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Application of genetic algorithms to the design of microstrip antennas, wire antennas and microwave absorbers

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Copyright
by
Hosung Choo
2003
The Dissertation Committee for Hosung Choo certifies that this is the approved
version of the following dissertation:
Application of Genetic Algorithms to the Design of Microstrip
Antennas, Wire Antennas and Microwave Absorbers
Committee:
Hao Ling, Supervisor
Mircea D. Driga
Edward J. Powers
Ross Baldick
Robert Rogers
Application of Genetic Algorithms to the Design of Microstrip
Antennas, Wire Antennas and Microwave Absorbers
by
Hosung Choo, BS, MSE
Dissertation
Presented to the Faculty of the Graduate School of
the University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
May 2003
UMI Number: 3110761
Copyright 2003 by
Choo, Hosung
All rights reserved.
________________________________________________________
UMI Microform 3110761
Copyright 2004 ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
____________________________________________________________
ProQuest Information and Learning Company
300 North Zeeb Road
PO Box 1346
Ann Arbor, MI 48106-1346
To my wife Seongsin
Acknowledgements
First of all, I would like to especially thank my supervisor Professor Hao
Ling, who greatly supported me during my five years of graduate study. He has
done an outstanding job of supplying me with suggestions and encouragement
during my studies at the University of Texas at Austin. He was always helpful and
gave me plenty of new ideas. Without his support, this work would have been
impossible to pursue.
I would also like to extend my gratitude to Dr. Robert Rogers, who has
been a valuable source of information in my small antenna research. His deep
knowledge of electrically small antenna theory and practice enabled me to
complete the work presented in this dissertation.
I also appreciate Professor Mircea Driga for graciously giving his time to
serve as the chair of my dissertation committee.
I am grateful to Professor Edward Powers and Professor Ross Baldick for
serving on my dissertation committee and kindly reviewing this work.
I would also like to thank all the current and former group members: Dr.
Hyeongdong Kim, Dr. Tao Su, Dr. Yuan Wang, Mr. Yong Zhou and Mr.
Sungkyun Lim. Special thanks to Dr. Luiz C. Trintinalia and Adrian Hutani, my
friends as well as my co-workers, for providing me with valuable numerical
simulation code and for sharing their expertise in setting up the antenna
measurement and fabrication systems.
Finally, I would like to thank my wife and my parents for their endless
support and love, which I will never forget.
Application of Genetic Algorithms to the Design of Microstrip
Antennas, Wire Antennas and Microwave Absorbers
Publication No._____________
Hosung Choo, Ph.D.
The University of Texas at Austin, 2003
Supervisor: Hao Ling
This dissertation explores the general methodology for designing
electromagnetic (EM) systems by combining genetic algorithms (GA) with
computational electromagnetic (EM) simulations. The EM problems investigated
are broadband and multi-band microstrip antennas, low-profile microwave
absorbers and electrically small wire antennas.
It is shown that optimized
performance can be designed and realized using simple shape control.
In
addition, novel GA approaches are investigated for more challenging multiobjective problems. The developed methodology is also used to explore
performance bounds in complex EM systems.
First, the use of GA to design microstrip antenna shapes for broadband
and multi-band applications is investigated. A full-wave electromagnetic solver is
vii
employed to predict the performance of microstrip antennas with arbitrary patch
shapes. A GA with two-point crossover and geometrical filtering is implemented
to optimize the patch shape. For broadband application, the optimized patch
antenna achieves a four-fold improvement in bandwidth when compared to a
standard square microstrip. For multi-band application, the optimized patches
show that arbitrary frequency spacing ranging from 1:1.1 to 1:2 can be achieved.
Tri-band and quad-band microstrip shapes are also generated and the resulting
designs show good operations at the designated frequencies.
Second, the use of GA for designing optimal shapes for corrugated
coatings under near-grazing incidence is examined. Optimized coating shapes
depending on different polarizations are generated. Physical interpretations for the
optimized structure are discussed, and the resulting shape is compared to
conventional planar and triangular shaped designs. This problem is also extended
from the single to multi-objective optimization using the Pareto GA. Optimization
results using two different objectives, the height (or weight) of the coating versus
absorbing performance, are presented.
Finally, this dissertation reports on the use of GA in the design
optimization of electrically small wire antennas, taking into account of bandwidth,
efficiency and antenna size. To efficiently map out this multi-objective problem,
the Pareto GA is implemented with the concept of divided range optimization. An
optimal set of designs, trading off bandwidth, efficiency and antenna size, are
viii
generated. Several GA designs are built, measured and compared to the
simulation. Physical interpretations of the GA-optimized structures are provided,
and the results are compared against the well-known fundamental limit for small
antennas. Further improvements using other geometrical design freedoms are also
discussed.
ix
Table of Contents
List of Figures ..................................................................................................xii
Chapter 1 Introduction .........................................................................................1
Chapter 2 Design of Broadband Microstrip Antennas on High-Dielectric
Substrate Using a Genetic Algorithm...........................................................7
2.1 Introduction .........................................................................................7
2.2 GA Optimization ...................................................................................9
2.3 Broadband Microstrip Antenna Design .............................................. 15
2.4 Summary ........................................................................................... 24
Chapter 3 Design of Multi-Band Microstrip Antennas Using a Genetic
Algorithm.................................................................................................. 25
3.1 Introduction ....................................................................................... 25
3.2 Miniaturized Dual-Band Microstrip Antenna Design ........................... 26
3.3 Multi-band Microstrip Without Slots-cut ............................................. 35
3.4 Summary............................................................................................. 43
Chapter 4 Shape Optimization of Corrugated Coatings Under Grazing
Incidence Using a Genetic Algorithm ...................................................... 44
4.1 Introduction ....................................................................................... 44
4.2 Approach............................................................................................. 46
4.3 GA-Optimized Coating Shape ............................................................. 50
4.4 Multi-Objective Optimization.............................................................. 61
4.5 Summary............................................................................................. 68
Chapter 5 Design of Electrically Small Wire Antennas Using a Pareto Genetic
Algorithm.................................................................................................. 70
5.1 Introduction ....................................................................................... 70
5.2 Pareto GA Approach ........................................................................... 73
5.3 GA-Optimized Results ........................................................................ 78
5.4 Comparison to Fundamental Limit ...................................................... 85
x
5.5 Further Improvement on GA Designs .................................................. 89
5.6 Summary............................................................................................. 90
Chapter 6 Conclusions ....................................................................................... 94
Appendix A ....................................................................................................... 98
Bibliography.................................................................................................... 112
Vita
............................................................................................... 118
xi
List of Figures
Figure 1.1: Flow chart of the genetic algorithm.
Figure 2.1: 2-D encoding of patch shapes for broadband shape optimization.
Figure 2.2: Calculation time of single shape depends on the chromosome size.
Figure 2.3: Coarse-to-fine 2 dimensional chromosome.
Figure 2.4: Median filtering.
Figure 2.5: Two-point crossover with three chromosomes.
Figure 2.6: Convergence rates of the conventional one-point crossover scheme
and our two-point crossover scheme. A population of 30, a crossover
rate of 0.8 and a mutation rate of 0.1 are used. In addition,
geometrical filtering is applied. The results are averaged over 10
trials.
Figure 2.7: Parallel processing for a GA application.
Figure 2.8: (a) GA-optimized microstrip antenna using 40� resolution within
a 72mm�mm area. The gray pixels are metal and the white dot
shows the position of the probe feed. (b) A picture of the microstrip
antenna built on FR-4 circuit board. (c) Return loss (dB) of the GAoptimized antenna from simulation (------) and measurement (?
??).
Figure 2.9: Bandwidth enhancement is achieved by two-mode operation and
ragged edges.
Figure 2.10: (a) Current distribution on the surface of the GA-optimized patch at
1.98GHz. (b) Current distribution on the surface of the GAoptimized patch at 2.1GHz.
Figure 2.11: (a) Measured radiation pattern at 1.98GHz (?=?45o plane). (b)
Measured radiation pattern at 2.1GHz (?=45o plane).
Figure 2.12: Bandwidth enhancement is achieved by two-mode operation and
ragged edges.
xii
Figure 3.1: Encoding scheme for the dual-band design.
Figure 3.2: (a) GA-optimized slot shape for dual-band operation. (b) Return loss
(dB) of the GA-optimized antenna from simulation (
) and
measurement (?
??).
