittivity characteristics of dielec
tric slabs have recently been shown to be achievable by embedding randomly oriented
thin conducting wires (inclusions) within a host slab [101]. A wide variety of electro
magnetic behaviors have been observed by adjusting the wire length and doping level
(wire density). In addition to wires, thin conducting patches or small particles of var
ious geometries may also be used as the dopant and, in the case of wires and patches,
either embedded or limited to the surface of the dielectric. The focus of this study is to
investigate the utility of random doping as a method for obtaining consistent reflection
and transmission frequencydependent profiles. Additional consideration is given to the
feasibility of fabricating such materials.
4.2 Numerical Modeling
To determine the reflection and transmission characteristics of a doped dielectric
slab, a numerical procedure is necessary. Many numerical methods including Finite
71
Element Method (FEM) (and hybrids) [101?109], Method of Moments (MoM) [110?113],
and Finite Difference Time Domain (FDTD) [114?122] have been applied to similar
problems with great success. However, edgebased FEM is most efficient for the problem
at hand because of its ability to easily account for inhomogeneous materials as well as
its suitability for closed region problems (i.e., waveguides, cavities, etc.) [123].
The computational domain of the FEM is created by placing the dielectric slab in
side a waveguide as shown in Fig. 4.1 and discretizing the geometry using a tetrahedral
elements. In this manner, the reflection and transmission properties of the material can
be obtained in a computationally efficient manner. A modematching (MM) scheme,
which allows for the development of TE and TM modes, is used to truncate the com
putational domain and permits the faces of the waveguide to be placed very close to
the slab [123,124]. Further reduction in computational intensity is achieved by utilizing
a multipoint Asymptotic Waveform Evaluation (AWE) [123] to quickly determine the
frequency response of the material over a wide band. The matrix solution of the FEM
problem is used to calculate the reflection (S11) and transmission (S21) coefficients which
define the filter?s response.
To further simplify the geometry and mesh generation as well as reduce solution
time, the following methodology is employed to generate the doped medium. In order to
model thin wires or conducting triangular patches, assume that the locations of the wires
or patch edges occupy the same locations as tetrahedral edges within the discretized slab
(see Fig. 4.2). Those edges where wires or patches exist can simply be assigned tangential
electric field values of zero. Since the unknowns in an edgebased FEM are the tangential
electric fields along the tetrahedral edges, this is equivalent to removing the appropriate
72
x
y z
a
b
Doped
Slab
2
1
3
?
?
?
Figure 4.1: Loaded waveguide geometry with doped slab. Boundary notations for FEM
formulation are indicated.
73
rows and columns of the FEM matrix equation which results in a reduced set of linear
equations. Additionally, by this method, several different cases may be considered us
ing the same matrix equation. All that is required is the appropriate removal of rows
and columns of the matrix, and, hence, no matrix reassembly is required. This results
in a computationally efficient procedure, especially beneficial to optimization problems
requiring many simulations. Although the edge locations within a given mesh are fixed,
modern unstructured mesh generators provide enough variety in edge orientation that
the wires/patches may still be assumed to be randomly placed [101]. Also notice that,
the wires are assumed to be infinitesimally thin elements and there is no provision to
include wire radius in the current formulation. To better illustrate this process, consider
the following example. To generate a doped slab with 100 wires, a random number
generator is used to select 100 numbers from a list numbering 1 to Nedges, the number
of edges within the discretized slab. This list maps to the corresponding edge numbers
which are then removed from the matrix equation as described. Generating a list of
edges corresponding to conducting patches is performed in the same manner with the
additional requirement that the three connecting edges composing a tetrahedral face
form one patch. Therefore, a doping level of 100 triangular patches would correspond to
the removal of 300 rows and columns from the FEM matrix equation.
4.3 Finite Element Formulation
The FEM presented here is formulated to solve for the Sparameters from a loaded
waveguide (see Fig. 4.1). Inside the waveguide, electromagnetic wave propagation is
governed by Maxwell?s equations. From these equations, the vector Helmholtz equation
74
X58 X59
X5A
Figure 4.2: Discretized slab illustrating wires lying along tetrahedral edges. Notice some
edges join to form long, erratically shaped wires. Wires may also be located on interior
edges but are not shown here.
75
for the electric field can be derived and is given by
????1?? vectorE ??2?vectorE = 0 (4.1)
where vectorE is the electric field inside the waveguide, ? is the radian frequency of the wave,
and ? and ? are the permeability and permittivity, respectively, at any given point in the
domain. Within the waveguide, the electric field is subject to the boundary conditions
?n? vectorE = 0 on ?1 (4.2)
?n??? vectorE +??n?
parenleftBig
?n? vectorE
parenrightBig
= vectorU on ?2,3 (4.3)
Eqn. (4.2) forces the tangential electric field to be zero along the waveguide walls.
Eqn. (4.3) is referred to as a boundary condition of the third kind. Boundary con
ditions of this type are necessary to accurately truncate the computational domain of
the problem, and a discussion of their parameters will be given in a later section. Phys
ically, (4.3) represents a relationship between the tangential electric and magnetic fields
along the boundary.
As with most numerical methods, the FEM procedure begins with a discretization
of the geometry into elements. Although there are many elements available, tetrahedral
elements, as shown in Fig. 4.3, are usually chosen due to their ability to conform to
arbitrarily shaped structures. Inside each element, the electric field can be approximated
by an expansion such as
vectorEe =
6summationdisplay
j=1
vectorNj (x,y,z)Ej (4.4)
76
where vectorEe is the electric field inside the eth element, vectorNj (x,y,z) is the vector basis function
associated with the jth tetrahedral edge, and Ej is defined as the electric field value at
the center of edge j projected along that edge?s direction. The vector basis functions
used in this work are termed the zerothorder vector basis functions and given by
vectorNj = ?j (Lj1?Lj2 ?Lj2?Lj1) (4.5)
where ?j is the length of edge j and Lj? is the linear interpolation function (volume
coordinate) for the ?th node bounding edge j (see Fig. 4.3) [123]. The necessity of edge
based (vector) elements as opposed to more traditional nodebased elements has been
discussed extensively in the literature [6,103,125?155].
Following Galerkin?s procedure [123], the weighted residual of (4.1) is minimized
according to
Rei =
integraldisplay
?e
vectorNi ?parenleftBig????1?? vectorE ??2?vectorEparenrightBig d?e (4.6)
where Rei is the weighted residual corresponding to edge i of element e; vectorNi, defined
by (4.5), is the weighting function for edge i; and ?e represents the domain of element
e. Using the first vector Green?s theorem, (4.6) can be cast in the weak form which is
given by
Rei =
integraldisplay
?e
?? vectorNi ??? vectorE ??2??vectorNi ? vectorE d?e ?
contintegraldisplay
?e
parenleftBigvector
Ni ??? vectorE
parenrightBig
? ?n d?e (4.7)
77
1
2
3
4
I
II
III
IV
V
VI
Figure 4.3: Tetrahedral element showing edge and node numbering.
