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Synthesis of spatial filters and broadband microwave absorbers using micro-genetic algorithms

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The Pennsylvania State University
The Graduate School
College o f Engineering
SYNTHESIS OF SPATIAL FILTERS AND BROADBAND MICROWAVE
ABSORBERS USING MICRO-GENETIC ALGORITHMS
A Thesis in
Electrical Engineering
by
Sourav Chakravarty
© 2001 Sourav Chakravarty
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
May 2001
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W e approve the thesis o f Sourav C hakravarty.
D ate o f S ign atu re
Raj Mittra
Professor o f Electrical Engineering
Thesis AdvisoK'ChairJbf Com m ittee
^ / . (aI siamQ/^
3 /a?/of
Werner
Ass
Professor o f Electrical
Engineering
V * 1 / 0 1 ___________
Lynrf A. Carpente
A ssociate Professor o f Electrical
Engineering
Professor.of Electrical Engineering
M ^ d U 2 - 5 , 2 -QO'
ikhfles"tiLcjJiftWkia
^^pfessonsf Engineering Science and
echanics
t
Vijay K. (jraradan
Distinguished A lum ni Professor o f Engineering
Science and M echanics and Electrical
Engineering
3 J z 6/a/
W. Kennet,h,Jenkins
Professor o f Electrical Engineering
Head o f the Department o f Electrical
Engineering
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ABSTRACT
Over the years, Frequency Selective Surfaces (FSSs) have found frequent use as
radomes and spatial filters in both commercial and military applications. The FSSs are
ofren embedded within dielectric or magnetic materials that may be lossy or lossless. Past
research has concentrated on the synthesis of broadband microwave absorbers and spatial
filters using multilayered dielectrics via the application of Genetic Algorithms (GAs).
Typically,
these problems can be categorized,
relatively
speaking,
as
being
computationally inexpensive. A comprehensive research effort has not been made to
apply the GA to synthesize broadband microwave absorbers and spatial filters with
embedded FSS screens, which can be termed as a computationally expensive problem.
To enhance the computational efficiency of the GA, a variant of the Conventional
Genetic Algorithm (CGA), referred to as the Micro-Genetic Algorithm (MGA), is
introduced here. The MGA is applied to optimize various parameters, viz., the thickness
and relative permittivity or permeability o f each layer, the FSS screen design and
materials, the x- and y-periodicities of the FSS screen, and their placement within the
dielectric composite. The result is a multilayer composite that simultaneously provides a
maximum reflection or transmission of both TE and TM waves for a prescribed range of
frequencies and incident angles, while automatically placing an upper bound on the total
thickness o f the composite. Three basic types of problem geometries have been
considered to illustrate the numerical efficiency of the MGA: (i) composites with no FSS
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screens embedded in them; (ii) composites with a single embedded FSS screen; and
(iii) two FSS screens embedded in the composite.
For the first problem, a recursive formulation for layered inhomogeneous media is
utilized to calculate the reflection and transmission coefficients of the composite. The
second problem is analyzed by using a frequency domain Electric Field Integral Equation
(EFIE) formulation in conjunction with a spectral domain Method of Moments (MoM)
solution. Finally, for the third problem, the Generalized Scattering Matrix (GSM)
approach is used to cascade multiple FSS screens and generate reflection and
transmission coefficients.
MGA, which is a variant of the CGA, uses a very small population base for the
optimization process, and this serves as the key to its numerical efficiency. CGAs
perform poorly with small population sizes due to insufficient information processing and
they converge prematurely to non-optimal results. The MGA has two major advantages
over the CGAs: (i) it works with a small population base for each generation; and (ii) it
reaches near-optimal regions faster than the CGAs that work with a large population
base. The general choice o f population size for the CGAs can range between 100 and
10000, while the MGAs typically work with a population size between 5 and 50.
Numerical experiments show that using the MGA can decrease the computational run
time by 50%, even for “best-case” problems for the CGAs.
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V
TABLE O F CONTENTS
LIST OF TABLES........................................................................................................
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LIST OF FIGURES......................................................................................................
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ACKNOWLEDGEMENTS............................................................................................
xiv
Chapter I INTRODUCTION.........................................................................................
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1.1 Research Objective and General Technique...............................................
1.2 Thesis Outline...............................................................................................
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Chapter 2 THE OPTIMIZATION ALGORITHM.......................................................
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2.1 Genetic Algorithm Fundamentals................................................................
2.1.1 GA Terminology...........................................................................
2.1.2 Overview o f Genetic Algorithms................................................
2.1.3 GA Operators................................................................................
2.1.3.1 Selection Operator........................................................
2.1.3.2 Crossover Operator.......................................................
2.1.3.3 Mutation Operator........................................................
2.1.4 Selection Schemes........................................................................
2.1.4.1 Population Decimation................................................
2.1.4.2 Roulette Wheel Selection............................................
2.1.4.3 Tournament Selection...................................................
2.1.5 Crossover Schemes.......................................................................
2.1.5.1 Single Point Crossover................................................
2.1.5.2 Dual Point Crossover....................................................
2.1.5.3 Uniform Crossover.......................................................
2.1.6 Mutation Schemes........................................................................
2.1.7 Elitism............................................................................................
2.1.8 Fitness Function............................................................................
2.1.9 Classification o f Genetic Algorithms.........................................
2.2 The Conventional Genetic Algorithm (CGA)...........................................
2.2.1 Population Sizing for the CGA....................................................
2.2.2 Drawbacks o f the CGA.................................................................
2.3 Micro-Genetic Algorithm (MGA)..............................................................
2.3.1 The MGA Optimization Procedure............................................
2.3.2 Advantages of the MGA...............................................................
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Chapter 3 SYNTHESIS OF COMPOSITES COMPRISING ONLY DIELECTRIC
AND MAGNETIC MATERIALS..................................................................................
3.1 Problem Geometries.....................................................................................
3 .2 Formulation o f Reflection and Transmission Coefficients for
Inhomogeneous Layered Media........................................................................
3 .3 MGA Formulation for the Spatial Filter and Broadband Microwave
Absorber Problems.............................................................................................
3.3.1 Spatial Filter Synthesis.................................................................
3.3.2 Numerical Results for Spatial Filter Design...............................
3.3.3 Synthesis o f Broadband MicrowaveAbsorbers........................
3 .3 .4 Numerical Results for the Synthesis o f Broadband Microwave
Absorbers...............................................................................................
3 .3 .5 Numerical Results for the Synthesis o f Broadband Microwave
Absorbers Using Lossy Carbon Fiber and Lossless Dielectric
Materials.................................................................................................
Chapter 4 SYNTHESIS OF COMPOSITES COMPRISING A SINGLE
FREQUENCY SELECTIVE SURFACE (FSS) EMBEDDED IN DIELECTRIC
AND/OR MAGNETIC MEDIA.....................................................................................
4.1 Problem Geometries.....................................................................................
4.2 Analysis o f Doubly-Periodic, Planar, Infinite FSS Screens Embedded in
Inhomogeneous Layered Media........................................................................
4.2.1 Numerical Solution Employing the Method of Moments
(MoM)....................................................................................................
4.2.2 Evaluation of Reflection and Transmission Coefficients
4.3 MGA Formulation for the Spatial Filter and Broadband Microwave
Absorber Problems.............................................................................................
4.3.1 Broadband Microwave Absorber Synthesis...............................
4.3.1.1 Numerical Results for Single FSS Screen Embedded
in Lossy or Lossless Dielectrics...............................................
4.3.1.2 Numerical Results for Single FSS Screen Embedded
in Dielectric and Magnetic Media that can Either be Lossy
or Lossless..................................................................................
4.3.1.3 Numerical Results for a Single FSS Screen
Embedded in Lossy Carbon Fiber and Lossless Dielectric
Material.......................................................................................
4.3.2 Spatial Filter Synthesis.................................................................
4.3.2.1 Numerical Results for Spatial Filter Synthesis
4.3.2.2 The Domain Decomposition Approach Applied to
the Design o f Spatial Filters.....................................................
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Chapter 5 SYNTHESIS OF COMPOSITES COMPRISING MULTIPLE
FREQUENCY SELECTIVE SURFACES (FSSs) EMBEDDED IN DIELECTRIC
AND/OR MAGNETIC MEDIA....................................................................................
5.1 Problem Geometry.......................................................................................
5.2 Analysis of Multiple FSS Screens Embedded in Dielectric and
Magnetic Medium..............................................................................................
5.3 MGA Formulation for Spatial Filters and Broadband Microwave
Absorber Designs..............................................................................................
5.3.1 Broadband Microwave Absorber Synthesis...............................
5.3.1.1 Numerical Results for Two FSS Screens Embedded
in Lossy or Lossless Dielectrics...............................................
5.3.1.2 Numerical Results for Multiple FSS Screens
Embedded in Dielectric and Magnetic Medium that is Either
Lossy or Lossless......................................................................
5.3.2 Spatial Filter Synthesis................................................................
5.3.2.1 Numerical Results for Spatial Filter Synthesis
Chapter 6 THE PARALLEL IMPLEMENTATION OF THE MICRO-GENETIC
ALGORITHM (MGA)..................................................................................................
6.1 Reasons for Parallelization.........................................................................
6.2 Micro-Grain MGA.......................................................................................
6.3 Design of Polarization Selective Surfaces (PSSs) for Dual Reflector
Applications........................................................................................................
Chapter 7 VALIDATION OF THE RESULTS GENERATED BY THE MICROGENETIC ALGORITHM (MGA)................................................................................
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7.1 The FDTD Method......................................................................................
7.2 The Direct Method of Moments (MoM) Approach to Analyze Multiple
FSS Screens........................................................................................................
7 .3 Application of the FDTD Method to Analyze Single Screen FSS
Embedded in Inhomogeneous Media...............................................................
7.4 Application of the Direct-MoM to Analyze Multiple FSS Screens
Embedded in an Inhomogeneous Composite...................................................
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Chapter 8 SUMMARY, CONCLUSIONS, AND FUTURE WORK.........................
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8.1 Improving the Efficiency of the Optimization Process.............................
8.2 Accuracy o f the MGA Optimized Results..................................................
8.3 Suggestions for Further Research...............................................................
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BIBLIOGRAPHY...........................................................................................................
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LIST OF TABLES
3-1
GENETIC ALGORITHM PARAMETER SEARCH SPACE
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3-2
e\ AS A FUNCTION OF FREQUENCY FOR THE LOSSY
DIELECTRIC MATERIAL..........................................................................
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e”
r AS A FUNCTION OF FREQUENCY FOR THE LOSSY
DIELECTRIC MATERIAL..........................................................................
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3-4
GA PARAMETER SEARCH SPACE.........................................................
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3-5
MATERIAL DISTRIBUTIONS AND THICKNESS (MM) OF
COMPOSITE FOR THE THREE C A SES...................................................
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3-6
PERFORMANCE COMPARISION OF CGA AND MGA
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4-1
PARAMETERS SELECTED BY THE GA FOR THE 7 CA SES
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4-2
MGA PARAMETER SEARCH SPACE.......................................................
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5-1
MGA PARAMETER SEARCH SPACE FOR ONE SUB-COMPOSITE
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5-2
PARAMETERS SELECTED BY THE MGA FOR THE TWO CASES
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3-3
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LIST OF FIGURES
Single point crossover with p ^ Peross ....................... .....................................
Single point crossover withp ^ Pcross. ..............................
Dual point crossover withp > pcro« . .............................................................
Dual Point Crossover with p < Pcross. .............................................................
Uniform Crossover with p > / w * . ................................................................
Procedural flowchart o f the CGA...................................................................
Procedural flowchart o f the MGA..................................................................
Synthesis o f spatial filters with lossless dielectrics. A/-layered composite
with a TE polarized uniform plane wave incident at an arbitrary angle.
For TM polarization the E and H fields are interchanged along with a
phase shift of 180 degrees introduced in the H field...................................
3.2
Synthesis o f broadband microwave absorbers with lossy or lossless
dielectrics and lossy magnetic materials, ^-layered composite with a TE
polarized uniform plane wave incident at an arbitrary angle. For TM
polarization the E and H fields are interchanged along with a phase shift
o f 180 degrees introduced in the H field.......................................................
3.3
Three-layered medium with TE polarized wave incident at an arbitrary
angle.................................................................................................................
3.4
Low pass filter design. The weights used for the MGA were 1.3 for 12.2
to 18.0 GHz and 1.0 for the rest of the band. The desired cutoff is at 13 .0
GHz..................................................................................................................
3.5
High pass filter design. The weights used for the MGA were 1.3 for 14.0
to 18.0 GHz and 1.0 for the rest of the band. The desired cutoff is at 14.0
GHz..................................................................................................................
3.6
Band stop filter design. The weights used for the MGA were 1.5 for 1.0
to 11.0 GHz and 15 .0 to 18.0 GHz, and 1.0 for the rest of the band. The
desired cutoffs are at 10.0 and 15.0 GHz......................................................
3.7
Band pass filter design. The weights used for the MGA were 1.3 for 1.0
to 11.0 GHz and 15 .0 to 18.0 GHz, and 1.0 for the rest of the band. The
desired cutoffs are at 10.0 and 15.0 GHz......................................................
3 .8
Real part o f permittivity o f magnetic material................................................
3.9
Imaginary part of permittivity of magnetic material.....................................
3.10
Real part of permeability of magnetic material..............................................
3.11
Imaginary part o f permeability o f magnetic material....................................
3.12 (a) Worst- and best-case reflection coefficient levels for the composite with
only lossless and lossy dielectric materials..................................................
3.12 (b) Design o f composite with only lossy and lossless dielectric materials
3.13 (a) Worst- and best-case reflection coefficient levels for the composite with
only lossless dielectric and lossy magnetic materials..................................
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
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3 13(b) Design o f composite with only lossy magnetic and lossless dielectric
materials..........................................................................................................
3.14 (a) Worst- and best-case reflection coefficient levels for the composite with
interspersed lossy or lossless dielectric and magnetic materials................
3 .14 (b) Design of composite with both lossy and lossless magnetic and dielectric
materials..........................................................................................................
3.15
Evolution of the average and best fitness values of the CGA and MGA
for Case-1........................................................................................................
3.16
Evolution of the average and best fitness values of the CGA and MGA
for Case-2........................................................................................................
3.17
Evolution of the average and best fitness values of the CGA and MGA
for Case-3........................................................................................................
3.18
The permittivity o f carbon fiber material as a function of frequency: (a)
real part; (b) imaginary part...........................................................................
3 .19
Case-1: (a) composite design; (b) reflection coefficient vs. frequency
3.20
Case-2: (a) composite design; (b) worst- and best-case reflection
coefficient vs. frequency................................................................................
4.1
A'-layered composite with a TE polarized uniform plane wave incident at
an arbitrary angle. For TM polarization the E and H fields need to be
interchanged along with a phase shift of 180 degrees introduced in the H
field.................................................................................................................
4.2
Synthesis of broadband microwave absorbers with a single FSS screen
embedded in lossy or lossless dielectric and magnetic material with PEC
backing. jV-layered composite with a TE polarized uniform plane wave
incident at an arbitrary angle. For TM polarization the E and H fields
need to be interchanged along with a phase shift of 180 degrees
introduced in the H field................................................................................
4.3
The composite structure to be analyzed: (a) side view; (b) top view
4.4
The original problem is split into two sub-problems.....................................
4.5
Surface equivalence principle applied to Problem-II: (a) equivalence
principle applied to Region-I; (b) perfect Magnetic conductor (PMC)
used to short out the surface magnetic current density; (c) image theory
applied to the Region-I problem...................................................................
4.6
The transmission line equivalent of the FSS scattering problem for
Region-I..........................................................................................................
4.7
The admittance seen by the current induced on the FSS screen for R
superstates......................................................................................................
4.8
x- and y-components of the unit amplitude rooftops.....................................
4.9
FSS cell discretized into 16 x 16 pixels with the shaded part
corresponding to metal...................................................................................
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Grouping the infinite summation over 0 ^ into M segments of M
4.11
elements. The like terms o f each segment are summed, forming an array
A, composed o fM elements. In this caseM =M=4....................................
Multilayered composite with single embedded FSS screen and uniform
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4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
4.23
4.24
4.25
4.26
4.27
4.28
4.29
4.30
4.31
4.32
4.33
4.34
4.35
4.36
5.1
5 .2
5.3
5.4
A 16 x 16 pixel FSS unit cell design represented in the form of “ l ’s” and
“0’s” ................................................................................................................
Design for Case-1.........................................................................................
Design for Case-2.........................................................................................
Design for Case-3.........................................................................................
Case-3: Reflection coefficient as a function of the elevation angle with
frequency as a parameter...............................................................................
Design for Case-4.........................................................................................
Design for..Case-5..........................................................................................
Design for..Case-6.........................................................................................
Case-6: Reflection coefficient as a function of the elevation angle with
frequency as a parameter...............................................................................
Design for..Case-7..........................................................................................
Fitness value vs., the number of generations for Case-7..............................
Population distribution of the MGA in the 1st, 40th, and 75th generations
for Case-7........................................................................................................
Frequency response o f the composite shown in Fig. 4.21 with a FSS
screen having two-fold or top-bottom symmetry embedded in it...............
Frequency response o f dielectric composite with a single FSS screen
possessing eight-fold symmetry embedded in it...........................................
Worst- and best-case reflection coefficients over the frequency band for
composite with FSS screen in Fig. 4.25.......................................................
Design for composite with both dielectric and magnetic materials
Population distribution for the optimization carried out for the case with
lossy magnetic layers in the composite.........................................................
Fitness value vs. the number of generations for case with magnetic losses
in the composite.............................................................................................
The permittivity of carbon fiber material as a function of frequency: (a)
real part; (b) imaginary part..........................................................................
Design o f carbon fiber composite with FSS screen embedded in it: (a)
FSS unit cell design (white - metal, black - free space); (b) composite
design; (c) frequency response of composite in (b).....................................
Idealized response of spatial fi Iter..................................................................
Spatial filter design with single FSS screen embedded in lossless
dielectric medium..........................................................................................
Sensitivity study of spatial filter design with single FSS screen embedded
in it...................................................................................................................
64 x 64 pixel modified DSL FSS screen embedded in dielectric
composite........................................................................................................
Frequency response o f composite in Fig. 4.35..............................................
Problem geometry............................................................................................
Schematic representation of a two-port system.............................................
The scattering matrix structure and contents for composites comprising
multiple FSS screens embedded in layered media........................................
System division into sub-systems...................................................................
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5.5
5.6
5.7 (a)
5.7(b)
5.8 (a)
5.8 (b)
5 .9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5 .21
5.22
5.23
5.24
5 25
6.1
6.2
6.3
6.4
7.1
Composite system...........................................................................................
FSS unit cell with either two-fold, four-fold, or eight-fold symmetry
Composite design for Case-1.........................................................................
Composite design for Case-2.........................................................................
FSS unit cell design for Case-1......................................................................
FSS unit cell design for Case-2......................................................................
Frequency response o f composite in Fig. 5.7 (a)..........................................
Worst- and best-case reflection coefficients of composite in Fig 5.7 (b):
(a) TE polarization; (b) TM polarization; (c) TE and TM polarization
Performance o f the MGA: (a) Case-1; (b) Case-2.......................................
Population distribution w.r.t. total thickness of composite and reflection
coefficient for Case-1.....................................................................................
Population distribution w.r.t. total thickness of composite and reflection
coefficient for Case-2.....................................................................................
Invariance o f the frequency response with azimuthal angle for composite
inCase-1.........................................................................................................
Invariance o f the frequency response with azimuthal angle for composite
in Case-2: (a) 0 = <J>= 0 degrees, 0 = 0 and <J>= 45 degrees; (b) 0 = 45 and
<(>= 0 degrees, 0 = <J>= 45 degrees.................................................................
Effect of the change in the number o f harmonics included in the
scattering matrix on the frequency response: (a) Case-1, 0 = 45, <f>= 0;
(b) Case-2, 0 = 45, <J>= 0 ...............................................................................
Plot of the worst-case higher order harmonics vs. frequency for Case-1:
(a) TE polarization; (b) TM polarization...................................................
Plot of the worst-case higher order harmonics vs. frequency for Case-2:
(a) TE polarization; (b) TM polarization...................................................
FSS unit cell design for the two FSS screens embedded in the composite
in Fig. 5.20......................................................................................................
Synthesized composite for two FSS screens embedded in layered
media...............................................................................................................
Worst- and best-case reflection coefficient of composite in Fig 5.20: (a)
TE polarization; (b) TM polarization; (c) TE and TM polarization
Population distribution with evolving generations........................................
MGA performance to optimize composite in Fig. 5.20.................................
Idealized response of spatial filter..................................................................
Spatial filter synthesis: (a) composite design; (b) FSS unit cell design; (c)
frequency response.........................................................................................
Pseudocode for a micro-grain micro-genetic algorithm................................
Problem geometry for PSS optimization........................................................
FSS unit cell for PSS design application........................................................
Frequency response of PSS: (a) 0 = 0, <J>= 90; (b) 0 = 10, <j>= 90; (c) 0 =
20, (j>= 90; (d) 0 = 30, 4» = 90; (e)0 = 40, <f>= 90.......................................
The Yee cell with the six electric and magnetic field components all
offset by half a space step..............................................................................
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7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
FSS screen for example in Fig. 4.25: (a) MGA-generated FSS unit cell;
(b) after processing......................................................................................
Comparison between FDTD and MoM results for example 1 in Fig.
4.25
MGA-generated spatial filter design: (a) composite; (b) FSS unit cell
design...............................................................................................................
Comparison between FDTD and MoM results for spatial filter design
with four lossless dielectric layers and one FSS screen...............................
Comparision between the GSM-MGA and the Direct-MoM for Case-1:
(a) TE polarization; (b) TM polarization......................................................
Comparision between the GSM-MGA and the Direct-MoM for Case-2:
(a) TE polarization; (b) TM polarization......................................................
Comparision between the GSM-MGA technique and the Direct-MoM
for Case-3: (a) normal incidence-TE polarization; (b) normal incidenceTM polarization; (c) 45° from broadside-TE polarization; (d) 45° from
broadside-TM polarization............................................................................
Comparision between the GSM-MGA and the Direct-MoM for Case-1
with 9 harmonics included in the scattering matrix: (a) TE polarization;
(b) TM polarization.........................................................................................
Comparision between the GSM-MGA and the Direct-MoM for the
composite with low-loss dielectric layers and two FSS screens.................
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ACKNOWLEDGEMENTS
I am dedicating this thesis to my parents, for their patience, support and
unbounded love throughout my graduate studies. Without their encouragement over the
years, the completion of this thesis would not have been possible. 1 would like to also
thank my master’s thesis adviser Dr. M. N. Roy for encouraging me to pursue doctoral
studies.
1 am indebted to my adviser. Dr. Raj Mittra for his valuable guidance and support
throughout the course of this research. I also thank my committee members for patiently
reviewing my thesis and administering the Comprehensive and Final Oral Examinations
My deepest appreciation for the support and advice from all the members of the
Electromagnetic Communications Laboratory at Penn State, especially Dr. Wenhua Yu,
Dr. Ji-Fu Ma, Dr. Nader Farahat and Dr. Slava Bulkin. Their helpful suggestions and
assistance are gratefully acknowledged.
Finally, I would like to thank Dr. Neil R. Williams at W. L. Gore and Associates
Inc., and Dr. V. K. Varadan for providing me with the lossy dielectric and magnetic
material values.
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Chapter 1
INTRODUCTION
In recent years, Genetic Algorithms (GAs) [1], which are robust and efficient
optimization techniques, have received widespread attention of the engineering
community. Since its inception by Holland [2], GAs have come a long way towards
solving real world problems, especially where other conventional optimization schemes
appear to have failed. GAs and its variants have been applied to solve many design
problems in electromagnetics, involving spatial filters [3-5], array antenna configurations
[6-7], and wire antennas [8], Though Conventional GAs (CGAs) [1] have been applied to
many optimization, synthesis, and inverse problems with enormous success, only a few
of these can be categorized as computationally intensive. To date, the problems solved by
the CGAs have either been formulated by using simple analytic functions [3-5], or by
employing numerical Green’s function for fixed geometries. Furthermore, CGAs have
not been applied to computationally intensive problems, since they require a large
number of objective function evaluations to achieve convergence. This is because the
CGAs are inherently stochastic search processes and they perform the search in a large
solution space [1].
The objective of this effort is to synthesize broadband microwave absorbers and
spatial filters using a composite comprising a combination of dielectric and magnetic
materials with or without doubly periodic, multiple, Frequency Selective Surfaces (FSSs)
[9] embedded within it. Throughout this thesis, only isotropic materials are considered.
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FSS structures, also referred to as spatial filters, are counterparts of filter circuits used in
the microwave frequency range and they find widespread applications as radomes [10]
and reflectors [11]. Considering the synthesis problem as stated above, it is evident that
with no FSS screens within the composite, the objective function has a simple recursive
form [12], and evaluating it is computationally inexpensive. However, in the presence of
FSS screens, we require the Method of Moments (MoM) to solve for the unknown
current coefficients on the metalized portions of the FSS structures, and this is a very
time-consuming, computationally intensive procedure. Since the CGAs typically search
from a large space o f potential solutions, and the matrix solution has to be repeated for
each iteration step, this leads to an impractical run time to achieve convergence, and
prompts one to seek more advanced and numerically efficient optimization techniques.
A possible solution to decreasing the run-time is to reduce the size of the search
space of the potential solutions. However, the caveat is that the CGAs cannot be
employed as an optimizer if this is done, because they exhibit very poor convergence to
global extrema with reduced search spaces. Hence, to work with such reduced search
spaces, we need a version o f the GA whose convergence to global extrema is not affected
as the search space is reduced. A good candidate for this type of GA is the Micro-genetic
Algorithm (MGA) [13], which is a small-search-space-based GA with good convergence
properties.
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1.1 Research Objective and General Technique
The objective of this thesis is to develop a numerically efficient optimization
technique for synthesis of spatial filters or broadband microwave absorbers. The generic
problem geometry of interest is a composite comprising of multiple FSS screens
embedded in lossy or lossless dielectric or magnetic materials, with or without a Perfect
Electric Conductor (PEC) backing. The known quantities for the synthesis problem are
the frequency response, the allowed total thickness of the composite, the range o f values
for relative permittivities and permeabilities o f lossless dielectric/magnetic material, and
the frequency dependent real and imaginary parts of relative permittivity and
permeability o f lossy dielectric/magnetic material. The synthesized quantities are the
thickness o f each layer, their constitutive parameters, the design and material of the FSS
screen, the position o f this screen within the composite, and the total thickness of the
composite.
The MGA is initially applied to the synthesis of composites without FSS screens.
This is done to familiarize oneself with the various parameters of the MGA and their
effect on the optimization results. A recursive formulation for heterogeneous layered
media [12] is employed to calculate the reflection coefficient of the composites. Both
spatial filters and broadband microwave absorbers are designed. The MGA optimizes the
thickness and constitutive parameters for each layer resulting in a multi-layered
composite that provides maximum absorption o f both TE and TM waves simultaneously
for a prescribed range of frequencies and incident angles. A vast improvement in both the
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4
computational efficiency and convergence is observed in the MGA as compared to the
CGA for both of these sets of problems.
The first problem geometry that is expected to be computationally intensive
involves the optimization o f a single lossy or lossless FSS screen embedded in dielectric
or magnetic materials that may be lossy or lossless. Periodicity in the FSS screen is
handled by applying Floquet’s theorem [14], according to which the surface currents
induced over the infinite FSS due to an incident plane wave can be expressed in terms of
the current on a single unit cell. Next, the spectral domain MoM [15] technique is applied
to evaluate the unknown current coefficients on the metalized portion of the FSS unit
cell. Once the induced current on the infinite FSS screen is known, the electric field
scattered from the FSS screen is easily evaluated [15]. Finally, transmission line
techniques in conjunction with the spectral immitance approach [16] are employed to
determine the reflection coefficient of the FSS-dielectric combination. The MGA
optimizes the thickness and constitutive parameters for each layer, the design, material
and position of the FSS screen within the composite. This results in a multi-layered
composite that provides maximum absorption of both TE and TM waves simultaneously
for a prescribed range of frequencies and incident angles.
For designs requiring optimization at very high frequencies, e.g., 95.0 GHz, the
FSS unit cell resolution has to be increased to account for the fine geometrical features.
This leads to a large number of unknowns to be solved for in the MoM, and results in a
drastic increase in the computation time for a single objective function evaluation,
prompting us to seek an alternate approach to the optimization process. The domain
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5
decomposition [10] technique is employed to handle these problems, wherein the
dielectric composite is synthesized by using the MGA, while conventional FSS screen
design techniques are employed to design the FSS screen. Finally, the two separate
designs are combined to satisfy the design criteria.
The second problem geometry considered consists of multiple FSS screens
embedded within the composite, it has been found [39] that two or more layered FSS
screens are needed to provide the necessary degrees of freedom for meeting stringent
design specifications, which requires the mutual coupling between the screens to be
modeled accurately. This, in turn, requires that the surface currents on each screen be
generated, and then translated to scattered electric fields, which have to be made incident
on all other FSS screens within the composite to generate the total scattered fields.
However, using the direct method to model the currents on each FSS screen leads to a
large number o f unknowns to be solved for, and this precludes the application of the
MGA to the synthesis problem. Consequently, we employ an alternate “circuit theory”
approach called the Generalized Scattering Matrix (GSM) [17-18] technique.
In this method we derive, as a first step, the generalized scattering matrices of the
individual screens by using the MoM, and of the dielectric layers by following the
procedure we described in the previous page. These matrices can be subsequently used to
generate a composite scattering matrix for the entire system. Both spatial filters and
broadband microwave absorbers were designed successfully with the MGA using the
GSM technique.
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6
1.2 Thesis Outline
Chapter 2 provides a background for the optimization technique employed for the
synthesis. First, the basic GA fundamentals are discussed and a brief discussion is
presented on the stochastic operators used, the fitness or objective function employed,
and the classification of the GA. Next, the CGA is described with the aid of a flowchart,
and a discussion on population sizing as well as the drawbacks of the CGA is presented.
Finally, the MGA is described with the help of a flowchart along with a discussion on
population sizing, population restart strategy, and the advantages of the MGA.
The proposed MGA technique is validated in Chapter 3 by applying it to certain
composites without any FSS screens. First, the problem geometry is defined, and this is
followed by the formulation for the reflection coefficient of inhomogeneous-layered
media. Next, the details of the application of the MGA to design high-pass, low-pass,
band-pass, and band-stop filters [19] are discussed. Finally, the MGA is applied to design
three types of broadband microwave absorbers, viz., composites comprised of either
dielectric or magnetic materials only, or a combination of dielectric and magnetic
materials. This is followed by a detailed discussion and interpretation of the results.
Chapter 4 deals with the synthesis of composites comprising a single FSS screen
embedded in dielectric or magnetic materials, which may be lossy or lossless. First, the
problem geometry is defined, followed by the formulation for the spectral domain MoM
to evaluate the induced electric currents on the metalized portions of the FSS screen and
the reflection coefficient o f the FSS/dielectric combination. Next, the details of the
application of the MGA to the design of spatial filters and broadband microwave
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absorbers are discussed followed by an interpretation o f the results. Finally, the domain
decomposition approach is described for geometries that require the problem to be
partitioned into two or more domains to maintain numerical efficiency.
Chapter S deals with the synthesis of composites comprising multiple FSS screens
embedded in lossy/lossless dielectric/magnetic materials. First, the problem geometry is
defined, followed by the formulation and limitations of the Generalized Scattering Matrix
technique, which extends the solution from single to multiple screens. Finally, the details
of the application of the MGA to the design of spatial filters and broadband microwave
absorbers are discussed, followed by an interpretation of the results generated by the
MGA.
In Chapter 6 a parallel MGA is implemented to decrease the computational
expense o f the serial implementation. First the reasons for parallelization are outlined.
Next, the Micro-grain MGA model is discussed and a psuedocode is presented. Finally,
the computational expenses involved in the serial and parallel implementations of the
MGA are compared by considering the problem of designing a polarization selective
surface (PSS) for dual reflector applications.
In Chapter 7, specific composites designed by the MGA in Chapters 4 and 5 are
selected, and the results are validated by using the Finite Difference Time Domain
(FDTD) technique and the direct Method of Moments (MoM). First, the limitations of the
MoM and FDTD technique for a single FSS screen are discussed, and this is followed by
an identification of drawbacks on using the scattering matrix approach vis-a-vis the MoM
applied to multiple FSS screens.
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Chapter 8 presents a summary and conclusions of the research and suggests
future avenues o f investigation.