Figure 3.3: Simulated total metal loss (dBm) of the GA-optimized dual-band
microstrip (?
??), a square microstrip of 72mm�mm (------) and a
square microstrip of 36mm�mm (-?-?-?-?).
Figure 3.4: (a) Current distribution on the surface of the GA-optimized patch at
1 GHz. (b) Current distribution at 2 GHz.
Figure 3.5: (a) Measured radiation patterns at 1 GHz (?=0 plane). (b) Measured
radiation pattern at 2 GHz (?=0 plane).
Figure 3.6: Shapes of three GA-optimized dual-band microstrip antennas, and
the resulting return loss from simulation (------) and measurement
(?
??). (a) Frequency ratio of 1:1.3 (1.8GHz and 2.34GHz). (b)
Frequency ratio 1:1.6 (1.8GHz and 2.9GHz). (c) Frequency ratio of
1:1.9 (1.8GHz and 3.42GHz).
Figure 3.7: Shapes of dual-band microstrip antennas. Frequency ratio from 1:1.1
to 1:2.
Figure 3.8: Shape of the GA-optimized tri-band microstrip antenna that operates
at 1.6GHz, 1.8GHz and 2.45GHz, and the resulting return loss of the
antenna from simulation (------) and measurement (?
??).
Figure 3.9: Shape of the GA-optimized quad-band microstrip antenna that
operates at 0.9GHz, 1.6GHz, 1.8GHz and 2.45GHz, and the resulting
return loss of the antenna from simulation (------) and measurement
(?
??).
Figure 4.1: Geometry of the corrugated absorber.
Figure 4.2: Encoding of corrugated absorber into a binary chromosome.
Figure 4.3: (a) Before geometrical filter. (b) After 7-point sliding window filter.
(c) After the descending order filter.
xiii
Figure 4.4: (a) GA-optimized shape for the vertical polarization. (b) Reflection
coefficient (dB) versus frequency at 30 degrees from grazing.
Figure 4.5: (a) GA-optimized shape for the horizontal polarization. (b)
Reflection coefficient (dB) versus frequency.
Figure 4.6: Performance comparison of the planar absorber (------), triangular
shaped absorber (?
??), and GA-optimized absorber (-?-?-?-?) for the
horizontal polarization.
Figure 4.7: Effect of cost definitions on the optimization results for the
horizontal polarization. (a) Cost definition in (4.1). (b) Minimax
cost definition. (c) Average power reflection coefficient cost
definition.
Figure 4.8: (a)Performance sensitivity to variations in the shape of the profile for
the horizontal polarization. (i) Original GA-optimized design. (ii)
GA-optimized shape with RMS error of 0.4 mm. (iii) Smoother
shape after 6:1 undersampling. (b) Reflection coefficient (dB) versus
frequency at 25o, 30 o and 35 o from grazing.
Figure 4.9: (a) GA-optimized shape taking into account of both the vertical and
horizontal polarizations. (b) Reflection coefficient (dB) versus
frequency.
Figure 4.10: (a) Planar absorber with horizontal polarized wave incidence. (b)
Rectangular profile with horizontal polarized wave incidence.
Figure 4.11: Convergence of the Pareto front as a function of the number of
generations for absorbing performance versus absorber height. (a)
Initial population. (b) After 5 generations. (c) After 20 generations.
(d) After 200 generations.
Figure 4.12: Final converged Pareto front of absorbing performance versus
absorber height. The insets show 4 sample designs on the Pareto
front.
Figure 4.13: Final converged Pareto front of absorbing performance versus
absorber weight. The insets show four sample designs on the Pareto
front.
xiv
Figure 5.1: Achievable bandwidth in terms of antenna size and some examples
of small antennas.
Figure 5.2: Configuration of the multi-segment wire antenna used in the GA
design.
Figure 5.3: Encoding of the wire configuration into a binary chromosomes.
Figure 5.4: Divided range multi-objective GA approach.
Figure 5.5: Convergence of the Pareto front as a function of the number of
generations in terms of bandwidth, efficiency and antenna size. (a)
Initial generation. (b) After 200 generations. (c) After 1000
generations.
Figure 5.6: Three samples from the Pareto front (a) kr=0.34, (b) kr=0.42 and (c)
kr=0.50.
Figure 5.7: (a) Photo for antenna B, which has an antenna size of kr=0.42. (b)
Return loss and (c) efficiency versus frequency of antenna B. The
efficiency measurement was done using the Wheeler cap method.
Figure 5.8: Pareto front of the GA designs after convergence. The surface is
generated using a least squares fitting to best fit the GA results
shown as dots.
Figure 5.9: (a) Projection of the Pareto front onto the size and efficiency plane.
(b) Projection of the Pareto front onto the size and bandwidth plane.
(c) Projection of the Pareto front onto the bandwidth and efficiency
plane. (d) through (f) show the corresponding fundamental limit
based on (5.4).
Figure 5.10: Small antenna performance using the definition of ? = (Eff � BW) /
Theoretical BW Limit. (a) Original 7-wire configuration. (b)
Variable input impedance. (c) Multi-arm configuration. (d) Multiple
wire radii.
Figure A.1: Input resistance of a standard square-shaped (36mm�mm)
microstrip built on FR-4 before (------) and after (?
??) using the
Wheeler cap size of 10cmm�cmm�m. (a) Measured (b)
ENEMBLE simulated.
xv
Figure A.2: Wheeler cap measured efficiency based on parallel circuit model (-----), Wheeler cap measured efficiency based on series circuit model,
efficiency by Wheeler cap simulation and efficiency by gain
simulation. (a) microstrip (36mm�mm) built on FR-4 substrate (b)
microstrip (17.5mm�mm) build on Duroid.
Figure A.3: Measured input resistance of a standard square-shaped (36mm x
36mm) microstrip built on Duroid before (------) and after (??)
using (a) the Wheeler cap size of 17cmm x 17cmm x 8.5cm. (b)
using the Wheeler cap size of 17cm x 17cm x 2cm.
Figure A.4: Interior cavity modes spectrums for two Wheeler cap size. The upper
spectrum is for the Wheeler cap size of 17cm x 17cm x 8.5cm.The
lower spectrum is for the Wheeler cap size of 17cm x 17cm x 2.0cm.
Figure A.5: (a) A photo of one of the sample miniaturized microstrip. Wheeler
cap measured efficiencies for the microstrips (b) built on FR-4
substrate in terms of % from the regular size and (c) the microstrips
built on Duroid.
Figure A.6: EB Product against physical microstrip patch size for microstrip built
on three different substrates. (a) FR-4 (?
??), Duroid (------)and Air
(-?-?-?-?).
xvi
Chapter 1
Introduction
The optimization of a real-world electromagnetic (EM) system is in
general a challenging problem, as it typically involves a large number of degrees
of freedom. In addition, the cost function of a complex EM problem often
contains many local minima and may include non-differentiable regions. These
factors make it difficult to find a global optimum using deterministic optimization
methods such as the quasi-Newton method or the conjugate gradient method [1].
Genetic algorithm (GA) is a stochastic search method based on the
principles of natural selection and evolution [2]. GA is classified as a global
optimizer instead of a local optimizer, so the solutions are less dependent on the
initial values. For this reason, GA has shown highly efficient optimization
capability in various engineering applications. In particular, GA is more attractive
for finding an approximate global optimum in a problem that has a very highdimensional space with many local minima.
The idea of GA was developed by modeling the natural processes of
evolution and adaptation. In 1975, Holland first applied a GA to the design of an
artificial system [3]. Since then, GA has proven to be a powerful tool in solving
various kinds of design and optimization problems ranging from building heating
1
systems and job scheduling to VLSI layout and cell planning for communication
systems. The block diagram of a basic GA is shown in Fig. 1.1. The GA process
can roughly be divided into three parts: initialization, evaluation, and
reproduction. The algorithm starts with an initial population of possible solutions.
The solutions are encoded as binary or real-valued chromosomes. In the
evaluation phase, the performance of each solution in the population is predicted
using a simulation tool.
Initial Solutions
Reproduction
Crossover
Performance
Simulation
Mutation
Geometrical
Filtering
Evaluate
Cost
Optimized Solution
Fig. 1.1 Flow chart of the genetic algorithm.
2
Then the cost (or fitness) of each solution is evaluated using the proper cost
function defined in terms of the design goals. According to the cost, the
chromosomes are refined into the next generation through a reproduction process
that includes crossover, mutation and geometrical filtering. This series of
processes is repeated until the cost is minimized, meaning an optimum solution
has been found.