78
From a vector identity, the integrand of the surface integral in (4.7) can be rewritten
in two different ways as shown by
parenleftBigvector
Ni ??? vectorE
parenrightBig
? ?n =
parenleftBig
?n? vectorNi
parenrightBig
?
parenleftBig
?? vectorE
parenrightBig
= ?vectorNi ?
parenleftBig
?n??? vectorE
parenrightBig
(4.8)
Since Galerkin?s method calls for the weighting and basis functions to be identical,
vectorNi must satisfy the same boundary conditions as the electric field. Therefore, the second
term in (4.8) shows that the integrand is zero for the portion of the surface integral over
?1. Using the third form in (4.8), the weight residual can be rewritten as
Rei =
integraldisplay
?e
?? vectorNi ??? vectorE ??2??vectorNi ? vectorE d?e +
integraldisplay
?e2,3
vectorNi ?parenleftBig?n??? vectorEparenrightBig d?e2,3 (4.9)
4.3.1 Absorbing Boundary Condition
Many domain truncation schemes exist as outlined in [123]. A simple firstorder
absorbing boundary condition (ABC) method is presented here followed by a more accu
rate ModeMatching (MM) technique. Substituting (4.3) for the integrand of the surface
integral in (4.9), the weighted residual is given by
Rei =
integraldisplay
?e
?? vectorNi ??? vectorE ??2??vectorNi ? vectorE d?e +
integraldisplay
?e2,3
vectorNi ?parenleftBigvectorU ???n? ?n? vectorEparenrightBig d?e2,3 (4.10)
The remaining task is to determine appropriate values for ? and vectorU. Assuming the
surfaces ?2 and ?3 are sufficiently far from the discontinuity in the waveguide, it can be
assumed that only the TE10 mode is propagating (no higher order or evanescent modes).
79
It is also assumed that a unit amplitude TE10 mode wave is incident on ?2. The total
field on ?2 may then be written as
vectorE = vectorEinc + vectorEref = sinparenleftBig?x
a
parenrightBig
e?jkzz1 +Rsin
parenleftBig?x
a
parenrightBig
ejkzz1 (4.11)
whereRisthereflectioncoefficient, kz isthepropagationconstantoftheemptywaveguide,
and z1 is the position of ?2. Applying (4.11) to the first term of the lefthand side of (4.3),
the following is determined:
?n??? vectorE = ??z ??? vectorE = ?jkz vectorEinc +jkz vectorEref = jkz vectorE ?2jkz vectorEinc (4.12)
Equating this with (4.3), it is apparent that ? = jkz and vectorU = ?2jkz vectorEinc since the total
field is assumed to be in the form of the TE10 mode. Along ?3, a similar analysis can
be performed. The field along ?3 can be expressed as
vectorE = vectorEtr = T sinparenleftBig?x
a
parenrightBig
e?jkzz2 (4.13)
where T is the transmission coefficient and z2 is the location of ?3. Applying the same
procedure, it is easily determined that, along ?3, ? = jkz and vectorU = 0.
Substituting these results into (4.10), the weighted residual becomes
Rei =
integraldisplay
?e
?? vectorNi ??? vectorE ??2??vectorNi ? vectorE d?e
?
integraldisplay
?e2,3
vectorNi ???n? ?n? vectorE d?e2,3 +
integraldisplay
?e2
vectorNi ? vectorU d?e2
(4.14)
80
which can be rewritten as
Rei =
integraldisplay
?e
?? vectorNi ??? vectorE ??2??vectorNi ? vectorE d?e
+
integraldisplay
?e2,3
??n? vectorNi ? ?n? vectorE d?e2,3 ?
integraldisplay
?e2
?n? vectorNi ? vectorU ? ?n d?e2
(4.15)
Substituting (4.4) into (4.15), the final form of the residual is obtained and given
by
Rei =
6summationdisplay
j=1
Ej
integraldisplay
?e
?? vectorNi ??? vectorNj ??2??vectorNi ? vectorNj d?e
+
integraldisplay
?e2,3
??n? vectorNi ? ?n? vectorNj d?e2,3 ?
integraldisplay
?e2
?n? vectorNi ? vectorU ? ?n d?e2
(4.16)
Following the matrix assembly procedure outlined in [123], the final matrix equation
can be obtained and is written as
([S]?[T]+[Q]){E} = {b} (4.17)
The entries in the local element matrices are, therefore, given by
Seij =
integraldisplay
?e
?? vectorNi ??? vectorNj d?e
Teij =
integraldisplay
?e
?2??vectorNi ? vectorNj d?e
Qeij =
integraldisplay
?e2,3
??n? vectorNi ? ?n? vectorNj d?e2,3
bei =
integraldisplay
?e2
?n? vectorNi ? vectorU ? ?n d?e2
(4.18)
81
4.3.2 ModeMatching Domain Truncation
The above domain truncation method requires at least one wavelength separation
of ?2 and ?3 from the discontinuity in the waveguide [123]. This method is sufficient
when the frequency is low or when the waveguide load does not require a significant
number of unknowns. For complex structures or high frequencies, the ABC method is
somewhat inefficient. Therefore, an expansion of the unknown field at the boundary in
terms of the orthogonal waveguide modes (eigenfunctions) presents a means by which
the computational domain may be significantly reduced.
For the MM technique [6], the electric field along ?2 can be expressed by
vectorE = vectorEinc +
?summationdisplay
m=0
?summationdisplay
n=0
amnvectorETEmne?mnz1 +
?summationdisplay
m=1
?summationdisplay
n=1
bmn
bracketleftBigvector
ETMmn + ?zETMzmn
bracketrightBig
e?mnz1 (4.19)
where amn and bmn are expansion coefficients and ?mn is the propagation constant of
the specified mode in the empty waveguide. The terms vectorETEmn, vectorETMmn , and ETMzmn are the
normalized waveguide modes and are given by
vectorETEmn =
??
m?n
nmn
parenleftBign
b cos
parenleftBigm?x
a
parenrightBig
sin
parenleftBign?y
b
parenrightBig
?x? ma sin
parenleftBigm?x
a
parenrightBig
cos
parenleftBign?y
b
parenrightBig
?y
parenrightBig
(4.20)
vectorETMmn = 2
nmn
parenleftBigm
a cos
parenleftBigm?x
a
parenrightBig
sin
parenleftBign?y
b
parenrightBig
?x+ nb sin
parenleftBigm?x
a
parenrightBig
cos
parenleftBign?y
b
parenrightBig
?y
parenrightBig
(4.21)
ETMzmn = 2k
2c
??mnnmn sin
parenleftBigm?x
a
parenrightBig
sin
parenleftBign?y
b
parenrightBig
(4.22)
where ?? is Neumann?s number, nmn =
radicalBig
n2ab +m2 ba, and kc is the cutoff wavenumber
of the specified mode. Note that (4.204.22) are orthogonal according to the following
82
relationships:
bintegraldisplay
0
aintegraldisplay
0
vectorETEmn ? vectorETEm?n?dxdy = ?mm??nn? (4.23)
bintegraldisplay
0
aintegraldisplay
0
vectorETMmn ? vectorETMm?n?dxdy = ?mm??nn? (4.24)
bintegraldisplay
0
aintegraldisplay
0
vectorETEmn ? vectorETMm?n?dxdy = 0 (4.25)
bintegraldisplay
0
aintegraldisplay
0
ETMzmnETMzm?n?dxdy = ?mm??nn? k
2c
?2mn (4.26)
Therefore, the coefficients of (4.19) may be expressed according to the following deriva
tion.