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Chapter 2
THE OPTIMIZATION ALGORITHM
2.1 Genetic Algorithm Fundamentals
Genetic Algorithms (GA) fall under a special category of optimization techniques
that are robust stochastic search methods, loosely modeled on Darwin’s principles of
natural selection and survival of the fittest. The GA - as an optimizer - is useful for
solving complex combinatorial problems. It is particularly effective in searching for
global extrema in a multi-dimensional and multi-modal functional domain. The GA
excels when the problem can be cast in a combinatorial form. It simultaneously processes
a population of points in the optimization space, and uses stochastic operators to
transition from one generation of points to the next, resulting in a decreased probability
of them being trapped in local extremas.
2.1.1 GA Terminology
Several definitions, borrowed from the descriptions in the natural world are
frequently utilized in the literature on GA optimization, and are also used throughout the
thesis. Some o f these words are defined here for the sake o f clarity, and are further
embellished during the course of this thesis in subsequent chapters.
Allele. A single binary bit that can be either a zero or a one.
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10
Gene: A sequence o f bits or alleles. This represents an encoded parameter in the
optimization space. The number of parameters to be optimized determines the
dimensionality of the optimization problem.
Chromosome: Also referred to as an individual, is a sequence o f genes/parameters. Each
individual represent a potential design/solution.
Cardinality: Number of possibilities in a single allele, e.g., for a binary bit (allele) the
cardinality is 2, as it can assume two values, either 0 or 1.
Population: A group of individuals or chromosomes that the GA utilizes to search for the
optimum solution.
Generation: Each GA iteration is called a generation. The application o f GA operators
continues until a new generation replaces the original generation.
Parents: The individuals that are selected for reproduction to create offspring/children are
called parents. The selection process can be either deterministic or probabilistic.
Children: Simple stochastic operators are applied to the selected parents to generate
individuals that are termed “children”. They constitute the population of the next
generation and replace the current generation.
Fitness: The function that defines the optimization goal is called the fitness or objective
function. The objective function is applied to each decoded individual to generate a real
number, which provides a measure of the “goodness” of the trial solution. This is the only
link between the physical problem and the GA optimization process.
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2.1.2 Overview of Genetic Algorithms
Numerical search techniques can be divided into two broad groups, viz., local and
global. Conjugate Gradient (CG) methods can be considered as good candidates for the
former technique, while the GAs fall under the latter category. The local and global
techniques can be distinguished from each other by the fact that the former produce
results that are highly dependent on the initial guesses, while the latter are largely
independent of the starting points. Local techniques are tightly coupled to the solution
domain, resulting in fast convergence to local extremums. Furthermore, the tight coupling
to the solution space also places constraints, such as differentiability and continuity in the
solution domain, which are difficult to handle.
In contrast to the local techniques, the global ones are loosely coupled to the
solution domain and place very few constraints on it. This means that these techniques
are better equipped to deal with solution spaces that have constrained parameters,
discontinuities, large number of dimensions and large number of potential local extrema.
The GA-based combinatorial optimization technique offers several advantages over
existing approaches: (i) it succeeds in designing broadband microwave absorbers and
spatial filters consisting o f only a few layers, and, therefore, almost always leads to a
physically-realizable structure; (ii) it is considerably simpler to implement than gradientbased search procedures [20-21],
The binary GA operates on a coding of the parameters to be optimized. The coded
representation o f the problem geometry consists of sequence of bits or alleles that contain
information regarding each parameter, represented by a string of bits called the gene,
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12
whose length is determined by the allowed range of real values and the discretization step
to be implemented. Potential design geometries can be represented by the concatenation
of all the genes in the design space and are referred to as chromosomes or individuals.
Each individual represents a separate design, and a group of them constitutes a GA
population.
The GA, which optimally chooses each parameter, is an iterative optimization
procedure. It starts with a randomly selected population of potential solutions, and
gradually evolves towards improved solutions via the application of the genetic operators.
These genetic operators mimic the processes o f procreation in nature. The GA begins
with a large population Po, comprising of an aggregate of sequences, with each sequence
consisting o f a randomly selected string o f bits. It then proceeds to iteratively generate a
new population P,~i, derived from P„ by the application of selection, crossover, and/or
mutation operators.
2.1.3 GA Operators
The GA utilizes three stochastic operators to transition from one generation of
points to the next, viz., selection, crossover and mutation. The first two are the primary
operators, while the last one is considered a secondary operator.
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2.1.3.1 Selection Operator
For successful procreation, individuals belonging to a particular generation need
to be selected from the population base to act as parents and produce children that
become members o f the next generation. The process of selecting particular individuals
while rejecting others is called selection. Different schemes have been proposed to carry
out this process. Popular among them are population decimation, roulette wheel selection
and tournament selection. These processes are discussed further in later sections.
2.1.3.2 Crossover Operator
Crossover is a primary operator that creates offspring from selected parents
Several methods have been proposed to generate offspring from parents, viz., single
point, multi-point, and uniform crossover [20], The effect of crossover is to produce fitter
individuals with better genetic make-up than the parent individuals. The commonly used
schemes for crossover are discussed in a later section.
2.1.3.3 Mutation Operator
Mutation is a secondary operator that aids the GA in exploring possible solution
spaces that are not included within the genetic makeup of the present generation. In the
case o f binary coded GAs, the mutation operator randomly changes a bit from “ 1” to " 0 ”,
to prevent the GA from converging prematurely to local maxima. The probability of
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14
mutation is usually kept very low to ensure that it does not prohibit the beneficial actions
of selection and crossover.
2.1.4 Selection Schemes
With research on GAs moving at a rapid pace, several schemes have been
proposed for performing the selection. Three popular schemes are discussed below.
2.1.4.1 Population Decimation
In this selection scheme, individuals are ranked according to their fitness values.
A cutoff for fitness is decided either arbitrarily or by tailoring it to the problem, and all
individuals satisfying the cutoff requirements are considered parents. Crossover and
mutation are applied repeatedly to the selected parents until the population quota is filled
for the next generation. This is a deterministic selection scheme with simplicity as its
advantage. However, the disadvantage of this scheme is that important genetic traits of an
individual are lost forever due to the decimation of individuals, resulting in a loss of
diversity, which occurs long before the beneficial properties of the decimated individual
are recognized by the evolutionary process. The unique characteristics of a decimated
individual can be reintroduced into the evolutionary process by using mutation, which
adds new genetic materials, and assists the GA to explore portions of search space that
otherwise it would not have. However, this is a very inefficient method for adding a
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15
specific genetic material, and the detrimental effects of premature losses o f beneficial
characteristics has led to the development of stochastic selection strategies.
2.1.4.2 Roulette Wheel Selection
This scheme is also called proportionate selection. It is a stochastic scheme and
works on the principle of generation replacement. For a particular generation G;, (j =
I
Ngen), where Mgen is the number of generations, the probability of selection for each
individual is calculated as follows [20 ]:
a „
£ / ( parent,)
i=i
where / ( parent,) is the fitness of the i,h parent in the population base and N pop is the
number o f individuals in a generation. From (2.1) it is evident that the probability of
selection is dependent on the normalized fitness of each individual. The larger the value
of pseiect, the higher the probability of an individual being selected as a parent for
reproduction. The obvious distinction between population decimation and Roulette
Wheel selection is that the latter provides a finite probability for currently unfit
individuals to be selected for the reproduction phase; hence it maintains the diversity in
the genetic makeup. Proportionate selection is prone to stochastic errors for small
population sizes. The drawbacks of this method are as follows: (i) problems involving
large run-times for single function evaluation require enormous computational resources
if a large population base is selected; (ii) highly fit individuals tend to dominate the
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16
population base at the initial stage of optimization; (iii) at the final stages of the
optimization the scheme is unable to differentiate between best individuals, leading to
convergence to local extremums; (iv) the fitness value has to be maintained positive.
The Roulette Wheel selection scheme can be implemented in three steps. First, a
random number is generated between 0 and 1. Next, the normalized fitness value of each
individual is calculated by using (2.1). Finally, a loop aggregates the normalized fitness
values until the sum exceeds the value of the random number selected in step 1. At this
point, the loop counter variable gives the number of the selected individual. This process
is repeated until the required number of individuals is selected.
2.I.4.3 Tournament Selection
The most effective o f the three schemes for a wide range of applications is
tournament selection [22]. In this scheme, a sub-population of M individuals is randomly
selected from the population base. Next, these individuals are made to compete against
each other on the basis o f their fitness. Finally, the individual with the best fitness is
judged to be the winner of the tournament, and is selected as a parent. The other M-I
individuals are placed back into the general population, and this process is repeated until
the population base to be used for crossover is filled. The number M can be arbitrarily
chosen. In this thesis, binary tournament selection is applied with M = 2.
First, a random number is generated between 1 and the number of individuals
constituting the population base. This number indicates the individual to participate in the
tournament. For a tournament size of M, this process is repeated M times until the sub­
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17
population of M individuals is filled. Next, a simple comparison sort of individual fitness
filters out the best individual, which is selected as a parent. Finally, the remaining M -l
individuals are placed back in the original population. The process is repeated until the
tournament sub-population is filled. This scheme does not rule out the possibility of
multiple pairing, the probability increasing with convergence to an optimal solution.
Tournament selection shows a better convergence at the initial stages o f the optimization,
and has a faster execution time than proportionate selection. The execution time of
tournament selection is O(n), as compared to 0(n2) for proportionate selection.
2.1.5 Crossover Schemes
With the selection of parent individuals completed, offspring or children need to
be created to replace the current generation with the next generation. Children are created
from parents by employing crossover with probability / W j- Upon applying crossover,
the genes rearrange with the objective of producing better combinations, thus resulting in
fitter individuals. The probability of crossover depends strongly on the crossover scheme
employed as discussed below.
2.1.5.1 Single-Point Crossover
The single-point crossover scheme was first suggested by Holland [2]. In this
scheme, first a random number p with a value lying between 0 and 1 is generated. As
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18
shown in Fig. 2.1, if p > p Cross, crossover is applied by selecting random locations in both
parents-1 and -2. The alleles to the left of the selected locations in parent-1/parent-2 are
copied into the corresponding positions in child- l/child-2 , respectively, while alleles to
the right of the selected location in parent- 1/parent-2 are copied into the corresponding
positions in child-2/child-l, respectively. If p
< p cr0ss,
then, as shown in Fig. 2.2, child-1
and child-2 are made exact copies of parent-1/parent-2, i.e., crossover is not applied. This
process is repeated for each selected parent individual. For most optimization problems a
value between 0.6 and 0.8 for pcross has been found suitable [2 0 ],
Parent-1
Parent-2
Chi d -1
Chi d -2
Crossover point
Fig. 2 . 1. Single point crossover with p
> p cr0ss.
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19
Parent-1
1
0
Parent-2
1
II 0
1
0
0
1
1
1
1
0
1
1
\
0
Chi «
1
.
I
0
V
I
1/ / 0
A
J
\
J
V
J
W
Chi d-2
0
1
1
Crossover point
Fig. 2.2. Single point crossover with/?
pcross.
2.1.5.2 Dual-Point Crossover
In the dual-point scheme, we have two crossover points instead of a single one.
Similar to the single point crossover, a random number p with a value between 0 and 1 is
generated. As shown in Fig. 2.3, ifp > p cr0ss, the alleles between the two crossover points
of parents-1 and -2 are copied into child-2 and -1, respectively. The remaining alleles are
copied into the corresponding positions from parents- 1 and -2 to child- 1 and -2
respectively. If p < p cr0ss, then as shown in Fig. 2.4, parents-1 and -2 are copied into
child- 1 and -2 , respectively.
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20
Parent-1
Parent-2
C h ild -1
Child-2
crossover point-2
crossover point- 1
Fig. 2.3. Dual point crossover forp > p c
Parent 2
Parent 1
1
0
0
1
1
0
Chi
1
0
°\
1
>
crossover point- 1
4
0 1 1 0 i
^
/
/
Chi «
X X
*0/
'/
i
crossover point-2
Fig. 2.4. Dual point crossover forp <Pc
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i
i
21
2.1.5.3 Uniform Crossover
The logical extreme o f the dual point crossover is the uniform crossover [23],
First, a random number p with value lying between 0 and 1 is generated. Next, a mask
comprising alleles and having the same chromosome length as the parent individual is
created, the alleles being distributed in accordance with a uniform probability
distribution. Finally, for p > p Cmu, a suitable boolean function, viz., AND, OR, NOR, is
applied to the mask and the parent chromosome to generate child chromosomes as shown
in Fig. 2.5; while, if p
< p cross,
the parent chromosome is copied into the child similar to
single and dual point crossovers. Conventionally, the probability of uniform crossover is
maintained either at 0.5 or 1.0.
parent chromosome
1
0
0
1
1
0
child chromosome
Boolean
Operator
0
1
1
0
1
1
1
1
1
1
0
0
Mask chromosome
with a certain
probability
distribution
Fig. 2.5. Uniform crossover for p > p cr0ss.
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22
2.1.6 Mutation Schemes
The application o f the mutation operator in GAs is secondary in nature. It
provides the means to explore those parts of solution surfaces that otherwise would not
have been searched by introducing a new genetic makeup into the current population. It
also prevents the GA from convergence to local extrema. There is one scheme that is
predominantly used for binary-coded GAs and can be implemented as follows: A random
number p is generated, and compared with the stored value of p mut (probability of
mutation). Ifp > p mut, a random allele is selected in the parent chromosome and inverted,
i.e., 0 to 1 or I to 0, while, mutation is not applied if p < p mut. The process is repeated for
all the selected parent individuals. The mutation probability is usually kept low, between
0.01 - 0 . 1, becuase a higher value typically hampers convergence and overshadows the
beneficial effects o f crossover and selection.
2.1.7 Elitism
De Jong [24] introduced the concept of elitism as an extension to the basic GA.
Being stochastic in nature, the GAs do not assure that the best individual in one
generation betters its counterpart in the preceding generation, in the same manner as the
elitist operator, which is an extension of the basic GA. If the best individual in the current
generation has a fitness value lower than its counterpart in the preceding generation, then
we can proceed one of the following three ways: (i) the best individual in the current
generation is replaced with its counterpart in the preceding generation; (ii) the best
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23
individual is passed on from one generation to the next as the (N* J)th individual; and,
(iii) the best individual in the preceding generation replaces the least fit individual in the
current generation. Elitism ensures that a monotonic increase in the individual fitness is
maintained throughout the optimization process.
2.1.8 Fitness Function
Also termed as the objective function, the fitness function is the only link between
the physical problem being optimized and the GA. To utilize the fitness function, each
individual inthepopulation
proportional to the
base is assigned a fitness value, which
istypically
goodness o f the trial solution. Fitness functions can be broadly
categorized as single-objective and multi-objective types. In the former, the contribution
to the value o f fitness comes from one objective function being evaluated; in a multi­
objective function, it is from more than one such function that are being evaluated
simultaneously. The following equations are an example for single- and multi-objective
fitness functions:
fitness = m in(g)
(2 .2 )
fitness = m in(g + a t)
(2.3)
where, g and t are the two objective functions and a is a weight coefficient for the latter.
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24
2.1.9 Classification of Genetic Algorithms
Depending on the type o f optimization, Genetic Algorithms can be broadly
classified into two groups, viz., constrained or unconstrained optimization. Each of these
can be further classified as stationary and non-stationary, depending on the type of
function being optimized. The modality of the function results in a final classification
into unimodal and multimodal functions.
Suppose we have a function
y = f { x ) for <5; < x< 8,
(2.4)
and we need to find the global minimum ofy. We can define an objective function as,
obj = m in ^ ] for Sx < x <
(2.5)
Equation (2.4) is an example o f an unconstrained function optimization. An example for
a constrained optimization function is
obj = minf vl for 8, <x< 8,
r 1
‘
= max [y] for 8Z < x < 8l
(2 6)
where 61 < 82 < 83. In this research proposal, spatial filter design comes under the
category of constrained optimization, while broadband microwave absorber design is
considered as an unconstrained optimization problem.
Stationary functions are well-defined entities that do not change at a rate faster
than the GA can reach a global optimum. The functions dealt with in this work fall into
this category. Contrastingly, the non-stationary functions can be defined as those that
evolve at a rate faster than the GA can reach a global maximum. A typical example of
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25
non-stationary function is pursuit and evasion, i.e., the function to be optimized is
redefined depending on what the evader does.
Finally, the modality classification can be explained with the following examples
[25], A unimodal function is one that has a single maximum, as for instance, the
following:
/ ( x , y ) = (x + y - l l ) : +(x + .y -7 ):
(2.7)
This function is called Himmelblau’s function, the search space is confined to x > 0 andy
< 6 , a single minimum point is at (3,2) with a function value of zero.
In contrast, a multi-modal function may be written as
1 = 10
/ (*|
*lo) = 10O + 2 > r - 10 cos ( 2 ;rx,)
1=1
( 2 .8 )
This iscalledRastrigin’s function in which each variable lies in the range (-6 , 6 ), and has
a globalminimum
at x, = 0, where the function value also equals tozero.
All the
functions optimized in this work are categorized as massively multi-modal functions.
2.2 The Conventional Genetic Algorithm (CGA)
The Conventional Genetic Algorithm (CGA), also referred to as the Simple
Genetic Algorithm (SGA), is the first and simplest of GAs proposed by Holland [2]. It
utilizes a serially implemented binary-coded GA with three basic operators, viz.,
selection, crossover and mutation. The CGA is illustrated via a flowchart in Fig. 2.6 [20],
First, a random set of individuals that constitutes the initial population is created. Then
the selection operator is applied to generate a sub-population o f individuals that act as
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parents. Next, the crossover and the mutation operator are repeatedly applied with
probability p cross and p mut respectively, to pairs of individuals in the sub-population to
generate offspring constituting the population base o f the next generation. This process is
repeated until convergence is reached. Finally, the termination criterion is applied, which
is arbitrary and mostly problem-specific for a GA. Any of the schemes described in the
earlier sections can be applied for selection, crossover and mutation. The choice of
parameters for the CGA is usually based on studies by De Jong [24] and Grefenstette
[26], The termination criterion can be a fixed number of generations, or convergence of
the fitness function to a predetermined value.
2.2.1 Population Sizing for the CGA
The question of what is an appropriate population size needs to be addressed
before any GA calculations can be run. For CGAs, sizing the population is problemspecific and a strong function of the length of the chromosome and the cardinality of the
allele. An estimate of the population size for the CGA is made using the following
equation [27-28]:
n pop
where
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(2.9)
27
Initialize Population
Evaluate Fitness
Selection
r
i
Until
Crossover
^
T
A
population^
full
Replace Population
Evaluate Fitness
n
Termination
criteria met
Yes
Fig. 2.6. Procedural flowchart of the CGA.
/ = length of the chromosome or individual
k=
1 V"
V G, ; Npar = number of parameters or genes; G, = length of the ilh gene
Npar tr
X = number of possibilities for each allele or bit = 2 for binary coding
2.2.2 Drawbacks of the CGA
Equation (2.9) leads us to conclude that the sizing of population for the CGA is
dependent on the length of the chromosomes, the number of parameters or genes that
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28
determine the dimensionality of the optimization problem and the type of encoding
employed. The number o f alleles in each gene is dependent on the range of parameters
and the implemented step-size. A multi-dimensional search space leads to a large number
of parameters to be optimized by the CGA, and results in an increase in the chromosome
size for each individual. Utilizing (2.9) and estimating the population size under the
conditions mentioned above leads to a large population base, which requires several
generations to achieve convergence, and hence places a considerable burden on
computational time and resources.
To reduce the computational burden, population sizes less than that estimated can
be employed by the CGA. However, this results in a premature convergence to nonoptimal results due to insufficient information processing by the CGA.
2.3 Micro-Genetic Algorithm (MGA)
The drawbacks o f the CGA in handling computationally intensive problems
prompted the search for a small-population-based GA, coupled with efficient
convergence properties. The ideal candidate is the Micro-Genetic Algorithm (MGA) [13],
which is a non-mutation-based GA, i.e., it does not employ the mutation operator, instead
uses a population restart strategy to avoid local extremas. It also makes clever use of the
elitist operator to ensure the migration of the best individual from one generation to the
next. The MGA is a variant o f the SGA, wherein a small population size is utilized for
the optimization procedure.
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29
The poor performance o f GAs for small population sizes can be attributed to
insufficient information base and convergence to local extremums. As the population size
increases, so does the number of function evaluations. In real-world problems, more often
than not, the computation time for a single function evaluation is excessive, and this
necessitates the use o f enormous computational resources to complete the optimization.
The need to minimize the number of function evaluations while maintaining the global
optimality o f the GA has led to the MGA as one possible solution. Outlined below is a
scheme proposed for GAs with small population sizes [28]:
1. Randomly generate a small population.
2. Perform genetic operations until nominal convergence is reached (as
measured by bit wise convergence or some other reasonable measure).
3. Generate a new population by transferring the best individuals of the
converged population to the new population and then generating the
remaining individuals randomly.
4. Go to step-2 and repeat.
2.3.1 The MGA Optimization Procedure
Based on the scheme outlined above, the MGA procedure utilized in this thesis is
presented below and is illustrated by the flowchart shown in Fig. 2.7.
1. Randomly generate N individuals to constitute the first generation. A random
number generator with uniform probability distribution is utilized.
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30
2. Evaluate the fitness of each individual by applying the problem-specific
objective function. Select the individual with the best fitness value and place it
as the N01 individual in the next generation. Next, generate the remaining N-1
individuals for the next generation.
3. To select the individuals constituting the mating pool, binary tournament
selection is applied as explained in Sec. 2.1.4.3. Hence, (N-l) 2 individuals
with better fitness values are selected as parents and are passed on to the next
generation.
4.
In step-3, prior to copying the parents selected to the next generation, uniform
crossover (Sec. 2.1.5.3), is applied to generate one offspring from each
selected parent with pcross = 0.5. This generates the remaining (N-l) 2
individuals required to fill the next generation.
5.
The process is repeated for each generation until convergence is reached.
Convergence is achieved when every individual in a particular generation
differs by less than 5% - on a bit-by-bit comparison - with the best individual
in that generation.
6. Once convergence is achieved, the best individual is replicated and passed to
the next generation. The remaining N -l individuals are generated randomly.
This is the start-restart or population restart strategy employed by the MGA.
7. Repeat process from step-2 until termination criteria are met.
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31
The key factor that contributes to the ability of the MGA to solve a
computationally intensive problem efficiently is its employment of a small population
base, while the population restart strategy ensures that the algorithm does not get trapped
in local extrema.
2.3.2 Advantages of the MGA
In this thesis the MGA is employed to solve a computationally intensive design
problem. In the past, CGAs have met with minimal success in solving such problems at
least within a practical time frame. A design problem is typically categorized as
computationally intensive when a single function evaluation takes an appreciable amount
of computation time. Some o f the design problems dealt with in this proposal involve the
analysis o f scattering from FSS screens, which is achieved by using the Method of
Moments (MoM). The main contributors to the large computation time for a single
function evaluation are the matrix fill time and inversion, which are dependent on the
complexity of the FSS screen design. The number of function evaluations in a single
generation depends on the population base used for the optimization.
To attain a high numerical efficiency, we need to work with a small population
base and this is provided by the MGA, which also has good numerical convergence
properties. Furthermore, the MGA reaches near-optimal regions quicker than the largepopulation-based CGAs. The general choice of population size for CGAs can range
between 100 and 10000, compared to the MGAs, which typically utilize a population size
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32
ranging between 20 and 50. It is easily estimated [33] that even worst-case considerations
lead to a decrease by 50% in computational run time.
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33
Initialize Population
Selection
Evaluate Fitness
*
Select best
individual
(Elitism)
Crossover
Evaluate
Fitness
Evaluate
Fitness
Until temporary
population is full
Restart
population
Select best
individual
(Elitism)
Replace Population
Check
convergence
Termination
criteria met?
Select best
individual
(Elitism)
Fig 2.7. Procedural flowchart of the MGA.
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Chapter 3
SYNTHESIS OF COMPOSITES COMPRISING ONLY DIELECTRIC AND
MAGNETIC MATERIALS
Equipped with the Micro-Genetic Algorithm (MGA), which was described in the
previous chapter, we proceed now to apply it to a variety of synthesis problems in this
chapter. For instance, we consider multilayer composites consisting of dielectric,
magnetic or combinations of interspersed dielectric and magnetic materials. First, we
synthesize spatial filters by using lossless dielectrics to obtain specified frequency
responses. Next, we study the effect of combining dielectric and/or magnetic materials,
which may be lossy or lossless, to design broadband microwave absorbers. Finally, we
synthesize a broadband absorber consisting of lossy carbon fiber and lossless dielectric
materials.
3.1 Problem Geometries
Figure 3.1 shows a /V-layered composite structure whose parameters we wish to
optimize with a view to realizing a specified frequency response for spatial filters, while
Fig. 3.2 shows the same structure except for a Perfect Electric Conductor (PEC) backing
that must be included in the synthesis o f broadband microwave absorbers. We assume a
plane wave to be incident on the composite at any arbitrary angle and polarized as either
Transverse Electric (TE) or Transverse Magnetic (TM).
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35
S \, d \
Fig. 3.1. Synthesis o f spatial filters with lossless dielectrics. Nlayered composite with a TE polarized uniform plane wave
incident at an arbitrary angle. For TM polarization the E and H
fields are interchanged along with a phase shift of 180 degrees
introduced in the H field.
Given the range of allowed values for the constitutive parameters, in this
case s ' (real part of the relative permittivity), the minimum thickness of each layer and the
total allowed thickness o f the composite, the MGA synthesizes a AMayer spatial filter that
has the desired frequency response for a given incident angle and polarization.
To synthesize broadband microwave absorbers the MGA optimizes the thickness
of each layer, and the type of material in the layer, which may be either lossy dielectric,
lossy magnetic or lossless dielectric, such that the composite exhibits a low reflection
coefficient for a prescribed set of frequencies and incident angles simultaneously for both
TE and TM polarizations. The MGA accesses the values of the frequency
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36
dependents’ (imaginary part o f the relative permittivity) a n d ( i m a g i n a r y part of the
relative permeability) of lossy dielectric and magnetic materials via a lookup table.
£\,
UNt d \
----------- ^► PEC backing
Fig. 3 .2. Synthesis of broadband microwave absorbers with lossy
or lossless dielectrics and lossy magnetic materials. yV-layered
composite with a TE polarized uniform plane wave incident at an
arbitrary angle. For TM polarization the E and H fields are
interchanged along with a phase shift of ISO degrees introduced
in the H field.
3.2 Formulation of Reflection and Transmission Coefficients for Inhomogeneous
Layered Media
Consider an inhomogeneous, A'-layered medium, as shown in Figs. 3.1 and 3.2
with the material inhomogeneity along the z-direction. To ease the effort of developing an
expression for the reflection and transmission coefficients of the AMayered media a
simpler three layered media as shown in Fig. 3.3 is considered. Due to the recursive
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nature of the expressions to be developed, the formulation can be easily extended from
the J-layered to ^-layered media [12]. Alternate formulations for the reflection and
transmission coefficients for an AMayered inhomogeneous media [29-30] can also be
employed.
Region-1
Region-2
Z = -d 2
Region-3
Fig. 3.3. Three-layered medium with TE
polarized wave incident at an arbitrary angle.
Considering Fig. 3.3, the Ey component in Region-1 can be expressed as the sum
of a downward- and upward-going wave as follows:
(31)
The upward-going wave has contributions from both the wave reflected from the first and
the second layers, and transmitted through the first layer. In (3.1) R is referred to as the
generalized reflection coefficient, which includes the effect of reflections from the first
interface and the subsurface.
Similarly, the fields in the Regions-2 and -3 can be expressed as
(3.2)
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38
E,y = A ,e lk- \
(3.3)
where R 23 in (3.2) is the Fresnel reflection coefficient expressed as
p < N, q < N +1
Mpkq: + V i *
-1
p = N ,q = N + \
k p z = k po cos 0 p i,kn
po
(3.3a)
= G)Ju
\ r * Ps P
for a iV-layered medium, kpz is the wave number along z in layer p, and pp and ep are the
permeability and permittivity, respectively, of the p lh layer. The second equation in (3 .3a)
is used when the composite is terminated with a Perfect Electric Conductor (PEC).
Equation (3.3) has only one term since Region-3 extends to infinity in the -z direction,
and, hence, there is no reflected wave.
The four unknowns Ai, A
A 3, and /^,in (3.1), (3.2), and (3.3) need to be
evaluated to arrive at the expression for the reflection and transmission coefficients for
the three-layered medium. The wave traveling downward in Region-2 is the sum of two
waves; (i) transmitted from Region-1 to Region-2; and, (ii) reflected from Region-1.
Thus, at z = -d], we have the following equality:
A2e,k'-A = Ate:k'A Ti: + R2lA2R2/ - ,k'-A'-,k'-A
(3.4)
where i?,, is the Fresnel reflection coefficient as defined in (3.3a) and 7J, is the Fresnel
transmission coefficient which can be expressed as
7* = ! + /?„,
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(34b)
39
Similarly, the upward traveling wave in Region-1 can be written as a combination of the
wave reflected of Region-1 while incident from Region-1 and the wave transmitted
through Region-1 while incident from Region-2. This can be expressed as follows:
A,R,ze'k'A = Rl2Ale*lA +
(3.5)
From (3.4), A: can be solved for in terms o f A/, yielding
T A
~k'- ld'
,36)
Substituting Aj into (3.5) we arrive at the expression for the generalized reflection
coefficient for the three-layered medium, which includes the effect of subsurface
reflections as well as the reflection from the first interface. It can be expressed as
(3.7)
Rr = R r +
: 11
If a fourth layer is added to the original three then the only change required in
(3.7) will be to change the Fresnel reflection coefficient R?s to the generalized reflection
coefficient incorporating subsurface re fle c tio n s^ . Therefore, for a general ^-layered
medium (3.7) can be extended as follows:
I R
(
R
*
Furthermore, using (3.4b) and the identity Rn = -R v , (3.8) can be simplified to
R,,.=
R, . , + £ ,
i ,n
5
J-------r
dj)
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(3.9)
40
A similar expression can be derived for the TM polarized wave. For the special
case of a PEC backed composite, the reflection from the PEC backing (see (3.3a)) is
taken to be - I , and the recursive formulation given in (3.9) can still be employed to
determine the overall reflection and transmission coefficients of an iV-layered
inhomogeneous composite.
3.3 MGA Formulation for the Spatial Filter and Broadband Microwave Absorber
Problems
The MGA is applied to optimize various parameters pertaining to the two design
problems of interest. Binary tournament selection (see Sec. 2.1.4.3) is employed along
with uniform crossover (see Sec. 2.1.5.3) with a crossover probability of 0.5. Elitism is
utilized along with a population restart strategy that is applied when the bitwise
difference between the best and other individuals/chromosomes in the current generation
is less than 5%.
3.3.1 Spatial Filter Synthesis
For the synthesis o f spatial filters [19] we are given the range of allowable values
of the real part o f permittivity for the lossless dielectric layers, the minimum thickness of
each layer and the total permissible thickness of the composite. The MGA needs to
synthesize a composite incorporating jV layers of dielectric such that it has the desired
frequency response for a given incident angle 0, The design process therefore entails
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41
making choices for the optimal values of e o i each layer and its thickness. For the present
problem, we define the fitness function to be:
j=l
'
' + £;=1a - K
^ f ’
(3I0)
where M and N are the number of frequencies, respectively, for which the transmission
and reflection are maximized, and T is the reflection coefficient evaluated using (3.9). In
(3.10), a, s are the coefficients that can be used to weigh the relative importance of the
filter performance for different frequencies, and
or
is a parameter that controls the trade­
off between the ripple in the stop and pass bands, as well as the rejection ratio. The
relative values o f the weight coefficients can be tailored to yield the desired frequency
response and can be used to trade-off the filter performance in one part of the frequency
spectrum against that in another part. A smaller ripple in the transmission band can be
achieved at a cost of increased reflection, by assigning larger weight coefficients to the
sampling points in the transmission band, and vice versa.
The number of sampling points also plays an important role in the optimization
procedure. Especially near the edges of the frequency band of interest, it may be
preferable to choose several points with moderate weight coefficients, rather than a few
points with large coefficients. This could result in a design that matches the desired
characteristic at the specified frequency points perfectly, but has a large ripple in the
neighborhood o f the transition. Specification of the desired response at more points in the
transition region leads to designs with smaller ripples. However, this increases the
computational cost associated with a single function evaluation.
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42
The MGA operates on a chromosome defined in (3.14), which represents a
combination o f the parameters in a coded form. The coded representation of the coating
consists of a sequence of bits that contain information regarding each of the parameters.
Given the allowed range o f values and the choice of step size to be implemented for the
real part of permittivity o f the layers in the composite, the number of possibilities for a
single layer, (NP), can be represented as follows:
(f,.)
-(fr)
r* v l/mui
(£r‘
(3.11)
The material choice for layer / is represented by a sequence M, composed of y bits, as
shown below:
2s <(NP) <2r
(312)
M , = mlm :.......... m;
The higher power o f 2, i.e., y determines the number of bits in the parameter.