Recently, GA has been applied to a number of EM design and inverse
problems [4]. For instance, Alatan et al. [5] employed a GA to design circularly
polarized (CP) microstrip antennas. Johnson and Rahmat-Samii in [6] showed that
it is possible to obtain microstrip shapes for broadband and dual-band applications
by using a GA. Haupt reported on the use of GA in designing antenna arrays [7].
Altshuler and Linden examined the use of GA in designing wire antennas [8,9].
Michielssen et al. applied a GA to find optimal thicknesses for multi-layer planar
coatings for radar cross section (RCS) reduction [10]. GA has also been applied to
inverse scattering problems to reconstruct the shape of an unknown target from its
scattered field data [11,12].
While the application of GA to EM problems has met with initial success,
much more research is still needed to apply GA to real-world EM design
problems. In addition, the adaptation of advanced GA techniques is also needed in
order to address more challenging problems. In this dissertation, GAs are applied
to the design of antennas and absorbers. The objectives of the dissertation are
3
threefold. The first objective is to formulate general design methodologies that
combine GAs with state-of-the-art computational EM simulations to create novel
EM designs.
The second objective is to investigate more advanced GA
techniques such as the Pareto GA for multi-objective problems.
The third
objective is to explore the use of GA for mapping the fundamental performance
bounds of a given EM system.
Three classes of EM design problems are
investigated in this dissertation: microstrip antennas, shaped microwave absorbers
and electrically small wire antennas.
This dissertation is organized as follows. In Chapters 2 and 3, the design
of microstrip antennas using GA for broadband and multi-band applications is
addressed.
Because of their low profile and ease of fabrication, microstrip
antennas are a very popular choice for many antenna applications. However, a
well-known drawback of the microstrip is that it is an intrinsically narrow-band
device. The goal is to improve the frequency characteristics of microstrip
antennas by exploring arbitrary patch shapes. Therefore a design methodology
based on a GA in conjunction with a fast EM solver is developed to find unique
patch shapes for achieving optimal performance. For the broadband application
discussed in Chapter 2, the design methodology is applied to explore the
maximum achievable bandwidth of microstrips. Basic antenna measures such as
return loss, current distributions and far field radiation patterns are examined to
provide a physical interpretation of the optimized design. In Chapter 3, the design
4
methodology is applied to search for patch shapes for dual-band, tri-band and
quad-band operations.
Chapter 4 discusses the design of shaped microwave absorbers using GA.
Microwave absorbers are often used to reduce the RCS of a target. An effective
absorber needs to suppress the reflection over a wide range of frequencies. In
addition, the profile of the absorber needs to be small for practical constraints.
Thus the goal is to design low-profile microwave absorbers with broadband
frequency characteristics. These low-profile and broadband requirements are
usually not compatible with each other. The Pareto GA is an attractive technique
of efficiently mapping out this multi-objective problem. The Pareto GA is applied
to examine the optimized absorber shapes in terms of the height (or weight) of the
coating versus its absorbing performance. This chapter also discusses a physical
interpretation of the optimized structure and compares the GA-designed shapes to
conventional planar and triangular shaped designs.
Chapter 5 examines the design of electrically small wire antennas using
GA. Many antenna applications require physically small antennas. However, the
size of an antenna usually needs to be on the order of a wavelength of the
operating frequency in order for the antenna to operate efficiently. In 1948, Chu
derived a formula showing that the fundamental limit of the bandwidth
performance of an antenna is a function of the antenna?s electrical size [13]. The
smaller the antenna size, the narrower the bandwidth. In addition to bandwidth,
5
antenna efficiency is also very important in the design of small antennas. The
goal is to use GA to design electrically small wire antennas, taking into account
bandwidth, efficiency and antenna size. Because smaller-sized solutions are
particularly difficult to optimize, the Pareto GA is enhanced by the concept of
divided range optimization to efficiently map out this challenging multi-objective
problem. The GA generates an optimal set of designs, trading off bandwidth,
efficiency and antenna size. The performance of the designs is analyzed by
comparing it against the well-known fundamental limit for small antennas. This
chapter also discusses further improvements to the design using other geometrical
design freedoms.
Finally, Chapter 6 concludes the dissertation and discusses directions for
future research. Appendix A is included to provide some new insights on the
Wheeler cap method for measuring antenna efficiency, which is a central issue
considered in several chapters in the dissertation.
6
Chapter 2
Design of Broadband Microstrip Antennas on High-Dielectric
Substrate Using a Genetic Algorithm
2.1 Introduction
Microstrip antenna has many favorable characteristics such as low profile,
conformal shape and ease of fabrication. In many antenna applications the use
microstrip antennas is preferred over other types of antennas. However, a wellknown drawback of the microstrip is that it is an intrinsically narrow-band device.
Increasingly, wireless devices are being asked to handle larger amounts of data at
faster speed, which requires the antenna to be broadband. Much research has been
carried out to improve the bandwidth of microstrips, and a number of different
techniques have been published including adding parasite patches, adopting multilayer structures, using thick air substrate and adding shorting posts as reactive
loading [14,15]. However, these techniques are usually accompanied by an
increase in overall size and/or manufacturing cost. In the past several years, some
researchers have applied GA to the design of microstrip antennas. Alatan et al. [5]
employed a GA to design circularly polarized (CP) microstrip antennas by
optimizing the corners of a square microstrip. Johnson and Rahmat-Samii in [6]
showed that it is possible to obtain novel shapes for broadband and dual-band
7
applications by using a GA. They used air as substrate material by suspending a
patch above the ground plane. The attractiveness of GA shape optimization is that
improved performance can be achieved without increasing overall volume or
manufacturing cost.
In this chapter, the use of GA for broadband applications is examined. In
contrast to the work of Johnson and Rahmat-Samii, standard FR-4 is employed as
the substrate, since it is the most commonly used material in wireless devices.
Microstrips built on high dielectric material substrate such as FR-4 (dielectric
constant of 4.3) have narrower bandwidth. Consequently, it is more challenging to
obtain the desired frequency characteristics without sufficient degrees of freedom
in the design process. The approach is to employ a full-wave EM patch code to
predict the performance of an arbitrary shaped microstrip. A GA is implemented
to optimize the patch shape that is encoded into a 2-D chromosome. A two-point
crossover scheme with three chromosomes is used as a crossover operator to
achieve faster convergence. In addition, geometrical filtering is adopted to create
more realizable shapes. This methodology is applied to broadband patch antenna
design.
This chapter is organized as follows. In Section 2.2, the EM simulation
tool and details of the GA algorithm are described. In Section 2.3, the application
of the GA algorithm to broadband microstrip antenna design is described. A
summary of this chapter is given in Section 2.4.
8
2.2 GA Optimization
To predict the performance of each patch shape, an EM simulation code
adapted from a code for analyzing frequency selective surfaces is used [17,18].
The code uses the electric field integral equation (EFIE) formulation in
conjunction with the periodic Green?s function for a layered medium. Rooftop
functions are used as basis functions to represent the currents on the patch. To
speed up the computation of the matrix elements, the fast Fourier transform (FFT)
is employed. To further reduce the matrix fill time during GA, the matrix
elements for all the possible basis positions are computed once and stored before
the GA process. Consequently, the moment matrix for any patch shape can be
easily assembled without any computation.
To achieve broadband operations, GA is implemented as an optimizer for
the microstrip patch shapes. The algorithm starts with an initial population of
shapes that are encoded as chromosomes. After these chromosomes are evaluated
by the EM simulation code, a cost function is computed. According to the cost
function, the chromosomes are refined into the next generation through a
reproduction process that includes crossover, mutation and geometrical filtering.
This series of processes is repeated until the cost function is minimized. In this
GA process, each patch shape is represented as a binary bitmap. In the resulting
2-D chromosome [19], ones represent the metallized areas and zeros represent the
areas without metal (Fig. 2.1).
9
2-D Chromosome
Shapes
Substrate
Metallic
Patch
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
1
1
1
1
0
0
1
1
1
1
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
r =4.3
FR-4
Fig. 2.1
2-D encoding of patch shapes for broadband shape optimization.