Firstly, the reflected electric field is written as
vectorEref =
?summationdisplay
m=0
?summationdisplay
n=0
amnvectorETEmne?mnz1 +
?summationdisplay
m=1
?summationdisplay
n=1
bmn
bracketleftBigvector
ETMmn + ?zETMzmn
bracketrightBig
e?mnz1 (4.27)
from which amn may be obtained by taking a vector dot product on both sides by
vectorETEm?n?e??m?n?z1 and integrating over ?2 as is shown by
integraldisplay
?2
vectorETEm?n?e??m?n?z1 ? vectorErefd?2 =
bintegraldisplay
0
aintegraldisplay
0
vectorETEm?n?e??m?n?z1 ? vectorErefdxdy =
?summationdisplay
m=0
?summationdisplay
n=0
bintegraldisplay
0
aintegraldisplay
0
vectorETEm?n?e??m?n?z1 ?amnvectorETEmne?mnz1dxdy
+
?summationdisplay
m=1
?summationdisplay
n=1
bintegraldisplay
0
aintegraldisplay
0
vectorETEm?n?e??m?n?z1 ?bmnbracketleftBigvectorETMmn + ?zETMzmnbracketrightBige?mnz1dxdy
(4.28)
83
which results in
amn =
integraldisplay
?2
vectorETEmne??mnz1 ? vectorErefd?2 (4.29)
Similarly, bmn is found to be
bmn =
integraldisplay
?2
vectorETMmn e??mnz1 ? vectorErefd?2 (4.30)
Following a procedure similar to that of the previous section, (4.12) becomes
?n??? vectorE = ?n??? vectorEinc
+
?summationdisplay
m=0
?summationdisplay
n=0
amn?mnvectorETEmne?mnz1 ?
?summationdisplay
m=1
?summationdisplay
n=1
bmn k
20
?mn
vectorETMmn e?mnz1
(4.31)
which by substitution of (4.29) and (4.30) into (4.31) can be written in terms of the total
field and incident field as
?n??? vectorE = vectorP + vectorU (4.32)
where
vectorP =
?summationdisplay
m=0
?summationdisplay
n=0
?mnvectorETEmn
integraldisplay
?2
vectorETEmn ? vectorEd?2
?
?summationdisplay
m=1
?summationdisplay
n=1
k20
?mn
vectorETMmn
integraldisplay
?2
vectorETMmn ? vectorEd?2
(4.33)
vectorU = ?n??? vectorEinc ?
?summationdisplay
m=0
?summationdisplay
n=0
?mnvectorETEmn
integraldisplay
?2
vectorETEmn ? vectorEincd?2
+
?summationdisplay
m=1
?summationdisplay
n=1
k20
?mn
vectorETMmn
integraldisplay
?2
vectorETMmn ? vectorEincd?2
(4.34)
84
Similarly, on ?3, (4.32) is used with vectorU = 0. Therefore, (4.9) can now be expressed by
Rei =
integraldisplay
?
?? vectorNi ??? vectorE ??2??vectorNi ? vectorE d?
+
integraldisplay
?2
vectorNi ?parenleftBigvectorP + vectorUparenrightBig d?2 +
integraldisplay
?3
vectorNi ?parenleftBigvectorPparenrightBig d?3
(4.35)
where vectorP and vectorU are evaluated on the appropriate surfaces.
When expanded, (4.35) is given by
Rei =
integraldisplay
?
?? vectorNi ??? vectorE ??2??vectorNi ? vectorEd?
+
?summationdisplay
m=0
?summationdisplay
n=0
?mn
integraldisplay
?2,3
vectorNi ? vectorETEmnd?2,3
integraldisplay
?2,3
vectorETEmn ? vectorEd?2,3
?
?summationdisplay
m=1
?summationdisplay
n=1
k20
?mn
integraldisplay
?2,3
vectorNi ? vectorETMmn d?2,3
integraldisplay
?2,3
vectorETMmn ? vectorEd?2,3
+
integraldisplay
?2
vectorNi ?parenleftBig?n??? vectorEincparenrightBigd?2
?
?summationdisplay
m=0
?summationdisplay
n=0
?mn
integraldisplay
?2
vectorNi ? vectorETEmnd?2
integraldisplay
?2
vectorETEmn ? vectorEincd?2
+
?summationdisplay
m=1
?summationdisplay
n=1
k20
?mn
integraldisplay
?2,3
vectorNi ? vectorETMmn d?2,3
integraldisplay
?2,3
vectorETMmn ? vectorEincd?2,3
(4.36)
85
Substituting (4.4) yields
Rei =
6summationdisplay
j=1
Ej
integraldisplay
?
?? vectorNi ??? vectorNj ??2??vectorNi ? vectorNjd?
+
?summationdisplay
m=0
?summationdisplay
n=0
?mn
integraldisplay
?2,3
vectorNi ? vectorETEmnd?2,3
integraldisplay
?2,3
vectorETEmn ? vectorNjd?2,3
?
?summationdisplay
m=1
?summationdisplay
n=1
k20
?mn
integraldisplay
?2,3
vectorNi ? vectorETMmn d?2,3
integraldisplay
?2,3
vectorETMmn ? vectorNjd?2,3
+
integraldisplay
?2
vectorNi ?parenleftBig?n??? vectorEincparenrightBigd?2
?
?summationdisplay
m=0
?summationdisplay
n=0
?mn
integraldisplay
?2
vectorNi ? vectorETEmnd?2
integraldisplay
?2
vectorETEmn ? vectorEincd?2
+
?summationdisplay
m=1
?summationdisplay
n=1
k20
?mn
integraldisplay
?2,3
vectorNi ? vectorETMmn d?2,3
integraldisplay
?2,3
vectorETMmn ? vectorEincd?2,3
(4.37)
which can be written in the matrix form of (4.17). Now, however, the elements Qij and
bi are expressed by
Qij =
?summationdisplay
m=0
?summationdisplay
n=0
?mn
integraldisplay
?2,3
vectorNi ? vectorETEmnd?2,3
integraldisplay
?2,3
vectorETEmn ? vectorNjd?2,3
?
?summationdisplay
m=1
?summationdisplay
n=1
k20
?mn
integraldisplay
?2,3
vectorNi ? vectorETMmn d?2,3
integraldisplay
?2,3
vectorETMmn ? vectorNjd?2,3
(4.38)
86
bi = ?
integraldisplay
?2
vectorNi ?parenleftBig?n??? vectorEincparenrightBigd?2
+
?summationdisplay
m=0
?summationdisplay
n=0
?mn
integraldisplay
?2
vectorNi ? vectorETEmnd?2
integraldisplay
?2
vectorETEmn ? vectorEincd?2
?
?summationdisplay
m=1
?summationdisplay
n=1
k20
?mn
integraldisplay
?2,3
vectorNi ? vectorETMmn d?2,3
integraldisplay
?2,3
vectorETMmn ? vectorEincd?2,3
(4.39)
The resulting matrix equation can be solved using any single matrix inversion scheme.
4.3.3 Scattering parameters
After a matrix solution is obtained, the Sparameters of the waveguide section must
be obtained. The scattering coefficients along the waveguide crosssections can be written
as
S11 =
vectorEref
vectorEinc
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglez=z
1
=
vectorE ? vectorEinc
vectorEinc
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglez=z
1
=
vectorE
vectorEinc
vextendsinglevextendsingle
vextendsinglevextendsingle
vextendsinglez=z
1
?1 (4.40)
S21 =
vectorEtrvextendsinglevextendsinglevextendsingle
z=z2
vectorEincvextendsinglevextendsinglevextendsingle
z=z1
=
vectorEvextendsinglevextendsinglevextendsingle
z=z2
vectorEincvextendsinglevextendsinglevextendsingle
z=z1
(4.41)
where the electric fields expressed are the average field over the waveguide crosssection.