The thickness o f the ith layer can be encoded by applying (3 .11) with er replaced
with d, and represented by a sequence D, consisting of /? bits as
Dt = d)d;.......... d f
(3.13)
The entire Allayer composite can be represented by a sequence C, which
represents the ilh chromosome, as given below:
C, = (A/,A/;
M SL ,L
l s. )_
(3.14)
Each quantity in (3.14) is defined in (3.12) and (3.13) and each sequence C, consists of
(y-P) bits.
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43
3.3.2 Numerical Results for Spatial Filter Design
The MGA has been successfully applied to the design of various filters in the
frequency range o f 8.0 - 18.0 GHz. The number of dielectric layers was fixed at five and
is surrounded by air, though the algorithm is generally applicable to any number of layers
and to any type of termination (e.g., PEC backing), as will be explained in the next
section when we deal with broadband microwave absorbers. The angle of incidence is 45
degrees and the optimization is performed only for the TE polarized incident wave. Four
basic types of filter design, viz., low-pass, high-pass, band-pass and band-stop are
attempted. The parameter a employed in (3.10) is chosen to equal 1.0 for all the cases
investigated. Table 3-1 shows the GA parameter search space for this particular problem.
The reflection coefficient in dB vs. frequency for both TE and TM polarizations is plotted
for the synthesized filter in Figs. 3.4 through 3.7. The reflection coefficents for the TMpolarized wave are obtained by using the MGA optimized parameters for the TEpolarized wave in (3 .9). The cutoff frequencies for these filters can be easily identified in
the reflection coefficient plots. While only lossless dielectrics have been considered in all
the cases, the method itself can easily be used to handle lossy dielectrics with both
electric and magnetic losses, as is shown in the next section.
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44
TABLE 3-1 GENETIC ALGORITHM PARAMETER SEARCH SPACE
Parameters
1st layer thickness (cm)
2nd layer thickness (cm)
3rd layer thickness (cm)
4th layer thickness (cm)
5th layer thickness (cm)
er of 1st layer
sr of 2nd layer
er of3rd layer
sr of 4th layer
er of 5th layer
Range of
Increment
parameters
0.35-3.0
0.01
0.35-3.0
.0.01
0.35-3.0
0.01
0.35-3.0
0.01
0.35-3.0
0.01
1.03-10.0
0.1
1.03-10.0
0.1
1.03-10.0
0.1
1.03-10.0
0.1
0.1
1.03-10.0
Number of
possibilities
512
512
512
512
512
128
128
128
128
128
Number of
binary digits
9
9
9
9
9
7
7
7
7
7
Total# of bits
80
3.3.3 Synthesis of Broadband Microwave Absorbers
The MGA is employed to optimize various parameters of the composite, viz., the
thickness of each layer, and the type of material in the layer, which may be either a lossy
dielectric, lossy magnetic or lossless dielectric. We assume that we are provided with a
set of lossy dielectric materials Em with frequency dependent permittivities s,(f), (i=
I
E ^ , as well as with a set of lossy magnetic materials M m, with frequency
dependent permittivities e,(f), (i= I
(i = 1
Mm), and frequency dependent permeabilities
M m).
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45
-10
<D-15
TM
TE
£T* 1 .4 S ,.d ^ A » .
-35
-40
Frequency in GHz
Fig. 3.4. Low pass filter design. The weights used for the MGA were
1.3 for 12.2 to 18.0 GHz and 1.0 for the rest of the band. The desired
cutoff is at 13.0 GHz.
TM
TE
E -15
. Ef5?
'd *r L 5 .' • ‘ •
C -25
-30
-35
-40
Frequency in GHz
Fig. 3.5. High pass filter design. The weights used for the MGA were
1.3 for 14.0 to 18.0 GHz and 1.0 for the rest of the band. The desired
cutoff is at 14.0 GHz.
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TM
TE
-10
-20
-25
-35
-40
-45
Frequency in GHz
Fig. 3.6. Band stop filter design. The weights used for the MGA
were 1.5 for 1.0 to 11.0 GHz and 15.0 to 18.0 GHz, and 1.0 for the
rest o f the band. The desired cutoffs are at 10.0 and 15 .0 GHz.
TE
TM
00
•o
c
§ -10
o
£u
o
/
W -15
c
o
o
a
^ *20
00
-25
a
9
10
11
12
13
14
15
16
17
18
Frequency in GHz
Fig. 3.7. Band pass filter design. The weights used for the MGA
were 1.3 for 1.0 to 11.0 GHz and 15.0 to 18.0 GHz, and 1.0 for the
rest o f the band. The desired cutoffs are at 10.0 and 15.0 GHz.
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47
For the composite shown in Fig. 3.2, the MGA determines the following: (i) type
o f material for the N layers; (ii) material parameters of N layers; and, (iii) thickness of the
N layers such that the composite exhibits a low reflection coefficient for a prescribed set
of frequencies f (/ = I
Fj) and incident angles 0, (i = I
Ad, simultaneously
for both the TE and TM polarizations. In the context of the present problem, the
magnitude o f the largest reflection coefficient is minimized for a set of angles, for both
the TE and TM polarizations, and for a selected band of frequencies. Hence the fitness
function can be written as [31]
F {m ^d u
where
ms ,d s ) = -m ax jr*1*™ (0,, / , )J
(3.15)
miand dt are the material and thickness parameter of the ith layer
in the
TE.
,
,
composite, respectively, and T n{ (0,, f s ) is the reflection coefficient. Thelatter is a
function o f polarization, incident angle and frequency, and is computed using (3 .9).
The MGA operates on a chromosome, as defined in (3.19), which represents a
combination o f the parameters in a coded form. The coded representation of the coating
consists o f a sequence of bits that contain information regarding each of the parameters.
Given a database containing Lm= 2 ^ different materials, the material choice for layer / is
represented by a sequence M, of
bits as follows:
M t - m\mf............................(3.16)
The thickness o f the iih layer can be encoded by a sequence D, o f
bits and represented
as shown below:
D ,= d]d;................................................................(3.17)
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48
Additional bits are needed to aid the MGA in selecting lossy dielectric layers,
lossy magnetic layers and lossless dielectric layers. A two-bit sequence is needed to
represent the three choices. Hence, the type of material for the ilh layer can be encoded by
a sequence (CL), made up of two bits and represented as
(.CL\=(cl)](cl );
(3.18)
The entire AMayer composite can be represented by the sequence C,, which is the
ith chromosome or individual, given below:
C, ={(A/,A/: ..... M SL ,L
L ,( C L \
(CZ)v)}_
(3.19)
Each quantity in (3.19) has been defined in (3.16) through (3.18) and each sequence in
(3.19) consists of (2~Lmb~Ldb) bits.
3.3.4 Numerical Results for Synthesis of Broadband Microwave Absorbers
The reflection coefficient evaluation procedure given in Sec. 3.2 is used in
conjunction with the MGA algorithm described in Sec. 3.3 to synthesize broadband
microwave absorbers in the frequency range of 19.0-36.0 GHz, with the number of layers
fixed at four. These layers are surrounded by air on one side and terminated on the other
side by a PEC backing as shown in Fig. 3.2. The measured values of s ’ and s ’ of ten
lossy dielectric materials and s r , s ’ ,
and n ’ of six lossy magnetic materials are
considered and a database of these values as a function of frequency is created. Figures
3.8 through 3.11, plot the parameters of the lossy magnetic material. The real and
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49
imaginary parts o f the permeability and permittivity of the lossy dielectric material are
not plotted because, unlike the lossy magnetic materials, they are relatively slowly
varying functions of frequency, and they are catalogued in Tables 3-2 and 3-3 instead.
Linear interpolation is employed to generate the values of s 'r and e" as functions
o f frequency for odd-numbered material types, while the even-numbered material types
are averaged. For example, values of the “Type 2” material at 19.0 GHz is obtained by
averaging the material values of “Type 1” and “Type 3” at 19.0 GHz, and the same
procedure is repeated for the lossy magnetic materials. Thus, in effect, we use 19 types of
lossy dielectric materials and 11 types of lossy magnetic ones. Small values of losses are
added to the layers designated as lossless, because it is not practically feasible to fabricate
a perfectly lossless dielectric. Hence, the e ’ value of the first layer is usually 0.01 if it is
tagged as lossless by the MGA, and fixed at 0.1 for the rest of the lossless layers. The
MGA simultaneously optimizes the absorber for elevation angles varying from 0 to 60
degrees (increment of 15 degrees) for both TE and TM polarizations over the frequency
band ranging from 19.0 to 36.0 GHz. The population size and the number of generations
for all the three cases are 50 and 1000 respectively. The following material combinations
are investigated.
(i)
Case-1: lossless or lossy dielectric
(ii)
Case-2: lossless dielectric or lossy magnetic
(iii)
Case-3: lossless or lossy dielectric, or lossy magnetic.
The parameter search space for the MGA is illustrated in Table 3-4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
The MGA-optimized composites and their frequency responses are shown in Figs.
3.12 through 3.14 for the three cases, respectively. The results can be explained as
follows: over the band of frequencies, the range of elevation angles of interest, and for
both the TE and TM polarizations, the maximum and the minimum values of the
reflection coefficients correspond to the worst and the best results expressed in dB.
Mathematically, this can be expressed as
, T ™ J (in dB) (worst case)
(in dB) (best case)
(3.20)
(3.21)
where Ff and A$ give the number of frequencies and elevation angles over which the
MGA optimization is carried out. We observe that, for Cases-1 and -2, the MGA fails to
maintain the largest reflection coefficient below —15 .0 dB over a broad range of elevation
angles, whereas it succeeds in maintaining the worst-case reflection coefficient below -15
dB for Case-3. Thus Case-3 provides an improvement of -2.6 dB over Case-1, and -5.0
dB over Case-2. Table 3-5 lists the types of material selected by the MGA for each case.
The parameters used in the CGA are: population size = 500; number of
generations = 1000; probability of creep mutations = 0.0153; probability of jump
mutations = 0.002. Uniform crossover is applied along with Elitism and Niching [32],
We note that the population size used in the CGA optimization is ten times larger, than
that employed by the MGA, because the CGA does not converge efficiently for smaller
population sizes, typically handled by the MGA. This results in a ten-fold increase in the
number of function evaluations, and increases the run time appreciably. Table 3-6
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
compares the performance o f the MGA and CGA algorithms. It is seen that the CGA is
not able to reach the best fitness value attained by the MGA. This leads us to deduce an
important conclusion that the CGA needs more individuals in its population base than
does the MGA to converge to an optimal solution. Depending on the chromosome size as
shown in Table 3-4, an estimate of the population size can be made for this particular
problem by using (2.9) in Sec. 2.2.1.
From Table 3-4, we see that, for binary coding we have / = 93, x = 2 and the
average length o f the schema (this is the average length of the genes that make up one
parameter) of interest is (l/12)(10+13+l3+13+2+9+2+9+2+9+2+9) = 8. Using (2.9), we
estimate the population size for this case to be (93/8)(28) = 2976, although we have only
used a population size which is l/6th of the estimated one. This could be one reason for
the poor performance of the CGA.
On the other hand, if we do use the estimated
population size in the CGA, the computational expense to achieve the same results as
derived by the MGA would be enormous. This leads us to conclude that it is preferable to
use the MGA over the CGA, to take advantage of its superior convergence properties and
lower computational expense.
Figures 3.15, 3.16, and 3.17 illustrate the average and best fitness values for the
MGA and the CGA for the three cases. In Fig. 3.15, we note that there are well-defined
undulations in the curve for the average fitness value until we reach the 450th generation
for the MGA, and this is attributable to the population restart strategy (Sec. 3.3). The
random variations in the curve for the average fitness value after the 450th generation is
due to the inability o f the MGA to restart the population when following the 5% criteria.
The same characteristics are observed in Figs. 3.16 and 3.17. Thus, Table 3-6 and the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
curves in Figs. 3.15 through 3-17, leads us to definitely conclude that the convergence of
MGA is superior to that of the CGA, and that the former requires less computational time
as well.
15
—« —
•••■A--
14
aS^B'
a|>
^B 13
m■
m*
typel
type3
typeS
type7
type9
type11
i 12
o
£■11
0
1
CL
75
o
*
*
9
8
18
20
22
24
26
28
30
32
34
36
F re q u e n c y (GHz)
Fig. 3.8. Real part of relative permittivity of magnetic
material.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
- #
-W
■ typel
type3
type5
type7
- - type9
typel 1
6.4
.§*6.2
>
p 6.0
E 5.8
8 . 5,6
O 5.4
t:
5.2
&
!5.0
g 4.8
'
5 ) 4.6
(0
E 4.4
4.2
18
20
22
24
26
28
30
32
34
36
F re q u e n c y (GHz)
Fig. 3.9. Imaginary part of relative permittivity of magnetic
material.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
1-6r
-W —
Real part of p e rm e a b ility
1.5 -
typel
type3
type5
type7
type9
typel 1
1.4 1.3 -
1.2 h
1.1
1.0
0.9
_ l __________ I__________ I__________ I__________ I__________ I__________ I__________ I__________ I__________ L_
18
20
22
24
26
28
30
32
34
36
F re q u e n c y (GHz)
Fig. 3.10. Real part of relative permeability of magnetic material.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
Imaginary
pert of p e rm e a b ility
0.7
#
typel
type3
type5
—
“ • * type7
- W - - type9
typel 1
,...A...
0.6
0.5
0.4
0.3
0.2
’"A-.
0.1
0.0
18
20
22
24
26
28
30
32
34
36
F re q u e n c y (GHz)
Fig. 3.11. Imaginary part of relative permeability of magnetic
material.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
-1Z
I
■ I------------------1
♦
I
T
*~
'
~ 1-----------------1
-----------1 -
t
Reflection coefficient in dB
-14*4'*'......... ......
-1 6 -
+
-16-
+
t
+
+
-2C +
-
♦
+
22-
worst case
best case
4*
'2 18
20
*
1
22
24
26
28
Frequency in GHz
i
30
i
32
34
Fig. 3.12. (a) Worst- and best-case reflection coefficient levels
for the composite with only lossless and lossy dielectric
materials.
Sr = 1.29, Sr = 0.1,
= 1.0, |ir =0.0
d = 3.65 mm
Sr = 4.85, Sr = 1.526, fir = 1.0, M-r =
0.0, d = 1.25 mm (Type 2, lossy
dielectric material)
Sr = 21.37, Sr"= 16.77, Hr' = 1.00, (ir " =
0.0, d = 0.55 mm (Type 18, lossy
dielectric material)
Sr = 1.03, Sr =0.1, Hr = 1.0, |ir =0.0,
d = 0.05 mm
Fig. 3 .12. (b) Design o f composite with only lossy and lossless
dielectric materials
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
w
-1Q
♦
«
t
♦
♦
<f►
..............1►
........... t"
.
♦
+
►
Reflection coefficient in dB
-15♦
<► ♦
H
► ♦
4
--
♦
+
-2C -
+
■b
f
+
4-
-25 ‘
F.............
+
-
+
H-
-3C -
•+ •
•3? T
20
22
24
26
♦
+
+
1
28
1
30
worst case
best case
*
32
»
34
36
Frequency in GHz
Fig. 3.13. (a) Worst- and best-case reflection coefficient levels
for the composite with only lossless dielectrics and lossy
magnetic materials
Er = 1.74, er = 0.1, pr = 1.0, |ir =0.0
d = 2.54 mm
Er = 8.25, Sr = 5.09, |ir = 1.016, Hr =
0.11, d = 0.85 mm (Type 6, lossy
magnetic material)
Er = 9.5 1, Er = 5.35, |ir = 1.007, (ir =
0.25, d = 0.05 mm (Type 10, lossy
magnetic material)
Er = 2.44, Er = 0.1, pr = 1.0, (ir = 0.0,
d = 0.05 mm
Fig. 3.13. (b) Design of composite with only lossy magnetic
and lossless dielectric materials
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
I
-14T
±
-16'
I
I
V
I
I
♦
+
♦
.* ...
♦ t
Reflection coefficient in dB
I
♦ ;
♦ f
* r
* ’
4-
~f....
-18'
.. 4 -
+
-
20
'
+
-
22
'
-24'
♦
+
T"
worst case
best case
-26.........
-28
f
-30'
4
8
20
22
24
26
28
30
32
34
Frequency in GHz
Fig. 3.14. (a) Worst- and best-case reflection coefficient levels
for the composite with interspersed lossy or lossless dielectric
and magnetic materials.
Br
= 1.59, E r = 0.1, | ! r = 1.0, |ir =0.0
d = 2.75 mm
Sr
Er
= 4.85, E r = 1.52, ^lr = 1.0, f i r =
0.0, d = 1.25 mm (Type 2, lossy
dielectric material)
= 24.77, E r = 18.98, ( i r = 1.0, (ir =
0.0, d = 0.35 mm (Type 19, lossy
dielectric material)
8r = 9.0, Er = 6.25, |ir = 1.0, (ir =
0.039, d = 0.27 mm (Type 11, lossy
magnetic material)
Fig. 3.14. (b) Design of composite with both lossy or
lossless magnetic and dielectric materials.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
59
-0.1
-0 .2 ' r
Average fitness MGA
IX
best fitness MGA
Average fitness CGA
best fitness CGA
-0.
Number of Generations
Fig. 3.15. Evolution of the average and best fitness values of the
CGA and MGA for Case-1.
-
0.2
_
Average fitness MGA
' ~ best fitness MGA
Average fitness CGA
hest fitness CGA
0
I
100
I_____ L .. ■■ .1
1
1
200 300 400 500 600 700
Number of Generations
L
»
800
900
1000
Fig. 3.16. Evolution o f the average and best fitness values of the
CGA and MGA for Case-2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
Fitness value
-
0.2
-o.e
-0.7
Average fitness MGA
best fitness MGA
Average fitness CGA
best fitness CGA
-0.9
800
900
Number of Generations
Fig. 3 .17. Evolution of the average and best fitness values of the
CGA and MGA for Case-3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1000
61
TABLE 3-2 e’r AS A FUNCTION OF FREQUENCY FOR THE LOSSY DIELECTRIC
MATERIAL
Material
type
F (GHz)
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
Type 1
Type 3
Type5
Type 7
Type 9
4.48257
4.48299
4.48331
4.48353
4.48367
4.48377
4.48383
4.48388
4.48390
4.48392
4.48393
4.48394
4.48395
4.48395
4.48395
4.48396
4.48396
4.48396
5.21093
5.21152
5.21190
5.21216
5.21232
5.21244
5.21251
5.21256
5.21259
5.21261
5.21262
5.21263
5.21264
5.21265
5.21265
5.21265
5.21265
5.21265
5.83968
5.84028
5.84070
5.84098
5.84117
5.84130
5.84138
5.84143
5.84147
5.84149
5.84151
5.84152
5.84153
5.84153
5.84154
5.84154
5.84154
5.84154
7.08171
7.08190
7.08231
7.08264
7.08288
7.08304
7.08314
7.08321
7.08326
7.08329
7.08331
7.08333
7.08334
7.08334
7.08334
7.08335
7.08335
7.08335
22.41061
22.43457
22.44027
22.44174
22.44219
22.44237
22.44247
22.44252
22.44256
22.44258
22.44260
22.44261
22.44262
22.44262
22.44262
22.44263
22.44263
22.44263
Material
type
Type
ii
Type 13
Type 15
Type 17
Type 19
F (GHz)
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
9.83625
9.83703
9.83769
9.83816
9.83848
9.83870
9.83884
9.83893
9.83899
9.83904
9.83906
9.83908
9.83909
9.83910
9.83911
9.83911
9.83912
9.83912
11.86623
11.86821
11.86921
11.86979
11.87017
11.87041
11.87057
11.87068
11.87075
11.87080
11.87083
11.87085
11.87086
11.87087
11.87088
11.87088
11.87088
11.87088
12.73006
12.73291
12.73412
12.73477
12.73516
12.73541
12.73557
12.73568
12.73575
12.73580
12.73583
12.73585
12.73586
12.73587
12.73588
12.73588
12.73589
12.73589
17.96287
17.96867
17.97069
17.97164
17.97217
17.97249
17.97271
17.97285
17.97295
17.97300
17.97305
17.97308
17.97309
17.97310
17.97311
17.97312
17.97312
17.97312
24.76098
24.76884
24.77163
24.77292
24.77365
24.77411
24.77440
24.77459
24.77473
24.77481
24.77487
24.77491
24.77493
24.77495
24.77497
24.77497
24.77497
24.77497
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
TABLE 3-3 < AS A FUNCTION OF FREQUENCY FOR THE LOSSY DIELECTRIC
MATERIAL
Material
type
F (GHz)
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
Type 1
Type 3
Type 5
Type 7
Type 9
1.84577
1.86013
1.86536
1.86726
1.86794
1.86819
1.86828
1.86831
1.86832
1.86833
1.86833
1.86833
1.86833
1.86833
1.86833
1.86833
1.86833
1.86833
1.18222
1.18308
1.18340
1.18351
1.18355
1.18357
1.18358
1.18358
1.18358
1.18358
1.18358
1.18358
1.18358
1.18358
1.18358
1.18358
1.18358
1.18358
1.65893
1.65913
1.65922
1.65925
1.65927
1.65928
1.65928
1.65928
1.65928
1.65928
1.65928
1.65928
1.65928
1.65928
1.65928
1.65928
1.65928
1.65928
2.31634
2.31605
2.31597
2.31595
2.31594
2.31594
2.31594
2.31594
2.31594
2.31594
2.31594
2.31594
2.31594
2.31595
2.31595
2.31595
2.31595
2.31595
8.03474
8.03710
8.03804
8.03841
8.03856
8.03862
8.03865
8.03866
8.03866
8.03867
8.03867
8.03867
8.03867
8.03867
8.03867
8.03867
8.03867
8.03867
Material
type
Tvpe
11
Tvpe
13
Type
15
Type 17
Type 19
F (GHz)
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
4.98416
4.96560
4.95882
4.95636
4.95547
4.95515
4.95504
4.95500
4.95499
4.95498
4.95498
4.95498
4.95498
4.95498
4.95498
4.95498
4.95498
4.95498
9.74786
9.73054
9.72429
9.72205
9.72125
9.72097
9.72087
9.72084
9.72083
9.72082
9.72082
9.72082
9.72082
9.72082
9.72082
9.72082
9.72082
9.72082
8.17483
8.14305
8.13155
8.12741
8.12593
8.12540
8.12521
8.12515
8.12512
8.12512
8.12511
8.12511
8.12511
8.12511
8.12511
8.12511
8.12511
8.12511
14.62385
14.59117
14.57935
14.57511
14.57359
14.57306
14.57287
14.57281
14.57279
14.57278
14.57277
14.57277
14.57277
14.57277
14.57277
14.57277
14.57277
14.57277
19.05948
19.01088
18.99331
18.98702
18.98477
18.98397
18.98369
18.98359
18.98356
18.98355
18.98354
18.98354
18.98354
18.98354
18.98354
18.98354
18.98354
18.98354
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
TABLE 3-4 GA PARAMETER SEARCH SPACE
Parameters
Total thickness (mm)
1st layer thickness (mm)
2nd layer thickness (mm)
3rd layer thickness (mm)
Choice of type o f material
for 1st layer
er of 1st layer
Choice of type of material
for 2nd layer
er of 2nd layer
Choice o f type of material
for 3rd layer
er of 3rd layer
Choice o f type of material
for 4th layer
er of 4th layer
Number of
possibilities
590
5950
5950
5950
Number of
binary digits
10
13
13
13
1
3
2
0.1
497
9
1
3
2
0.1
497
9
1
3
2
0.1
497
9
1
3
2
0.1
497
9
Range of
Increment
parameters
0.01
0.1-6.0
0.05-6.0
0.001
0.05-6.0
0.001
0.001
0.05-6.0
0-3
1.03-10.0
0-3
1.03-10.0
0-3
1.03-10.0
0-3
1.03-10.0
Total# of bits
93
TABLE 3-5 MATERIAL DISTRIBUTIONS AND THICKNESS (MM) OF
COMPOSITE FOR THE THREE CASES
Total
thickness
Dielectric Layer Types
Case 1
Case 2
Case 3
Layer 1
Layer 2
Layer 3
Layer 4
Lossless
dielectric
d = 3.65
Lossless
dielectric
D = 2.54
Lossy
dielectric
d = 1.25
Lossy
magnetic
D = 0.85
Lossy
dielectric
d = 0.55
Lossy
magnetic
D = 0.05
Lossless
dielectric
d = 0.05
Lossless
dielectric
D = 0.05
Lossless
dielectric
d = 2.75
Lossy
dielectric
d = 1.25
Lossy
dielectric
d = 0.35
Lossy
magnetic
d = 0.27
5.5
3.49
4.62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
TABLE 3-6 PERFORMANCE COMPARISION OF CGA AND MGA
Cases
Case 1
Case 2
Case 3
MGA (populaltion size = 50)
Best fitness
Computation
value reached
time (secs)
after 1000
generations
-0.234
299.0
-0.314
199.0
-0.174
232.0
CGA (population size = 500)
Best fitness
Computation
value reached
time (secs)
after 1000
generations
3312.0
-0.32
2294.0
-0.33
2336.0
-0.297
3.3.5 Numerical Results for the Synthesis of Broadband Microwave Absorbers
Using Lossy Carbon Fiber and Lossless Dielectric Materials
In this section we optimize a composite consisting of either lossy carbon fiber or
lossless dielectric material. Figs. 3 .18 (a) and (b) plot the real and imaginary parts of the
permittivity o f the five (m=lg, 3g, 7g, 9g and 11.25g) carbon fiber materials as a function
of frequency. The MGA optimization is performed in the X-band (8.2-12.4 GHz) and
linear interpolation is utilized for the carbon fiber materials, in a manner similar to that
applied to the lossy dielectric and magnetic materials in the previous sections. The two
cases investigated are as follows:
(i) normal incidence - TE and TM polarization
(ii) 0 varying from 0 to 45 degrees - TE and TM polarization
The population size and the number of generations are fixed at 5 and 1000, respectively,
and the frequency resolution is chosen to be 0.42 GHz. The results of the MGA
optimization o f the reflection coefficient in dB are presented in Figs. 3 .19 and 3 .20. For
both the cases the azimuthal angle is fixed at zero degrees.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
Carbon Fiber Composite
•— • — • — • — • — • — •— • — • — • — •
18
9
10
11
12
13
12
13
Frequency (GHz)
Carbon Fiber Composite
10
11
Frequency (GHz)
Fig. 3.18. The relative permittivity of carbon fiber material as a
function of frequency , (a) real part; (b) imaginary part.
We see from Fig 3.19 that the frequency response obtained by using the MGA
resembles, as expected, the familiar filter response derived by using the Tchebyscheff
equal ripple optimization procedures. For Case-1, the worst-case reflection coefficient is
maintained below -25.0 dB. Figure 3.20 shows the worst- and best-case reflection
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66
coefficients realized by the MGA, while optimizing simultaneously for both the TE and
TM polarizations overthe range of elevation angles of interest (0=0.0 to45.0degrees).
Imposing an additional burden of optimizing over a range of elevation angles, causes the
worst-case reflection coefficient value to degrade further to -16.4 dB. The total
thicknesses o f the composites are maintained at 6.0 and 5.96 mm for the two cases,
respectively.
er’=4.55, er” =0.1, d = l.55 mm
m = 5g, d= 1.45 mm
m = 11.25g, d = 2.25 mm
m = 11,25g, d = 0.75 mm
(a)
-24
-26
-28
S -30
c -32
K -34
-36
-38
8
8.5
9
9.5
10.5
10
Frequency in GHz
11
11.5
12
12.5
(b)
Fig. 3.19. Case-1: (a) composite design; (b) reflection
coefficient vs. frequency.
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67
er’=6.0, er” =0.1, d=1.15 mm
m = 2g, d= 1.45 mm
m = 11.25g, d = 2.85 mm
m = 11.25g, d = 0.51 mm
(a)
-16
-18
Reflection coefficient in dB
-20
-22
-24
-26
-28
-30
worst case
best case
-32
-34
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
Frequency in GHz
(b)
Fig. 3.20. Case-2: (a) composite design; (b) worst- and best-case
reflection coefficient vs. frequency.
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Chapter 4
SYNTHESIS OF COMPOSITES COMPRISING A SINGLE FREQUENCY
SELECTIVE SURFACE (FSS) EMBEDDED IN DIELECTRIC AND/OR
MAGNETIC MEDIA
In Chapter 2 we have successfully demonstrated the enormous advantages of the
MGA over the CGA when applied to problems that are computationally non-intensive in
nature. The optimization problems in Chapter 2 were investigated with the sole purpose
of gaining insight into the way the MGA performs, and in the subtle manner by which the
MGA parameters affect its efficiency and convergence properties. Armed with this
knowledge, we now proceed to the next step, viz., that of tackling a computationally
intensive optimization problem via the application of the MGA. The first problem we
consider is that of synthesizing a broadband microwave absorber comprising a single FSS
screen embedded in dielectric and/or magnetic media-that may be lossy and/or lossless to operate simultaneously for both TE and TM polarizations and for a specified range of
incidence angles [33]. Next, we turn to the problem of designing a spatial filter with a
single FSS screen embedded in lossless dielectric layers that exhibits good performance
for different angles o f incidence and polarizations. Finally, the domain decomposition
method is applied to synthesize a dual-band radome consisting of a single FSS screen
embedded in lossless dielectric medium, and operating at a high off-normal incident
angle [10],
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69
4.1 Problem Geometries
Figure 4.1 shows a AMayered composite structure whose parameters we wish to
optimize with a view to realizing a specified frequency response for spatial filters. Figure
4.2 depicts the same structure except for a Perfect Electric Conductor (PEC) backing,
which is introduced to test the effectiveness of broadband microwave absorbers designed
for shielding. We assume a plane wave is incident upon the composite at an arbitrary
angle and is polarized either Transverse Electric (TE) or Transverse Magnetic (TM)
Sn
Mn
V
?
-
W
z
dn
Lossy or lossless
FSS screen
Fig. 4.1. iV-layered composite with a TE polarized uniform plane wave
incident at an arbitrary angle. For TM polarization the E and H fields
are interchanged along with a phase shift of 180 degrees introduced in
the H field.
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70
PEC
backing
Lossy or lossless
FSS screen
Fig. 4.2. Synthesis of broadband microwave absorbers with a single
FSS screen embedded in lossy or lossless dielectric and magnetic
material with PEC backing. AMayered composite with a TE polarized
uniform plane wave incident at an arbitrary angle. For TM polarization
the E and H fields are interchanged along with a phase shift of 180
degrees introduced in the H field.
To synthesize both spatial filters and broadband microwave absorbers, the MGA
optimizes the thickness and constitutive parameters of the layers in the composite, and
also determines the FSS screen design, its material content, position within the
composite, and the x- and y-periodicities. An upper bound is imposed on the total
thickness o f the composite. As mentioned in Chapter 2, the MGA selects the values of
frequency dependent e r and /ur of lossy dielectric and magnetic materials via lookup
tables shown in Tables 3-2 and 3-3.
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71
4.2 Analysis of Doubly-Periodic, Planar, Infinite FSS Screens Embedded in
Inhomogeneous Layered Media
In this section, we present a quantitative analysis of planar FSS systems [15],
[17,18], [34] to illustrate the computationally intensive nature of the problem. The
geometry to be analyzed is shown in Fig. 4.3. The structure consists o f N stratified planar
layers, infinite in both the x- and y-directions, with arbitrary constitutive parameters and
thicknesses s„, fun, and d, (i=l, 2........ N), respectively, for each layer in the composite,
which also has an infinite, planar, doubly periodic, and infinitesimally thin FSS screen
embedded in it. The assumptions regarding the thickness and infinite nature of the FSS in
the x-y plane has been made to significantly reduce the computational complexity of the
problem. We assume that there are R superstrates and S substrates in the structure, i.e.,
R~S = N, where N denotes the total number of layers in the composite.
FSS Screen
SrR+S
SrR+S-l
SrR+l |
£r2
Sri
Y
MrR+S
MrR+S-1
dR+S
dR+S-l
-z
• • •
fJrRH 1 1 ftrR
■
dR+i |
d,R
• • •
Vr2
d2
d,
1
(a)
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72
▲y
Reference
cell
arbitrary cell
(b)
Fig. 4.3. The composite structure to be analyzed: (a) side view; (b) top
view.
The steps for analyzing the composite structure, shown in Fig. 4.2, are now given.