Additionally, the coarse-to-fine concept is adopted to save computation time. To
get better design, it is necessary to use large size of chromosome. However, the
use of large size chromosome causes a considerable computational cost. Fig.2.2 is
calculation time depend on number of genes in chromosome. The calculation time
for one shape scales O(N3) where N is the number of genes in the chromosome. In
order to use large size of chromosome without that much computational burden,
coarse-to-fine size variant chromosome is introduced in this GA.
10
-- Computation time by Pentium III 550Mhz
Fig. 2.2
Calculation time of single shape depends on the chromosome size.
The basic idea of the coarse-to-fine chromosome is to begin the initial generations
with coarse resolution and increase the resolution forward the later generations.
Fig. 2.3 describes how coarse-to-fine chromosome works. It starts with coarse
resolution of chromosome. Without that much time, GA finds the result that has
the lowest cost for this resolution. If the cost does not satisfy the design goal, then
the resolution is increased to twice as much as before. In transition from coarse
to fine, shapes are already partially optimized with the coarse resolution.
Therefore, at higher resolutions, GA only needs to slightly tune the results. This
variable resolution of chromosome from coarse to fine makes it possible to save
much computation time as compared to starting with very fine resolution. Since
it is more desirable to have a well-congregated patch shape in the final design, a
11
2-D median filter [20] is applied to the chromosomes at each generation of the
GA process.
YES
Convergence
Check
Converge?
Increase the
resolution
NO
Stay in current
resolution
Fig. 2.3
Coarse-to-fine 2 dimensional chromosome.
Fig. 2.4 shows a sample chromosome before and after the 2-D median filter
operation. Before median filtering, the chromosome shows many isolated patches.
After median filtering, most of the isolated patches in the chromosome are gone,
and the overall shape of the chromosome becomes more gathered.
For the crossover operation, a two-point crossover scheme using three
chromosomes is used to boost the GA convergence rate. Fig. 2.5 depicts the
crossover scheme. It starts by selecting three chromosomes as parents and divides
each chromosome into three parts.
12
After
Before
Median
Filtering
Fig. 2.4
Median filtering.
Then the next generation is made by intermingling the three parent chromosomes.
This crossover scheme exhibits a more disruptive characteristic for regeneration
than the conventional one-point or two-point crossover. It counteracts against
the median filtering effect and is found to result in better convergence rate. A
comparison of the results between the two-point crossover and the conventional
one-point crossover is shown in Fig. 2.6. Each crossover scheme is applied to the
broadband patch antenna design problem to compare the convergence rate. The
two-point crossover scheme results in faster convergence.
13
Parent
Children
Fig. 2.5
two-point crossover with three chromosomes.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Fig. 2.6
10
20
30
40
50
Convergence rates of the one-point crossover scheme (------), and the two-point
crossover scheme (?
??). A population of 30, a crossover rate of 0.8 and a
mutation rate of 0.1 are used. In addition, geometrical filtering is applied. The
results are averaged over 10 trials.
14
2.3 Broadband Microstrip Antenna Design
2.3.1 GA Optimization for Broadband Application
In this section, the design goal is to broaden the bandwidth of a microstrip
by exploring arbitrary patch shapes through GA. To achieve the design goal, a
cost function is defined as follows:
Cost =
1
N
N
n =1
Pn where Pn =
S 11 (dB ) + 10 dB
if S 11 (dB ) ? ? 10 dB
0
if S 11 (dB ) < ?10 dB
(2.1)
The cost function in (2.1) averages those S11 values (in dB) that exceed ?
10dB within the desired frequency range. In the GA process, the cost function is
an important factor not only for the final design but also for the overall GA
convergence time. To reduce the computational burden, an adaptive cost function
is used. In the first stage of the GA process, a small frequency range compared to
the eventual design goal is used. Then the frequency range is gradually increased
whenever the cost converges to a minimum. This process is repeated until the
eventual design goal is achieved. In general, the number of frequency calculations
required in this scheme is much less than if the entire design frequency band is
used throughout the GA iterations.
In this GA, the size of the population is 30. A crossover probability of
80% is used, while the probability of mutation is set to 10%. The frequency range
15
is set between 1.9GHz and 2.1GHz. A square design area (72mm�mm) and
FR-4 substrate with a thickness of 1.6mm are used. The design area is discretized
into a 40 � 40 grid. Since lots of repeated EM computations are necessary at each
generation, cost calculations for the population are distributed to multiple
computers using the parallel processing concept as shown in Fig. 2.7. The total
time of the design process is about 24 hours on ten Pentium 4 (1.7GHz) machines
running in parallel. Fig. 2.8(a) is the resulting shape of the GA-optimized
microstrip. The white dot represents the location of the probe feed. Note that the
position of the feed is a part of the parameters to be optimized, since the patch has
the freedom to be located anywhere within the total design area.
Calculation
results
Chromosome
Information
Fig. 2.7
Parallel processing for a GA application.
16
For experimental verification, a prototype of the GA-optimized microstrip patch
using aluminum tape is built (copper tape was also tried with nearly identical
results). A photo of the tested patch is shown in Fig. 2.8(b). Fig. 2.8(c) shows
the return loss comparison between the measurement (solid line) and the
simulation (dashed line) results. Good agreement is observed. The bandwidth is
found to be 8.5% by simulation and 8.1% by measurement. It should be noted that
the same GA methodology using coarser 16� grids and 32� grids within the
same design area have also been applied [16]. As the design resolution of the grid
increases from 16� to 32�, the bandwidth was improved by about 30% as
shown in Fig. 2.9. However, the higher resolution 40� grid shows only a slight
bandwidth increase over that of the 32� grid. The gain of the GA-optimized
microstrip relative to the reference square microstrip under the same construction
was also measured. Nearly negligible gain loss is observed (0.9dB gain loss at
1.98GHz, and 1.0dB at 2.1GHz).
17
72mm
40�
(b)
(a)
-5
S11 (dB)
-10
-15
-20
1.90
1.96
2.02
2.08
2.14
2.20
Frequency (GHz)
(c)
Fig. 2.8
(a) GA-optimized microstrip antenna using 40� resolution within a
72mm�mm area. The gray pixels are metal, and the white dot shows the
position of the probe feed. (b) A picture of the microstrip antenna built on
FR-4 circuit board. (c) Return loss (dB) of the GA-optimized antenna from
simulation (------) and measurement (?
??).
18
72mm
72mm
� 1.3
.
16x16
32x32
(a)
(b)
72mm
� 1.05
40x40
(c)
Fig. 2.9
Bandwidth enhancement is achieved by two-mode operation and
ragged edges.
19
2.3.2 Operating Principle of the GA Design
Next, an interpretation on the operating principle of the GA-designed
antenna is provided. The two frequency dips in Fig. 2.8(c) show that the antenna
supports two operating modes very close to each other in frequency. To verify
the two operating modes, the current distributions on the patch around these
frequencies are analyzed. Figs. 2.10(a) and 2.10(b) show the current plots at
1.98GHz and 2.1GHz respectively. At 1.98GHz, current flows predominantly in
the lower-right direction, while at 2.1GHz, current flows predominantly in the
upper-right direction. The currents corresponding to the two modes are nearly
perpendicular to each other. The two-mode operation can be seen even more
clearly in the measured radiation pattern plots. Figs. 2.11(a) and 2.11(b) show the
radiation patterns at the two frequencies along the two dashed cuts. It is observed
that at 1.98GHz the dominant polarization is in the ?=?45o plane while at 2.1GHz
the dominant polarization is in the ?=45o plane, as expected from the current
plots. This two-mode operation improves the bandwidth of a square microstrip by
a factor of about three. Note that the enhancement of bandwidth in this antenna
comes at the price of polarization purity, which was not a constraint in the GA
process. For some applications where the depolarization from the propagation
channel is dominant, polarization purity might not be a strong design
consideration.
20
(a)
(b)
Fig. 2.10
(a) Current distribution on the surface of the GA-optimized patch at
1.98GHz. (b) Current distribution on the surface of the GA-optimized
patch at 2.1GHz.
21
90
120
-30
60
-20
150
30
-10
180
0
o
?=-45
210
330
E
E
240
300
270
(a)
90
120
60
-30
?=45o
150
30
-20
-10
180
0
210
330
240
300
E
E
270
(b)
Fig. 2.11
(a) Measured radiation pattern at 1.98GHz (?=-45o plane). (b) Measured
radiation pattern at 2.1GHz (?=45o plane).