If only the TE10 mode is assumed to exist, this formulation of the Sparameters is
sufficient. However, in the case of the MM formulation, many modes may exist at the
boundary. Therefore, the Sparameter definition must be augmented to remove any
modes other than the TE10 mode. The Sparameters for this situation are expressed by
S11 =
integraltext
?2
vectorETE10 ? vectorEd?2
integraltext
?2
vectorETE10 ? vectorEincd?2 ?1 (4.42)
87
S21 =
integraltext
?3
vectorETE10 ? vectorEd?3
integraltext
?2
vectorETE10 ? vectorEincd?2 (4.43)
4.3.4 Asymptotic Waveform Evaluation
The computation of filter?s response requires the solution of (4.17) over a broad range
of frequencies. This can be very timeconsuming since the matrix must be inverted at
each discrete frequency point. When a filter has rapid variation over a narrow band, more
points are needed to accurately sample the response which only compounds the matrix
solution problem. In this section, a method for quickly evaluating (4.17) over a broad
frequency range is presented. This technique is referred to as Asymptotic Waveform
Evaluation [156]. In AWE, the matrix equation is expanded using a Taylor series and is
then converted to a rational Pad?e function. The procedure is as follows.
Consider a matrix equation of the following form
Ax = b (4.44)
In general, A, the square matrix, is frequency dependent as well as the unknown x and
the source term b. The AWE procedure begins with the expansion of x into a Taylor
series [156] as
x(f) =
Qsummationdisplay
n=0
mn (f ?f0)n (4.45)
88
where the mn are unknown coefficients. A and b are then expanded into Taylor series as
well. For simplicity, consider only a two term expansion where
x(f) = m0 +m1 (f ?f0)
A(f) = A(f0)+A?(f0)(f ?f0)
b(f) = b(f0)+b?(f0)(f ?f0)
(4.46)
wheretheprimesindicatederivativeswithrespecttofrequency. Substituting thisinto(4.44)
yields
A(f0)m0 +A(f0)m1 (f ?f0)+A?(f0)m0 (f ?f0)
+A?(f0)m1 (f ?f0)2 = b(f0)+b?(f0)(f ?f0)
(4.47)
Matching like powers of (f ?f0) then gives the set of equations
A(f0)m0 = b(f0)
A(f0)m1 +A?(f0)m0 = b?(f0)
(4.48)
from which the moments of x may be solved recursively as
m0 = A?1 (f0)b0
m1 = A?1 (f0)bracketleftbigb?(f0)?A?(f0)m0bracketrightbig
(4.49)
More generally, the solution of an arbitrary number of moments is given as
m0 = A?1 (f0)b0
mn = A?1 (f0)
bracketleftBigg
b(n) (f0)
n! ?
nsummationdisplay
i=1
A(i) (f0)mi?1
i!
bracketrightBigg (4.50)
89
where A(i) is the ith derivative of A and b(n) is the nth derivative of b.
The Taylor series expansion has an inherently limited bandwidth. A prohibitive
number of terms are necessary to achieve practical bandwidths [123]. Therefore, x can
be represented as a wellbehaved rational Pad?e function of the form
x(f) =
Lsummationtext
i=0
ai (f ?f0)i
1+
Msummationtext
j=1
gj (f ?f0)j
(4.51)
where L+M = Q and the best performance is obtained when M = L. Substituting (4.51)
into (4.45) yields the equations for gj and ai after some manipulation. They are given
by
?
??
??
??
??
??
??
?
mL mL?1 mL?2 ??? mL?M+1
mL+1 mL mL?1 ??? mL?M+2
mL+2 mL+1 mL ??? mL?M+3
... ... ... ... ...
mL+M?1 mL+M?2 mL+M?3 ??? mL
?
??
??
??
??
??
??
?
?
???
???
???
???
?
???
???
???
???
?
g1
g2
g3
...
gM
?
???
???
???
???
?
???
???
???
???
?
= ?
?
???
???
???
???
?
???
???
???
???
?
mL+1
mL+2
mL+3
...
mL+M
?
???
???
???
???
?
???
???
???
???
?
(4.52)
ai =
isummationdisplay
j=0
gjmi?j 0 ? i ? L (4.53)
Theoretically, more and more terms can be added to (4.45) to achieve higher band
width. However, there are several practical issues to consider. First, AWE is memory
intensive since the storage of the derivatives of A is required. Therefore, to maintain
reasonable memory limits, the number of terms must usually be kept low. Additionally,
solution of the higher order moments is difficult since these equations are illconditioned.
90
Therefore, it may be necessary to perform the AWE procedure at multiple points within
the frequency band (A must be inverted each time). For this case, a multipoint AWE
procedure is needed where a few frequencies are judiciously chosen to maximally utilize
the bandwidth obtainable from a given level of expansion. In this work, an 8th order
expansion is employed (L = M = 4) as well as the Complex Frequency Hopping [157]
to automatically select the AWE frequencies. In other words, this method is a binary
search where initially the minimum and maximum frequencies of interest are chosen as
expansion points. After evaluation of these AWE solutions, the error between them is
checked at a number of points (2 evenly spaced points in this work). If the error is
below a certain tolerance, the solution is obtained. If not, the AWE is performed again
at the center frequency of the band. Once again, the error is checked at a number of
points between [fmin,fcenter] and [fcenter,fmax]. This procedure continues to divide the
frequency band as necessary until the error is below the tolerance. It was found that for
the problems considered here, around 5 AWE evaluations were required.
4.4 Method Verification
To verify the accuracy of each formulation presented, the Sparameters for an X
band waveguide loaded with a 1 cm thick slab (?r = 2.2) were calculated over the full
X band range. After running several test cases, the ABC formulation was determined
to be unsuitable for this study since the size of the computational domain necessary to
achieve reasonable accuracy was prohibitively large (number of unknowns). Using the
MM formulation, the slab was discretized as shown in Fig. 4.4. For the undoped slab, ?2
and ?3 were chosen to coincide with the faces of the dielectric. The resolution shown by
91
Figure 4.4: Waveguide mesh used for method verification.
92
9 10 11 120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
S11
Frequency (GHz)
Discrete Sweep
AWE Sweep
Theory
Figure 4.5: Comparison of FEM and theoretical S11 for the loaded waveguide.
93
9 10 11 120.8
0.85
0.9
0.95
1
S21
Frequency (GHz)
Discrete Sweep
AWE Sweep
Theory
Figure 4.6: Comparison of FEM and theoretical S21 for the loaded waveguide.
94
the mesh is far greater than what is actually necessary but is used anyway for illustrative
purposes. The matrix solution (9458 unknowns) was carried out using UMFPACK [158],
an efficient direct solver. Figs. 4.5 and 4.6 show the magnitude of S11 and S21, respec
tively, as obtained from theory and the numerical procedure. Two numerical solutions
are shown: one which employs the AWE and one which is a traditional discrete sweep.
In both cases, the results obtained are indistinguishable from the theoretical result. The
discrete sweep consumed 835.08 seconds of CPU time whereas the AWE sweep needed
only 84.22 seconds. Therefore, for optimizations presented later, the AWE will be used
in all work presented in the remainder of this dissertation.