A uniform plane wave incident on the structure induces currents on the metalized
portions of the screen that reradiate to form the scattered fields. Knowledge of the
currents induced on the screens can be used to calculate the scattered fields. These, in
turn, can be utilized to calculate the reflection and transmission coefficients of the entire
structure. In view of the infinite and periodic nature of the FSS screen, the currents
induced on any one unit cell will be identical to those on any other, with the exception of
a phase progression introduced by the incident plane wave. This is the basic premise of
Floquet’s theorem, which is invoked in the analysis of periodic structures, to
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73
mathematically relate the induced current density J { x , y ) in an arbitrary cell to the
current J 0 (x,_y)in the reference cell (see Fig. 4.3). Utilizing this theorem, we can write
[35]:
£
Y . S ( x - m T x) S ( y - n T y) e ^ z' ^ >]
(4.1)
where, m and n are arbitrary integral multiples of the x- and y-periodicities of the FSS
screen;
= /?0 sin 0 cos^ and p '* = p o sin 0 sin
where /?„ = (OyJ/unstl is the free
space wave number; 0 and ^are the angles in elevation and azimuth respectively; and ®
is the convolution operator. Equation (4.1) implies that the surface current densities need
be calculated only for a single unit cell and this drastically reduces the computational
burden o f the problem at hand. We should mention that all the FSS geometries dealt with
in this work are periodic along a rectangular grid and (4.1) is valid only for this case,
though it can be generalized for a skewed grid [15],
Next, we present the formal derivation employed to obtain the currents induced
on the metallic portions o f the screen and the scattered fields in the entire composite. We
begin by splitting the original problem into two sub-problems (see Fig. 4.4), viz.. the
calculation o f the incident and scattered fields. The former exist in the structure in the
absence o f the metallic screen, while the latter arise from the currents induced on the
screen, and summing the two yields the total fields. The first part of the problem
(Problem-I) evaluates the incident fields in the absence of the metallic screen by
following a procedure similar to that described in Sec. 3.2, while for the second pan
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74
(Problem-II) an integral equation is formulated in terms of the unknown surface currents.
The procedure for deriving this equation is outlined below.
E t,
Z = Zr
Q
Z -!Z r
z
= Zr
t
Et, H t
E ,H
Original problem,
total fields
Problem-I,
incident fields
Problem-II,
scattered fields
Fig. 4.4. The original problem is split into two sub-problems.
Z = ZR
Region-1
Region-0
z = Zr
Region-I
j Region-0
z = Zr
Region-I
Region-0
Js
E=0
0 = 0
Es
Os
pmc
Ms
(a)
lI
i
i
t
t
(b)
(c)
Fig. 4.5. Surface equivalence principle applied to Problem-II: (a)
equivalence principle applied to Region-I; (b) perfect magnetic
conductor (PMC) used to short out the surface magnetic current density;
(c) image theory applied to the Region-I problem.
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75
First, the Surface Equivalence principle [29] is applied to Problem-II by replacing
the metallic conductor with equivalent electric and magnetic current densities as shown in
Fig. 4.5 (a). These currents produce the original fields Es and H s in Region-I and are
expressed as
Js = zxH s
(4.2)
M , = 0 on metal surfaces
. = - z x , otherwise
(4.3)
Next, Region-II is replaced with a Perfect Magnetic Conductor (PMC), which
shorts out M s as shown in Fig 4.5 (b). Finally, to eliminate the presence of the PMC, we
employ the image theory [29] to derive an equivalent problem shown in Fig. 4.5 (c),
leaving only the unknown electric currents radiating into Region-I.The same procedure
is repeated
to evaluate the unknown electric currents radiating
into Region-II.
Superposition o f the currents radiating into Region-I and -II yields the total currents from
which the scattered fields in the whole composite is evaluated.
The scattered field in the media above and below the plane containing the FSS
screen can be expressed [29] in terms of the magnetic and electric vector potentials, A
and F , respectively, as
Es = - j ( o A — ^ - V ( v i ) - - ( V x F )
tofie v
7 sv
7
(4.4)
Hs =-ja)F— ^ - V ( V F ) + - ( V x i)
iofis v
M
(4.5)
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76
For thescatteredfieldwe select the z-component of the vectorpotentials A =zA(x,y,z)
andF = z F(x,y,z). Thesepotentials satisfy the source-free,
scalarHelmholtz
wave
equations
V 2A; +/?2A; =0
(4.6)
V 2FZ+ 0 2F: =O
(4.7)
where /? = coV^s is the wave number in the medium.
The spectral domain approach [36] has been found to be very useful for solving
infinite, planar, periodic geometries. To apply this approach to our problem we define the
Fourier transform pair
(4.8)
/(A . A ) = I J /(*■>’)
=7rV J / f(P „ P r) > A^ ' M dP,Pr
[
27Cj
(4.9)
-cc
Applying the method of separation o f variables, we express the solution of (4.6) and (4.7)
in the spatial domain as Transverse Electric (TE) and Transverse Magnetic (TM) E and
H fields as:
Este
=
r
^
a
e dy
xr
- -
j c tA z . h
(o/ ie dxdz
te
co/us dxcz
H m _ _ cA.
M cy
jj d2A:
d A . . ^TE _
j dutm
1 cA.
(4.10)
’
JV
f \_
1 TV
o)/u£ cydz
o)fi£ cydz
/J. c x
f ^2
(
m
J
C
02
E t J= 0 - E :‘ ‘ = ^ t + p A e -.h ? =0
onus dz
J
eofds c:
£Te _ 1 dF.
s dx
£.ni _
’
TV
Substituting / (x ,y ) as F:(x,y) and Az(x,y) in (4.8) and (4.9), we get
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77
~jPA(P^Py^)
aft (A. A-*)
-jPf:(P*,Py^)
(4.11)
~JP,A: ( P ^ P ^ - )
dx
cA: ( p x,Py,z)
dy
- j P y A : (Px,Py,z)
Using (4.11) in (4.10), the E and H fields in the spectral domain for TE and TM modes
are expressed as
e tJ (A. A-*) = ^ f t (A.A.--);£“ (A. A.--) =
e
■
cojus
cz
(A. A.--)=- hco/j£
- cA A hczih A
£? (A .A.*)=-—
ft (A. A
e
J
( c2
£?(A.A.-) = 0;£"'(A,A.-) = - —
-rr-A J A(A,A.-')
ry/ic^cz
(4.12)
«?(A . A--) =
<y//f
fl? (A. A
'
-
(A ,
cz
)
A?(A.A.-) = —COfXS
■ r *'
%CZ - :?W
A.--) =
A(A. A.-)
(A. A- -') ° ■
— A (A. A--)
M
|ft(A.A-•);«“ (A.A.-) =0
In (4.10) and (4.12) the scattered fields are decomposed into TE components
(transverse to z -directed electric field) that are expressed in terms o f F only, and TM
components (transverse to z -directed magnetic field) that are written in terms of A
only. To calculate the total scattered fields, the superposition theorem is applied to sum
the TE and TM components.
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78
Referring to Fig. 4.5(c), and using (4.2) and (4.12), the transforms o f the x- and ycomponents o f the surface current at z = zRcan be related to the transforms of the vector
potentials as
(4 13)
(Ofxs dz
cojue dz
It is evident
n
—
n
(4.14)
that the transforms of the x- and y-componentsof the surface
representcoupled
equations. To
current
decouple these equations, thefollowingcoordinate
transformation, as suggested by the spectral domain immittance approach [16], is applied
v = xcos#+> 'sin 0 ; x - vcostf+wsin#
u = jrsin# —>>cos0 ; y = v sin 0 -w co s0
(4.15)
sinfl = - 7=-^-— , cos 6 =
This results in a new Cartesian coordinate system rotated by an angle 6 relative to the
original x-y coordinates. It is evident from the last equation in (4.15) that the rotation
angle 6 is different for each spectral component.
Applying the transformation in (4.15) to (4.13) and (4.14), we get
J.
(4.,6)
(DUE
* L ,
.
while inserting it into (4.12) yields
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(417)
We note that (4.16), (4.17), and (4.18) are decoupled and the solutions to (4.6) and (4.7)
in each of the homogeneous layers of the FSS structure can now be obtained.
It is possible to derive an alternate representation for the stratified structure by
using the transmission line analogy, as portrayed in Fig. 4.6, and applying the
transmission line solution to the transform of the electric and magnetic vector potentials.
Both A. and F; can be written, as shown below in (4.19) and (4.20) as superpositions of
negative- and positive-going wave potentials associated with the individual layers within
the composite:
A. = A~e' y: + A . e ' r:
(4.19)
F = F . V r- '+ F .V y;
(4.20)
where
chosen to satisfy the radiation condition.
Equations (4.19) and (4.20) can be written as
(4.21)
(422)
where, T u (z) = - — e2y: and TE (z) = — e~r:
A'
F~
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80
J S|K.V)
J S lu .v i
I
Fig. 4.6. The transmission line equivalent of the FSS scattering
problem for Region-I.
Taking the first derivative o f (4.21) and (4.22) we get
dA
5z
(4.23)
and
dz
= - y F : e ^ [ \ - Y E(z)]
(4.24)
Using (4.18), (4.22), and (4.24) we obtain the impedances for the TE modes,
which may be expressed as
Z E(z) = ^ = - ^
V ' H.. Y f
Likewise, the expression for
i+rE(z)
1- r E(z)
(4.25)
impedances for the TM modes can be derived by using
(4.18), (4.21), and (4.23). It reads
7 u (-\=
^
-
H..u
1
Y:u
0
'l +r"(r)‘
1-r"(r)
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(4.26)
81
V
J/i)P
where, Y E = —— and Tjw = - — , are the characteristic impedances of the individual
ja>M
Y
transmission line segments. Employing the relationships given above, and using the fact
that the tangential components o f the electric field at the interface between the layers are
continuous above and below the FSS screen position, we can evaluate the reflection
coefficients and the transmission line admittances at any point along the line. To evaluate
the scattered fields in Region-I, we need to find the admittances Y ' [E'Kt), as seen from the
left o f the current source at z = zr, (see Fig. 4 .7).
*(E. \1),
no*)
( zr )
no-)
free
space
Line
Line 2
Line R -1
Line R
z=0
z = Z\
z = Z;
z-
Z r -2
Z - Z r -1
z-
Zr
Fig. 4.7. The admittance seen by the current induced on the FSS
screen for R superstrates.
A recursive procedure, outlined below, may be employed to calculate the
admittance seen from the left o f the current source. The procedure is outlined below.
(i) Initialize the admittance at z = 0 as
r (£. u ) = ^ . u ,
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(427)
82
(ii) Calculate
I+ r <
1( ; = ,v , ) coth { r ,
{Y, - V ,)}+ r 'SM'(: = -V.)
Yf, = - r 1— ; c = —
) Z- • r, =
jm p
Ya
y
■
'
. Mr, =
. . - JA,.
(4.28)
recursively, until p = R.
Once the impedances for the TE and TM modes are known, the transverse
components of the scattered fields can be easily related to the surface currents as follows:
e ; =(z‘'-7;)•-•A
(4.29)
£; =(z“v;)
where the (+) superscript indicates the currents and impedances of the Region-I problem.
A relationship which is similar to (4.29), between the surface currents and the scattered
electric fields in Region-II, can be expressed as
e; =(z*
I ).
(4.30)
e;
= (z “v ;)
To evaluate the electric and magnetic impedances for Region-II, we apply the same
procedure as we did for Region-I.
The total surface current on the screen at z
= zr
is given by
•7“
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H-31*
83
where J (* v) and J,~ v) are the currents that produce the fields in Regions-I and -II,
respectively. The total currents can be related to the scattered fields above and below the
FSS screen at z = zr as follows:
(4.32)
where, Z(EM1 =
*
U)_ is a parallel combination of the upper- and lower-half
impedances.
Using the coordinate transformations of (4.15) in (4.32), and then implementing
some mathematical manipulations, we arrive at the following expressions for the
transforms of the scattered electric fields in terms of the transforms of the currents:
Ex =Gxx J x +Gxy J v
(4.33)
Ey =G va: J x +G» ' J v
where
-
fcZ " + p ;Z E
P l +P;
-
p ;Z E+ p;Z M
-1 T W ~
^
(434)
M r (2 " - 2 ‘ )
Lr„. = Lr,_ = --------- —;--- ----------
P;+P;
The G ’s in (4.34) can be interpreted as elements of a Green’s tensor that is symbolically
represented as G .
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84
Since the expressions in (4.33) are in the spectral domain, we need to apply the
inverse Fourier transform to revert to the spatial domain, so that we can impose the
boundary condition Es - - E x on the metalized portions of the FSS screen. This leads to
the following Electric Field Integral Equation (EFIE):
|
-E'*(x,y)
-OC *<E
i& lL
JX
e ^ (0'x^ yy)d p xd p v
(4.35)
G >y
G >*
where the incident fields - E™[ x , y ) and -E'™(x,y), the dyadic Green’s functions Ga ,
Gn , G„ , and Giy, are known, and
Jx and J v,are the unknowns.
From (4.1), we note that the surface currents on a periodic screen can be
expressed in terms of the currents on a single unit cell. Utilizing the fundamental
properties of Fourier transforms [37], viz., frequency shifting and the Fourier transform of
a periodic train of delta functions spaced Tx centimeters apart, consists of another set of
delta functions weighted by the factor I Tx and regularly spaced I Tx radians/cm apart
along the spectral axis, we arrive at the following expression for the induced surface
current on the FSS screen,
-x
—X
X l7t) 1
T T m=-<cn=-cc 1
x
v
.( 2 a-)
•/,(a . a W ,.( a ,a )VTx Tfv - »1r= —* 1n = - «
-
a .)
(4.36)
5(a - a .M a - a .)
where
(4.37)
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85
are the Floquet wave numbers. Substituting (4.36) into (4.35) we get
j
Tx Ty
m=~<c
n=-co
A, (A . A )
I In=-<x
■ U A .A )
(4.38)
where, Tx and Ty are the periodicities o f the FSS screen in the x- and y-directions,
respectively. We note that the dyadic Green’s function or the Green’s tensor as given in
(4.34) is no longer a continuous function but assumes discrete values due to the periodic
nature of the geometry. Equation (4.38) is the final form of the EFIE that is well suited
for numerical computations. It is useful to point out that: (i) only the currents on the unit
cell are used to evaluate the scattered feilds, which makes the computation tractable; (ii)
the Green’s function is dependent upon P ^ and p
, which are functions of the incident
field, and this implies that the associated EFIE matrix must be re-computed for each
incidence angle.
Equation (4.38) is enforced at s =
zr,
but the incident fields, E'"c (*,>’) and
£^(x,_y)are known only at z = 0. To evaluate the incident fields at z =
zr,
we utilize
transmission line techniques as shown in Figs. 4.6 and 4.7, along with the recursive
procedure as outlined in (4.27) and (4.28) for the N =R~S layers. To facilitate this step,
we need to separate the TE and TM components of the incident fields. Towards this end
we express the electric and magnetic vector potentials for the incident fields as
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86
I =eA^
y)e ^ -
(4.39)
p =eAtr*-t>?y)er«;
Substituting (4.39) into (4.10) we obtain the x- and y-components of the electric fields for
the TE and TM modes
J fiy
g tn c T E _
p
. pm cTE _
j Px
p
e
(4.40)
inc n in e
~incTM _
Px
/
(OHS
ineTSt
a :- e :
A.
coh£
Having derived the expressions for the incident fields and the dyadic Green’s
function, we can now apply the Method of Moments (MoM) to (4.38), and evaluate the
unknown induced surface currents on the metallic portions of the FSS screen.
When the screen has a finite conductivity, the boundary condition on its surface
maybe written as
= Es - ZSJ S, where a surface of finite thickness and given loss
tangent is approximated by an infinitely thin surface with complex surface
impedanceZ s measured in
.
For the finite conductivity case (4.38) becomes
I
xv
m=*<®n~-<o
<U a . a )
(4.41)
AP>mX~
where the additional term on the right side corresponds to the product of the complex
surface impedance of the FSS screen and the induced surface currents in the spatial
domain.
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87
4.2.1 Numerical Solution Employing the Method of Moments (MoM)
To apply the MoM [38] to (4.38) and (4.41), we need to represent the unknown
currents in terms of a complete set of basis functions as follows:
M
.V
P = --< F ~
"-I
u
(4.42)
■>
s
p= -— 1 = ~
where the c^ ’s are the unknown complex coefficients of the known basis set
b.\"i)*/Qi, (*p, q, x, y)*. The inner product integral used in Galerkin’s testing procedure is
defined as
Ty
T,
where, t(x rt)( r,j,x ,y )is the testing function which has the same form as the basis
function b{Xay )( p , q, x, y) and * is the complex conjugation operator. Substituting (4.42)
into (4.38) and applying (4.43), we arrive at a set of M x N equations in terms of the M x
N unknowns, which can be expressed in a matrix form as
v f ,nrjx,,(r,s)
v^nr-™>(r,s)
Z~ { P ^ S)
Zytip, q>r ’s )
Zv i W ' s)
Zyy( p, q, r , s )
(4.44)
Cy0 ( P ’ <?)
where
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88
_ , a!nc
' ? m = J £ £2 - K ( r' s< K '’P ? )
s
tnc nine
(ope
v
- '
inc a r n c
C w = —cOfie
^ - > ^ (' r , * . / r . / > r )7
|
*<c *-<c
Z xx{ p , q , r , s ) = - — ^
' Z G BC{ m , n ) b Xa( p , q , m , n ) ^ ( r , i ,m ,/ i )
x v m=-<c«=-«
1
Zxy{p,q, r, s) = - — '£t £ Gr ,(w,/i) 6Vo(/>,?,m,n) £ ( r ,5,m,«)
x v w=-<e /r=-<c
=
*vC
1
-<c
Z G A m ' n ) b* ( P ' £l ' m ' n ) i ^ { r , s , m , n )
m=-xi n--«
J -<x «t
Zw ( p, q, r, s) = — Z Z ^vr K " ) \
x
v
x
v m = n= -< c
(p,q,/n,n) Ka {r,s,m,n)
(4.45)
U
M ., --------<
N o , and,
andA p, q, r, andA s are integers whose
ranges a r e M < p , r < ----1
?
?
?
AT
5 < — - 1 . When the FSS screen has a finite conductivity, we modify (4 44) as follows:
v^TE.nt)^ ^
C r f n n (r,s)
^ (p ,< 7 ,r,5 )
Z „( p, q, r , s )
Z„( p, q, r , s )
Zyy(p, q,r,s)
(4.46)
Z« (/>,?,/■, 5 )
0
0
Z[y ( p , q , r , s )
where the new matrix on the right side of (4.46) is called the adjunct impedance matrix.
Its elements can be expressed as
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89
.h-JL
1 ■>
zUP’<i’r’s)= J J c^ R ) K ( p ^ x'y) C(r’s’Jf’>’) ^
Jr-h .
2 2
J±,l1
2
(4.47)
2
^ (P, R,r,s) = | J c>o (p, q) AVn(/>, <7, x,.y) t’yo (r, s,x ,y) dxdy
Ijl-L.
2 2
The integrals in (4.47) can be evaluated analytically since they are simple multiplication
operations
Z^.
instead
of
convolutions.
The
expressions
for
Z a’ (p,q,r,s) and
will be provided after the sub-domain basis functions have been defined.
With the formulation o f the EFIE for the unknown surface currents now complete,
we proceed next to define an appropriate basis set. The basis functions should be chosen
to ensure that they portray the physical behavior of the surface currents accurately and
efficiently. Depending on the FSS-screen geometry, we either choose the entire domain
or the sub-domain basis functions. However, the former are tailored to specific unit cell
geometries for which the physical form o f the currents are known a priori, and their use
is not convenient for us because the MGA carries out the optimization of the FSS screen
in real time. For this reason we opt for the sub-domain basis functions and discretize the
currents to span a small fraction of the domain. These functions can accurately represent
the currents in arbitrary unit cell geometries; hence, they are versatile. However, the
drawback of using these functions is that, in many cases, a very large number of basis
functions are needed to accurately represent the surface currents.
It is useful to select the basis set whose Fourier transform can be derived
analytically, which is needed in (4.45). Furthermore, the transforms of the basis set
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90
should be such that the product, G(m,n) b ( p , q , m, n ) t ' ( r , s , m , n ) , decays rapidly for
increasing orders o f the Floquet’s harmonics,
and /? . This allows us to truncate the
double summation in (4.45) by retaining only the lower order of harmonics, improving
the rate of convergence o f the summations, which is the principal factor in determining
the speed with which we solve the EFIE.
x = pAx
x = pAx
x = (p+l)Ax
Ay
y = qAy-
Ay
Ax
•y = qAy
4
Ax
bx(p,q,x,y)
Ax
by(p,q,x,y)
Fig. 4.8. x- and y-components of the unit amplitude rooftops.
The selected sub-domain basis functions, called “rooftops”, are shown in Fig. 4.8
and expressed in the spatial domain as
bg( p, q, x, y) = \ ( p , x ) I l ( q , y )
by { p ^ x , y ) = n { p , x ) \ { q , y )
(4.48)
where
u-
Au
A (/, m) =
Am
n(/,w) =
..
Am
..
3Am
/Am --------< m < /Am h-----2
2
elsewhere
Am
A//
/Am -------- < m < /Am 4---2
2
elsewhere
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(4.49)
91
In the spectral domain the transforms of the rooftop basis functions in (4.48) and (4.49)
are given by
\r
bx{p,q,m,n) = (AxAy)sine2 & — sine ^
2 )
-j(ft.m
p^v)
(4.50)
s, {p,q,m,n) = ( AxAy)sinc2
where sinc(//) =
\
f
0^
^ J s in c [ i
\
Ax
^ e
2j
,R
Av
1 f t >n ^
sin(w)
To apply the rooftop basis functions just defined, the FSS unit cell needs to be
discretized into an M x N grid, as shown in Fig. 4.9 for a square patch and M = N =16.
Next, we place x - and y -directed basis functions everywhere on the metalized portions
of the FSS screen. If necessary, the density of the grid can be increased to provide a more
accurate representation of the surface currents.
In (4.44) and (4.46), the doubly infinite summations can be efficiently handled
when the FSS screen is represented as described above. Since these summations cannot in
general be evaluated in closed forms, they must be computed numerically; after
truncation. The rooftop basis functions possess very good convergence properties [15]
and their transforms are well suited for summation using the Fast Fourier Transform
(FFT) algorithm. Utilizing the FFT and a special partitioning scheme for the summations,
a large number of terms can be summed in a highly efficient manner. Substituting (4 .50)
into the last four equations in (4.45) we get
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92
Fig. 4.9. FSS cell discretized into 16 x 16 pixels with the
shaded part corresponding to metal.
-<x
Za ( p, q, r, s) = £
*ac
' £ G a f ; P ]( p, q) P; (r, s)
*l=-<xi n = -< c
Z„{p, q, r, s) = £
nt—
Z yr(w ,s)= £
;L ^ Iyf xf yP\ { P^) P' {r , s ) e
—<c
£ 6 >x/ x/ ^ (/>,?)/}*(/•,*)<?
(B=-<Cffr-H®
Zvy( p ,^ ,r,5 )= £
£ G ^ /; ^ ( p ,< 7) ^ * (r ,j)
where
f
/
Ax"
,
AxAy . ,
/ x= —
sin e P*m
sine
T TV
V
V
^ J
/
ArAy . 2
Ax"
Ay"*
sine
sine
A.
/v =
A
.
T ;
7T
T,
V
V
* X *
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(4.51)
93
The infinite summation in (4.51) can be rearranged as follows. First, we subdivide
the spectrum into M segments consisting of M harmonics along /&, and N segments
consisting of N harmonics along $ , where M and N are the number of segments into
which the FSS unit cell has been divided. Upon doing this, we effectively truncate the
summation to ( M x M) x ( N x N) elements. Next, we generate an array consisting of the
sum o f corresponding elements from each of the M and N segments (see Fig. 4.10), and
use (4.37) to express the summations in (4.51) as
±L,
(4.52)
where
, Ax
-jZ.T - l [ p - r
Av; ( < ? - 11
!- —
Fz = e
Segment I
Segment 2
Segment 3
A2
A3
Segment 4
-10 -9
A
A1
Fig. 4.10. Grouping the infinite summation over
A4
into M
segments of M elements. The like terms o f each segment are
summed, forming an array A, composed o f M elements. In this
case M =M=-f
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94
Following the same procedure, we can also express Zxy, Z>x, and Zyy to define the
T
T
remaining terms in the impedance matrix of (4.46). Next, by letting Ax = — and Ay = — ,
M
N
- jZ x
we can show that P{(m'M,n'N) = 1 and Px{m",n’) - e
mV
' — {
n
p - r y — {$-*)
u
v
Applying these
expressions to (4.52) we get
•"-i
X X<5J ;
Z„ = FFT
m
\1
(4.53)
, .V
n- —
where the FFT is used to implement the discrete Fourier transform of the type
T -1
^
/ ( f'*7)= S
m.
T->
M
'« •
«• '
F i m ^ n ^ e 1 -u v'
(4.54)
.V
n. =—
The summation in (4.54) can be efficiently evaluated by using the FFT for a large
number o f matrix elements. To achieve the highest efficiency, the values of M and N
must be powers o f 2. We should also point out that the use of the FFT also requires that
the unit cell geometry be discretized in a rectangular grid.
With the basis functions and its transforms defined in (4.48) through (4.50), we
can now derive expressions for the elements o f the adjunct matrix. Substituting (4.48)
and (4.49) into (4.47) we obtain the following equations [34] for these elements:
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95
-S{p-r)S(q-s)
= AxAy
S ( p - r - l ) S ( q - s ) / + S ( p - r +\ ) S ( q - s ) f 2
+
(4.55)
y £ (p -r)< % -.s)
Z w = AxAy
+
8 ( p - r ) 8 { q - s - \ ) f ^ 8 { p - r ) S { q - s +\)fi
where
for p =
-1 and r = -
M
(4.55a)
otherwise
for p ~
fz
M
M
and r = ------ 1
=
0
/,=
otherwise
e
y for q = ----- 1 and 5= ----2
2
otherwise
0
e
(4.55b)
y y for q =
2
/> =
0
and 5 = ----- 1
2
(4.55c)
(4.55d)
otherwise
The phase terms in (4.55a) through (4.55d) are necessary because the basis
functions are selected such that at the edge of the unit cell they overlap the adjacent unit
cell.
With all the tools ready to compute the matrix elements efficiently and accurately,
(4.44) and (4.46) can be expressed as
v = Zc
r_
_
_ .-i
v = | Z + ZSZ Ic
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(4.56)
96
The solution to (4.56) requires the inversion of a matrix to obtain the unknown current
coefficients, which is typically performed by using the LU decomposition algorithm with
partial column pivoting.
4.2.2 Evaluation of Reflection and Transmission Coefficients
Once we have computed the surface currents, the reflection and transmission
coefficients can be easily calculated from the tangential scattered fields at r = 0 and z =
:r~ s,
expressed in the spectral domain. The scattered fields are calculated from (4.38) or
(4.41), depending on whether the FSS screen is a PEC or has a finite conductivity, and
then transferred to z = 0 and z = zr-s by repeatedly applying (4.27) and (4.28) The
incident fields can be evaluated by using (4.40) and similarly transferred to z = 0 and z =
zr-s .
The sum o f the scattered and incident fields yields the total fields at z = 0 and z =
Z r-s-
The scattered field at z = 0 can be expressed [39] as a superposition of Floquet’s
harmonics:
(4.57)
where
-fa
.
The scattered fields can also be written in terms of the vector potentials A1 and F 5as
Es = -ja )A s
=£— V ( V - / H - — ( V x F 1)
co ^sa
v
7
v
where the potentials are expressed as
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(4 58)
97
•<r
wr
I O V « * = < >
p=-<0
(4.59)
F ’ =z £ f
p=-«?=-
at : = 0
_ t?j(v * V ) e "*. Next, applying (4.10) to (4.59), we get
=
and
WO
jP y *
«: = I E
•CO
—x
\
»<o
p --< t3 C p
d te
, 'K
'V
ym
.
D r.u
y
e°
jP*,
dte
W
*<>
-X
^
+ Pyjm p m
, . 1. . r.
W
■o
¥ pqi
(4.60)
V pq
y
Multiplying both sides of (4.60) by ^*„ and integrating over the unit cell, we obtain
•<C
j
<bdy = X
umt ctil
-WO
*■<*• ***
I
1
j3
umt cell
JHx? ftTE +
w
B y
w gr\t
w • f
p = -* c j= -
umt ctil
VpqVm dxdy
j
PH
'0
v.
P =~,c-
f
JPy. Dnr , P xp. Y pm ^r.u
Z
(4.61)
V py mn dxdy
umt cell
The left sides o f (4.61) are the Fourier transforms of E\ and E*y evaluated at fi and /? .
The double summations appearing on the right sides can be eliminated by using the
orthogonality properties of Floquet’s harmonics. Applying these properties we get
C
= je n
Py.
{£ K
’ Py. ) +
} ~ P , { % { P , •Py. ) + ^ S4
Pl+Pl.
(4.62)
Rmr = 0 )^en
A {£ (a . . a..)-1-£;*_} +a {*; (a . ■a . ) ■*5 a ,}
/-(A + A )
where ^
is equal to unity only for m = n = 0, which corresponds to the reflected field
term. Following the same procedure, the scattered fields at 2 = zR~s can be derived, and
the following expressions for the transmission coefficients can be obtained:
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98
TTE
mn = 7is
*0
P i * o ly«
(4.63)
'rTM
mn
A . { a ( A ..A .K A .} + ^ . { 5 K A ) + f iA ,} j
~ ^ M l^ O
y~.{oi+oi.)
In (4.62) and (4.63) the reflected and transmitted fields are
(£; ) TE =- ^ -k ,(£; )™=
£•„
( £ ;) rc J -h a - r ^ e ; )™ = ^ ^ L
mA>
\ TE
(S) -
-/?
<y//n£-0
r
(4.64)
jfiy 0 j e -rn\R-S) ( r< \ n i _
0,1
■Te
G)^0£0
-y^R-S)
Qi^uSn
4.3 MGA Formulation for the Spatial Filter and Broadband Microwave Absorber
Problems
In this section we apply the MGA to optimize various parameters pertaining to the
two design problems of interest. Binary tournament selection (Sec. 2.1.4.3) is employed
along with uniform crossover (Sec. 2.1.5.3) with a crossover probability of 0.5. Elitism is
utilized along with a population restart strategy that is applied when the bitwise
difference between the best and other individuals/chromosomes in the current generation
is less than 5%.
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99
4.3.1 Broadband Microwave Absorber Synthesis
Figure 4.11 shows a multi-layered composite structure whose parameters we wish
to optimize with a view to realizing a specified frequency response.
PEC backing
FSS screen
Fig. 4.11. Multilayered composite with single embedded FSS screen
and uniform plane wave incident at an arbitrary angle.
We assume that we are given a set of different materials M m with frequency
dependent permittivities s,(f) (/ = 1,
M m). Our design goal is to determine a coating
consisting of N different layers, an FSS cell design generated by the MGA, the FSS cell
periodicity, its position within the dielectric composite and the FSS screen material, such
that the coating exhibits a low reflection coefficient for a prescribed set of frequencies f
(/ = / ,
Ff) and incident angles 9, (i = 1
A q/, for both the I E and TM
polarizations. In the context of the present problem, the magnitude of the largest
reflection coefficient is minimized for a set of angles, for both TE and TM polarizations,
and for a selected band of frequencies. Hence the fitness function can be written:
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100
F (mx, dx, ....... ms , ds , 5 ^ , tx, ty, 5 ^ ) = - max {r^™ (0,, / ; ) J
(4.65)
where, m, and d, are the material parameter and thickness of the i,h layer, respectively;
S'j" is the unit cell o f the periodic FSS screen generated by the MGA; tz and ty are the xand y-periodicities of the FSS screen;
TE,
/
is the position of the FSS screen within the
v
dielectric composite; and, T nt [ 0 , , / j ) is the reflection coefficient, which is a function
of polarization, incident angle and frequency.
The MGA operates on a coding of the parameters. The coded representation of the
coating consists of sequence of bits that contain information regarding each parameter.
Given a database containing Mm = 2Klmt different materials, the material choice for layer /
is represented by a sequence L, o fM ^ bits as follows:
L, =/,'/;.................................................................... (4.66)
The thickness of the i,h layer can be encoded by a sequence D, of
bits and represented
as
A
..................................................................(4.67)
The periodicity o f the FSS screen can be encoded depending on its allowed range of
values and the step-size (see Table 4-2) utilized by the MGA. It can be represented by a
sequence Tx of M t b bits for the x-directed periodicity, and similarly by Ty of M t J>bits for
the y-directed one as given below:
Tx =t1
/ ; ............ r y
w
T = /‘r
*y
y y ............ t * v»
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(4.68)
101
For an N layer composite, the FSS screen can be placed in the composite at N-l
positions. For example, for a four-layer composite, the FSS screen can be placed at the
following three positions: (i) between the first and the second layers; (ii) between the
second and third layers; and, (iii) between the third and fourth layers. For this case, we
would need 2 bits to encode the three possibilities. Thus, the position of the FSS screen
within the composite can be encoded by a sequence P of two bits as
P = p lp 2
(4.69)
The MGA designs the FSS cell structure automatically The pan of the code
analyzing the FSS screen embedded in dielectric media utilizes a 16 x 16 discretization
(32 x 32 and 64 x 64 discretizations can also be handled) of the periodic structure unit
cell in the form o f 1’s (one) and 0’s (zeros); the l ’s corresponding to PEC or lossy metal
and the 0’s to free space. The MGA randomly generates this 16 x 16-gridded structure
filled with l ’s and 0’s as explained next. Figure 4.12 shows a 16 x 16 matrix filled with
l ’s and 0’s.