22
An additional important bandwidth enhancement effect is also observed
through the ragged edge shape. When the patch is restricted to single-mode
operation by symmetry constraints, the ragged edges in the GA-optimized shape
enhance the bandwidth by about 30% compared to the reference square
microstrip.
72mm
72mm
� 1.3
(a)
(b)
� 4.0
72mm
� 3.0
(c)
Fig. 2.12
Bandwidth enhancement is achieved by two-mode operation and
ragged edges.
23
This is shown in Figs. 2.12(a) and 2.12(b). The ragged edges cause the
broadening of the resonant frequency by introducing multiple resonant lengths
between the two radiating edges on the two sides of the patch. Therefore, this
ragged edge shape in conjunction with the two-mode operation in the GAoptimized design in Fig. 2.12(c) results in the broadest bandwidth possible (�0
compared to Fig. 2.12(a)).
2.4 Summary
Optimized patch shapes for broadband microstrip antennas on thin FR-4
substrate have been investigated using the genetic algorithm. The optimized shape
showed a four-fold improvement in bandwidth compared to a standard square
microstrip. This result has been verified by laboratory measurement. The basic
operating principle of the optimized shape can be explained in terms of a
combination of two-mode operation and ragged edge shape.
The number of iterations needed for GA to converge is in general quite
large in this implementation. This results in long computation time during the
design. The convergence rate could potentially be reduced by hybridizing GA
with other optimization algorithms such as local search [23,24] or Tabu search
[25].
24
Chapter 3
Design of Multi-Band Microstrip Antennas Using a Genetic
Algorithm
3.1 Introduction
In the previous chapter, a design method for broadband microstrip antenna
was developed. Arbitrary microstrip patch shapes with the broadest bandwidth
were found using a GA. The designed antenna operates at the frequency band
around 2GHz. However, with the growing demand in wireless applications,
microstrip antennas that can operate at more than one frequency band have
become an area of great research interest.
This chapter examines the use of GA to design optimal shapes for
microstrip antennas to achieve multi-band operation. Various multi-band methods
for microstrip antennas have been proposed to date. For example, multi-layered
structures, parasitic patches and shorting posts are some of the well-known
techniques for achieving multi-band operation [14,15].
However, these
techniques usually lead to an increase in antenna size or manufacturing cost. The
design of dual-band microstrip antennas using GA was first addressed by Johnson
and Rahmat-Samii [6]. The attractiveness of the GA design over the
aforementioned methods is its ability to achieve the desired performance using a
25
single, unique patch shape. In their study, they used air as the substrate martial.
Just as in the broadband design in Chapter 2, we focus on FR-4 as the substrate
material here, as it is the most commercially viable material in wireless devices.
First, dual-band microstrips are designed by optimizing the size of vertical
slots on the metallic patch. Unfortunately, these antenna designs exhibit poor
antenna efficiency. To solve the low efficiency problem, dual-band microstrips
are designed without slots cut, but with an emphasis on optimizing the overall
shape. The optimized patches can achieve arbitrary frequency spacing ranging
from 1:1.1 to 1:2. Furthermore, tri-band and quad-band microstrip shapes are
also generated and the resulting designs show good operation at the designated
frequencies. All results are verified by laboratory measurements on FR-4
substrate.
3.2 Miniaturized Dual-Band Microstrip Using Slots-Cut
3.2.1 GA and Cost Function for Dual-Band Design
GA is applied to the design of a dual-band microstrip using slots cut on the
metallic patch. This dual-band design uses the same substrate material and
substrate thickness as used in the broadband design. The design goal is to produce
a good impedance match at the frequencies of 1 and 2GHz. In addition, the size of
the patch is constrained to 42.5mm�mm. This patch size is 40% smaller than
that of a standard square microstrip working at the frequency of 1GHz. It is
26
known that slots in microstrips have the effect of lowering the resonance
frequency [21]. Therefore, a slot-encoding scheme as shown in Fig. 3.1 is adopted
to encode each shape into a chromosome.
Substrate
Metal
Feed
Position
0
0
1
Fig. 3.1
0
1
0
1
1
0
0
1
1
0
1
0
1
0
0
1
1
1
1
0
1
1
1
0
1
1
1
0
1
1
0
1
1
1
0
0
1
0
1
1
1
1
1
0
0
1
1
0
Encoding scheme for the dual-band design.
27
1
1
0
0
1
1
0
1
1
0
1
1
Five equally spaced vertical lines are chosen in the patch area. These pre-selected
lines are encoded into a binary chromosome where ones denote metal, and zeros
denote non-metal. In addition, three more bits are added to the chromosome to
represent the position of the feed along the dashed line on the patch. Since
significant power dissipation is expected due to the introduction of the slots, metal
loss is taken into account by adding a second term to the cost function. The first
part of the cost function accounts for the impedance mismatch, in the same way as
in (2.1). The second part of the cost function accounts for the metal loss generated
by the current flowing on the patch. The conductivity of aluminum (?=3.82�7
S/m) is used.
Cost =
1
N
N
( Pn
n =1
where Pn =
Qn =
+ Qn )
(3.1)
S11 (dB) + 10dB if S11 (dB) ? ? 10dB
if S11 (dB) < ?10dB
0
PJ (dB) + 70dB if PJ (dB) ? ? 30dB
if PJ (dB) < ? 30dB
0
Rs
2
J s ds (dB)
2 s
w�
Rs =
2?
PJ =
28
3.2.2 Simulation and Measurement Results
Fig. 3.2(a) shows the GA-optimized microstrip for dual-band operation.
The white dot shows the position of the feed, and it resides 12.5mm from the left
edge. Fig. 3.2(b) shows the measurement and the simulation results of the return
loss. Other than a slight shift in the resonant frequencies, the graph shows good
agreement between the measurement and the simulation. The bandwidths at the
two frequencies of 1 and 2GHz are 1.2% and 1.37%, respectively. The measured
relative gain of this microstrip shows a gain loss of ?7 dB at 1GHz and ?7.5 dB
at 2GHz when compared to square microstrips operating at those two frequencies.
To explain the gain loss, Fig. 3.3 shows the calculated metal loss (dBm) of
the microstrips. The solid line is the metal loss of the GA-optimized microstrip,
and the dashed lines are the losses of two reference square microstrips. At the
low-band frequency (1GHz), the GA-optimized microstrip shows a metal loss 5
dB higher than the square microstrip of size 72mm�mm, while at the high-band
it shows a metal loss 6 dB higher than the square microstrip of size 36mm�mm.
These values show that the main cause for the gain loss comes from metal loss. It
should be noted that a dual-band microstrip without any constraint on the metal
loss has been also optimized, and the resulting design shows an even larger metal
loss. Thus the GA design achieves better radiation efficiency by taking the metal
loss on the patch into consideration.
29
40mm
y
x
42.5mm
(a)
S11 (dB)
-5
-10
-15
-20
0.80
1.08
1.36
1.64
1.92
Frequency (GHz)
2.20
(b)
Fig. 3.2
(a) GA-optimized slot shape for dual-band operation. (b) Return loss (dB) of
the GA-optimized antenna from simulation ( ) and measurement (?
??).
30
30
25
Loss (dBm)
20
15
10
5
0
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Frequency (GHz)
Fig. 3.3
Simulated total metal loss (dBm) of the GA-optimized dual-band microstrip
(?
??), a square microstrip of 72mm�mm (------) and a square microstrip
of 36mm�mm (-?-?-?-?).
3.2.3 Operating Principle of the GA Design
To explain the miniaturization principle of the GA design, it should be noted that
a microstrip of size 42.5mm�mm without any slots has a resonant frequency of
1.7GHz for the TM10 mode, and a resonant frequency of 3.4GHz for the TM20
mode. Fig. 3.4 shows the current plots for the GA-optimized microstrip with slots
at the resonant frequencies of 1 and 2GHz. At the low band, the surface current
distribution resembles the TM10 mode.
31
(a)
(b)
Fig. 3.4
(a) Current distribution on the surface of the GA-optimized patch at 1 GHz. (b)
Current distribution at 2 GHz.
32
The excited surface current is forced through the narrow channel near the feed.
Since the current path has to meander around the slots, the total length of the
current path is increased. Therefore, the resonant frequency is reduced by a factor
of about 1.7 compared to the microstrip without slots.
At the high frequency band, the patch current resembles the TM20 mode.