4.5 Results and Discussion
All results discussed in this section refer to a host dielectric material (?r = 2.2,
?r = 1, d = 1 mm) placed in an X band rectangular waveguide (dimensions a = 2.29
cm, b = 1.02 cm) as shown in Fig. 4.1. The longitudinal dimension of waveguide was set
to 5 mm with the slab located at the center. The MM scheme was applied to the faces
of the waveguide and convergence of the solution was tested for a variety of situations
(e.g., undoped and doped slabs). In all cases, the solution was found to converge when
the MM scheme allowed up to the first 40 TE modes (and corresponding degenerate TM
modes).
To verify the accuracy of the FEM, the model was compared to the analytical solu
tion for undoped slabs presented in Chapter 2 and found to be in excellent agreement.
The FEM results for doped cases were compared to models created in HFSS, a com
mercially available FEM software tool [159]. For these comparisons, the dopants were
95
manually inserted into the HFSS model. In all cases, agreement with the FEM was found
to be excellent as shown in Fig. 4.7.
Presently, fabrication techniques do not exist that would allow control of the place
ment and orientation of dopants embedded within a host medium. Therefore, for ran
domly doped materials to be useful they must exhibit nearly identical frequency responses
(Sparameters) for a given set of doping conditions (e.g., doping level, feature size, etc.).
To determine the consistency of the frequency response when using wire dopants, 20
different random wire distributions were selected for each doping level simulated and
their results compared. The discretized geometry detailed previously contained a total
of 1921 edges (average edge length 1.25 mm). Doping levels of 50, 100, 150, and 200
wires were each simulated (80 total simulations).
The results of these simulations clearly showed that randomly doped dielectric slabs
can, for some cases, exhibit notch filter behavior. However, the results lacked any signifi
cant consistency, with number of notches, notch frequency, and level of attenuation being
unpredictable. Fig. 4.8 shows two profiles generated using 150 wires and illustrates the
extreme variation in the obtained responses. Upon further investigation, the cause of the
inconsistency was determined to be the wire selection process. During this procedure, no
mechanism was in place to prevent wires from connecting. Consequently, in many cases,
random selection of wire locations resulted in the presence of long, erratically shaped
wires an example of which is illustrated in Fig. 4.2. Therefore, a second set of simulations
was conducted with the algorithm appropriately modified to prohibit wire connection.
For this condition, only negligible notch behavior was developed in most cases. How
ever, if a notch did appear, there was noticeable consistency in notch frequency (fnotch
96
9 10 11 12?50
?40
?30
?20
?10
0
Frequency (GHz)
S 21
(dB)
HFSS
FEM
Figure 4.7: Comparison of magnitude of S21 for triangular patch doping using present
method and HFSS.
97
= 12.3 GHz) among responses for a given doping level as well as a correlation between
doping level and notch frequency. Furthermore, there was no consistency in the level of
attenuation for a given doping level as shown in Fig. 4.9. Therefore, since neither the
allowance or prevention of wire connections would be controllable during fabrication of
such a material, it must be concluded that random doping of a dielectric slab with short
wires does not result in a material with any predictable and useful properties.
Even though randomly embedded wires in the dielectric showed no promise of pro
ducing a useful material, random wire placement on the dielectric surface has shown some
utility. Fabrication of such a material is also relatively easy using printed circuit and
microelectronic fabrication techniques. To easily characterize the behavior of surface
doped dielectrics, valid wire locations were limited to vertical (ydirected) edges on the
front face of the slab with no wire connections permitted. The limitation of vertical
wires on the slab face was motivated by the fact that this particular orientation of wires
would have the most dramatic effect on the wave propagation. Doping levels of 10, 20,
35, 50, 65, and 150 wires (selected from Nsurface = 380 total vertical edges) were each
simulated 20 times to examine the variation in the frequency responses. In all cases, the
reflection and transmission coefficients were found to be nearly identical to the undoped
slab?s response. Therefore, for this particular frequency range, 1 mm wires did not result
in any form of substantial frequencydependent behavior.
In order to investigate the response?s dependence on wire length, 2 mm wires were
generated by connecting two vertically adjacent edges. Again, doping levels of 10, 20,
35, and 50 2 mm wires were each simulated 20 times to determine the utility of such
materials. In this case, all 10 wires cases produced responses nearly indistinguishable
98
9 10 11 12?40
?35
?30
?25
?20
?15
?10
?5
0
Frequency (GHz)
S 21
(dB)
Figure 4.8: A wide variation of S21 responses is shown for the cases of randomly embed
ded wires.
99
9 10 11 12?20
?15
?10
?5
0
Frequency (GHz)
S 21
(dB)
Figure 4.9: Illustration of the wide range of notch behaviors observed even when wire
connections within the dielectric are not allowed.
100
from the undoped slab?s response. For the 20, 35, and 50 wire cases, a few of the
responses exhibited a notch filter behavior as shown in Fig. 4.10.
A series of 3 mm wire simulations were also performed which included 20 simulations
for each doping level of 10, 20, and 35 wires. Results for the 10 and 20 wire cases generally
showed a notch filter response in the 1012 GHz range (see Fig. 4.11). In the 35 wire
simulations, however, the response typically showed the presence of two notches, one
between 910 GHz and another between 1112 GHz (see Fig. 4.11).
In the final analysis, we can conclude that, for the cases studied, short wires do not
yield especially useful reflection and transmission properties for the doped dielectrics.
Although, their utility may be questioned, dielectrics with wires only on the surface are
certainly realizable using current printed circuit technology. Therefore, the consistency
of the random wire configurations? responses no longer presents the same fabrication
problem observed in the embedded wire case. However, these conclusions may not be
valid for finite wire radius doping.
Noting the above behavioral trends seen in the thin wire cases, the response of
thin triangular patches (?1 mm edge length) placed on the dielectric surface was also
investigated. For these simulations, no restrictions were placed on the possibility of
patches connecting to form larger conducting shapes since the fabrication is relatively
simple for any arbitrary surface pattern. Twenty simulations were performed for each
of the 10, 20, 30, 40, 50, and 75 patch doping levels (120 simulations total). For the
case of 10 patches, no significant deviations from the undoped response were observed.
At the higher doping levels, a large variation in the notch frequency, bandwidth, level of
attenuation, and number of notches was found. Fig. 4.12 shows several of the possible
101
9 10 11 12?30
?25
?20
?15
?10
?5
0
Frequency (GHz)
S 11
, S
21
(dB)
S21
S11
Figure 4.10: Magnitude of S11 and S21 for typical case of a dielectric slab doped with 50
2 mm wires.
102
9 10 11 12?50
?40
?30
?20
?10
0
Frequency (GHz)
S 21
(dB)
10 wires
20 wires
35 wires
Figure 4.11: Comparison of magnitude of S21 for typical cases of a dielectric slab doped
with 10, 20, and 35 3 mm wires.
103
9 10 11 12?60
?50
?40
?30
?20
?10
0
Frequency (GHz)
S 21
(dB)
75 patches
30 patches 20 patches
Figure 4.12: Comparison of magnitude of S21 for cases of a dielectric slab doped with
20, 30, and 75 triangular patches.
104
types of responses realizable using the higher doping levels. As expected, these results
indicate that surface patterning based on triangular patches allows a wider variety of
notch transmission profiles than thin wires. Therefore, should a certain type of notch
behavior be desired, it is necessary only to find the right combination of patches to realize
the required response. This problem is addressed in the next chapter.