The MGA considers each row in the FSS cell as a parameter. For each row, the
MGA generates a random number between 0 and 2Ucb-J, where MCb is the number of
columns in the FSS cell matrix. These random numbers are then converted to a binary
format for each row. These binary numbers are combined into an array, which is ready to
be analyzed by the FSS code.
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102
R, 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0
R2 0 1 1 1 1 1 0 0 1 1 I 1 1 1 0 1
R3 0 1 1 1 0 0 1 1 1 I 1 1 1 1 1 0
1100110011001000
1 1 0 0 1 1 00 1 I 00 1 0 0 0
01 1 I 0 0 1 1 1 I I 1 1 1 1 0
0 I 1 1 1 1001 1 1 1 1 10 1
0 0 0 0 10 1 1 1 I 1 10 1 10
0 0 0 0 10 1 1 1 I 1 10 1 10
01 1 1 1 1 0 0 1 1 1 1 I 1 0 1
0 1 1 1 00 I 1 I I 1 1 1 1 1 0
1 1 00 I 1 00 1 1 00 1 0 0 0
1100110011001000
01 1 1 0 0 1 1 1 I 1 1 I 1 1 0
0 I 1 1 1 10 0 1 1 1 1 1 10 1
Ri6 0 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0
y A
Fig. 4.12. A 16 x 16 pixel FSS unit cell design
represented in the form of “ l ’s” and “O’s”.
Two-fold symmetry is introduced into the FSS structure by making the rows 5
through 8 mirror images o f the rows 1 through 4, and rows 9 to 16 the images of rows 1
through 8. This effectively reduces the number of MGA parameters needed to design the
FSS cell from 16 to 4, and makes it considerably more efficient than when no symmetry
is imposed. Thus the FSS cell can be designed by encoding each row into a sequence
{CD)j of M Cb bits as
(iCD)j = (CD)\ ( C D ) ' ................. (C D )"-
(4.70)
where j is the number of rows considered as parameters by the MGA; j max is the total
number of rows in the FSS cell structure.
Additional bits are needed in the MGA optimization process if we select lossy or
lossless dielectric layers, and a lossy or PEC FSS screen. As each selection offers two
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103
choices, we need one bit for each, i.e., a total of two bits. These two parameters can be
represented by the sequences (CL), o f 1 bit and (CS) of 1 bit as follows:
-(CL)I
(4.71)
(CS)=(CS)'
If a lossy FSS screen is selected, the real and imaginary parts of the complex sheet
impedance appear as two new parameters, to be optimized by the MGA. The range of
these parameters can be encoded depending on their permissible range of values and the
step-size (see Table 4-2) introduced by the MGA. The real and imaginary parts of the loss
can be represented by a sequence (FLR) of A//W)bits and (FLI) of M lhbbits, respectively,
yielding
(FLR)-(flr) (fir)'
(Jlrf'
(4.72)
(F L I)-(flij(jliy
(ftif»
The entire composite can be represented by a sequence C, which is referred to as a
chromosome:
C = L,....Ls Dv . . . D „ ( C L \ . . . . ( C L ) J J yP(FLR){FLI)
(CS)(CD\....(CD)j
(4.73)
Each quantity in (4.73) has been defined in (4.66) through (4.72) and each sequence in
(4.73) consists of (2~Mmb- M db~Mt t b ~2~j*Mcb-Mfirb~Mfi,b) bits.
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104
4.3.1.1 Numerical Results for Single FSS Screen Embedded in Lossy or Lossless
Dielectrics
The MGA has been successfully applied to the synthesis of broadband microwave
absorbers in the frequency range o f 19.0 - 36.0 GHz. The number of dielectric layers was
fixed at four, was surrounded by air on one side, and terminated on the other side by a
PEC backing. Though only dielectric layers with electric loss are considered in this
section, we show later that the method can be extended to handle both electric and
magnetic losses. The measured values of s ’ and e* of ten different lossy materials were
considered and a database of these values as a function of frequency was created, as
shown in Tables 3-2 and 3-3, respectively. A linear interpolation was used to generate the
values of s ’ and s ' as functions of frequency for intermediate material types to obtain
the values of the real and imaginary parts of the permittivity for even numbered material
types at a particular frequency, while the material parameters of the odd-numbered ones
were averaged. For example, values for the “Type 2” material at 19.0 GHz was obtained
by averaging the material values for “Type 1” and “Type 3” at 19.0 GHz. Thus,
effectively, a total of 19 different types of materials were used in this study. For lossless
layers, small values of losses were added, since it is not feasible in practice to fabricate a
perfectly lossless dielectric. Hence, if the MGA selects the first layer as lossless, its er
value is deliberately fixed at 0.01, while for all the other layers the corresponding value
o f ^ is 0.1.
A total of seven cases, listed below were investigated
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105
(i) Normal Incidence (0 = 0.0,<J>=0.0), TE polarization
(ii) Oblique Incidence (0 = 45.0, <j>= 0.0), TE polarization
(iii) 0 varying from 0 to 45 degrees, TE polarization
(iv) Normal Incidence (0 = 0.0,<J>=0.0), TM polarization
(v) Oblique Incidence (0 = 45.0, <J>= 0.0), TM polarization
(vi) 0 varying from 0 to 45 degrees, TM polarization
(vii) 0 varying from 0 to 45 degrees, TE and TM polarizations
The population size and the number o f generations were fixed at 50 and 405,
respectively. The periodicity of the FSS screen in the x- and y-directions were made
equal. The loss in the FSS screen material was restricted to real values by forcing the
imaginary part to be zero and the frequency resolution was chosen to be 1.0 GHz. To
obtain the values of reflection coefficients at intermediate frequency points, a spline
interpolation in frequency was carried out by using MATLAB™. The optimized
parameters from the MGA output were used in a separate code, which handled FSS
screens embedded in lossy dielectric media. To verify the intermediate values of the
reflection coefficients, the FSS code was run with the same frequency resolution as the
spline interpolation. If the reflection coefficient values at the intermediate frequencies
generated by these two methods were found to disagree within a certain tolerance, the
MGA was re-run at a finer frequency resolution, using the population of the last
generation, to obtain the correct values.
The results of the MGA optimization o f the reflection coefficient in dB are
presented in Figs. 4.13 through 4.21. In the FSS cell structure white represents metal,
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106
while black corresponds to free space. For all the seven cases, a two-fold symmetry is
introduced in the FSS unit cell design and the azimuthal angle is fixed at zero degrees (<J>
= 0.0, x-z plane). The parameters selected by the MGA for each case is listed in Table 41.
We see from Figs. 4.13, 4.14, 4.17 and 4.18 that the frequency response obtained
by using the MGA resembles, not unexpectedly, the familiar filter response derived by
using the Tchebyscheff equal ripple optimization procedures. For all the four cases, the
worst-case reflection coefficient is maintained below -18.0 dB. In Figs. 4.15 and 4.19,
the worst and best-case reflection coefficient can be interpreted as follows. For a single
frequency, over the elevation angles of interest (0 = 0.0 to 45.0 degrees), the maximum
and the minimum values of the reflection coefficients correspond to the worst and the
best results expressed in dB. Imposing an additional burden of optimizing over a range of
elevation angles, causes the worst-case reflection coefficient value to degrade further.
Figures 4.16 (a) through (d) and 4.20 (a) through (d) present the behaviors of the
reflection coefficient as functions of the elevation angle for the TE and TM polarization,
respectively, with frequency as a parameter. Figure 4.21 shows the worst- and best-case
reflection coefficients realized by the MGA, while optimizing simultaneously for both the
TE and TM polarizations over the range of elevation angles o f interest (0=0.0 to 45.0
degrees). We see that the MGA maintained the worst-case reflection coefficient below 15 .0 dB for this case.
Figures 4.22 and 4.23 illustrate certain details of the MGA optimization procedure
for Case-7, which may be regarded as a representative example. Figure 4.22 depicts the
variation o f the fitness value as given by (4.65), vs. the number of generations. Two
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
107
important observations can be made about this result: (i) the MGA restarts the population
at the 50th generation as seen from the sudden dip in the curve for average fitness value;
(ii) the best value of fitness (which is not necessarily the global maximum, but satisfies
our design requirements) is achieved by the 75th generation. Figure 4.23 shows the
population distribution at the 1st, 40th, and 75th generation, respectively.
Table 4-2 lists the search space of the MGA parameters for this problem. If the
MGA selects a PEC FSS screen with lossy dielectric layers, the number of parameters and
bits in the chromosome correspond to 18 and 168, respectively. However, if the MGA
selects a lossy FSS screen together with lossy dielectric layers, the number of parameters
increases to 19, and the chromosome length becomes 194 bits. These two cases constitute
the best- and worst-case scenarios in terms of the population sizes for the problem at hand.
It is possible to obtain an estimate of the population size for a CGA by using (2 .9) given in
Sec. 2.2.1.
From Table 4-2 we observe that for the best-case we have / = 168, x = 2 and the
average length o f the schema of interest was (1/18)(64+10+52+18+2+1+1+20) = 9. This is
the average length of the gene that makes up one parameter
Correspondingly, for the
worst-case, the parameters were: / = 194, x = 2, while the average length of the schema of
interest was (l/19)(64+10+52+18+2+l+l0+l+36) = 10. Applying equation (2.9), the
population sizes for the best- and worst-cases is estimated to be 9560 and 19866,
respectively. The relative advantage of the MGA over the CGA in terms of the required
population size provides adequate justification of its use to address our design
optimization problem. It is evident that we gain substantial savings in computational time
and resources when the MGA is used.
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108
\'
o * — :______
...... V
...........................................................
Original points
Interpolated points
-40
-42 -------- 1-----------1-----------1—
18
20
22
24
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c -32
-r
r
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T3
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28
30
32
34
36
Frequency in GHz
er = 1.03, d = 0.45
s r - 1.65, d = 1.85
Type 1, d == 1.25
K
Type 2, d = 0.6
Fig. 4.13. Design for Case-1.
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109
£-22
2 *26
5 -28
o
Orieinal Doints
Interpolated points
•r
Freauencv in G H z
er = 1.03, d = 1.65
er = 2.38, d = 1.75
Type 14, d = 0.25
Type I, d = 0.77
Fig. 4.14. Design for Case-2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
110
i
-12
i
-14
T
CQ
-o
c -16
i
♦
♦
■r
i
i
30
32
t
♦
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c
<u -18
o
fc
8u
-20
c
o -22
o
4>
c
-24
it
-26
♦
-28
18
20
22
worst case
best case
24
26
28
34
36
Frequency in GHz
Er =
1.03, d = 3.15
6r =
1.65, d = 0.05
Er
= 2.09, d = 1.75
Type 16, d = 0.6
Fig. 4.15. Design for Case-3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reflection coefficient in dB
111
-10
-15
-25
-30
Angle Of Incidence m degrees
19.0GHfe -
* 20.0GHz — * —21.0GHz — * -2 2 .0 G H z — * -23.0G H :
Reflection coefficient in dB
(a)
-10
-15
-20
Angle Of Incidence in degrees
24.0 Gffe— ■— 25.0 GHz-
~tr - 26.0 GHz — * -2 7 .0 GH-— * - 28.0 Ghfe
(b)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Rellcction coeflicienl in dB
112
-10
-15
-20
-25
-30 ^
Angle Of Incidence in degrees
■29.0 Gto — ■— 30.0 GHz-
-tr -31.0GI-te — * -3 2 .0 GHz—• * -3 3 .0 GHz
Reflection coefficient in <JB
(C)
-20
-25
Angle Of Incidence m degrees
■34.0 Ghte — ■— 35.0 GHz- -*- -36.0 GHz
(d)
Fig. 4.16. Case-3: Reflection coefficient as a function of the elevation
angle with frequency as a parameter.
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113
-20
i ;
CQ -21
T3
t :
\\ 1;
I:
_C
cu -22
'3
£
gu ' 23
:
c
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t
/
;
A-
■ x
< 2
°
~b~
-25
20
t
-
W
-26
18
•4
Original points
Interpolated points
22
24
26
28
30
32
34
36
Frequency in GHz
s r = 2.67, d —1.45
Type 1, d = 0.05
Type 7, d = 0.95
Type 16, d = 3.55
Fig. 4.17. Design for Case-4.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
O'
-36 er
18
I
I
i
1
20
22
24
26
O
11
28
O riainal nom ts
tnieroolaied
noinis .
...........................
30
32
34
36
Frenuenov in G H 7
er = 2.67, d = 1.35
Type 1, d == 1.45
Type 11, d = 1.05
Type 17, d = 0.96
Fig. 4.18. Design for Case-5.
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115
I
I
I
!
♦ ♦
CO
T3
c-20
♦
+
♦
c
V
E- 25
♦
worst case
best case
c
•2-30
u
c:u
^-351
18
20
22
24
26
28
30
32
34
36
Freauencv in GHz
Er = 2.67, d = 1.25
er = 2.5, d = 0.35
Type 6, d = 0.95
c
Type 15, d = 0.75
Fig. 4.19. Design for Case-6.
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116
10
15
25
20
30
40
45
Reflection coefficient in dB
-5
-10
-15
-40 - ...........................................................................................................
Angle o f Incidence m degrees
19.0GHz —■*— 20.0 Ghte - -Ar -2 1 ,0GHz — * - 2 2 .0 GHz — * -23.0G H z
(a)
Reflection coefficient in dB
-10
-15
-20
-25
-30
-35
-40
-45
Angle o f Incidence in degrees
♦ — 24.0 Ghfe— » — 25.0 GHz - -A r - 26.0 Ghfe— * -2 7 .0 Gffe— * - 28.0 GHz
(b)
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50
117
.1 -15
a
a
» -20
e2
5a -25
-30
-35
-40
Angle of Incidence in degrees
'29.0 GHz - —• • ' ‘‘•30.0 GHz ‘
31.0 GHz —
32.0 GHz
* " 33.0 GHz
(C)
e
-io
C
I -15
1
-20
Ia *
a
ca
£ -30
-35
-40
Angle o f Incidence in degrees
*34.0 GHz'
“ 35.0 GHz “ * * 36.0 GHz
(d)
Fig. 4.20. Case-6: Reflection coefficient as a function of the elevation angle with
frequency as a parameter.
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118
15 ■
ffl
-a
c
♦
+
16 ■
worst case
best case
17
c
u
'3 18
£
8o 19
c 20
o
*■3
ou
c:u 21
a:
+
22
•23
18
20
22
24
26
28
+•
+
30
32
34
36
Freauencv in GHz
Fig. 4.21. Design for Case-7.
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119
-0.1
-
0.2
Fitness value
-0.3
average fitness value
best fitness value
-0.4
-0.5
-
0.6
-0.7
-
0.8
-0.9
0
50
100
150
200
250
300
350
400
Number of Generations
Fig. 4.22. Fitness value vs., the number of generations
for Case-7.
SsnsKHtpn
Generation 40
G e n e ratio n 7.->
-12
-10
-8
-6
-4
Reflection coefficient in dB
Fig. 4.23. Population distribution of the MGA in the 1st,
40th, and 75th generations for Case-7.
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120
TABLE 4-1
PARAMETERS SELECTED BY THE MGA FOR THE 7 CASES
Dielectric Layer types
FSS screen properties
Layer 1
Layer 2
Layer
3
Layer 4
Case-1
Lossless
Lossless
Lossy
Lossy
Case-2
Lossless
Lossless
Lossy
Lossy
Case-3
Lossless
Lossless
Lossle
ss
Lossy
Case-4
Lossless
Lossy
Lossy
Lossy
Case-5
Lossless
Lossy
Lossy
Lossy
Case-6
Lossless
Lossless
Lossy
Lossy
3rd and
4th
1st and
2nd
Case-7
Lossless
Lossless
Lossy
Lossy
1st and
-jnd
Position
between
Cell size
(mm)
3rd and
4th
3rd and
4th
2nd and
3rd
1st and
2nd
17.11 x
17.11
11.26 x
11.26
11.16 x
11.16
9.92 x
9.92
31.78 x
31.78
26.70 x
26.70
8.767 x
8.767
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Surface
Resista
nee
Q
square
137.08
164.45
394.16
988.5
646.36
980.67
447.93
121
TABLE 4-2
MGA PARAMETER SEARCH SPACE
Range of
parameters
Step size
Number of
possibilities
(nearest 2n)
Number of
binary digits
4
0 -6 5 5 3 5
1
65536
16 bits x 4
rows = 64
1
0.1 -6 .0
0.01
1024
10
di, d2, d3, d»
4
0 .0 5 -6 .0
0.001
8192
Txi, Tyi
2
1.0-50.0
0.1
512
1-4
1
4
13 bits x 4
layers = 52
bits
9 bits x 2 =
18 bits
2
1-3
1
2
1
10.0- 1000.0
1
1024
10
1
2
1
1.03-6.0
0.01
512
9 bits x 4
layers = 36
1 - 19
1
32
5 bits x 4
layers = 20
One row
Total 4 rows
Total
thickness of
composite
(mm)
FSS position
1
Choice of
FSS screen
1
material
Re(Zs)
1
Choice of
layer 1
1
material
Erl-, Sr2i Sr3i Sr4
4
for lossless
dielectric
Lossy
dielectric
4
for all four
layers
Lowest total parameters
Highest total parameters
o
r»-i
1
O
Number of
parameters
Parameters
18
19
Lowest total bits = 168
Highest total bits = 194
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122
We note from Figs. 4.13 through 4.21 that the MGA introduces a two-fold (or
top-bottom) symmetry in the FSS screen designs. As explained earlier, the MGA
optimizes, simultaneously, for a range o f frequencies, elevation angles, and for both the
TE and TM polarizations. However, it is necessary to maintain the invariance of
reflection coefficient not only with frequency and the elevation angle 9, but also with the
azimuthal angle <j>. For composites that are formed by using only dielectric layers, the
reflection coefficient is independent of the azimuthal angle <j>and is not considered during
the optimization process. However, for composites with an FSS screen embedded within
the dielectric layers, the reflection coefficient is a strong function of <f>and has to be
accounted for during the process of optimization.
The ^-variation can be introduced into the optimization process either explicitly
or implicitly. In the former case, we incorporate an additional loop into the numerical
process, which accounts for each azimuthal angle to be optimized. For the implicit case
we introduce higher orders of symmetry into the FSS screen design to reduce the strong
dependence of the reflection coefficient on <f>. The drawback of the explicit optimization
scheme is that it leads to an increase in the computational time and resources. In contrast,
as shown below, the implicit scheme carries out the optimization with respect to the <fb
variation efficiently, and without additional burden on the computational expense. For
this reason, the implicit scheme is preferable to the explicit one in our work.
For the implicit scheme, the MGA incorporated eight-fold symmetry in the FSS
screen design. Case-7 was first optimized for a composite with an FSS screen possessing
two-fold symmetry. Thereafter, the same case was repeated with an FSS screen
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
123
incorporating eight-fold symmetry. The composite containing the FSS screen, shown in
Fig. 4.21, is simulated for 0= 45, <j>= 0 and 45 degrees, and for both the TE and TM
polarizations. The variation of the reflection coefficient vs. frequency is plotted in Fig.
4.24 for the four cases mentioned above. It is evident that the frequency response shows
an considerable amount o f variation as we change the azimuthal angle <p.
-1Q
-15
CQ
-o
'C
eu
o
sa
8 -2 5 ....
c
\
0
\
1
H
-3C........
aL
-35
++
TE - theta = 45, phi = 0
TM - theta = 45, phi = 0
TE - theta = 45, phi = 45
TM - theta = 45, phi = 45
-4(7
Frequency in GHz
Fig. 4.24. Frequency response of the composite shown in Fig. 4.21
with a FSS screen having two-fold or top-bottom symmetry
embedded in it.
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124
Reflection coefficient in dB
-161
TE-phi=0
TM-phi=0
TE-phi=45
TM-phi=45
-24
Frequency in GHz
e r - 1.02, d = 0.35
e r = 2.272, d = 1.85
Type 6, d = 1.05
Type 14, d = 2.28
Fig. 4.25. Frequency response of dielectric composite with a single
FSS screen possessing eight-fold symmetry embedded in it.
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125
-19
♦
4
4►
♦
-
4
♦
*
.....▼....
18-
H
c
♦
+
-k
+
+
:
+
ce
u -2C '
'5
£
8u 2 2
I
u
'S -24
-
t
+•
CQ
■o
-
> ♦
♦
►
t
+
’
4-
+
♦
+
w o rs t
best
.................J L........... 4-.........
+
26T
-
-2 ff
8
20
22
24
26
28
30
Frequency in GHz
32
34
36
Fig. 4.26. Worst- and best-case reflection coefficients over the
frequency band for composite with FSS screen in Fig. 4.25.
Figure 4.25 depicts the frequency response of a dielectric composite with an FSS
screen possessing eight-fold symmetry embedded in it. The x- and y-periodicities of the
FSS screen are equal to 1.671 mm, and its surface resistance is 31.5 Q/square Comparing
Figs. 4.24 and 4.25, we observe that the frequency response for the latter varies relatively
little with 0. Thus, we have achieved the invariance of reflection coefficient with 0 via
the implicit scheme without placing additional burden on the computational resources or
time. We point out that the dependence of the reflection coefficient on the azimuthal
angle can be eliminated completely only if we incorporate radial symmetry in the FSS
screen design and choose the x- and y-periodicities to be equal. Figure 4.26 shows the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
126
worst- and best-case reflection coefficients for the composite and FSS screen shown in
Fig. 4.25.
4.3.1.2 Numerical Results for Single FSS Screen Embedded in Dielectric and
Magnetic Media that can Either be Lossy or Lossless
In the previous section, we designed a composite structure comprising either lossy
or lossless dielectric layers, with a single FSS screen embedded within the layer. In this
section we will optimize a composite consisting of dielectric or magnetic layers that
could either be lossy or lossless. The magnetic materials, if selected by the MGA, are
always assumed to be lossy. The lossy dielectric materials that we choose from are
unchanged from the previous section, and Figs. 3.8 through 3.11 show the frequency
dependence o f £r, s r’ ,jur, a n d //’ for the six lossy magnetic materials utilized. Linear
interpolation is utilized for the magnetic materials, in a manner similar to that applied to
the lossy dielectric materials in the previous section. All the optimization parameters
remain the same as in the previous section, and Case-7 is repeated with the lossy
magnetic materials included in the optimization, and with an eight-fold symmetry
incorporated in the FSS screen design. Figure 4.27 shows the composite with an FSS
screen whose x- and y-periodicities are both 8.767 mm. For a PEC FSS screen, the figure
also shows the worst- and best-case reflection coefficients over the frequency band of
interest. Comparing Figs. 4.26 and 4.27, we see that the worst-case reflection coefficient
in both the cases are within 1 dB of each other, and that the composite thickness is 5.53
mm for the former design, and 2.95 for the latter.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
I
Reflection coefficient in dB
-151
11"
'
I
♦
4-
I
+
♦
-2C '
+
-25
+
-3C ’
t
+
-
+
+
w o rs t
b est
+
■+*
-35
-40
8
20
22
24
26
28
30
Frequency in GHz
32
34
e r = 1.873, e = 0.01, d = 1.95
H r = 1.0, H r = 0.0, lossless
dielectric material
" r
e r = 8.19, e r = 4.8, d = 0.85
H r = 1.0093, h’V = 0-195 (36.0
GHz), Type-7-lossy magnetic
material
8r = 1.75, 8 r = 0.1, d = 0.05
H r = 1.0, H r = 0.0, lossless
dielectric material
er = 5.543, e r = 0.1, d = 0.107
H r = 1.0, H r = 0.0, lossless
dielectric material
Fig. 4.27. Design for composite with both dielectric and magnetic
materials.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
128
As expected, the employment of lossy magnetic material leads to a thinner
composite [40], albeit at the expense of increased weight due to the high concentration of
iron in the lossy magnetic materials.
Figures 4.28 and 4.29 illustrate the population distribution and the fitness values,
respectively, as the generations evolve during the MGA optimization. We note that at the
beginning of the optimization the population is randomly distributed with respect to the
total thickness of the composite. At the 40th generation the individuals are all lined up at a
constant value o f the total thickness. Eventually, at the 75th generation, the effect of the
Elitist operator is observed, as only one individual having the same thickness as the
others but with a superior value of reflection coefficient is filtered out of the population.
1
-----------------
1
O
c •
D
°
?15
c/1
c/1
O
O
O O.
-5
0
O
Generation 40
■
5‘
o
o o
o
1
cn
u
e
u
Is
H
-10
O
°o$%
I
i
aooo
me
o
-5
o
E
u
c
o
Generation 1
0
1
O
Generation 75
-
5o
~
oo
®
o
o
--------------------------------- 1-------- -------------------------1---------------------------------
-15
-10
-5
Reflection coefficient in dB
Fig. 4.28. Population distribution for the optimization carried out
for the case with lossy magnetic layers in the composite.
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129
-0.1
§ -0.4
J
>
(A A
g -0.5
c
u.
-
A verage
b e st
-0.9
3
100
Number of Generations
150
Fig. 4.29. Fitness value vs. the number of generations for case with
magnetic losses in the composite.
4.3.1.3 Numerical Results for a Single FSS Screen Embedded in Lossy Carbon Fiber
and Lossless Dielectric Material
In this section, we optimize a composite consisting of either lossy carbon fiber or
lossless dielectric material with a single FSS screen embedded within the layer. Figures
4.30 (a) and (b) plot the real and imaginary parts of the permittivity o f the five carbon
fiber materials as a function o f frequency. The MGA optimization is performed in the Xband (8.2-12.4 GHz) and linear interpolation is utilized for the carbon fiber materials, in a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
130
manner similar to that applied to the lossy dielectric and magnetic materials in the
previous sections. Case-7 is repeated with the carbon fiber materials included in the
optimization, and with an eight-fold symmetry incorporated in the FSS screen design.
Carbon Fiber Composite
18
9
10
11
12
13
Frequency (GHz)
Carbon Fiber Composite
-e— • — • — e
8
9
10
11
12
13
Frequency (GHz)
Fig. 4.30. The permittivity o f carbon fiber material as a function of
frequency: (a) real part; (b) imaginary part.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
Figure 4.31 depicts the composite with an FSS screen whose x- and yperiodicities are both 4.44 cm. For a PEC FSS screen, the figure also shows the best- and
worst-case reflection coefficients over the frequency band of interest. We observe that the
worst-case reflection coefficient is maintained below —14.0 dB and the total thickness of
the composite is equal to 5.23 mm.
sr’=4.34, Er” = 0 .1, d=2.25 mm
m = 11.25g, d=2.85 mm
m = 5g, d = 0.05 mm
sr’=1.03, 8r” =0.1, d=0.08 mm
(a)
-12
(b)
• r•
»
r
-14
♦
CD -16
■o
<u
a
£O)
o
u
e
0
1
c«
0£
i
♦;
i
■i
t
^
-t
■1■" ■
♦
♦
-18
♦ worst case
best case
-20
-22
+
•r
i
-24
'r *
-26
-28
-30
1+
— - *■
8.5
»
9
t
i
9.5
10
10.5
11
11.5
____+
12
12.5
Frequency in GHz
(C)
Fig. 4.31. Design o f carbon fiber composite with FSS screen embedded
in it: (a) FSS unit cell design (white - metal, black - free space); (b)
composite design; (c) frequency response of composite in (b).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
132
4.3.2 Spatial Filter Synthesis
To synthesize spatial filters, we follow the same procedure as described in Sec.
(3.3.1), except that we now embed a single FSS screen within dielectric layers that are
lossless, and remove the PEC backing. The FSS unit cell periodicities, its position within
the composite, its design, and material content are encoded in binary by the MGA and are
given by equations (4.68) through (4.71), respectively. The fitness function employed can
be expressed as
*4j £ a; (max(r(<ffi),rraln(tffi)))
(4.74)
F = -
(dB)))
where T and T are the reflection and the transmission coefficients, respectively, A and B
are additional weight coefficients that are problem-specific and are introduced to equalize
the two summation terms. Also, M and M are the number of frequencies for which the
reflection and transmission coefficients are minimized to fixed, problem-specific values,
corresponding to
and Tmin, respectively.
4.3.2.1 Numerical Results for Spatial Filter Synthesis
The objective of the spatial filter synthesis is to design a composite comprising
lossless dielectric layers and a single FSS screen that satisfies the following
specifications:
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133
(i) incident wave polarization: TM
(ii) pass band: 960 MHz - 1.215 GHz for 0 = 70 degrees and <J>= 0 and 90 degrees
simultaneously.
(iii) stop band: 2.0 - 18.0 GHz for 0 = 70 degrees and 4> = 0 and 90 degrees
simultaneously.
(iv) pass-band transmission: Transmission better than -0.5 dB.
(v) stop band attenuation: Transmission less than —15.0 dB.
(vi) total composite thickness: 12.7 mm.
(vii) minimum layer thickness: 0.05 mm.
Four layers o f dielectric are employed in the filter design along with a single FSS screen.
Small losses are introduced in the lossless dielectric layers by assigning s r =0.1 to
simulate practical materials. The MGA is given the choice of selecting a lossy or lossless
material for the FSS screen.
0 dB
■20 dB
start
fend
Fig. 4.32. Idealized response of spatial filter.
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134
The desired frequency response for the transmission coefficient is displayed in
Fig. 4.32, where f c
u t i , f c u t 2 , f sta rt,
and f
end
are 1.2, 2.0, 0.8, and 18.0 GHz, respectively.
The frequency increments are 0.2 GHz and 0.025 GHz for frequencies ranging between
0.8-2.0 GHz and 2.0-18.0 GHz, respectively. The values of M and N are 6 and 40,
respectively, while,
rmin=Tmin= -20.0
dB. Finally, the weighting coefficients are
selected as follows: A = 5 0 0 ,5 = 1 , and
1.0 forj=l,2
a,= 1.1 forj=3, 4, 5,6
1.2 forj=7 to 47
The number of generations and the population size for this filter design are 300
and 5, respectively. Figure 4.33 shows the optimized composite, the FSS screen design,
and the frequency response of the filter. Except for a single peak near the 8.0 GHz point,
the transmission coefficient values over the frequency band meets the design criteria.
The FSS screen has a periodicity of 13.18 mm in both the x- and y-directions. The
loss in the FSS screen has identical resistive and inductive components equal to 1.73
Q/square. It is to be noted that the MGA was optimized at 0 = 70 and <j>= 90 degrees. The
design criterion requires that the frequency response of the composite be identical at both
<j>= 0 and 90 degrees. This is achieved by employing the implicit scheme mentioned in
Sec. 4.3.1.1, and also by observing that at <J>= 0 and 90 degrees the FSS screen appears
the same to the incident plane wave due to the imposed eight-fold symmetry.
Finally, a sensitivity study of the designed composite is performed to check the
feasibility of fabricating the MGA optimized design in practice. The frequency responses
o f the four cases studied are listed below.
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135
(i) composite in Fig. 4.33 with the er = 0 instead of 0.1 - solid line in Fig. 4-34.
(ii) original composite in Fig. 4.33 - dashed line in Fig. 4-34.
(iii) material values changed to that practically available - dotted line in Fig. 4-34.
(iv) material values changed to that practically available but the thickness of the
first layer is decreased by the same ratio (square root) by which the er increases dotted-dashed line in Fig. 4-34.
•o
-1 C
-15
-25
-30
Frequency in GHz
Fig. 4.33. Spatial filter design with single FSS screen embedded in lossless
dielectric medium.
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136
We observe that the frequency response does not change appreciably for any of
the four designs even with an appreciable variation in the constitutive parameters and
thicknessess o f the layers. This leads us to conclude that the MGA optimized design for
the spatial filter is robust, dependable, and can be fabricated in practice.
CQ
•a
c
.1
o
-
10 -
s
8
c
_o
’35
ai
E
c -20t2
E-25
-3Q
Frequency in GHz
Fig. 4.34. Sensitivity study of spatial filter design with single FSS
screen embedded in it.