The currents meander around the two large vertical slots near the two vertical
radiating edges on the patch. In contrast to the slots near the feed, these slots
lower the resonant frequency for the high band. These two effects have been
verified through numerical simulation. If the length of the large slots near the two
radiating edges is slightly reduced, the resonant frequency of the high band is
increased. If the length of the slots near the feed is reduced, the resonant
frequency of the low band is increased. These results clearly show that the
miniaturization of the GA-optimized design is achieved by the frequency
lowering effect of the slots to both the low and high frequency bands of the
microstrip.
Figs. 3.5(a) and (b) illustrate the measured radiation patterns at the
frequency of 1GHz and 2GHz respectively. The radiation plots exhibit broadside
radiation and show less than ?20dB cross polarization level at both frequencies. In
addition, the radiation patterns are almost identical at the two frequencies.
33
90
120
-10
60
-20
150
30
-30
180
0
210
330
240
300
E
E
270
(a)
90
120
60
-10
-20
150
30
-30
180
0
210
330
240
300
270
E
E
(b)
Fig. 3.5
(a) Measured radiation patterns at 1 GHz (?=0 plane). (b) Measured radiation
pattern at 2 GHz (?=0 plane).
34
3.3 Multi-Band Microstrip Without Slots-cut
In Section 3.2, a GA-designed dual-band microstrip on high-permittivity
substrate based on slots cut into the patch was attempted. However, metal loss
was found to be significant due to the use of the slots. Also only a particular
dual-band frequency spacing ratio of 1:2 (1GHz and 2GHz) was investigated.
This section examines the design of optimal patch shapes for multi-band
operation. To reduce the metal loss, the patch shape without slots cut is
investigated. Unlike the fixed spacing of dual-band operation, as reported in
Section 3.2, the general set of frequency spacing between the two frequency
bands is investigated in this section. The optimized patches achieve arbitrary
frequency spacing ranging from 1:1.1 to 1:2. In addition to dual-band operation,
tri-band and quad-band microstrip shapes are also generated.
3.3.1 GA and Cost Function for Multi-Band Design
The GA methodology similar to one reported earlier for broadband
application in Chapter 2 is implemented to optimize the patch shape. In this GA
implementation, a 2-D chromosome is used to encode each patch shape into a
binary map [19]. The metallic sub-patches are represented by ones and the nonmetallic areas are represented by zeros. Since it is more desirable to obtain
optimized patch shapes that are well connected from the antenna efficiency and
35
manufacturing point of view, a 2-D median filter [20] is applied to the
chromosomes to create a more realizable population at each generation of the GA.
To evaluate the performance of each patch shape, a full-wave periodic
patch code [17,18] is used. The design goal is to maximize antenna bandwidth at
multiple frequency bands by changing the patch shape. To achieve the design
goal, the cost function in (3.2) is defined as the average of those S11 values that
exceed ?10dB (i.e., VSWR=2:1) within the frequency bands of interest.
Cost =
1
N
N
n =1
where Pn =
( Pn )
(3.2)
S11 (dB) + 10dB if S11 (dB) ? ? 10dB
if S11 (dB) < ?10dB
0
Based on the cost function, the next generation is created by a reproduction
process that uses crossover, mutation, and 2-D median filtering. A two-point
crossover scheme involving three chromosomes is used.
3.3.2 GA and Cost Function for Dual-Band Design
First, the dual-band design is carried out to achieve different frequency
ratios between the low and the high bands. For each microstrip, the low frequency
band is fixed at 1.8GHz, and the high frequency band is varied. The insets in Figs.
3.6(a), (b) and (c) are the GA-optimized designs for the frequency ratios of 1:1.3,
1:1.6 and 1:1.9, respectively. A 72mm � 72mm square design area on which the
metallic patch can reside is discretized into a 32 � 32 grid for the chromosome
36
definition. The thickness of the FR-4 substrate (dielectric constant of about 4.3) is
1.6 mm. The white dot shows the position of the probe feed. In the same figure,
the calculated return loss (S11 in dB) of the resulting microstrip is plotted as a
dashed line. It shows good dual-band operation at the designed frequencies. To
verify the GA design experimentally, the microstrip patches described above are
built and measured. Copper tape is used to construct the metallic patches and the
dimension of the ground plane is 15.3cm � 15.3cm. The measurements were
taken on an HP8753C network analyzer. The solid lines in Fig. 3.6 are the
measured return losses versus frequency.
Good agreement can be observed
between the measurements and simulations. The radiation patterns for these
microstrips are also measured. All three microstrips show broadside radiation
patterns at both operating frequencies, with linear polarizations that are nearly
orthogonal to each other. The measured realized gains for these three antennas
range from ?1.3dB to 2dB in the broadside direction. Due to the high loss tangent
of FR-4, the dissipation in the antenna (and thus the radiation efficiency) is
mainly dominated by dielectric loss.
37
S11 (dB)
-5
-10
72mm
-15
-20
1.65
2.03
2.41
2.79
3.17
3.55
Frequency (GHz)
(a)
S11 (dB)
-5
-10
72mm
-15
-20
1.65
2.03
2.41
2.79
3.17
3.55
Frequency (GHz)
(b)
Fig. 3.6
Shapes of three GA-optimized dual-band microstrip antennas, and the
resulting return loss from simulation (------) and measurement (??). (a)
Frequency ratio of 1:1.3 (1.8GHz and 2.34GHz). (b) Frequency ratio 1:1.6
(1.8GHz and 2.9GHz). (c) Frequency ratio of 1:1.9 (1.8GHz and 3.42GHz).
38
-10
-15
72mm
S11 (dB)
-5
-20
1.65
2.03
2.41
2.79
3.17
3.55
Frequency (GHz)
(c)
Fig. 3.6
(Cont'
d)
(c) Frequency ratio of 1:1.9 (1.8GHz and 3.42GHz).
This was verified by running the simulation with and without dielectric loss, and
found that dielectric loss causes a 4 to 8 dB loss in gain.
In addition to the three frequency spacings presented in Fig. 3.6, designs
for other frequency ratios ranging from 1:1.1 to 1:2 were also realized using GA.
Figs. 3.7(a) to (g) are optimized microstrip shapes for other frequency spacings.
39
(f) Ratio 1:1.8
72mm
(e) Ratio 1:1.7
72mm
72mm
(d) Ratio 1:1.5
(c) Ratio 1:1.4
72mm
(b) Ratio 1:1.2
72mm
72mm
(a) Ratio 1:1.1
72mm
(g) Ratio 1:2
Fig. 3.7
Shapes of dual-band microstrip antennas. Frequency ratio from 1:1.1 to 1:2.
40
S11 (dB)
-5
-10
72mm
-15
-20
1.40
1.68
1.96
2.10
2.52
2.80
Frequency (GHz)
Fig. 3.8
Shape of the GA-optimized tri-band microstrip antenna that operates at
1.6GHz, 1.8GHz and 2.45GHz, and the resulting return loss of the antenna
from simulation (------) and measurement (?
??).
All designed shapes showed good dual-band operation at the two design
frequencies. It was also numerically verified that these shapes could be scaled in
size to different operating frequencies of interest or to other substrate materials
with only minor modifications.
Next, tri-band designs having three operating frequencies at 1.6GHz
(GPS/L1), 1.8GHz (DCS) and 2.45GHz (ISM/Bluetooth) is attempted. Fig. 3.8
shows the optimized shape using the GA technique and the corresponding return
41
loss. It shows excellent tri-band operation at the design frequencies. The
measured result again shows close agreement with the simulation result. The
bandwidths obtained at these frequency bands are respectively 2.36%, 2.54% and
1.22% from simulation and 1.81%, 2.16% and 1.42% from measurement. Finally,
quad-band designs having operating frequencies at 0.9GHz (GSM900), 1.6GHz,
1.8GHz and 2.45GHz are tried. Reasonably good quad-band operation is
demonstrated in Fig. 3.9. Simulation shows a return loss of less than ?10dB (the
design goal) at all four bands.
S11 (dB)
-5
-10
144mm
-15
-20
0.80
Fig. 3.9
1.16
1.52
1.88
Frequency (GHz)
2.24
2.60
Shape of the GA-optimized quad-band microstrip antenna that operates at
0.9GHz, 1.6GHz, 1.8GHz and 2.45GHz, and the resulting return loss of the
antenna from simulation (------) and measurement (?
??).
42
The measured result shows a return loss of less than ?10dB at the first, third and
fourth band, while the second band has a slightly higher (?9.4dB) return loss.