4.6 Summary
An investigation into the possible use of randomly doped dielectrics in order to
obtain consistent and useful frequency response profiles has been presented. Using the
FEM, the types of responses and feasibility of fabrication of the doped dielectric slab is
discussed. The dopants considered include short wires embedded in the dielectric and
wires and triangular patches distributed on the surface of the dielectric. Generally, it was
found that random doping produces materials with random notch filter behavior. In the
case of embedded wires, it is impractical to fabricate such materials due to the sensitivity
of the frequency response to wire location and orientation. In other words, randomly
doped slabs do not exhibit any predictable or repeatable filter behavior and, therefore,
are unsuitable to practical use. However, printed circuit and microelectronic fabrication
techniques would easily allow the creation of useful dielectric slabs with wires or patches
printed on the surface. In the above studies, it was shown that wires less than 3 mm in
length (for X band) are likely to give transmission responses which are too narrowband
to be of any practical value. Alternatively, doping with thin triangular patches showed
a wide range of notch filter profiles of which many show potential utility.
105
Chapter 5
Waveguide Filter Optimization Using Surface Patches
5.1 Overview
ThischapterpresentsanoveloptimizationschemeutilizingahybridModeMatching/Finite
Element Method (MM/FEM) and Genetic Algorithm (GA) to evaluate the versatility
of transverse waveguide filter designs. The novelty of the method lies in its completely
autonomous framework. By specifying a desired transmission response in terms of scat
tering parameters, the GA optimizes the filter design using the MM/FEM discussed in
Chapter 4 and [7,160] without any further input from the user. The power of this method
is demonstrated through the optimization of a number of useful X band (8.2  12.4 GHz)
waveguide filters. Additionally, practical concerns related to the fabrication and ele
mental connectivity within the waveguide are addressed and necessary modifications are
discussed.
5.2 Numerical Procedure
5.2.1 Theory
The filter geometry used in this work is composed of a dielectric slab located within
an X band waveguide as represented in Fig. 5.1. Conducting patches printed on the front
face of the slab will perturb the transmission and reflection response (Sparameters) of
the slab possibly resulting in novel filter behavior within the frequency band of interest.
To characterize the response of such a geometry, a FEM forward solution technique is
106
x
y z
a
b
Filter
2
1
3
?
?
?
Figure 5.1: Rectangular waveguide with optimized filter.
107
employed. The details of threedimensional edgebased FEM formulations are discussed
in Chapter 4. A ModeMatching (MM) scheme is employed on ?2 and ?3 which effectively
truncates the FEM domain [6,7,160].
5.2.2 Optimization Technique
The front face of the dielectric slab located within the waveguide was discretized
into a regular grid as shown in Fig. 5.2. During the iterative process, sets of pixels are
selected to be conductors, and the responses of the resulting structures are determined.
Over the course of several iterations, a set of pixels is found which yields the optimal
response.
Given the discrete nature of the gridding, the GA is an advisable choice for carrying
out the optimization. An overview of the GA have been presented in Chapter 2. Fig. 5.3
shows a flowchart of the GA implemented in this chapter. The pixel material is chosen
as the optimization parameter such that the front face of the slab is encoded into a
binary string where ?1? represents a perfectly electrically conducting (PEC) patch and
?0? represents the dielectric (no conductor). For all the cases presented in Section 5.3, the
slab surface is divided into a 20 x 10 grid (x vs. y) resulting in 200 pixels. To reduce the
search space, 4fold symmetry is assumed as illustrated in Figs. 5.1 and 5.2. Therefore,
the search space is reduced to a 50 element binary string.
Initially, the typical elitism, crossover, and mutation operators are employed in the
GA. At each generation, the two best solutions are chosen as elite children and are
directly copied into the population of the next generation. Binary tournament selection
is used to choose members to be evaluated for crossover [38]. Singlebit and doublebit
108
mutation is implemented as outlined in Chapter 2. The crossover and mutation rates are
chosen to be 80% and 10%, respectively [38]. Once again the Total Redundancy Removal
GA (TRRGA) scheme is employed to more aggressively search the solution space. The
population size is set as 20, and the algorithm terminates after 500 generations if a fitness
tolerance (0.01) has not been met. In the examples presented here, S21 is the desired
filter characteristic, and the error function used to evaluate each population member is
given by
Err = 1N
Nsummationdisplay
i=1
parenleftBigvextendsinglevextendsingle
SFEM21 (fi)vextendsinglevextendsingle?
vextendsinglevextendsingle
vextendsingleSDesired21 (fi)
vextendsinglevextendsingle
vextendsingle
parenrightBig2
(5.1)
where N is the number of frequency points used and vextendsinglevextendsingleSFEM21 (fi)vextendsinglevextendsingle and vextendsinglevextendsingleSDesired21 (fi)vextendsinglevextendsingle are
the calculated and desired S21 parameters evaulated at frequency fi, respectively.
Once the best population member has a fitness value below a certain value (0.15
in this work), the redundancy removal scheme becomes inefficient. This tolerance level
represents an approximate point at which a ?finetuning? scheme is bettersuited to
further reduce the error than aggressive interrogation of the entire solution space. It
was found that switching to a high mutation rate (80%) as shown in Fig. 5.3 yielded
better performance since the search was then centered around the member with the best
fitness.
5.2.3 Practical Considerations
Ohira, et al. [72] cite a practical issue associated with FSS optimization. In their
work, a MoM code is employed to optimize an FSS element shape. They show that when
metal patches are joined only at the corners, the adopted MoM scheme does not allow
109
dx
dy
b
a
Connected
metallic patches
4fold symmetry enforced for optimization
Figure 5.2: Gridded substrate face showing metallization for fabrication and FEM sim
ulation.
110
Initialize
Population
Evaluate
Fitness
Population
Generation
Elitism
Selection
Crossover
Any redundancies
in population?
Eliminate extra
population
Fill population
with new
members
Geometry
Refinement
Exit
Yes
No
Exit
Condition
Best fitness
below tolerance?
Mutation
(10%)
Mutation
(80%)
No
No
Yes
Yes
Figure 5.3: Flowchart illustrating the GA procedure with geometry refinement, redun
dancy removal, and high mutation procedures.
111
current to flow between the patches. This represents a significant analysis difficulty in
terms of assuring that the optimized design would be realizable when fabricated. To
correct this deficiency, an automated geometry modification procedure known as the
Geometry Refinement Method (GRM) is required.
Fig. 5.4 shows two cornerconnected patches and two mesh triangles at the patch
junction. The traditional FEM vector basis functions suffer from a weakness similar to
that described in [72]. The electric field inside the triangular patches on either side of
the metallized patches are represented by the set of basis functions. However, when two
sides of the triangle have no tangential electric field, only the basis function associated
with the outer edge remains to describe the field in this corner region (in the case of
zerothorder basis functions). By definition, this function is zerovalued (both tangential
and normal components) at the corner where the PEC patches meet. Therefore, there
can be no electrical connection between the PEC patches (no surface charge is present).
Similar arguments hold even if the corner junction is discretized in a different fashion
such as when only one triangle edge rests along the patch or higher order functions are
used.
This issue is addressed by the GRM which is part of the GA as shown in Fig. 5.3.
After the generation of a new population member, the binary string is scanned for
cornerconnected patches. When this condition is encountered, one of the two empty
pixels is filled (random decision) so as to eliminate the corner connection. The process
is repeated until no corner connections are present on the substrate grid. In addition to
resolving the problem presented by the basis functions, this method also eliminates any
112
fabrication uncertainty as to whether or not a physical connection was actually made in
the measured sample.