4.3.2.2 The Domain Decomposition Approach Applied to the Design of Spatial
Filters
The domain decomposition approach is utilized to design both spatial filters and
broadband microwave absorbers when the required geometry resolution of the FSS unit
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137
cell is very fine, and a 16x16 pixel resolution, shown in Fig. 4.12, is unable to resolve the
geometrical details that are required to achieve the design specifications. This typically
occurs when the wavelength of operation lies in the millimeter regime and a larger pixel
resolution, viz., 32x32 or 64x64, is required to describe the FSS unit cell and accurately
model the surface current distribution. For the same geometry, the numerical procedure
for 16x16, 32x32, and 64x64 pixel resolutions requires about 0.65, 60, and 3780 seconds,
respectively, to compute the response for a single frequency. So, with the cpu time
increasing rapidly when a fine pixel resolution is used, it is not feasible in practice to
utilize the MGA in conjunction with the MoM code to design an FSS screen because of
the heavy burden imposed on the computational resources.
To circumvent this difficulty, a two-step approach is utilized to meet the design
requirements. The first step involves the development of a free-standing FSS screen,
which is later augmented by a multi-layer dielectric composite backing designed to
embellish the performance o f the FSS. The complexity of the task prompts us to employ
the MGA to develop the design o f the five-layer dielectric composite. In the second step,
the freestanding FSS screen is designed by using well-established concepts for single and
multiband FSS screen designs [11], [41-42]. Finally, the two separate designs are
combined to synthesize a composite structure that satisfies the design criteria.
The domain decomposition approach is applied to design a dual band radome
operating at high off-normal incidences using a composite with a single FSS screen
embedded in lossless dielectric. The radome simultaneously transmits frequencies
between 2.0 to 18.0 GHz and also at 95.0 GHz. The design to be presented has three
unique features: (i) broad passband ranging from S to Ku band (2.0 - 18.0 GHz) at the
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138
low frequency end of the spectrum of interest and a narrow millimeter wave band at 95.0
GHz; (ii) wide separation (77.0 GHz) between passbands; and (iii) good performance at
wide elevation angles, e.g., 70 degrees.
To design a single dual-band FSS screen that simultaneously transmits 2.0 to 18.0
and 95 .0 GHz, we require a variant of the Double-Square-Loop (DSL) patch element (see
inset in Fig. 4.36). This element exhibits resonances at two frequencies, determined
primarily by the dimensions of the larger (lower frequency) and the smaller (higher
frequency) loops. As shown in the inset of Fig. 4.36, the FSS cell consists of 9 square
loops. Two of these are small square loops, which resonate around 95.0 GHz, and are
inserted into a large square loop. The latter helps to pass through signals over a wide
frequency band ranging from 2.0 to 18.0 GHz. Three sets of these loops are stacked with
the ends of the three outer loops joined at the intersections. Accurate modeling of the
induced surface currents at 95.0 GHz requires the use of a high-resolution discretization
of the FSS element, comprising 64x64 pixels. This, in turn, increases the computational
burden substantially, but yields improved results over coarser discretizations, e.g., 16x16
or 32x32. However, the MGA is not directly applied to optimize the composite radome.
comprising a combination o f an FSS screen and a multilayer dielectric substrate, because
of the excessive computational expense as mentioned previously.
The MGA is used to optimize the thickness and relative permittivity of each layer
simultaneously, while maintaining the total thickness of the composite below a certain
limit. The upper limit for the total composite thickness is 1.75 cms. In virtually all multi­
layer radome designs, thick low-density layers i.e., layers with Sr <1.3 (foam or core
material) alternate with thin high density laminates (layers with Sr > 2.2) to provide for
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139
good bond lines. Bond lines are required between foam or core materials (er <1.3) to
provide a structural wall as materials with low density have relatively small bonding
surfaces. The range o f parameters used in the MGA optimization is governed by the
limits mentioned above. Due to mechanical considerations, the er of the first and second
layer is held constant at 3 .2 and 1.24, respectively. The thickness of the first layer is also
fixed at 0.1016 cms. Honeycomb or foam material (sr = 1.03) is used for the fourth layer.
The thickness and er o f the remaining layers are now optimized by the MGA to satisfy the
design criteria.
After designing the dielectric composite and the freestanding FSS, the last and
crucial step in the optimization procedure is to optimally position the FSS screen in the
dielectric composite. The placement of the FSS screen is important because it determines
whether combining the two degrades the individual responses of both the dielectric
composite and the FSS screen and leads to a poor response. Since mechanical
considerations rule out the possibility of introducing the FSS screen between either the 1st
and 2nd, or the 2nd and 3rd layers, there are only two allowed positions, viz., either between
the 3rd and 4th, or 4th and 5th layers. Both of these positions are individually analyzed by
using an FSS code capable o f handling the presence of both dielectric superstates and
substrates [15], Positioning the FSS screen between the 4th and the 5th layers was found to
give the best results.
Figure 4.35 shows the dielectric composite with the modified DSL/FSS screen
introduced between the 4th and 5th layers. The values of 6r and thickness of each dielectric
layer is determined by the MGA. The 64 x 64 pixel DSL/FSS screen is shown in the inset
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140
o f Fig. 4.36. The cell size o f the FSS screen is 0.18 x 0.43 square cms. Figure 4.36 plots
the frequency response of the FSS screen embedded in the dielectric composite for TE
and TM polarizations. Note that, in line with the specifications, the transmission
coefficient is better than -3.0 dB over the bands of interest. It is observed that the
insertion o f the FSS screen into the dielectric composite has a greater influence at the
higher passband (95.0 GHz) than it does at the lower one (2-18 GHz).
s r=3.2, d=0.1016 cm
er=l.24, d=l.O cm
sr=3.0, d=0.00508 cm
s r=1.03, d=0.489 cm
FSS superstrate
er=2.2, d=0.00508 cm
FSS substrate
Fig. 4.35. 64 x 64 pixel modified DSL
FSS screen embedded in dielectric
composite.
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141
0
Specular transmission coefficient in dB
-5
-10
-15
-20
-25
-30
-35
-40
-45
0
10
20
30
40
50
60
Frequency in GHz
70
80
90
Fig. 4.36. Frequency response of composite in Fig. 4.35.
TE
TM
Inset: Single cell structure of 64 x 64 pixel modified DSL FSS screen.
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100
Chapter 5
SYNTHESIS OF COMPOSITES COMPRISING MULTIPLE FREQUENCY
SELECTIVE SURFACES (FSSs) EMBEDDED IN DIELECTRIC AND/OR
MAGNETIC MEDIA
In the previous chapter we have dealt with the design of composite type of
absorbers that incorporated a single FSS screen embedded in various types of layered
dielectric media. The technique applied in that chapter is very efficient for analyzing
single screen FSSs, but suffers from some drawbacks when dealing with multiple screens,
owing to a rapid increase in the number of unknowns as we increase the number of
screens. For a single FSS screen with N unknowns, the size of the matrix to be inverted is
JV* For M screens the number o f unknowns increase to ( N i-N ? -......... ~N\0 and the size
of the matrix becomes ( N i-N ? -
rendering the direct method impractical for
all but a moderately small M. One solution to this problem is to use the Conjugate
Gradient (CG) or other iterative techniques to solve for the unknowns, thus
circumventing the need to invert a matrix. However, the disadvantage of using iterative
methods is that they frequently require an exceedingly large computer run time to yield
accurate solutions.
One approach to obviating these difficulties is to employ the Generalized
Scattering Matrix (GSM) [17-18] technique. In this method, also referred to as the
Scattering Matrix Technique (SMT), the solution to an FSS screen analysis is expressed
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143
in terms o f the generalized scattering parameters. In this approach, we derive, as a first
step, the generalized scattering matrices of the individual screens by using the MoM, and
of the dielectric/magnetic layers by following the procedure described in the previous
chapter for a single FSS screen. These matrices are subsequently used to generate a
composite scattering matrix for the entire system by using cascading techniques, as
explained in detail later.
Initially, broadband microwave absorbers with two FSS screens embedded in
lossy and/or lossless dielectric and/or magnetic media are synthesized for different angles
of incidence and polarizations. Finally, a spatial filter with two FSS screens embedded in
lossless dielectric layers is designed.
S.l Problem Geometry
A
’
Fig. 5.1 shows a structure incorporating S sub-composites, £ N, layers, where N,
i= I
denotes the number of layers in the ith sub-composite, and S corresponds to the number of
FSS screens embedded in it. The PEC backing is introduced to evaluate the effectiveness
of the absorber but is removed when we design a spatial filter, where transmission
through the filter is o f interest. The MGA is applied to optimize various parameters, for
instance the thickness and constitutive parameters of each layer, the FSS screen designs
and materials, their x- and y-periodicities and their placement within the composite. The
result is a multilayer composite that provides maximum absorption o f both TE and TM
waves simultaneously for a prescribed range of frequencies and incident angles. The
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144
MGA automatically places an upper bound on the total thickness of the composite. For
lossy dielectric and magnetic material values the MGA accesses the lookup tables in
Chapter 3.
1st FSS screen
1st sub-composite
with Ni dielectric
layers
mN,I’ d\,l
mt:, 4t;
<
ni22, 422
2nd FSS screen
2nd sub-composite
with Ni dielectric
layers
mN,2, d N%,
J
S'11FSS screen
S'11sub-composite
with Ns dielectric
layers
N ,- I ) S ’ U (N ,-1 |S
ZL
PEC backing
Fig. 5.1. Problem geometry.
X
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145
5.2 Analysis of Multiple FSS Screens Embedded in Dielectric and Magnetic Medium
From the fundamentals of circuit theory we know that a two-port system can be
represented schematically as shown in Fig. 5.2.
reference plane 1
reference plane 2
Z = Zi
Z = Z2
ai
a2
Two-port
system
bi
b2
Fig. 5.2. Schematic representation of a two-port system.
The complex amplitudes of the incoming and outgoing wave can be expressed in terms of
the currents and voltages at the two reference planes as [43]
a
=
x
-4- /
Z"
N
7
2f t
_ ^ ( r (1.2)’/ )
7 ^(*(1.2)’*)
2&
2 ^
( 5 j)
where V and I are the voltages and currents evaluated at the reference planes rn ;i,
respectively, / is the time, and
Zo
is the characteristic impedance of the network. The S-
parameters for the two-port network shown in Fig. 5.2 can be expressed in terms of the
a ’s and the b's as follows:
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146
A
— *1*^11 "^"*2*^12
(52)
b-. —
Alternatively (5.2) can be expressed in matrix form as b = S a, where
a=
a\I
fA l
,b = i
b2
az
,s = 'Sn
Ai
Spl
I_
s ::_
Using the linear systems approach we obtain
S„ =
V
& ,=
(5.3)
5,2 =
'A , '
'A '
s,,=
1*2 / J, =0
1*2 / ^=0
When extended to the w-port case, the above analysis results in an n x n scattering matrix
S with vectors a and b containing n elements.
As explained in Chapter 4, the spectral domain approach can be applied to
determine the electromagnetic fields at any plane z = z„ for a plane wave incident on a
single-screen FSS structure embedded in an inhomogeneous medium. The fields are
expressed in terms o f Floquet’s harmonics, where the nlh harmonic corresponds to the nth
port in an n-port system. A set of generalized scattering parameters incorporating the
vector nature o f the electromagnetic fields for an w-port system can be defined as follows
[17-18]:
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147
TE
S™TE( m ,n J ,j ) =
A x . ’ A>„ ’ A j ’ A y ,
1
*L
7 <-.TE
A x . . A y . - A t , >A y , 7 * £
Sun, (m ,n \i,j) =
J P n s ( iJ )
r r ( * .-
S™e ( m , n j , j ) =
inf (w ,/i;/,y) =
SgTE(m ,rr,ij) =
ym
T \t
V TE
TE
S\2TU
Ax.,Ay„,AV A y,,^
Ax.>Ay.,A,,Ay,,^
A x . ’ A y . ' A x, ’ A y , »-R
A x . ’ A y , ' A t, ' A y , ' “ R
~
iO i)
yi-.-)
S™E{m ,n\i,j) =
ym
A x . ' A v . >A t , ' A y , i - R
w
)
y i - .- l
yTX!
ini
A x .. A y . , A i . A y , ^ «
=
S™TB( m , n J j ) =
TE
y<-.~
Slmt { m j r , i j ) =
y TE
vm
A x . ’ A y . ’ A t!
1
A y , >- £
Ax.’Ay.’Ax^Ay,^
Ax..Av,,Av Ay,,-,
jT £ {i.J)
y \~ .~
S S k i{ m w ,j) =
ynr
A x . > A y . » A s > A v , ' Z £.
V o u )
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(5.4)
148
-ri
V ’ ’ ’-// ”
r ;- - y ..A .4 .A ,- - - « )
/
,//k R u )
(5.5)
JW o J )
^22TM
J)“
V o u j
In Fig. 5 .1 ,r = r/? and 2 = 2*. correspond to the upper and lower interfaces of the S'* sub­
composite, respectively, and
’K
. A .„ - ^
;
vn i " ’ ( A
. *A . - A
-
A, ' - ) = 7
.A ,
K
'* -A - ) ‘i-1 1
, : ) = a (m ,n , /, j , 2 )l "
\ l PTE ( m < « )
(56)
1 ,]Pn i (m,n)
are the normalized Floquet voltage waves. Referring to the coordinate system in Fig. 5 1
we observe that the (±,±) notation in (5.6) denotes the waves traveling in the positive
and negative-2-directions, respectively. The first element in the parentheses corresponds
to the scattered waves and the second to the incident. Thus,
can be
interpreted as the (m,n)‘hharmonic reflected along the +2 direction from the leftmost
reference plane when the (/,y)rt plane wave is incident from the -2 direction on this
plane. The other terms in (5.4) and (5 .5) can be interpreted along similar lines. From (5.6)
we see that the voltage waves are expressed in terms o f the transforms of the electric and
magnetic vector potentials, f ( m , n , i , j , z ) and a ( m ,n ,i,j,z ) , respectively. To compute
the scattering parameters these potentials need to be evaluated at the reference planes
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149
located at the extreme right and left of the FSS. Thus, the vector potentials f { m ,n , i ,j , z )
and a (m ,n ,i,j,z ) can be expressed as
/ (m, n, /, j , z)[z'z} = F (im, n, /, j ) 1z'~1e/m":
a ( m ,n ,ij,z ) ~ ~ = A (m ,n ,i,j)" ~ «?'-•
(57)
Xmn( : ) = - j K +P i - P i for o r ,
v _________
= +y l K +P k ~ P l for z < z R
(5.8)
where
and the sign o f the radical is chosen to satisfy the radiation condition. Two approaches
can be utilized to evaluate the scattering parameters of an FSS screen embedded in the
layered media. In the first approach, one computes the scattering matrix for each layer
and the FSS screen separately, and then cascades them by using the scattering matrix
technique to generate the scattering parameters for the entire composite. In the second
approach, one computes the scattering matrix for an FSS screen embedded within a
number of superstates and substrates, rather than just for the screen by itself. The
reference planes for the latter approach are taken to be at the extreme left and right of the
entire sub-composite. The secondapproach is oftenconsidered to be preferable because it
requires a smaller number o f harmonics to be included in the scattering matrix, especially
when individual layers are thin and the first approach becomes vulnerable to errors unless
a large number of harmonics are included in the matrix.
In (5.4) and (5.5) the normalization factors are expressed as follows:
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150
/ £ > . » ) = (A - A ) 1'™
« ( < » .» ) =
(A* A)
„
(59)
7M
mn
where
_ J®£
y mn
Ymn
JVM
and the other terms are defined in (4.37) and (4.1). The scattering matrix only includes
the propagating harmonics and they constitute a finite subset of the infinite spectrum of
Floquet’s harmonics.
Applying (4.10), (4.11) and (4.12) to (5 .7), and expressing it in terms of the total
electric fields in the spectral domain, we obtain [ 15]
MAA. - AAJ
(5.10)
E~\
can be found as follows:
Aa.(A.A)-C,.
£ ,jA .A ) -£ n
where E
' x -y>( r t f .tram )
- a Ma - a )
- a Wa - a )
(5.11)
are the reflected and transmitted electric fields in the layered media
with the FSS screens removed, evaluated at z =
J
zr
and z = zi (see Fig. 5.1). As the
reflected and transmitted fields in a layered media follow Snell’s law o f reflection and
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151
transmission, these fields are added to the total fields only when the incident harmonic is
equal to the scattered harmonic, as indicated by the Kronecker’s delta functions in (5.11).
Efxy) is the scattered harmonic evaluated at z = zr and z =
zl,
following the approach
outlined in Chapter 4, for analyzing a single FSS screen embedded in layered media.
Substituting (5.10), (5.9), and (5.6) into (5.4) and (5.5) leads us to the final
expressions for the scattering parameters of a single sub-composite. The expressions for
Sn are given below.
SurE( m , n j , j ) -
r— ------ ^ ,
-r
vK- p i + P i M ' K )
VK
•/»;)(*•*)
(5.12)
S ^ ( m ,n ,i ,j ) ^
r
/ mn
or.w /
•
1
1
v\ ymnT X lvYijT E
m „P
I17™'
ij
/• mn
\y™
I
mn
( P E~
+3
t >iam})
p l + X M + K )
The other scattering parameters can be derived in a similar manner. It is worthwhile at
this point to comment on the details o f the structure and the contents of the scattering
matrix. It is evident from the analysis in Chapter 4 that a periodic FSS screen scatters an
incident plane wave into a discrete spectrum of Floquet’s harmonics. In a multi-screen
system, the fields incident on one screen are the harmonics scattered from the other.
Thus, the scattering matrix must contain the scattered field representations due to an
entire spectrum o f incident field harmonics. The scattering matrix elements correspond to
the expressions given in (5.4), (5.5), and (5.12). First, the composite scattering matrix is
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152
divided into four quadrants, one each for Su, S 12 , S21 , and S 2 2 . Next, each quadrant is
further sub-divided into four quadrants, corresponding to the two co-polarized
( Sf* and S™ ) and cross-polarized ( S™, and S™ ) components. Each of these sub­
quadrants consists of a two-dimensional array of elements corresponding to the scattered
fields due to each incident harmonic. The complete description is pictorially presented in
Fig. 5.3.
c r£ , «
J U 7£-„
■
II
5=
te„
c
_
i>\,
.1
L -1
S .j
22
'
“
r» scat * 1
inc«l
r» sea l»1
^ m c ff2
r»jaaf®2
inc»l
a scat *2
me *2
•
sS
r u «»
nTe
E «„
l_ ,lr
•
Sc nt~
u rr.v/«.
-v,« ..
•
•
s sca'l'
ne*S
1
•
sc a t» .V
inc*\
Fig. 5.3. The scattering matrix structure and contents for
composites comprising multiple FSS screens embedded in layered
media
Theoretically speaking, we need to include an infinite number of harmonics in the
scattering matrix to derive rigorous results. However, this is not feasible in practice, and
we must truncate the matrix by retaining only a finite number of harmonics, say N. It is
not altogether too obvious how this number N is to be determined, so that we have a good
balance between accuracy and computational efficiency. In general, the scattering
parameters o f FSS structures decay as exponentially damped sinusoids for increasing
and Az . Consequently, the scattering parameters can be expressed in terms of the
envelope o f the decay (exponentials) as
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153
W - e " ' - 4*
From (5.13) we note that, for a fixed A r, we can find a value of ym„ = p
(5.13)
such that the
scattering parameters Smn are essentially negligible as are those for which ymn > y
Furthermore, we observe that y
decreases with an increase in Az, implying that fewer
harmonics need be included in the scattered field representation as we move further away
from the screen (obviously, the converse is true when Azis decreased). As mentioned
earlier, in the first approach used, the reference plane is often chosen to coincide with the
plane o f the FSS screen (Az=0) when creating the scattering matrix and to achieve
desirable accuracy levels it may be necessary to include an inordinately large number of
harmonics in the scattering matrix. In contrast, the second approach does not suffer from
this problem, and is therefore the preferred scheme for creating the scattering matrix.
The next question of interest is: what is the cutoff-level for |5mn| below which we
can approximate it as being numerically zero? A rule of thumb has been established on
the basis of several numerical experiments in which the composite solutions, generated
by using different number o f harmonics were compared with a benchmark version ([43],
[34]). This criterion can be stated as follows: if the magnitude of the higher-order
harmonic is -22 dB or higher, then it and all the lower-order harmonics - even if these
harmonics have a magnitude less than -22 dB - should be included in the scattering
matrix. Once the higher-order harmonics have permanently decayed to a value of -22 dB
or less, and none is greater than -22 dB, the matrix can be truncated.
After the formation o f the scattering matrix we need to combine the matrices to
form the composite scattering matrix. For this we need to follow the linear systems
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154
approach to cascading the sub-systems. Shown in Fig. 5.4 is a two-sub-system composite.
We will apply the cascading technique to derive the expressions for the scattering
parameters of the composite system shown in Fig. 5.5 in terms of the scattering
parameters of the individual sub-systems, where the ar’s and the b's correspond to those
in Fig. 5.2 with the superscripts denoting either the sub-system-1 or -2.
a\
a;
4
<----Sub-system-1
Sub-system-2
<------b:
—
b;
-----
<------b\
b;
Fig. 5.4. System division into sub-systems.
a;
------- ►
< -------b;
___ a.
Composite
System
----- ^
b\
Fig. 5.5. Composite system.
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155
The matrix form corresponding to sub-system-1, -2, and the composite system can
s 2 1 "«r
_S;,
S ;2 j _a\
' b\~
b:
IV
J n
Js ic: J a !
X .
s :c: j _a;
,
i j
t
1
~s2
So
S l j l a\
I
1
A
r„,
X
b\
where the expressions for
___ 1
r.,-i
i
be expressed, respectively, as:
(5.14)
in terms of S‘ and Sr need to be derived.
Also, from Fig. 5.4 we have the auxiliary relations
b\=a{,b; =a\
(5.15)
The first two matrix equations for the sub-systems in (5.14) can be re-expressed to read
A,1 = 5 '1a 1l + 5 ,‘:a i
(5.16)
b\ =
(5.17)
b2 = S 2xa; + S 22a;
(5.18)
b; = S;xa 2 + S;2a;
(5.19)
Substituting (5.15) and (5.18) in (5.17) we obtain
of = TS^al + TS\2S 22a:
(5.20)
where, T = [l - 5 ^ 5 , ] '. Using (5.20) in (5.19) leads to
b; = [5 ;17’5’i1]a 11+ [S;JS\2S;2 + S;2]a:
&
Following the same procedure an expression for A,1 can be derived as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.21)
156
(5.22)
■v
where, R = [l -
]
The matrix form for the composite system (last equation in (5.14)) can also be expressed
as
(5.23)
(5.24)
Comparing (5.23) with (5.22), and (5.24) with (5.21), the scattering parameters for the
composite system in terms of the scattering parameters of sub-systems-1 and -2 can be
expressed as
(5.25)
If more than two sub-systems make up the multi-layered structure, then the composite
system is formed by repeatedly cascading the additional systems to the composite one
until all the layers have been added.
For both the spatial filter and broadband microwave absorber designs, equal
periodicities for the FSS screens have been assumed. This restriction imposed on the
optimization process is due to the GSM technique and is not inherent to the MGA.
Unequal periodicities can be handled by the GSM technique only when we can identify a
common period in the entire system. More often than not, the system period is either
equal to the larger of the two sub-system periods or an integral multiple of the same. The
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157
FSS unit-cell design is also assumed to be identical for the two FSS screens in order to
reduce the number of parameters to be optimized by the MGA. Neither of these
restrictions compromised the optimization process, and the MGA successfully generated
composite designs that satisfied the specified design criteria quite well.
5.3 MGA Formulation for Spatial Filters and Broadband Microwave Absorber
Designs
We will now apply the MGA to optimize various parameters pertaining to the two
design problems of interest. Binary tournament selection (Sec. 2.1.4.3) will be employed
along with uniform crossover (Sec. 2.1.5.3) with a crossover probability of 0.5. Elitism
will be utilized along with a population restart strategy that will be applied when the
bitwise difference between the best and other individuals/chromosomes in the current
generation is less than 5%.
5.3.1 Broadband Microwave Absorber Synthesis
Figure 5.1 shows a multi-layered multi-screen composite structure whose
parameters we wish to optimize, with a view to realizing a specified frequency response.
The composite is divided into S sub-composites, each comprising of N, (number of layers
in the s'* sub-composite) dielectric layers. The parameters for each sub-composite are
generated separately by the MGA. Assume that we are given a set of different materials
M m with frequency dependent permittivities e,(f) (/ = /
M m). For any single sub­
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158
composite, the MGA determines the following: (i) material parameters of N layers; (ii)
design o f the FSS cell element; (iii) cell periodicity of the FSS; (iv) position of the FSS
screen within the dielectric sub-composite; and, (v) the FSS screen material. For S, such
sub-composites, the same process is repeated such that the combined sub-composites
exhibit a low reflection coefficient for a prescribed set of frequencies f ( i = / , ........... F,)
and incident angles 9, (i = / ,
A q), simultaneously for both the TE and TM
polarizations. In the context of the present problem, the magnitude of the largest
reflection coefficient is minimized for a set of angles, for both TE and TM polarizations,
and for a selected band o f frequencies. Hence the fitness function can be written as
^\\’
F
\
S
I >• • J » .Vl 1 ’
’ • ■•
S’
t
\
! ’ ••
t
’ ■ -‘ x j ’
t
\
S ’
t
J y , >
g , ••
S’p o j ,
°
1
Sp a j j
= - max {T ni («../,)}
<526>
’ • • •'J
where mtJ is the material parameter of the i,h layer in the f h sub-composite; d,} is the
thickness o f the ith layer in the f h sub-composite; SdeSi is the unit cell design of the FSS
screen embedded in the ilh sub-composite; t and tv are the periodicities of the FSS
screens in the x- and y- directions, respectively, for the ith sub-composite; S
determines
TE
the placement of the FSS screen in the ilh sub-composite; and, T ni (# ,,/,) is the
reflection coefficient as a function of polarization, incident angle and frequency.
The MGA operates on a binary coded version of the parameters. The coded
representation of the coating consists of a sequence of bits, containing information
regarding each parameter. Given a database of M m = 2Umb different materials, the material
choice for layer / in the f h sub-composite is represented by a sequence LtJ of A /^ bits as
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159
A = /;/,;
-C
(5.27)
The thickness of the ith layer in the f h sub-composite can be encoded and represented by a
sequence Dt] of
bits as shown below:
(5.28)
The periodicity of the ilh FSS screen can be encoded depending on its permissible range
of values and the step size (see Table 5-1) utilized by the MGA. It can be represented by
a sequence T of M t b bits for the x-directed periodicity, and similarly by T of M t b bits
for the y-directed one, as follows:
A =1^ .............................................................. (5 29)
(5 30)
For an N layer sub-composite, the FSS screen can be placed within the composite
at N -l positions. For example, for a four-layer sub-composite, the FSS screen can be
inserted at the following three positions; (i) between the first and the second layers; (ii)
between the second and third layers; and, (iii) between the third and fourth layers. Thus,
the position o f the FSS screen within the ith sub-composite can be encoded by a sequence
P, oiMpb bits and represented as
P,=P!P,2.............P :1*
(5.31)
The MGA designs the FSS cell structure automatically by choosing the geometry
pattern of the unit cell. The code analyzing the FSS screen embedded in dielectric media
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160
utilizes a 16 x 16 discretization (32 x 32 and 64 x 64 discretizations can also be handled)
o f the periodic structure unit cell in the form of l ’s (ones) and 0’s (zeros), where the 1’s
correspond to PEC or lossy metal, and the 0’s to free space. As explained next, the MGA
randomly generates this 16 x 16 gridded stricture filled with l ’s and 0’s. Figure 5.6
shows such a cell geometry.
0 0 0 1 0 1 1 1 1 1 1 0 110
0 11
1111110 1
111111I0
0 1
1 1 ON) 1 1 0 0 1 1 0 0 1 0 0 0
1 1 0 ONI 10 0 1 1 0 0 1 0 0 0
0 111 0\) 1 1 I 1 1 1 1 I 1 0
0 1
0 1 1 1 1 IX) 0 I I n
0 0 0 0 I 0 l \ |l 1 1
I0
b’o'ooToTiM 1110 1 0
o 1 1 1 1 0 oil 11 I 1 1 0 1
o 1 1 B ill 11 1 1 1 1 0
oil 1 0 0 1 0 0 0
11 o d
1 1 0 ( 1 1 0 oil 1 0 0 1 0 0 0
0 1 1 1 0 0 1 ill 1I 1 1 1 I 0
0 11111 ooll 11 I I 10 1
Rl6 0 0 0 0 1 0 1 1 jl 1 _ 1 ] 0 _ 1 1 Oj
ya
Fig. 5.6 FSS unit-cell with either two-fold,
four-fold, or eight-fold symmetry
The MGA considers each row in the FSS cell to be a parameter. For each row, the
MGA generates a random number between 0 and 2Ucb-l, where Mcb is the number of
columns in the FSS cell matrix. The random numbers are converted to binary format for
each row. These binary numbers are combined into an array, which is ready to be
analyzed by the FSS code. The number of columns considered depends on the type of
symmetry introduced into the FSS screen geometry. The regions A, B and C in Fig. 2
represent the sections considered for eight-fold, four-fold and two-fold symmetry,
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161
respectively. This reduces the effective number of MGA parameters needed to design the
FSS cell, (see Table 5-1) resulting in efficient optimization. Thus, the FSS cell in the i,h
sub-composite can be designed by encoding each row into a sequence (CD),, of MCb bits
as
(C D \= (C D )'ti(C D )l................. (C D )"-
(5.32)
where j is the number of rows considered as parameters by the MGA, and j max is the total
number o f rows in the FSS cell structure.
Additional bits are needed to assist the MGA in selecting lossy or lossless
dielectric layers and a lossy or PEC FSS screen. As there are two choices in each
selection, we need one bit for each, totaling two bits. These two parameters can be
represented by sequences (CL),, of 1 bit and (CS), of 1 bit as follows:
(CL)„ = (CLt,
(5 33)
(C S)r
(5.34)
and
( C S \'
If a lossy FSS screen is selected, the real and imaginary parts of thecomplex
sheet impedanceappear as two new parameters, whose values are to beassigned by the
MGA. These parameters can be encoded in accordance with their allowed range of values
and the step-size (see Table 5-1) utilized by the MGA. The real and imaginary parts of
the loss can be represented by a sequence (FLR)j of M Jlrbbits and (FU), of M m bits,
respectively, as shown below:
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162
( F L R l^ f lr l'iJ l r ) ;
(5.35)
(F U \
(.fli) / " “
(5.36)
The entire composite, can be represented by the sequence C, which is referred to as a
chromosome:
C = CmCDCTxCTyCPCCDCCLCcsCFWCFU
(5.37)
A ,=A>- ....Av, 1A 2.... -Av,: ..... A ,.... •AvjS
(5.38)
C o = A i .....A VA : ..... Av .... A s-
(5.39)
where
/>•..<
A V= A ..... r
(5.41)
(5.42)
11
(5.40)
•P
Q .= A ,..... A J
_!"0
............
C co=(C D )n . \ C D \ ( C D l r . ( C O ) ... .(CD)
ii
i CL),ACL)r.
•
(C C ) ,,:'
£
II
.... (CS)„
Cra?=(FZJ?),
( FLR)S
=(*£/),(FU\
(5.45)
(5.46)
(5.47)
Each term in (5.37) has been defined in (5.38) through (5.47) and each sequence in (5.37)
consists of
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163
l A ,, J , (W ..+ A /,» + 2)
(5.48)
+ [(S )* (M Iit +M lyl
+
+A-/,.,)]
bits, where all terms are as defined in (5.27) through (5.36).
5.3.1.I Numerical Results for Two FSS Screens Embedded in Lossy or Lossless
Dielectrics
The MGA has been successfully applied to the synthesis o f broadband microwave
absorbers in the frequency range of 19.0 - 36.0 GHz [44], The composite comprising of
two (the algorithm can handle any number o f layers) sub-composites is surrounded by air
on top and terminated by a PEC backing at the bottom. The number of dielectric layers in
each sub-composite is fixed at four for this exercise, though this number is flexible. The
FSS cell design and periodicities are maintained constant for both the sub-composites.
Though only dielectric layers with electric loss are considered, the method can be further
extended to handle both electric and magnetic losses as will be shown in the next section.