The results demonstrate that it is possible to use this GA approach in designing a
multi-band microstrip that requires very specific frequency bands of operation.
3.4 Summary
Optimized patch shapes for multi-band microstrip antennas have been
investigated using GA. For the dual-band design with slots cut, the optimized
shape showed good operation at both frequencies with a 40% reduction in size
compared to a standard microstrip. The measurement result matched well with the
numerical prediction. The operating principle of the optimized dual-band
microstrip can be explained by the frequency lowering effect from the narrow
slots for the different modes operating at the two frequency bands. Gain loss due
to the presence of the slots, however, is intrinsic to the miniaturized design. It is
only partially alleviated by the GA process.
For the multi-band design without slots cut, it has been shown that
frequency ratios between the two bands ranging from 1:1.1 to 1:2 can be achieved
using the GA methodology. Tri-band and quad-band microstrip shapes have also
been generated, and the resulting antennas showed good operation at the design
frequencies. All results have been verified by laboratory measurements on FR-4
substrate.
43
Chapter 4
Shape Optimization of Corrugated Coatings Under Grazing
Incidence Using a Genetic Algorithm
4.1 Introduction
In this chapter, GA is utilized to design corrugated coatings to reduce the
forward scattering under grazing incidence. Lossy material coatings are
commonly used to reduce scattering from conducting bodies. In general, design of
coatings should meet multiple criteria including low reflection, small volume and
light weight. These design goals often conflict with one another. Multi-layer
planar coatings have been studied extensively for their wideband absorbing
characteristics [27,28]. GA have been applied with success in finding optimal
thicknesses for multi-layer coatings in either planar or cylindrical configurations
[29-30]. Corrugated coatings with non-planar profiles offer additional degrees of
freedom and have been studied in [31-33]. In particular, it was shown in [34,35]
that single-material corrugated coating can be exploited to alleviate polarization
dependence and improve the absorption performance over a wide range of
frequencies at near-grazing incidence. However, only a few simple shapes were
considered. In this chapter, GA is used to explore more arbitrary coating shapes in
an attempt to achieve better absorber performance. With more degrees of freedom
44
in the design, arbitrarily shaped coatings may give rise to better absorbing
performance. At the same time, finding an optimal shape is more challenging as
the design parameter space is much larger.
In this approach, a full-wave EM simulation code is used to evaluate the
absorbing performance of each shape. GA is implemented to optimize the shape
of the coating, which is encoded into a binary chromosome. A two-point
crossover scheme involving three chromosomes is used to achieve fast
convergence. In addition, geometrical filtering is adopted to create more
realizable shapes. This study focuses on the near-grazing incidence case.
A
single-layer MAGRAM material [36] is used for the coating. This method is first
applied to achieve optimal shapes under various polarization constraints. The
physical interpretation of the optimized structures is discussed and their
performance is compared to the baseline results obtained from conventional
planar and triangular shaped designs. Next, the Pareto GA [2,4] is employed to
map the more general set of optimal solutions trading off coating thickness (or
weight) versus absorbing characteristics.
This chapter is organized as follows. Section 4.2 describes the EM
simulation code utilized and other details in this GA implementation. Section 4.3
describes the application of the GA to the design of optimal coating shapes under
various geometrical and polarization constraints. In Section 4.4, multi-objective
optimization is applied to the design of coating, taking both absorbing
45
performance and coating thickness into account. Finally, Section 4.5 summarizes
this chapter.
4.2 Approach
4.2.1 EM Simulation Code
The geometry considered in this chapter is shown in Fig. 4.1. The shaped
grooves in the coating have a period of p along the x direction and extend to
infinity along the z direction. The bottom of the coating is backed by a conducting
ground plane. A plane wave is obliquely incident upon the infinite grating with ?el
and ?grating.
y
?grating
?el
x
z
Fig. 4.1
Geometry of the corrugated absorber.
46
To evaluate the performance of each shape for the coating, a full-wave
EM code based on a boundary-integral equation formulation [35] is used. The
formulation entails dividing one cell of the grating into different homogeneous
regions according to the material layers. A homogeneous Green?s function is first
used to calculate the moment matrix. Boundary integral equations are then
obtained for each region. Field continuity at region interfaces and periodic
boundary conditions at cell boundaries are then enforced. The fields in the top
half-space are expanded into a sum of Floquet harmonics and are matched to the
fields in the lower region so that the reflection coefficients can be found. The
code has previously been validated by comparing the simulation results to
measurement data in [34, 35].
4.2.2 GA Optimization
GA is employed to optimize the shape of the coating profile. In this GA
implementation, each possible absorber shape is encoded into a binary
chromosome, as shown in Fig. 4.2. The period of the absorber is divided into M
points. The height of the coating at each point along x is represented as a binary
number. A symmetry constraint is applied in the x-direction so that only the right
half of the absorber is encoded into the chromosome. In order to obtain coating
shapes that are not too complicated from the manufacturing point of view and to
speed up convergence of the GA, a geometrical filter is applied to the
chromosomes at each generation of the GA.
47
8mm
1
0
0
0
1
0
1
1
0
0
1
1
0
1
0
1
1
0
0
1
1
0
1
0
1
0
0
1
1
1
1
0
1
1
1
0
1
1
1
0
1
1
0
1
1
1
0
0
1
0
1
1
1
1
1
0
0
1
1
0
1
1
0
0
1
1
0
1
1
0
1
1
Period = 2.032mm
Fig. 4.2
Encoding of corrugated absorber into a binary chromosome.
Two different geometrical filters were tried: a 1-D sliding window filter [38] and
a descending order filter. Figs. 4.3(a) and (b) show the shapes before and after the
1-D 7-point sliding window filter, which is a low-pass, moving-average filter. As
expected, the surface shape after the filtering looks smoother without any sharp
peaks. From the results of using this filter, it is found that the GA-optimized
profiles consistently had shapes that monotonically decreased from a central peak.
Therefore, an alternative ?descending order filter? was also tried. This filter
simply rearranges the heights of the absorber at each of the M points so that the
highest point is at the center and all other points are placed in descending order.
48
Sliding Window Filter
(b)
Descending
Order Filter
(a)
(c)
Fig. 4.3
(a) Before any geometrical filters. (b) After the 7-point sliding window filter.
(c) After the descending order filter.
49
Fig. 4.3(c) shows the shape after the descending order filter. Note that this filter
preserves the sharp edges in the design while making the shape less oscillatory. It
is found that optimized shapes from the descending order filter gave better
performance than that from the 1-D sliding window filter. Therefore, all the
results presented in this chapter are generated by using the descending order filter.
After these chromosomes are evaluated by the EM simulation code, a cost
function related to the absorbing performance is computed. Based on the cost
function, chromosomes are refined into the next generation by a reproduction
process that involves crossover, mutation and geometrical filtering. For the
crossover operation, a two-point crossover scheme involving three chromosomes
is used. This series of processes is iterated until the cost function is minimized.
4.3 GA-Optimized Coating Shapes
This section investigates coating profiles that produce the best absorbing
characteristic for a given coating height. The design frequency band is chosen to
be from 8 GHz to 18 GHz, and the maximum height of the coating is restricted to
8mm. To avoid higher order diffraction, the period of the coating is set to
2.032mm. A MAGRAM material is used for the coating (the detailed absorption
characteristics can be found in Fig. 7 of [35]). The incident angle is ?el=30� and
?grating=0�.
50
To encode each possible shape of the coating into a binary string, the
period of the coating is discretized into 30 points. The height of the groove at each
point is described by a 6-bit number (i.e., in 64 steps) that ranges between 0 and
8mm. When the GA process converges to an optimal value, the discretization for
the period and the height is increased to 60 points and 8 bits, respectively, to
achieve a more refined coating shape. Associated with the design goal, the cost
function is defined as the average of those reflection coefficient values, ?(dB),
that exceed ?20dB within the frequency band of interest:
Cost =
1
N
N
n =1
where Pn =
( Pn )
?(dB) + 20 dB
0
(4.1)
if ?(dB) ? ? 20 dB
if ?(dB) < ?20 dB
In the GA, the population size is 30. A crossover probability of 0.8 is used,
and the probability of mutation is set to 0.1. The computational time is about 8
hours on a Pentium IV 1.7GHz machine for a typical design.