Similarly, for the algorithm presented here, no PEC patches are allowed along the
outer ring of the grid (Fig. 5.4). This ensures that the edge patches are electrically
isolated from the waveguide walls. Thus, this requirement further reduces the search
space to a 36 bit binary string.
5.2.4 Forward Solution
The fitness evaluation of any population member can be carried out via a forward
solution using the MM/FEM. This method was found to be an efficient means of deter
mining the Sparameters of the loaded substrate. For additional method verification, an
optimization was performed for a certain bandpass filter characteristic and then com
pared to the same geometry simulated in HFSS [159]. Measurements of the fabricated
filter were also conducted to ensure that both computational techniques returned accu
rate data. Fig. 5.5 shows the desired response compared to the optimized FEM response
as well as the results of HFSS and measurements. As indicated, the FEM scheme led
to errors in the computation of the fields. A probable explanation for these errors is
the mesh resolution. For the FEM to be a computationally efficient method, the mesh
generation was done before optimization so that one mesh, and therefore one matrix
could be used throughout the optimization. Also, each pixel was discretized as two tri
angles in the mesh generation scheme. However, in the vicinity of patch corners, the
fields vary rapidly and a linear basis functions cannot realize this type of behavior. Ef
forts were made to increase the resolution of the mesh, but the mesh generator available
113
X50X61X74X63X68X65X73X20X6EX6FX74
X61X6CX6CX6FX77X65X64X20X6FX6E
X73X75X62X73X74X72X61X74X65X20X65X64X67X65
X43X6FX72X6EX65X72
X63X6FX6EX6EX65X63X74X69X6FX6EX73
X6EX6FX74X20X61X6CX6CX6FX77X65X64
Figure 5.4: The lower left quarter of the waveguide filter is shown illustrating the
rules governing the geometry refinement process. The inset illustrates the problem with
cornerconnected patches.
114
was not sophisticated enough to do this efficiently. The matrix resulting from this high
resolution mesh was too computationally intensive to invert and maintain a reasonable
optimization time.
HFSS, however, returned results that were in excellent agreement with the mea
surements. Therefore, HFSS was integrated into the optimization scheme as the forward
solver. The accuracy of HFSS can be attributed to two things: (1) HFSS uses a higher
order basis function as opposed to the zerothorder basis functions used in Chapter 4,
and (2) HFSS utilizes an adaptive solver which refines the mesh as necessary in the
vicinities of rapid field variation.
HFSS employs a MM scheme slightly different from that presented here. Through
a hybridization procedure, the field within the waveguide structure and the unknown
waveguide eigenfunction amplitudes required for the impedance match at ?2 and ?3 are
solved simultaneously. The forward solution begins with a 2.5D eigenvalue solution at
each of the ports. Now the field variation for each mode is known at the ports and the
unknown amplitudes can be solved for during the FEM solution. Therefore, this has the
benefit of directly providing the magnitude and phase of the Sparameters (TE10 mode)
at the waveguide ports.
5.3 Results
In this section, four design examples are presented to demonstrate the utility of the
proposed optimization technique. For all examples, the filter patterns are optimized on
a substrate (?r = 2.2, ?r = 1, thickness = 0.7874 mm) placed in a section of WR90
waveguide as represented by Fig. 5.1. The model utilized in HFSS for the optimization
115
9 10 11 120
0.2
0.4
0.6
0.8
1
Frequency (GHz)
S
21

Ideal
FEM
Measurement
HFSS
Figure 5.5: Comparison of ideal data, MM/FEM, HFSS, and measurements. The
MM/FEM matches well with the ideal (requested) response. However, the geometry
generated during the process yields a different response in actuality as given by HFSS
and measurements.
116
procedure is shown in Fig. 5.6. The geometry is generated prior to the optimization, and
during the procedure, the algorithm simply selects which patches are to be conductors.
As previously mentioned, the GA is allowed to optimize the response until the RMS
error between the design goal and simulated responses is less than 0.01 or the algorithm
reaches 500 generations. Measurements of fabricated samples are carried out according
to the method presented in Section 3.3.
5.3.1 Notch Filter
As an initial test, a notch filter was specified with a center frequency of 10.4 GHz
and 3 dB bandwidth of 1.4 GHz. For the error function, only five points were evaluated:
the center frequency (S21 = 0), two 3 dB points (S21 = .5), and the ends of the X band
(S21 = 1). After optimization, the 5point RMS error between the simulated response
and the ideal response was 0.0933. The fabricated sample is shown as pattern A in
Fig. 5.7. Measurements showed excellent agreement with the simulation (RMS error =
0.0667) as well as the design goal. All three S21 responses for this pattern are shown
in Fig. 5.8.
5.3.2 Bandpass Filter
A bandpass filter was optimized with center frequency of 10.3 GHz and 3 dB band
width of 1.6 GHz. In this case, 51 points were evaluated for the error function. The
ideal response was specified as
S21 = exp
parenleftBigg
?
parenleftbigf ?10.3?109parenrightbig2
1018
parenrightBigg
(5.2)
117
Figure 5.6: HFSS model used for optimization
118
Figure 5.7: Fabricated samples produced using the patterns generated by the optimiza
tion procedure. A dime is shown for size reference.
119
X39 X31X30 X31X31 X31X32
X2DX35X30
X2DX34X30
X2DX33X30
X2DX32X30
X2DX31X30
X30
X46X72X65X71X75X65X6EX63X79X20X28X47X48X7AX29
X7CX53
X32X31
X64X42
X7C
X49X64X65X61X6C
X53X69X6DX75X6CX61X74X69X6FX6E
X4DX65X61X73X75X72X65X6DX65X6EX74X73
Figure 5.8: Comparison of S21 responses of ideal, simulated, and measured notch filter.
120
where f is the frequency. The resulting RMS error between the ideal and simulated
responses was 0.0482. The filter pattern is noted as B in Fig. 5.7. Measured results
were found to be in excellent agreement with the ideal and simulated results as shown
in Fig. 5.9. The RMS error between the measurements and simulation was calculated to
be 0.0252.
5.3.3 Low Pass Filter
The next example optimized was a low pass filter with 3 dB frequency of 11.1 GHz.
The ideal response was specified as shown in Fig. 5.10. Optimization yielded pattern
C shown in Fig. 5.7 for which the RMS error was 0.0685. Again, agreement was quite
good between the ideal, simulated, and measured responses (see Fig. 5.10) with the
measurement error being 0.0252.
5.3.4 High Pass Filter
Finally, a high pass filter with 9.1 GHz 3 dB frequency was desired as shown in
Fig. 5.11. The optimization yielded pattern D in Fig. 5.7 which gave an RMS error
of 0.0368. Fig. 5.11 shows the ideal, simulated, and measured responses, all of which
show excellent agreement with the exception of a small error near 12 GHz. This small
error resulted in a slightly higher measurement error of 0.134 although, clearly, overall
agreement is excellent.