Table 5-1 shows the parameter search space for the MGA for a single sub-composite It is
evident that the number o f parameters and the chromosome length is dependent on the
type of symmetry used for the FSS screen. The total number of parameters and
chromosome length (total number of bits) for the entire composite is 42 and 310,
respectively, when eight-fold symmetry is imposed. The measured values of er and s '
of ten different lossy dielectric materials are considered and a database o f these values as
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164
a function o f frequency are shown in Tables 3-2 and 3-3, respectively. Linear
interpolation is used to generate the values of £r and s ' as functions of frequency for
intermediate material types; thus we effectively use 19 types of materials. To obtain the
values o f the real and imaginary parts of permittivity for even numbered material types at
a particular frequency, the odd-numbered material types are averaged. For example,
values for material “Type 2” at 19.0 GHz are obtained by averaging the material values
for “Type 1” and “Type 3” at 19.0 GHz. For lossless layers, small values of losses are
added, as it is not practical to find a perfectly lossless dielectric. Hence, if the MGA
selects the first layer as lossless, its er value is fixed at 0.01, while for all the other layers
the corresponding value o ff, is 0.1. Two cases were investigated, as listed below:
(i) Oblique Incidence (0 = 45.0, <j>= 0.0), TE and TM polarization
(ii) Normal and Oblique Incidence (0 = 0.0 and 45.0, <J>= 0.0), TE and TM
polarization
The population size and the number o f generations were fixed at 50 and 100,
respectively. The periodicity of the FSS screen in the x- and y-directions were chosen to
be equal. Only the resistive component of the complex surface impedance of the FSS
screen was considered and the frequency increment was chosen to be 1.0 GHz in the
MGA optimization. Following the guideline mentioned in Sec. 5.2, the scattering matrix
for each sub-composite was truncated with 25 harmonics. This number was obtained after
several numerical experiments to maintain the trade-off between computational speed and
accuracy. The simulation showed that the magnitudes o f the higher order harmonics were
less than -40 dB; hence, our truncation criterion was met satisfactorily. Eight-fold
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165
symmetry was applied to the FSS cell design to take advantage of the rotational
symmetry in the azimuth plane.
The MGA-optimized composites are shown in Figs. 5.7 (a) and (b) for the two
cases o f interest. Figures 5.8 (a) and (b) show the MGA-generated FSS unit cell design.
For both Cases-1 and -2, the x- and y-periodicities for the FSS cells are 8.863 and 23 .44
mm, respectively. The reflection coefficient in dB is plotted vs. frequency in Fig. 5.9 for
Case-1. The dotted curve in Fig. 5.9 gives the worst-case reflection coefficient realized
by the MGA while optimizing simultaneously for both the TE and TM polarizations and
for each frequency in the band of interest at 0=45 degrees. The worst-case reflection
coefficient can be mathematically expressed as follows:
V rn ax (rrE, r m )(indB )
(5 49)
where Nf is the number o f frequencies in the band of interest.
For Case-2, the worst- and best-case reflection coefficient values in Figs. 5.10 (a),
(b), and (c) can be interpreted
as follows. Over the frequencyband,the range of elevation
angles of interest, and both the TE and TM polarizations, the maximum and the minimum
values of the reflection coefficients correspond to the worst and the best results expressed
in dB. Mathematically, the worst-case reflection coefficients can be written as
V m a x [rnr(0 = O ),rnr(0 = 45)] (indB)
s,
V m a x [r™ {0 = 0 ) ,r m {d = 45)] (indB)
V maxjY7^ (6 = 0 ) ,Tre(0 = 45),f™ (0 = 0 ),I™ (6 = 45)] (in dB)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.50)
166
where Nf is the same as defined earlier in (5.49). The worst-case reflection coefficients
for Cases-1 and -2 are -19.57 and -19.2 dB, respectively, and the total thickness of the
composites are 6.5 and 6.48 mm. Figures 5.11 (a) and (b) illustrate the performance of
the MGA search process by plotting the variation of average and best fitness value vs. the
number o f generations for the two cases. The dips in the curves for average fitness value
indicate the generations at which the MGA performs the population restart, as explained
in Chapter 2. The population distribution for Cases-1 and -2, are shown in Figs. 5.12 and
5.13, respectively, as we move from the first to the last generation. The additional burden
o f optimizing over a range o f elevation angles results in a slight degradation of the worstcase reflection coefficient values for Case-2. Case-1 and -2 have run times of 21 and 42
hours, respectively. It takes the MGA approximately 21 hours to optimize for a single
elevation angle, simultaneously, for both the TE and TM polarizations. In the figures
showing the FSS cell structure, white represents metal and black corresponds to free
space, and an eight-fold symmetry is evident in the FSS cell structures for both the cases.
(All dimensions are in millimeters). Table 5-2 lists the parameters selected by the MGA
for each case. For both the cases, the azimuthal angle is fixed at zero degrees (<|> = 0.0, x-z
plane) in the MGA optimization program.
From Figs. 5.14, 5.15 (a), and (b) we see that the frequency response of the
structure remains relatively invariant of the azimuthal angle. The angular independence is
achieved via the use o f eight-fold symmetry imposed on the FSS element. Figures 5.16
(a) and (b) show that, for 0 = 45 and <j) = 0.0 and for both Cases-1 and 2, increasing the
number o f harmonics from 25 to 49 in the MGA has little or no effect on the result. This
confirms that an adequate number of harmonics were included in the scattering matrix for
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167
the MGA optimization procedure. To verify the rule of thumb mentioned in Sec. 5.2, the
magnitude o f the reflection coefficient in dB, of the worst-case higher order harmonic,
for each frequency is plotted in Figs. 5.17 (a) - (b) and 5.18 (a) - (b), respectively, for the
two cases. We note that the magnitudes of the higher order harmonics are well below the
-22.0 dB limit.
e r = 1.75, e r = 0.01, d =1.825
FSS screen I
e r = 2.526, e r = 0.1, d = 0.525
V
Sub- composite I
Type 1, d = 0.475
s r = 4.65, e r - 0.1 , d = 0.493
Type 14, d =2.025
e r = 1.93, e r = 0.1, d = 0.125
FSS screen 2
Sub- composite 2
s r = 3.612, e r = 0.1, d = 0.475
Type 14, d = 0.56
PEC backing
Fig. 5.7. (a) Composite design for Case-1.
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168
e r= 1.99,e r = 0.01,d =1.875
FSS screen 1
Type 2, d = 0.975
\
Sub-composite 1
Type 12, d = 0.025
&r = 3.26, 6 r - 0.1 ,d = 0.119
Type 13, d =1.625
FSS screen 2
6 r = 1.685, 6 r = 0.1, d = 0.175
y Sub-composite 2
6 r = 3.34, 6 r = 0.1, d = 0.775
Type 8, d = 0.918
PEC backing
Fig. 5.7. (b) Composite design for Case-2.
Fig. 5.8. (a) FSS unit cell design for Case-1.
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169
Reflection coefficient in dB
Fig. 5.8. (b) FSS unit cell design for Case-2.
;yJ
•30-
TE and TM
TE only
TMonjy
32-
Frequency in GHz
Fig. 5.9. Frequency response o f composite in Fig. 5.7 (a).
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170
-20
co
■a
su
o
£
8o
s
oa
Cu
ae!
worst case
best case
-28
-30
-34
-36
18
20
22
24
26
28
30
Frequency in GHz
32
34
36
(a)
-18-
cn
•o
-
20 -
-
22
'♦ •*“
-
-24S
ZJ
3
£ -268o
so -28ou
co -30ac
-32-
worst case
best case
+
-34-
±
+
i . .......
•f
-36'
18
I_____
20
- i ------------------------- 1--------------------------1________________ i________________ ■
22
24
26
28
30
32
■
34
Frequency in GHz
(b)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
171
Reflection coefficient in dB
-18
-
20
-
-
22
-
♦
*
*. * .♦ ♦
t
± j
+
-24
-26-28♦
+
-30-
worst case
best case
-32-34
8
20
22
24
26
28
30
32
34
Frequency in GHz
(c)
Fig. 5. 10. Worst- and best-case reflection coefficients of
composite in Fig 5.7 (b): (a) TE polarization; (b) TM
polarization; (c) TE and TM polarization.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
172
Fitness value
-0.1
-0.4
-0.5
Average fitness
best fitness
- O .f f
100
Number of generations
(a)
-
0.1
1 -0.4
C/S
8
I -0.5
-
0.6 Average fitness
best fitness
-
0.8
100
Number of generations
(b)
Fig. 5.11. Performance of the MGA: (a) Case-1; (b) Case-2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
173
°
0
-18
-16
-14
-12
-10
-8
-6
-4
-2
on
nPVHDflP T1rifmnrnnr o
°
Generation 25
---------1--------- \---------1---------1--------- 1-----L ------1---------1-------- 1
-J£___ 4 * ___ A?___ A$____ -?
-?
-4
-?
O
Thickness in cm
o c p o q y p g p iL iiiii
Generation 1
O OD
—
)C3)
0.5
OO
Generation 50
t p — --4o
0.5
A?— -J4— -J£.
O O O
(3D
AfL
-4-
4-
CD M i i B f l f ) n m n i i m o
4~
0 (3 )
GfineratiQnJl
0
?p
4 8 ___ -A£___ 4 4 ___ A£.
40-
4 ------4 ------ 4 ------ 4 Generatinn 1001 -
0.5
-20
-18
-16
-14
-12
-10
-8
-6
-4
Reflection coefficient in dB
Fig. 5.12. Population distribution w.r.t. total thickness of
composite and reflection coefficient for Case-1.
11
1
|
T11
1
o
0.5
^20
0.5
-14
\
1
Generation 1
-18
-16
-12
-10
OO
G D D 4B H B OCBD O
-18
-16
-8
.. °
E
c
-2Q
-14
-12
-10
-6
-8
-4
-2
Generation 25 ■
- 6 - 4 - 2
.
w
O
O
J
n rn n m in n iii) n
Generation 50
■
g 0.5
O
VI
V]
0
fl---- =1£-----A£---- 44------lfL .
o
o
Generation 75
0.5
0
0----- -18
-14
-12
-10
1
0
-20
-18
-16
-14
-12
-10
-?
O O O OO O l'IW IM IIBIII
-8
1 )
0
1
0.5
-16
4
-8
-6
°
-4
-2
Generation lOf 1 .
-6
-4
-2
Reflection coefficient in dB
Fig. 5.13. Population distribution w.r.t. total thickness of
composite and reflection coefficient for Case-2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
174
-18
-20
Reflection coefficient in dB
-22
-24
-28-
-32
-34
-36
Frequency in GHz
Fig. 5.14. Invariance o f the frequency response with
azimuthal angle for composite in Case-1.
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175
-15
-20
-25
€
c
-30
TE .0 = 0. = 0
TM. e = 0. ♦ = 0
TE. 9 = 0. <t>= 45
TM. 9 = 0. ♦ = 45
-35
eo
a
eo
a
-40
-45
-50
(a)
-18
-20
-22
! -24
I -26
£
§
*28
os
- -30
e3
TE. 9 = 45. <t>= 0
TM. 9 = 45. 4>= 0
TE. 9 = 45. <t>= 45
TM. 9 = 45. * = 45
-34
-36
Frequency in GHz
(b)
Fig. 5.15. Invariance of the frequency response with azimuthal angle for
composite in Case-2: (a) 0 = <J>= 0 degrees, 0 = 0 and <J>= 45 degrees; (b) 0
= 45 and <t>= 0 degrees, 0 = <t>= 45 degrees.
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176
-18
-20
c -24
a
-26
§ -28
oa
-34
TE-25 harmonics
TM-2S harmonics
TE-49 harmonics
TVI-49 harmonics
-36
Frequency m GHz
(a)
-1 8 1
-20
c
e
a
-24
TE-25 harmonics
TV1-25 harmonics
TE-49 harmonics
TM-49 harmonics
a -30
-34
-36
Frequency in GHz
(b)
Fig. 5.16. Effect of the change in the number of harmonics included
in the scattering matrix on the frequency response: (a) Case-1, 0 =
45, (f) —0; (b) Case-2, 0 = 45, <J>= 0.
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177
-50
-55
Reflection coefficient (dB)
-60
-65
x
O
-70
sub-com posite 1
sub-com posite 2
-75
-80
O 0
0
0
-85
-90
©
-
-95
20
22
24
26
28
30
32
34
36
Frequency (GHz)
(a)
-4 5 ----------- 1
1 i---------- 1— ■■■ i----------- 1----------- 1----------- 1-----------
-50 - .......... ;................................
..
«
i^
X Xx X x
„
X
- ....
--............ *.....*......-.....
*.....-
x
x
*
,
*
X -
Reflection coefficient (dB)
-55 - ..........i...............................
X
- -............-............. -..............-
-60
•
x
oo
<> o
sub-composite-1
sub-composite-1
sub-composite-2
posite-2
sub-com
:
- 7 0 --------- JO...
® o
"
O
-75 ------------•..........
=
o
-
-............ -............. ...............-
©
*
o
-
°
-80 --- ------- •.............. •..............•.............................. - ...-
*.............................*
..C
l
O—
Q ---
—
O
o
-8 5 1
--------- i--------- --------- i---------- --------- --------- ---------- ---------r
1
20
22
24
26
1
1
1
1
28
30
32
34
Frequency (GHz)
(b)
Fig. 5.17. Plot o f the worst-case higher order harmonics vs.
frequency for Case-1: (a) TE polarization; (b) TM polarization.
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36
178
-40------------i---------- 1 ■
r ........... i
i
i
i
............................ A. ...
-45 -.......................................................................... - ..
i---------X
1
Reflection coefficient (dB)
-50 •....................................................................... - .............-............................- 5 5 .................
» '
X
-*•....................
X
■
-
••......... - ............................. -.............-.............-....................
* J
,
-60 -
*
-
« « « x
x
-
„
-60 -
-
-
x
O
- I
-70 -...............................................................
o ° o
sub-composite-1
sub-composite-2
-.... ••
0
-
• ......................... O oO oO o(
oo O
© °Q O
©.........................................................
-75 -.......................... ©
O
.Q O
18
1
i
20
22
O
O
oo
O
O
i---------------1
24
oo
oo
i
26
28
Frequency (GHz)
i
i
30
32
i--------------
34
36
(a)
-45
-50
'
*
*
x
x
x
x
x
x
Reflection coefficient (dB)
-55
-60
sub-composite-1
sub-composite-2
o
-65
-70
-75
o o o O o o o
-80
-85
18
20
22
24
26
28
Frequency (GHz)
30
32
34
36
(b)
Fig. 5.18. Plot of the worst-case higher order harmonics vs.
frequency for Case-2: (a) TE polarization; (b) TM
polarization.
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179
TABLE 5-1 MGA PARAMETER SEARCH SPACE FOR ONE SUB-COMPOSITE
Parameters
One row
Total 16 rows
Range of
Number of
Increment
parameters
possibilities
Two-fold symmetry (rectangular section in Fig. 2)
0 -2 5 5
1
Number of binary
digits
256
1 6x8= 128
four-fold symmetry (square section in Fig. 2)
One row
Total 8 rows
Row 1
Row 2
Row 3
Row 4
Row 5
Row 6
Row 7
Row 8
Total
thickness of
one composite
(mm)
d„
da
dji
d4i
Tx,
FSS position
Choice of FSS
screen
material
Re(Z.)
Choice of
layer 1
material
B,(U)
Choice of
layer 2
material
8*2,1)
Choice of
layer 3
material
£*3,1)
Choice of
layer 4
material
e*4,l)
0 -2 5 5
I
256
Eight-fold symmetry (triangular section in Fig. 2)
0 -2 5 5
1
256
0 -1 2 7
1
128
0 -6 3
1
64
0 -3 1
1
32
0 -1 5
1
16
0 -7
1
8
0 -3
1
4
0 -2
1
2
8 x 8 = 64
|
8
7
6
5
4
3
2
1
0 .0 -3 .0
0.01
512
9
0.025-3.0
0.025-3.0
0.025-3.0
0.025-3.0
1.0-50.0
1.0-4.0
0.001
0.001
0.001
0.001
0.1
1
2974
2974
2974
2974
512
4
12
12
12
12
9
2
1.0-3.0
1
2
1
10.0 - 1000.0
I
1024
10
1.0-3.0
1
2
1
1.03-6.0
0.01
512
9
1.0-3.0
I
2
1
1.03-6.0
0.01
512
9
1.0-3.0
1
2
I
1.03-6.0
0.01
512
9
1.0-3.0
1
2
1
1.03-6.0
0.01
512
9
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180
TABLE 5-2 PARAMETERS SELECTED BY THE MGA FOR THE TWO CASES
Sub-composite 1
Dielectric Layer types
Layer I
Case
1
Case
2
Layer 2
Layer 3
FSS screen 1 properties
Layer 4
Position
between
Lossless
Lossless
Lossy
Lossless
1* 2nd
Lossless
Lossy
Lossy
Lossless
1st, 2nd
Cell
size
(mm)
Surface
Resistance
<%>
8.863 x
8.863
23 .44 x
23.44
714.8
667.87
Sub-composite 2
Dielectric Layer types
Layer 1
Case
I
Case
2
Layer 2
Layer 3
FSS screen 2 properties
Layer 4
Position
between
Lossy
Lossless
Lossless
Lossy
2nd ^rd
Lossy
Lossless
Lossless
Lossy
1st, 2nd
Cell
size
(mm)
Surface
Resistance
8.863 x
8.863
23.44 x
23.44
<%>
842.85
956.23
5.3.1.2 Numerical Results for Multiple FSS Screens Embedded in Dielectric and
Magnetic Medium that is Either Lossy or Lossless
In the previous section we designed a composite comprising either lossy or
lossless dielectric layers with two FSS screens embedded in it. In this section we will
optimize a composite consisting of layers that could either be dielectric or magnetic
material and may be lossy or lossless. The magnetic materials, if selected by the MGA,
are always assumed to be lossy. The lossy dielectric materials used are the same as in the
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181
previous section, and, Figs 3.8 through 3.11 plot the frequency dependence of
er,e r,n r, and jur for the six lossy magnetic materials utilized. Linear interpolation,
similar to that applied to the lossy dielectric materials in the previous section, is again
utilized for the magnetic materials. All the optimization parameters remain the same as in
the previous section. Case-2 is repeated with the lossy magnetic materials included in the
optimization, and an eight-fold symmetry is incorporated in the FSS screen design.
Figure 5.19 shows the FSS unit-cell design (identical for both the screens), having x- and
y-periodicities of 5.03 mm. The surface resistances in the upper and lower FSS screens
are 658.1 and 330.62 ohms/square, respectively.
Fig. 5.19. FSS unit cell design for the two FSS
screens embedded in the composite in Fig. 5.20.
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182
e r - 1.5, e r = 0.01,
d =2.375 mm, lossless dielectric
FSS
screen 1
A
e r = 3.0, s r = 0.1,
d =0.275 mm, lossless dielectric
Sub-composite 1
Total thickness =
2.73 mm
e r = 15.35, e r = 11.35,
d=0.025 mm,Type 16, lossy
d ielectric
|*—4.77, E r 0.1,
d = 0.055 mm, lossless dielectric
6
FSS
screen 2
e r = 2.26, e r = 0.1,
d =1.175 mm, lossless dielectric
<
6 r=7.81,E r=5.33,d=0.525 mm,
p r= l . l l (p r= 0.027,Type 4,lossy
magnetic
>
e r = 5.57,e r = 0.1,
d = 0.025 mm, lossless dielectric
Sub-composite 2
Total thickness =
2.135 mm
e r = 9.0, e r = 6.25, d = 0.41 mm,
Hr = 1.0,(i r = 0.04,T ype9,lossy
magnetic
PEC
backing
j
Fig. 5.20. Synthesized composite for two FSS screens
embedded in layered media.
Figure 5.20 shows the composite with two FSS screens embedded in it. The total
thickness o f the composite is 4.865 mm - a 25% reduction in the overall thickness as
compared to 6.48 mm for Case-2 in the previous section, which dealt with a composite
consisting o f lossy and lossless dielectrics only. Figures 5.21 (a), (b), and (c), show the
MGA optimized worst- and best-case values o f reflection coefficient over the frequency
band o f interest for TE, TM, and both TE and TM polarizations. Comparing 5.21 (a), (b),
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183
and (c) with 5.10 (a), (b), and (c), we see that the inclusion of lossy magnetic materials in
the composite design results in an improvement of -4.0 dB in the worst-case reflection
coefficient while simultaneously reducing the overall thickness of the composite.
-24
-26” ♦
00
-a
c -26 ” +...+
••+
-
+
3 -3C ‘
4I>
o0 -32‘
|
1c -34”
o
cC
+
worst case
best case
-36”
-38 ”
-4Qr
8
20
22
24
26
28
30
Frequency in GHz
32
34
(a)
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36
184
-24
♦
26h
...
♦'....
.t
co 28
..
■a
e
3q— ......*....+--V'
0
+
1 -32h
± ....±
■+ - -
*
0 -34
1
i -36- -......*cu
O' -38.......... -
♦
+
worst case
best case
-40 ~
-42'
-441-------- 1-------- '-------- 1-------- 1-------- 1-------- 1—
18
20
22
24
26
28
30
32
34
36
Frequency- in GHz
(b)
-24
r***
-zq
1-------------* — + — !------------!------------- r
.
. - ♦
................. dir...
-i—
+
~r~
*
CO
■a -28 ■
f*
■g -30T
£1)
8 32"
I -34
!
+
f
i
+
;
*
+
worst case
best case
5=
U
at,
-36I"
-38-
+
+
i.
t
i---------1
i
-4C
8
20
22
24
26
28
30
32
34
36
Frequency in GHz
(c)
Fig 5.21. Worst- and best-case reflection coefficient of composite in Fig.
5.20: (a) TE polarization; (b) TM polarization; (c) TE and TM polarization.
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185
---- 1----
—
o
— 45—
------
o
a
B
o
0.5I
Thickness in cm
- ? n ____ — 4 5 —
C 1---
1
.o
n
Generation. 1
-------------
Generation 25
---------4 0 --------- ----- -S------------- -0
9
OOO OQDODBRD flDQDQDO CD u
Generation 50 .
eration 75
tmn 1Q0
-2 0
-1 5
-1 0
Reflection coefficient in dB
Fig 5.22. Population distribution with evolving generations.
-
0.1
-0.G-
-
Averaee fimess
best fimess
0.8
Number of Generations
Fig. 5.23. MGA performance to optimize composite in Fig. 5.20.
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100
186
5.3.2 Spatial Filter Synthesis
We employ the same procedure as described in Sec. (3.3.1) for designing spatial
filters, except we now embed multiple FSS screens in lossless dielectric layers and
remove the PEC backing. The FSS unit cell periodicities, its position within the
composite, its design, and the material content are encoded in binary forms by the MGA,
and are given in equations (5.29) through (5.34), respectively. The fitness function
employed can be expressed as
F =-
(551)
where T and T are the reflection and the transmission coefficients, respectively. A and B
are additional weight coefficients that are problem specific, and are introduced to
equalize the two summation terms. Also, M and N are the number of frequencies for
which the reflection and transmission coefficients are minimized to fixed, problemspecific values corresponding to r mn and Tmin, respectively.
5.3.2.1 Numerical Results for Spatial Filter Synthesis
We now present the results for a composite comprising lossless dielectric layers
and a single FSS screen that satisfied the following constraints:
(i) incident wave polarization: TM
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(ii) pass band: 960 MHz - 1.215 GHz for 0 = 70 degrees and <j>= 0 and 90 degrees
simultaneously.
(iii) stop band: 2.0 - 18.0 GHz for 0 = 70 degrees and <j> = 0 and 90 degrees
simultaneously.
(iv) pass-band transmission: Transmission better than -0.5 dB.
(v) stop band attenuation: Transmission less than -15.0 dB.
(vi) total composite thickness: 12.7 mm.
(vii) minimum layer thickness: 0.025 mm.
Eight layers of dielectric were employed, along with two FSS screens, to design the
composite. Small losses were introduced in the dielectric layers by assigning
£v=0.1,
as
fabricating a perfectly lossless dielectric is not feasible in practice. The MGA was given
the choice o f selecting a lossy or lossless material for the FSS screens.
0 dB
■20 dB
start
fend
f (G H z)
Fig. 5.24. Idealized response of spatial filter
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188
The ideal frequency response for the transmission coefficient is shown in Fig.
5.24, where fcuti, fcuti, /start, and f end are 1.2, 2.0, 0.8, and 18.0 GHz, respectively The
frequency increment was 0.2 GHz and 0.025 GHz for frequencies ranging between 0.8 2.0 GHz and 2.0 - 18.0 GHz, respectively. The values o f M and N were 6 and 40,
respectively, while,
= Tmn = -20.0 dB. Finally, the weighting coefficients were
selected as follows: A = 500, B = 1, and
1.0 for j= 1,2
a. =
1.1 forj=3, 4, 5, 6
1.2 forj=7 to 47
The number o f generations and the population size are 219 and 5, respectively.
Figure 5.25 (a), (b), and (c) shows the optimized composite, the FSS screen design, and
the frequency response of the filter.
e r = 3.1, e r = 0.1, d =0.4825
s r = 4.0, e r = 0.1, d = 0.0225
e r = 3.0, e r = 0.1, d = 0.0025
\
Sub- composite 1
FSS screen 1
e r = 6.6,8 r = 0.1, d = 0.0025
8 r = 2.3, 8 r = 0.1, d =0.5875
<
8 r = 6.0,8 r = 0.1, d = 0.0025
8 r = 4.0, 8 r - 0.1, d = 0.0025
y
Sub- composite 2
FSS screen 2
8 r = 6.8, e r = 0.1, d = 0.0025
(a)
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CQ
■o
Io
-io>
3
|
C/3
52 -15-
sC/3
i
-2G-
-2S
Frequency in GHz
(C)
Fig. 5.25. Spatial filter synthesis: (a) composite design; (b)
FSS unit cell design; (c) frequency response.
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190
The two FSS screens have a periodicity of 79.7 mm in both the x- and ydirections, and are lossless. The total thickness o f the composite is 11.05 mm, which is
less than the specified maximum. Comparing Fig. 5.25 (c) with the frequency response in
Fig. 4.30, we see that there is an overall improvement in the values of the transmission
coefficient, for both the pass and the stop bands, when two FSS screens are utilized
instead of one. However for the multiple FSS screen case, a lower minimum thickness is
needed for the dielectric layers, which may be difficult to realize in practice.
We note that the MGA was optimized at 0 = 70 and <t> = 90 degrees. The design
criterion requires that the frequency responses of the composite be identical at both <J>= 0
and 90 degrees. This was achieved by the implicit scheme mentioned in Sec. 4.3.1.1, and
also by observing that the eight-fold symmetry imposed on the FSS screen design makes
it appear to be identical to the incident plane wave at <t>= 0 and 90 degrees.
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Chapter 6
THE PARALLEL IMPLEMENTATION OF THE MICRO-GENETIC
ALGORITHM (MGA)
As described in Chapter 2, the genetic algorithm (GA) is a probabilistic search
technique, which navigates through the search space by making random choices, instead
of following a specific pattern. The steps we follow in the basic genetic algorithm is quite
simple: (i) begin with a random set (population) of points, where each point
(chromosome) represents a coding of a unique set of the problem's parameters; (ii)
evaluate the chromosomes individually to see how well they satisfy an object (fitness)
function; (iii) select individuals probabilistically to “breed” (selection) and exchange
portions o f their parameter encoding (crossover); (iv) occasionally mutate the
chromosomes to allow the exploration of new areas of the domain; finally, (v) repeat
steps (i) through (iv) until some member of the population satisfies a termination criteria
(i.e., the fitness is above a certain level) or a specified number of iterations (generations)
is reached.
GAs are highly successful in optimizing complex systems for several reasons.
First, they start with a population of initial guesses rather than a single one, and, hence,
they explore many regions o f the search domain simultaneously. This widespread search
also causes the GA to be less prone to (although not totally immune from) settling into a
local optimum during the search. GAs are a “weak” search method, meaning they do not
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192
rely on a knowledge o f the domain in performing the search, so the algorithm is less
likely to be led astray by inconsistent or noisy domain data as is common with hillclimbing or domain specific heuristics.
Although there are no rigorous guidelines available for deciding when to use a
genetic algorithm for a given optimization problem, there are several useful "rules of
thumb.” For example, a GA is most applicable when the search space of a problem is
large, multi-modal, and not necessarily smooth. A GA is also useful when the evaluation
(fitness) function is noisy or when a detailed knowledge of the characteristics of the
domain are not available. Perhaps the most useful application of the GA is to problems
that are multi-variate. Many heuristic algorithms are uni-variate, meaning they attempt to
optimize the solution one parameter at a time. In multi-variate problems, parameters
interact non-linearly, so a simple “one at a time” approach will not be successful. Another
important consideration is that a GA does not necessarily guarantee an optimal solution,
so a GA may not be suited to any problem where an optimal solution is strictly required.
Recently there has been a great deal of study on methods for parallelizing genetic
algorithms [45-46], The GA algorithm has a structure which is ideal for parallelization.
The process o f searching numerous points in the domain simultaneously can be carried
out very naturally in a parallel environment, rather than “one at a time” as is done in the
serial methods. Many o f the parallel GA techniques hold advantages beyond the simple
speedup of the basic GA algorithm.
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193
6.1 Reasons for Parallelization
There are many reasons for parallelizing the genetic algorithm. The most obvious
one, o f course, is the speed of computation. The GAs are computationally more
expensive than the more deterministic forms of search and optimization schemes. This is
due, in part, to the probabilistic approach taken by the GA, which requires it to explore
over a larger area than is needed in a deterministic algorithm. This is related to the fact
that numerous points in the domain are searched simultaneously, and this attribute makes
the GAs ideal for parallelization. In some cases, the GA can be parallelized such that
each individual chromosome is assigned to its own processor to perform necessary
computations. This essentially reduces the run time for each generation to that required to
perform the genetic operations on just one individual. Speed is not the only reason for
parallelizing a genetic algorithm. Some implementations o f parallel GAs have a
significantly higher cost than the serial ones, though the former increase the likelihood of
finding the optimal solution. A more intuitive argument for parallelizing a GA is simply
that if nature is parallel, then an algorithm that models systems in nature should also be
parallel. Genetic algorithms are built on the foundation that nature evolves solutions to
hostile environments all the time by recombining current patterns to make newer and
better patterns. If we are going to model a natural system, perhaps one that mimics nature
will be the best performer.
There are several additional mathematical reasons for parallelizing the GA. One is
to reduce the likelihood of premature convergence. Premature convergence occurs when
a few individuals with high fitness begin to dominate the population and cause all of the
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194
other chromosomes to become much too alike each other [45], When crossover is used
with such chromosomes, no new pattern is created, and this causes the search to grind to
a halt. Although, one purpose of the mutation operator is to prevent premature
convergence, this can still occur. This is due to the low probability of mutation and the
fact that mutated chromosomes may not necessarily survive the next selection process
Many of the parallel GA methods allow multiple populations to be kept, and this enables
them to explore different areas in the search domain. The concept of using multiple
populations has another saluting feature. In some cases, the objective function for a
problem is so complicated that attempting to simultaneously optimize the entire function
at once is difficult. Each sub-population can focus on a different area of the objective
function, and later combine them to reach the overall optimum.
There are several possible ways that one might run the genetic algorithm in a
distributed environment. For example, when speed is the most important factor, a global
GA is the simplest way is to distribute chromosomes without going through the effort of
segmenting the population and coordinating several separate GAs. Conversely, if an
optimal or near-optimal solution to a problem that has proven difficult for a simple
genetic algorithm is required, some of the more advanced techniques that use several
populations may be necessary. Similarly, some problems may be very sensitive to the
parameters of the GA. In this case, attempting to find the correct combination by trial and
error may be difficult, so a multi-layered GA that self-adjusts may be the only viable
solution. Other factors that influence the decision of which model to use is the number of
processors available and the processor to chromosome ratio. The type of parallel system
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195
available is also important {i.e., multi-processor - all processors on the same mother
board - versus a network o f physically distant machines).
It has already been demonstrated in Chapters 1 through 5 that the MGA is not
only a numerically efficient scheme, but that it also shows very good convergence
properties leading to solutions that satisfy the design criteria for most problems of
interest. However, we note that most of the FSS screens designed by using the MGA has
a pixel resolution of 16x16. Using a higher pixel resolution enables us to better
approximate of the induced surface current on the FSS screens. However, as mentioned
in Sec. 4.3.2.2, the time it takes to evaluate the reflection and transmission coefficients
increase dramatically with increasing resolution. To maintain the run times of the MGA
optimization such that they do not depend on the resolution of the FSS screen, a parallel
implementation of the MGA is required. Thus, as mentioned earlier, for simple speed-up
requirements, the global or Micro-grain GA [46] is the model best-suited for this
problem. We discuss this approach in the next section.
6.2 Micro-Grain MGA
The micro-grain micro-genetic algorithm (mgMGA), also referred to as a global
MGA, is probably the simplest form of parallel GA. Principally, a master process
maintains a single population. This master process generally performs the selection,
crossover and mutation operations while assigning all o f the fitness evaluations to other
processors acting as slaves. The task of evaluating a chromosome is usually more
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196
expensive than the others, allowing the majority of the work in the mgGA to be done
outside the master process. Ideally, there should be one processor for each individual in
the population, but the chromosomes can be distributed among any number of slave
processors. Speedup associated with the mgMGA for a population size of N and p
processors is given by:
S (p 'N ) = ^ ~ W
a +—
P
<6I)
where a represents the time required for the master process to perform selection,
crossover and mutation and b represents the amount of time required to perform an
evaluation by the slave process. Maximum speedup can only be attained when each of the
slaves receives an equal amount of work. In some instances, this means that sending an
equal number of individuals to each slave will result in sub-optimal performance, as some
evaluations may be more expensive than others in the same population. This is especially
true in our case because each composite has a different FSS screen design and this causes
the run times to be different. Sub-optimal performance could also occur when slave
processors do not run at a uniform speed, therefore, balancing the load among slave
processes is an issue with the mgMGA. The complete algorithm is given in Fig. 6.1.
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197
Master Process
population
nt
pop_size, generation;
float
p_cross, p_mutation;
. old_pop
4.
/* prob. o f crossover & m utation*!
= initial random population
2. for each sta ve p ro cess p
3.
I* current and n ext populations*!
old_pop, new_pop;
I* evaluate each chrom osom e *1
se n d a su b se t o/old_pop to p ro cess p
wait until all sla ves have returned their results
5. while(generation < MAX_GEN)
I* se le c t N m em bers o f old_pop *1
6.
new_pop = select(N, old_pop)
7.
new_pop = crossover(p_cross, new_pop)
8.
old_pop = mutate(p_mutation, new_pop)
9.
for each sla ve p ro cess p
10.
I* evaluate each chrom osom e *1
se n d a su b set <?Aold_pop to p ro cess p
11.
wait until all sla ves have returned their results
12.
generation = generation + 1
13. endWhile
14. re\um (individual with g rea test fitness)
Slave Processes
population
int
locaLpop;
local_size;
1. local_pop = receive one or m ore chrom osom es from m aster
2. evaluate(old_pop)
I* calculate fitn ess o f individuals *1
3. sen d locaLpop back to m aster
Fig. 6.1. Pseudocode for a micro-grain micro-genetic algorithm.
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198
The greatest advantage to the mgMGA method is its simplicity. This method does
not require a particular network topology, although a highly connected network is
desirable for reducing the communication overhead. The algorithm is roughly equivalent
to the serial MGA; hence, the serial MGA is usually a good indicator of whether a
particular problem will map well to the mgMGA. This is not generally the case with the
other distributed GA methods. Differing implementations of the GA network often cause
some methods to fail while others succeed. The mgMGA is generally a good method for
gaining speed with the genetic algorithm without introducing extra complexity issues.
There are also some disadvantages with the mgMGA compared to some of the
other distributed GA schemes. Perhaps the most important is that the mgMGA is not a
solution to the premature convergence problem, though this is not a problem we
encountered since the serial MGA always converged to solutions that satisfied the
specified design criteria. In our case only one population was maintained and selection
was performed by a single processor, so all o f the chromosomes interacted in exactly the
same manner as they did in the serial MGA.
Also, it should be mentioned that the distributed network environment plays a
significant role in determining the speedup associated with this method. For example, a
multi-processor may minimize the communication overhead between the master and the
slaves, but a network of physically-distributed machines may suffer from wasted cycles
because o f errors in synchronization and imperfect communication. The mgMGA relies
heavily on the master process; each slave communicates exclusively with the master, and
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199
each one passing through the MGA loop must be synchronized. This causes the master
process to become a bottleneck in the mgMGA model.
6.3 Design of Polarization-Selective Surfaces for Dual Reflector Applications
The serial and parallel MGA will now be applied to the design of a polarization
selective surface (PSS) for dual reflector applications. Figure 6.2 depicts the problem
geometry whose parameters were optimized by both the serial and parallel versions of the
MGA. The design required a freestanding FSS screen to transmit TE polarized waves to
within -0.01 dB and reflect TM polarized waves with isolation better than -20.0 dB, for
elevation angles ranging from 0 to 40 degrees, and for frequencies extending from 33.0 to
36.0 GHz. The problem was first optimized by the serial MGA with the population size
and number of generations equal to 5 and 100, respectively. The MGA designed the unit
cell of the freestanding FSS screen with 16x16 pixel resolution for 0 = 1 0 ° and <J) = 90°
over the frequency band of interest simultaneously for both the TE and TM polarizations.
The objective function employed by the optimization is shown below:
F ( ^ . < ,. < , ) = - m a x { ^ * r rc( » „ / , ) + a * T w (« „ /,)}
where A and B are the relative weight coefficients;
(6.2)
is the unit cell of the periodic FSS
screen generated by the MGA; tx and ty are the x- and y-periodicities of the FSS screen;
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200
and r , T are the reflection and transmission coefficients as a function of incident angle
and frequency.
The coding for the three parameters, viz., tx, ty, and Sdes are represented by (4.68)
and (4.70), respectively. The entire system is represented by a sequence C similar to that
in (4.73) as
C = TxTy {CD\
.(CD)j
(6.3)
Each quantity in (6.3) has been defined in (4.68) and (4.70), and each sequence in (6.3)
consists of (Mab~Mtyb~j*MCb) bits.
Air superstate
► FSS
Air substrate
Fig. 6.2. Problem geometry for PSS
optimization.
The serial MGA was executed on a 500 MHz DEC-ALPHA™ workstation with
512 MB RAM, and took 15 seconds to complete one generation. Figure 6.3 (white and
black represent metal and free space, respectively) shows the FSS screen design made of
PEC with unequal x- and y-periodicities. Figures 6.4 (a) through (e) plots the
transmission coefficient o f the freestanding FSS screen as a function of frequency with
elevation angle as a parameter for both the TE and TM polarizations. It was observed that
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201
the design constraints were satisfied for both the TE and TM polarized waves and the TM
response degraded for higher elevation angles.
The MGA formulation for the parallel MGA remained exactly the same as the
serial version and the optimized results obtained were identical. The parallel version of
the MGA was executed on a CRAY™ T3E, with 512 ALPHA™ processors, each with a
450 MHz cpu, local RAM o f 128 MB and a theoretical machine peak speed of 460
GFLOPS. For our optimization, six processors were utilized for the execution, with one
processor assigned as the master and the remaining five as slaves, thus assigning one
individual per slave processor. The execution time for one generation using the parallel
version of the MGA was 5.0 seconds, which was three times faster than the serial MGA.
Theoretically, a speed-up o f five times (100 % efficiency) was expected, as five
processors were utilized. However, this was not achieved in our case (60 % efficiency)
because the serial MGA code was only partly converted to a parallel MGA one. The
master process that performed the selection and crossover still remained a serial code.
Only with the complete parallelization of the serial code can higher efficiencies be
achieved as per Amdahl’s law [47], The speed up achieved using the parallel version of
the MGA provided the opportunity to generate FSS unit cell designs with higher
resolutions in the optimization process.
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Fig. 6.3. FSS unit cell for PSS design application.
o
TE
TM
■5
10
15
■20
•25
•30
•35
-40
-45
-50
33
34
35
36
37
38
39
40
41
42
43
Frequency in GHz
(a)
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203
TE
TM
c
E
S
o
m
c
7o(0>
c
£
-30
-35
-40
-45
Frequency in GHz
(b)
TE
TM
jj -15
9 -20
| -25
-30
-35
Frequency in GHz
(C )
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204
TE
TM
■ -
-10
-15
-20
-25
-30
33
34
35
36
37
38
39
Frequency in GHz
40
41
42
43
(d)
TE
TM
•o
S -10
-15
-20
-25
-30
33
34
35
36
37
38
39
Frequency in GHz
40
41
42
43
(e)
Fig. 6.4. Frequency response of PSS: (a) 0 = 0, <J>= 90; (b) 0 = 10,
4> = 90; (c) 0 = 20, <t>= 90; (d) 0 = 30, <t>= 90; (e) 0 = 40, <J>= 90.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 7
VALIDATION OF THE RESULTS GENERATED BY THE MICRO-GENETIC
ALGORITHM (MGA)
In this chapter, the results generated earlier by the MGA are validated by
comparing them against those obtained via the Finite Difference Time Domain (FDTD)
method for single embedded screens in Chapter 4. The MGA-generated results for two
FSS screens embedded in a composite, given earlier in Chapter 5, are verified against a
direct MoM solution for cascaded screens as opposed to the approximate Generalized
Scattering Matrix (GSM) technique used in the MGA optimization. A brief overview of
the two methods is presented along with some numerical results.
7.1 The FDTD Method
The FDTD method is a widely used electromagnetic modeling technique, first
proposed by Yee [48] in 1966. In this method the differential form of Maxwell’s equation
are transformed to central difference equations, and then discretized for computer
solution. The equations are explicit in nature, which implies that the field quantities at a
particular instant o f time are only dependent on their values at the preceding instant. The
equations can thus be solved by using a leapfrog scheme in which the electric fields are
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206
solved at one instant in time followed by the solution of the magnetic fields at the next
instant. The process is repeated over and over again.
The differential form o f Maxwell’s curl equations can be written as
5HZ
1
(dEy
cE.
dy
dHv
1
dt
My
l 5Z
( BE.
^ dx
1
(dEx
c t
/A
oEx
dz
dH.
dt
Mz
I
dEy
dx
dEx __
1
'dH.
CHy
ct
&
5Ey
1
ct
dE.
8 t
rcHz
fv y CZ
_
1 (
dHy
dx
CZ
cH.
dx
dHx
dy
A first step to applying the FDTD for numerically modeling a given problem is to
establish a computational domain and to discretize it. Yee [48] suggested that this be
done in the form of small cubes, commonly referred to as the “Yee cells” (see Fig. 7.1).
The material within each individual cell in the computational domain is then specified,
thus enabling one to model arbitrarily inhomogeneous structures. In approximating a
differential equation with central differences it is essential that the cell size be chosen
small enough to assure that the numerical inaccuracies introduced by the discretization
step is within the acceptable limits. In FDTD, this nominal cell size lies between A710 and
X/20, where X is the smallest wavelength in the computational domain. In certain cases,
the criterion mentioned above is superseded by one imposed by the geometrical features
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207
of the object being modeled. If a resolution lying between X/10 and X/20 is not sufficient
to accurately model the fine geometrical features of the object, then the cell size is
determined by the geometry instead.
z
Hi
Ex
Hx
O.i.K) .'l*
X
Fig. 7.1. The Yee cell with the six electric and magnetic
field components all offset by half a space step.
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208
Approximating the differential equations in (7.1) by central differences, and using
the Yee cell distribution of the fields, we can express (7 .1) as
At
*
v A '.J.k)
* r s ( u * ) - * r s (u * )+
E:(ij+ik)-E:(ij,k)
E :{i+ \j,k)-E :{ij,k)
My ( i J , k )
'
Ax
E zn { i J , k + l ) - E : ( i , j , k )
Ac
Erx { i j + \ , k ) - E nx { i j , k )
H !’ ~lz{ i J , k ) = H /n ' 2( i J , k ) +
E" (#' + \ , j , k ) ~ E”(/, y, k )
(7.2)
r Hn
: l'-{ij,k)-H :n' ' - { i j - i k )
E r(ij,k)= E :(u ,k)+
At
Ay
j
f Hn
;''-{ij,k)-Hn
; K- { i j , k - 1)
At
At
12 (/ - 1, y, at)
Ax
h
j,k)+
"; ' - ( i j . k ) - H : - ( i - u , k )
Ax
H ? ' - { i J , k ) - H ;n l' - ( i J - l k )
Ay
where At is the maximum time step whose value is determined by the Courant stability
[48] condition and can be expressed in terms of the cell sizes as
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209
r
•
(73)
\ ( to)! +(Av): V ' f
where v is the maximum velocity of propagation in any region within the computational
domain and Ax, Ay and Az are the cell sizes in the x-, y-, and z-directions, respectively. It
is necessary to impose the above stability condition in the FDTD algorithm because of its
explicit nature.
Once the computational domain and the properties o f the material media within
this domain have been established, we move on to specify the nature of the excitation
source. The source commonly used in FDTD is a Gaussian pulse with a specified 3-dB
frequency point. However, for our FDTD simulations we employ a sine-modulated
Gaussian pulse with specified 3-dB and modulating frequencies. Results valid over a
wide frequency band can be derived in a single simulation by using the FDTD, and
choosing the frequency characteristic of the source excitation appropriately. The source
can be an impinging plane wave, a current on a wire, or a voltage between two plates,
depending on the problem to be modeled. As the spectral-MoM used to analyze the
periodic structures assumes an incident plane wave, the FDTD simulations also utilize the
same type of excitation.
For numerical purposes, the computational domain must be truncated at some
point, even if we are dealing with an open region problem. In our case, recognizing the
periodic and symmetrical nature of the structure being modeled, we utilize appropriately
placed Perfect Electric Conductor (PEC) and Perfect Magnetic Conductor (PMC) walls to
truncate the x- and y-boundaries (assuming the FSS structure lies in the x-y plane), but
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210
employ either a Perfectly Matched Layer (PML) [49-50] type of Absorbing Boundary
Condition (ABC) or a Mur-type Outer Radiation Boundary Condition (ORBC) [51] at the
z-boundaries. Six layers o f unsplit-uniaxial-anisotropic-PML, proposed by Gedney [50]
are employed in our FDTD modeling. In certain cases, the PEC and PMC can be replaced
by a Periodic Boundary Condition (PBC) [52], which can be easily implemented by using
the conventional FDTD update equations, though only for the normal incidence case.
Thus we only validate our MoM results by comparing them to those obtained via the
FDTD for normal incidence, because, we only have access to a periodic-FDTD code that
can handle normal incidences. The material content of the conducting patches in the FSS
screens is assumed to be PEC in the FDTD simulation.
7.2 The Direct Method of Moments (MoM) Approach to Analyzing Multiple FSS
Screens
In Chapter 5, the Generalized Scattering Matrix (GSM) was introduced as an
alternate and approximate technique for analyzing multiple FSS screens embedded in a
composite. The numerically rigorous approach to simultaneously solving for the
unknown current coefficients for all of the FSS screens, though computationally
expensive, are nonetheless useful for validating the results generated by the MGA, which
incorporates the GSM technique to analyze multiple FSSs. The problem geometry is the
same as shown in Fig. 5.1. Equation (4.41) pertaining to a single FSS screen can be
expressed in a matrix form as
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211
i
-E T (x,yj
-Er(*-y)
=—
»<e
-**<o
<m
z z
fTT=—«3 m=-cc
a
,a
) <m
a
a
)'
•UA..A.)'
G„(A..A.) <UA„.A,.) A. (A.. A.)
’A^Py.y) - Z .
(7.4)
\{*>y)
Jyofry)
where the definition o f all the terms were given in (4.34) through (4.40) in Chapter 4. For
a multilayered FSS composite containing S conducting screens, the scattered fields or the
induced currents on each of the conducting surfaces are obtained by modifying (7.4) as
S •<*>
T Tv
*<c
Gi(A..A.)
g ;( a .,/? J
rr= -<e
g; ( a .. a .)
<%(a . . a .)
I I I
y = 1 m=
'K { P ^ P y .)
y.) ->(7.5)
J, ^ y )
J ,y» ( x ^ y )
where, the subscript / (i = 1, 2
S) corresponds to the i'h conducting surface. The left
hand side in (7.5) corresponds to the sum of the scattered fields due to the currents J, on
the j h conducting surface, where j varies from I through 5. Equation (7.5) is in a form
that is amenable to solution via the use of the spectral-Galerkin technique, which was
described earlier in Sec. 4.2.1.
7.3 Application of the FDTD Method to Analyze Single Screen FSS Embedded in
Inhomogeneous Media
The first example to be validated by using the FDTD is a composite whose
configuration is the same as shown in Fig. 4.25 in Chapter 4. Note that this composite
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212
contains lossy dielectric materials, whose accurate simulation over a broad band of
frequencies requires the use of a version of the FDTD that can handle dispersion. To
overcome this difficulty, the FDTD is simulated at six separate frequencies, i.e., 19.0,
22.0, 25.0, 28.0, 34.0, and 36.0 GHz and the appropriate values o f conductivity
corresponding to the lossy layers are used at each of these frequencies. To carry out a
good comparison between the two methods, viz., the MoM and the FDTD, for screens
designed by using the MG A, we need to ensure that their simulation environments are
close to each other, and this necessitates some changes in the unit cell design of the FSS.
First we note that, in the MoM technique we apply sub-domain basis functions only to
those regions in which the l ’s are consecutive in both the x- and y-directions.
Consequently, the current on isolated pixels that are ignored by the MoM formulation but
are included in the FDTD method, can lead to different results. To avoid this difficulty,
all single pixels from the FSS unit cell in Fig. 4.25 are removed when using the FDTD
simulation. Next, the sub-domain basis functions in the MoM are applied only in the xand y-directions, but not diagonally. Hence, the currents are not modeled in the MoM at
the point of intersection of the comers of pixels where the metallic patches make contact,
though they are in the FDTD. Figure 7.2 shows the FSS unit cell before and after
processing the geometry by removing all such intersecting metal comers. We also
observe in Fig. 7.2 that the FSS unit cell has metal patches penetrating into the x- and yboundaries. We find that for such structures the PEC and PMC truncation of the x- and yboundaries lead to a formation of psuedo-cavities that introduce late time ringing in the
FDTD simulations, and, in turn, affect the low frequency response of the composite. To
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213
overcome this difficulty, we use the PBC to truncate the x- and y-boundaries for all the
examples that have this attribute.
(a)
(b)
Fig. 7.2. FSS screen for example in Fig. 4.25: (a) MGA-generated
FSS unit cell; (b) after processing.
The boundaries at zmm and zmax are truncated with a PEC and the PML,
respectively. Also, one pixel (there are 16 pixels in both the x- and y-directions when the
discretization is 16x16) in the FSS unit cell in the MoM is regarded as being the
equivalent o f four cells in the FDTD. The FDTD cell sizes in the x-, y-, and z-directions
are 0.025, 0.025, and 0.0875 mm, respectively, which effectively results in a domain size
o f 64 x 64 x 64 cells. The FDTD is run for 16384 time steps in the absence o f any object
in the computational domain to provide the incident fields.
As mentioned earlier, the FDTD is simulated six times to cover the frequency
range to account for the change in conductivity with frequency due to the presence of
lossy materials in the composite. Figure 7.3 presents the comparison between the FDTD
and MoM results and they are seen to agree very well with each other.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
214
-14i
-16-
-
s
o
MoM
FDTD -
22-
-26-
-32-
Frequency in GHz
Fig. 7.3. Comparison between FDTD and MoM results for
example 1 in Fig. 4.25.
In the next example, the results for a spatial filter designed by the MGA are
validated by using the FDTD. The design requires that for TE and TM polarizations the
filter reflects all frequencies between 19.0 and 30.0 GHz with a reflection coefficient
greater than -1.0 dB, and transmits all frequencies between 31.0 and 40.0 GHz with a
transmission coefficient greater than -5.0 dB at normal incidence. The maximum
thickness o f the composite is restricted to be within 2.5 mm and four lossless dielectric
layers with a single FSS screen are used for the optimization. The population size and the
number of generations are 5 and 300, respectively. The MGA selects the FSS screen to be
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215
lossless with x- and y-periodicities chosen to equal 4.26 mm.
The MGA-generated
composite and the FSS unit cell are shown in Fig. 7.4.
er = 4.786, d = 0.55 mm
er= 11.57, d = 0.25 mm
er= 6.053, d = 0.75 mm
er= 2.125, d = 0.59 mm
(a)
ri
y
(b)
Fig. 7.4. MGA-generated spatial filter design: (a) composite; (b) FSS unit
cell design.
The parameters used in the FDTD simulation for this example are: Ax = Ay =
0.0665625, and As = 0.05 mm; domain size of 64 x 64 x 143 cells; PML absorbing
boundary condition at both zmm and zmax; and, a plane wave source located 20 cells away
from the PML. The FDTD simulation is run for 16384 time steps for both the incident
and total fields. Due to the frequency-independent (absence of lossy dielectric materials
in the composite) property o f this problem, the FDTD simulation for the total field is nan
only once to generate accurate results for all the frequencies of interest. The FDTD
results are shown in Fig. 7.5 along with those obtained from the MoM. We observe that
the frequency response obtained by using the FDTD shows a slight shift toward higher
frequencies, and the worst-case difference is about 1.4 %. The difference in the level of
the reflection coefficient at resonance can be improved by running the FDTD for a
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216
greater number o f time steps. Also, we found that the results converge towards the MoM
values [53] when the mesh density in the FDTD simulation is increased.
os
.£
c<u
*5-
1 10
-
"
8O
J -15"
O
0>
c:
<2 -2C ■
FDTD
MoM
-25"
Frequency in GHz
Fig. 7.5. Comparison between FDTD and MoM results for spatial
filter design with four lossless dielectric layers and one FSS screen.
7.4 Application of the Direct-MoM to Analyze Multiple FSS Screens Embedded in
an Inhomogeneous composite
The four cases studied in Chapter 5 will now be validated by comparing the
frequency responses generated by using the GSM-incorporated MGA method for
multiple FSS screens, and the direct-MoM as described in Sec. 7.2. The composite design
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217
and the structure o f the FSS unit cell are shown in Figs. 5.7 (a) and 5 .8 (a), respectively,
for Case-1. As explained in Chapter 5, 25 harmonics are included in the scattering matrix
for the GSM-MGA method. The parameters optimized by the MGA are utilized in a
separate code incorporating the direct-MoM to generate the frequency response of the
composites for the four cases. The frequency responses generated by the two methods are
compared for both TE and TM polarizations at 0 = 45°, and <j) = 0°, and are shown in Fig.
7.6.
-
20-
-
22-
-24
-26-28-3C
GSM Technique-25 harmonics
direct-MoM
-32
Frequency in GHz
(a)
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218
-ia
-
GSM Technique-25 harmonics
direct-MoM
20-
■o
-26-
-32-34
-3©
Frequency in GHz
(b)
Fig. 7.6. Comparision between the GSM-MGA and the direct-MoM
for Case-1: (a) TE polarization; (b) TM polarization.
The same comparison is performed for Case-2 for which the composite structure
and the FSS unit cell design are shown in Figs. 5.7 (b) and 5.8 (b), respectively. For this
case the MGA simultaneously optimizes for both TE and TM polarizations, at 0 = 0° and
45°. The frequency responses are compared only for 0 = 45°, for both the TE and TM
polarizations, and are presented in Fig. 7.7.
For Case-3, we have multiple FSS screens embedded in lossy or lossless dielectric
and magnetic media. The FSS unit cell and the composite design were earlier shown in
Figs. 5.19 and 5.20, respectively. The two methods are used to analyze both the TE and
TM polarization incidences with 0 = 0° and 45°, and <J>= 0°.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reflection coefficient in dB
219
GSM Technique-25 harmonic?
direct-MoM
______
Frequency in GHz
(a)
GSM Technique-25 harmonics
direct-MoM_______________
c
O
‘G
£ -26eo
oo
e
&
Frequency in GHz
(b)
Fig. 7.7. Comparision between the GSM-MGA and the direct-MoM
for Case-2: (a) TE polarization; (b) TM polarization.
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220
We arbitrarily choose the constitutive parameter values for the lossy magnetic
materials used in the validation at a frequency of 36.0 GHz. Thus, the reflection
coefficient values are accurate only at 36.0 GHz, and are maintained below the worstcase level optimized by the MGA in Chapter 5. To derive accurate results at other
frequencies in the band of interest, we must insert the corresponding constitutive
parameter values in the program. This is particularly true for lossy magnetic media,
whose constitutive parameters vary more rapidly with frequency than they do for lossy
dielectric media.
-12i
-14
cu
‘3
£
8
GSM Technique-25 harmonics
direct-MoM
-18-
-
20 -
§
S -22-
c:o
ac
-2 4 -26-
Frequency in GHz
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221
-12i
GSM Technique-25 harmonics
direct-MoM
-14
.2
8
-
18 -
-
20-
-
22-
-24
-26-
Frequency in GHz
(b)
-10.
—
GSM Technique-25 harmonics
direct-MoM
-1500
■o
c
S
u
0
-2 0 -
E
1 -25co
-35-
•4Qr
Frequency in GHz
(c)
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222
-1S
-
20-
25-
8 -30GSM Technique-25 harmonics
direct-MoM
-40-
Frequency in GHz
(d)
Fig 7.8. Comparision between the GSM-MGA technique and the
direct-MoM for Case-3: (a) normal incidence-TE polarization; (b)
normal incidence-TM polarization; (c) 45° from broadside-TE
polarization; (d) 45° from broadside-TM polarization.
We observe from Figs. 7.6, 7.7, and 7.8 that the agreement between the two
methods is excellent. This, in turn, leads us to conclude that an adequate number of
harmonics were included in the scattering matrices for each of the sub-composites. After
further study, we found that the 25 harmonics used by the MGA-GSM technique was
more than was necessary. This observation is based on the results shown in Fig. 7.9,
where only 9 harmonics were included in the scattering matrix for each of the sub­
composites in Case-1 and we see that the results agree very well with the direct-MoM.
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223
Obviously, reduction o f the number of harmonics included in the scattering matrix speeds
up the MGA optimization process for multiple FSS screens.
-18i
-2C -
-24
GSM Technique-9 harmonics
direct-MoM
-32*
Frequency m GHz
(a)
-18i
-2C - V
GSM Technique-9 harmonics
direct-MoM
-24
£
-26-28
-30
Frequency in GHz
(b)
Fig. 7.9. Comparision between the GSM-MGA and the directMoM for Case-1 with 9 harmonics included in the scattering
matrix: (a) TE polarization; (b) TM polarization.
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224
However, we note that it is possible to reduce the number of harmonics
significantly only when the composite is made up of materials with high losses, and
propagating higher order harmonics attenuate rapidly. As we will see in the next
example, such a reduction is not possible when the composite is fabricated either with
lossless or very low loss materials.
For the final example, we consider a spatial filter designed by the MGA discussed
earlier in Sec. 5.3.2. The composite and the FSS unit cell design are shown in Figs. 5.25
(a) and (b), respectively. The composite consists of low-loss dielectric materials and 49
harmonics were included in the scattering matrix for each sub-composite during the
MGA optimization. The results generated by the two methods are compared in Fig. 7.10
for the TM-polarized wave incident at angle of 0 = 70° off broadside, with <J>= 90°. It was
observed that although the frequency responses exhibit the same general trends, the
results do not match as well as they did for the previous three cases. We attribute this to
the fact that the number of harmonics included in the scattering matrix in the MGA-GSM
optimization process were inadequate. To achieve better accuracy, this number should be
increased, albeit with an increase in the computational expense of the optimization
process.
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225
Transmission coefficient in dB
GSM Technique-49 harmonics
direct-MoM
-1C
-15-
-2C
-2 9
Frequency in GHz
Fig. 7.10. Comparision between the GSM-MGA and the direct-MoM
for the composite with low-loss dielectric layers and two FSS
screens.
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Chapter 8
SUMMARY, CONCLUSIONS, AND FUTURE WORK
This thesis has presented a general stochastic technique for the efficient design
and optimization o f spatial filters and broadband microwave absorbers. In the past,
various stochastic optimization techniques have been utilized to successfully optimize
problems that were computationally non-intensive. The Micro Genetic Algorithm
(MGA) utilized in this thesis has successfully optimized designs that can be categorized
as computationally intensive. The optimization problem becomes computationally
intensive because o f the presence of doubly periodic, infinite Frequency Selective
Surfaces (FSSs) that must be analyzed by using the Method of Moments (MoM) in the
spectral domain.
8.1 Improving the Efficiency of the Optimization Process
There are two ways to decrease the computational expense of the optimization
process: (i) increasing the efficiency of the MoM matrix solution process, which plays a
key role in the optimization process; and, (ii) improving the efficiency and convergence
properties of the conventional genetic algorithms (CGA). The FSS screens can be
efficiently analyzed by working in the spectral domain to avoid costly convolution
operations (Chapter 4) and by using the Fast Fourier Transforms (FFTs) for efficient
evaluation o f the double summations present in the matrix elements (Chapter 4).
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227
Furthermore, highly efficient matrix inversion subroutines can be utilized to invert the
matrix (Chapter 4). To improve the efficiency of the optimization algorithm, we can use
the MGA instead o f the CGA, because it demonstrates superior convergence (Chapter 3)
properties for multimodal (dominant property of the problem at hand) problems.
Furthermore, the computational efficiency of the MGA can be attributed to its ability to
work with small population sizes (Chapter 3). Therefore, combining the MGA with the
efficient spectral-MoM enables us to optimize complicated geometries efficiently and to
maintain the computational expense within practical limits.
We need an alternate approach to optimize a design when FSS screens with
higher resolutions are embedded in the layered media,
because it becomes
computationally very expensive to deal with them by using conventional methods. Such
high resolution in FSS screens is required for designs in which the frequencies of interest
are in the millimeter wave band. To avoid excessive computational expenses, a domain
decomposition approach is utilized by splitting the problem domain into two, viz., the
dielectric composite without the FSS screen, and the freestanding FSS screen. The two
structures are designed separately by using the approach discussed in Chapter 4, and then
combined in a suitable manner to meet the design criteria.
Chapter 5 addressed the problem of designing composites with embedded
multiple FSS screens. The analysis for the single FSS screen can be extended to multiple
screens by using the MoM, as explained in Chapter 7. The direct-MoM can be
computationally expensive, but this difficulty can be circumvented by employing the
generalized scattering matrix (GSM) approach. In the latter method, we begin by dividing
the composite structure into sub-composites, each comprising a FSS screen embedded in
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228
layered media. Next, each sub-composite is assumed to be an Alport system and the
scattering parameters for each system is generated and assembled in a matrix. We have
demonstrated in Chapter 5 that it is necessary to include an adequate number of
harmonics in the scattering matrix to maintain the accuracy. Finally, we combine the
scattering matrices o f the sub-composites to generate the matrix for the composite from
which the reflection and transmission coefficients can be evaluated. As shown in Chapter
7, for composites incorporating high-loss materials the number of harmonics included can
be as low as 9, compared to 49 harmonics needed when the composite consists of lowloss or lossless materials.
To further enhance the efficiency of the MGA, a parallel MGA has been
implemented using the master-slave model (micro-grain MGA), and Message Passing
Interface (MPI) protocols. The parallel version has been executed on a CRAY™ T3E™, a
parallel architecture with 512 ALPHA™ processors working at clock speeds of 450 MHz.
The parallel MGA population was equally distributed on 5 processors, while the sixth
was used as the master process. The slave processors performed the reflection and
transmission coefficient calculations corresponding to each individual, while the master
process carried out the selection, crossover, and the population restarts. The parallel code
achieved a speed-up of three times, as compared to the serial version of the MGA.
8.2 Accuracy of the MGA Optimized Results
In Chapter 7, a few designs from Chapters 4 and 5 were selected to validate the
accuracy o f the results generated by the MGA-MoM optimization technique. The FDTD
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229
and the direct-MoM were employed to verify the results generated by the MoM/MGA
method for composites with single and multiple FSS screens embedded in layered media,
respectively. The FDTD was found to faithfully capture the trend of the MGA/MoM
results, except for a small shift in the frequency response characteristics. The results
generated by the direct-MoM and the MGA/ MoM-GSM methods agreed very well for
composites consisting of materials with high-loss, and closely reproduced the
characteristics o f the results for composites fabricated with low-loss materials.
8.3 Suggestions for Further Research
Improving the efficiency of the MGA-MoM to analyze FSS screens with higher
unit cell resolutions can extend the capabilities of the optimization procedure presented in
this thesis. Either improving the optimization algorithm or enhancing the matrix solution
procedure by using, for example, Model Order Reduction (MOR) [54] techniques to
effectively reduce the matrix size can possibly increase the efficiency. While a very
simple parallel MGA model has been presented in this thesis, several advanced models
[45-46] have been proposed in the literature that can further improve the efficiency of the
optimization algorithm. The most important advantage of the parallel GAs is that, in
many cases, they provide better performance than the single-population-based GA
algorithms, even when the parallelism is simulated on conventional serial machines. This
is because multiple populations permit speciation, a process by which different
populations evolve in different directions, i.e., towards different optima. Thus, parallel
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230
GAs are not only an extension o f the tradition sequential GA, but they represent a new
class o f algorithms in that they search the solution space differently.
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VITA
Sourav Chakravarty was bom in West Bengal, India on the 14th of April 1971. He
enrolled in the Electronics and Communications program at Regional Engineering
College, Kurukshetra, India in 1988 and received his B. Tech. degree in May 1992. From
1992 to 1995 he was employed as an Antenna Design Engineer at Superline Microwave
Pvt. Ltd., at Bangalore, India. He received his M. E. degree in Electronics and
Telecommunication in December 1997 at Jadavpur University. He began his doctoral
studies in Electrical Engineering at the Pennsylvania State University in fall of 1997 and
has held a research assistantship at the Electromagnetic Communication Laboratory at
Penn State. His current research interests are in the area of antennas, computational
electromagnetics with an emphasis on stochastic optimization techniques, numerical
methods, and the finite difference time domain (FDTD) technique.
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