5.4 Summary
A general optimization scheme for designing transverse rectangular waveguide filters
has been presented and shown to yield excellent results for a wide variety of useful filter
121
X39 X31X30 X31X31 X31X32X2DX36X30
X2DX35X30
X2DX34X30
X2DX33X30
X2DX32X30
X2DX31X30
X30
X46X72X65X71X75X65X6EX63X79X20X28X47X48X7AX29
X7CX53
X32X31
X20X64X42
X7C
X49X64X65X61X6C
X53X69X6DX75X6CX61X74X69X6FX6E
X4DX65X61X73X75X72X65X6DX65X6EX74X73
Figure 5.9: Comparison of S21 responses of ideal, simulated, and measured bandpass
filter.
122
X39 X31X30 X31X31 X31X32X2DX33X35
X2DX33X30
X2DX32X35
X2DX32X30
X2DX31X35
X2DX31X30
X2DX35
X30
X46X72X65X71X75X65X6EX63X79X20X28X47X48X7AX29
X7CX53
X32X31
X20X64X42
X7C
X49X64X65X61X6C
X53X69X6DX75X6CX61X74X69X6FX6E
X4DX65X61X73X75X72X65X6DX65X6EX74X73
Figure 5.10: Comparison of S21 responses of ideal, simulated, and measured low pass
filter.
123
X39 X31X30 X31X31 X31X32X2DX35X30
X2DX34X30
X2DX33X30
X2DX32X30
X2DX31X30
X30
X46X72X65X71X75X65X6EX63X79X20X28X47X48X7AX29
X7CX53
X32X31
X20X64X42
X7C
X49X64X65X61X6C
X53X69X6DX75X6CX61X74X69X6FX6E
X4DX65X61X73X75X72X65X6DX65X6EX74X73
Figure 5.11: Comparison of S21 responses of ideal, simulated, and measured high pass
filter.
124
responses. Additionally, this technique can be applied to waveguides of arbitrary cross
section while completely alleviating the burden of analytical design techniques. This
method employs a modified Genetic Algorithm to aggressively search a large discontinu
ous solution space until a low RMS error has been achieved. At this point the finetuning
mechanism becomes active. Experiments included optimization of notch, bandpass, low
pass, and high pass filters, all of which showed extremely good agreement between ideal,
simulated, and measured S21 responses.
125
Chapter 6
Conclusions
With the wireless communications industry and military demand for smaller, higher
performance devices, traditional electromagnetic design methods are no longer suitable.
A hybridization of computational tools and optimization algorithms has become neces
sary for the design challenges faced by today?s engineers. In response, this dissertation
presented several novel methods utilizing fast and efficient optimization techniques that
may lead to fully autonomous engineering design and inversion codes in the future.
In Chapter 2, a thorough characterization of optimization techniques and error
functions was conducted with regard to the extraction of complex permittivities from
multilayered dielectric slabs. Sequential Quadratic Programming (SQP), the Genetic
Algorithm (GA), and Particle Swarm Optimization (PSO) were applied to ideal and
measured Sparameter data, and the complex permittivities of each individual layer
were estimated. SQP was found to be a very fast method, but often suffered from
local minima trapping. GA and PSO, however, were able to accurately extract the
permittivities for multilayer materials (1, 2, and 3 layer materials discussed). After
application of the Total Redundancy Removal scheme to the GA, the convergence rate
was improved significantly, thereby establishing GA as the preferred method. Of the
three error functions considered, the function which utilized only magnitude information
achieved superior results in many cases. However, as the number of layers increased, the
inclusion of phase information within the error function was determined to be necessary.
126
In Chapter 3, the method of Chapter 2 was extended to include the extraction of
complex permeability. To the author?s knowledge, no other efforts to date have been
successful in extracting the full set of complex constitutive parameters from individual
layers of multilayered slabs. Since this problem results in a much higher dimensionality
search space, a novel MultiPoint SQP algorithm was developed and presented to exploit
the efficiency and accuracy of SQP while improving its ability to avoid local minima.
Overall, the algorithm showed better accuracy than the GA for single layer cases and
was over 20 times faster in finding the global minimum.
In Chapter 4, a novel methodology for realizing waveguide filters was presented.
In this method, small, thin conducting wires (or other conducting shapes) are used to
randomly dope a dielectric slab. With proper selection of wire length and density, it was
shown that novel filter behaviors can result. However, when practical constraints on the
doping methods were applied, the resulting doped slabs showed either no useful filter
behavior or erratic behavior not controllable by modern fabrication processes. When
conducting patches were used and restricted to the surface of the slab, a variety of novel
notch filter characteristics were observed. Using this approach, fabrication difficulties can
be avoided since standard printed circuit technology can easily be used to manufacture
the filters. To realize specific filter responses, optimization algorithms could be employed
to correctly locate the placement of the conducting patches.
In Chapter 5, a method is presented to automate the design of waveguide filters
using surface patches on dielectric slabs loaded into the guide. This technique utilizes a
combination of a FEM forward solver (HFSS) in the context of a GA. The GA presented
127
here was modified with a finetuning mechanism which becomes operational once the er
ror has dropped below a certain threshold. Also, some practical fabrication concerns are
addressed by employing the Geometry Refinement Method to eliminate cornerconnected
patches. The novelty and utility of this method were demonstrated by allowing the al
gorithm to optimize patch configurations for a number of filter responses. In all cases,
the resulting structure showed excellent agreement with the desired response and was
further verified by fabrication and measurements.
6.1 Future Work
6.1.1 Constitutive Parameter Extraction
A number of suggestions are given in this section regarding possible avenues for
future research regarding the constitutive parameter extraction technique. As previously
addressed, the availability of limited information over a band of frequencies determines
the number of layers that can be accurately handled. With the addition of more layers
and uncertainties in layer thicknesses or number of layers, significantly more information
must be provided to limit the number of local minima within the search space. This
problem is not only affected by the dimensionality of the search space but by the accuracy
of the measured data available. To increase the amount of information available to the
extraction algorithm, a free space measurement technique [52] is suggested in which
information over both frequency and angle of incidence can be provided. To realize such
a scheme, modification of the forward solver is all that is required. With the current
availability of numerical electromagnetics codes to handle free space scattering, it is likely
128
that this task could be implemented using FEM, Method of Moments, Finite Different
Time Domain, hybrid techniques, etc.
6.1.2 Waveguide Filters
Extensions to the waveguide filter design method of Chapter 5 are numerous. The
work can easily be extended to account for arbitrary waveguide cross sections since the
HFSS geometry is the only parameter requiring modification. Also, the method may
be extended to handle multiple layers or to also optimize dielectric thickness. These
modifications are trivial but would greatly increase the search space and, therefore,
computational intensity of the optimization. Also, it is not recommended, although
theoretically possible, to optimize for the dielectric constant of the slab since these
materials may not be readily available.
6.1.3 Extension To Antenna Design
The methodology utilized throughout this work may also be extended for the design
of antennas. Wire antenna shape can be optimized using the techniques presented here
to obtain novel radiation patterns or bandwidth characteristics. Similarly, the waveguide
filter design method can directly be applied to the automation of patch antenna design.
Simply changing the forward solver to an efficient Method of Moments code or modifying
the FEM code, the same optimization algorithm can be applied to design novel printed
circuit antennas.
In conclusion, the present work opens up new areas of research for accurate design,
simulation, and characterization of problems in a wide variety of fields. The inversion
129
techniques presented have enormous significance to an array of topics including well
logging, mineral location, unexploded ordnance discrimination, finance, etc. Also, the
hybridization of design methodologies and optimization techniques such as those pre
sented here will ultimately lead to fully autonomous engineering tools thereby rapidly
accelerating the pace of technological development.
130
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