First, only the reflection coefficient for the vertical polarization is used in
the cost function definition. Fig. 4.4(a) shows the resulting GA-optimized shape,
which closely resembles a triangular profile. Fig. 4.4(b) is a plot of the simulated
reflection coefficient (in dB) versus frequency for the optimized shape. The
reflection coefficient of the vertical polarization nearly meets the ?20 dB design
51
goal over the entire frequency band from 8GHz to 18GHz.
The horizontal
polarization is not optimized and shows a much higher reflection coefficient.
Next, the reverse situation when only the horizontally polarized reflection
coefficient is used in the cost function. Fig. 4.5(a) shows the resulting GAoptimized shape. The optimal shape of the corrugated coating resembles a
rectangular profile. Fig. 4.5(b) shows the associated reflection coefficient (in dB)
versus frequency for the optimized shape. In this case, the reflection coefficient of
the horizontal polarization meets the ?20 dB design goal for all the frequencies
above 10 GHz while the vertical polarization is higher. Further improvement in
the low-frequency performance will likely require a thicker coating.
Then this optimized shape is compared to the conventional planar and the
triangular shaped coatings that are also optimized using GA. The maximum
height of all three coatings is limited to the same 8mm thickness. Fig. 4.6 shows
the horizontally polarized reflection coefficients for the three coatings. The
dashed, solid, and dash-dotted lines are the respective reflection coefficients for
the GA-optimized planar, triangular, and arbitrarily shaped coatings. The planar
shaped coating (thickness of 1.19mm) shows a reflection of about ?5 dB within
the frequency range of interest.
52
8
7
Height (mm)
6
E
5
4
3
2
MAGRAM
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Period = 2.032 mm
(a)
0
H
| ? | (dB)
-10
-20
V
-30
10
12
14
Frequency (GHz)
16
18
(b)
Fig. 4.4
(a) GA-optimized shape for the vertical polarization. (b) Reflection
coefficient (dB) versus frequency at 30 degrees from grazing.
53
8
7
Height (mm)
6
E
5
4
3
2
MAGRAM
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Period = 2.032 mm
(a)
00
-5
| ? | (dB)
-10-10
V
-15
-20-20
H
-25
-30-30
-35
10
12
14
Frequency (GHz)
16
18
(b)
Fig. 4.5
(a) GA-optimized shape for the horizontal polarization. (b) Reflection
coefficient (dB) versus frequency.
54
0
0
| ? | (dB)
-10
-20
-30
10
Fig. 4.6
12
14
Frequency (GHz)
16
18
Performance comparison of the planar absorber (------), triangular shaped
absorber (?
??), and GA-optimized absorber (-?-?-?-?) for the horizontal
polarization.
Using the triangular shaped profile (base thickness of 0.03mm and
triangular height of 6.12mm) the reflection coefficient can be reduced to less than
?10dB. The GA-optimized arbitrarily shaped coating shows better absorbing
performance in terms of the cost definition in (4.1) than either of the conventional
designs.
55
8
(a)
7
0
(b)
6
5
4
3
2
-10
1
| ? | (dB)
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(b)
(c)
(c)
-20
(a)
-30
0
8
Fig. 4.7
9
10
10
11
12
12
13
14
14
Frequency (GHz)
15
16
16
17
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
18
18
Effect of cost definitions on the optimization results for the horizontal
polarization. (a) Cost definition in (4.1). (b) Minimax cost definition. (c)
Average power reflection coefficient cost definition.
Next, the sensitivity of the GA design is tested. To see the effects of
different cost definitions, the designs are optimized using two other cost
functions. The first alternative cost is the maximum reflection value across the
whole frequency range of interest (typically called the Minimax cost function).
The resulting performance is indicated by the dashed line in Fig. 4.7. The second
alternative cost function is the averaged power reflection coefficient (on a linear
scale) across the frequency band. This performance of design is indicated by the
56
dash-dotted line. Some difference in the overall performance is noted. However, it
is observed that the optimized shapes retain the overall feature of the original
design based on the cost definition in (4.1). To test the sensitivity of the GAoptimized shape to manufacturing tolerances, random RMS deviations of 0.4 mm
are introduced into the profile height. The resulting performance is shown by the
dashed line in Fig. 4.8(a). GA description of the profile is also intentionally
undersampled as the 60-point by a factor of 6, resulting in a more smoothed-out
profile. The performance is shown by the dash-dotted line. This shows that the
performance is not too sensitive to the deviation to the optimized profile. Fig.
4.8(b) shows the performance of the optimized coating for close-by incident
angles of 25o and 35o. The results indicate some degradation toward the smaller
grazing angles.
Optimizing the coating shape for both polarizations is also tried using the
average of the reflection coefficients from the horizontal and vertical polarizations
in the cost function. The resulting shape is shown in Fig. 4.9(a). As can be seen
from the previous examples, the design for the horizontal polarization is more
difficult than that for the vertical polarization. Therefore, in this case, the cost is
dominated by the horizontal polarization and the resulting GA-optimized shape is
not that different from that for the horizontal polarization shown in Fig. 4.5(a).
57
8
(ii)
(i)
7
6
0
5
4
3
2
-10
1
| ? | (dB)
0
-20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
(i)
8
9
0.4
0.6
0.8
1
1.2
1.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
(iii)
(iii)
-30
0.2
(ii)
10
11
10
12
13
12
14
15
14
16
17
16
18
18
0
0.2
0.4
1.6
1.8
2
Frequency (GHz)
(a)
0
| ? | (dB)
-10
25o
30o
-20
35o
-30
8
9
10
10
11
12
13
14
15
12
14
Frequency (GHz)
16
16
17
18
18
(b)
Fig. 4.8
(a) Performance sensitivity to variations in the shape of the profile for the
horizontal polarization. (i) Original GA-optimized design. (ii) GAoptimized shape with RMS error of 0.4 mm. (iii) Smoother shape after
6:1 undersampling. (b) Reflection coefficient (dB) versus frequency at 25o,
30o and 35o from grazing.
58
8
7
Height (mm)
6
5
4
3
2
MAGRAM
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Period = 2.032 mm
(a)
0
(V+H)/2
| ? | (dB)
-10
V
-20
H
-30
8
9
10
10
11
12
12
13
14
14
15
16
16
17
18
18
Frequency (GHz)
(b)
Fig. 4.9
(a) GA-optimized shape taking into account of both the vertical and horizontal
polarizations. (b) Reflection coefficient (dB) versus frequency.
59
Finally, an interpretation on the operating principle of the GA optimized
shape is attempted. The left side of Fig. 4.10 is a planar absorber. Typically it is
more difficult for the horizontal polarization to infiltrate an absorber near grazing
than the vertical polarization. However, if we look at the incident electric field on
the near-vertical sidewalls of the profile shown on the right side of Fig. 4.10, it
behaves more like the vertical polarization. Thus, the absorbing performance is
improved by effectively changing the horizontal polarization into the vertical
polarization.
This explains why the optimized design for the horizontal
polarization resembles a rectangular profile. Other researchers have discussed the
difference in coupling into corrugated profiles depending on polarization [39].
E
E
(a)
(b)
Fig. 4.10 (a) Planar absorber with horizontal polarized wave incidence. (b) Rectangular
profile with horizontal polarized wave incidence.
60
4.4 Multi-Objective Optimization
4.4.1 Pareto GA and Cost Function Definition
In addition to the absorbing performance of the coating, another design
criterion of interest is the coating volume, which is measured by the coating
height. An investigation of the absorbing performance versus the coating height
is studied. This can be done by repeatedly using the same methodology described
in Section 4.3 for various heights. However, it is much more efficient to cast this
problem into a multi-objective problem rather than using the conventional GA.
Pareto GA [2,4] is a useful tool for this problem. In the Pareto GA, a wide range
of solutions corresponding to more than one objective can be mapped by running
the optimization only once. In this GA implementation, two cost functions are
defined:
Cost 1 = Normalized coating height
Cost 2 = Normalized value of
where Pn =
1
N
N
n =1
(4.2)
( Pn )
(?? + ?| | ) / 2 (dB) + 20 dB
if (?? + ?|| ) / 2 (dB) ? ? 20 dB
0
if (?? + ?|| ) / 2 (dB) < ?20 dB
(4.3)
Cost 1 is determined by the coating height and Cost 2 is associated with the
reflection cost. Both costs are normalized to a value between zero and one. For
61
Cost 2, one denotes an average reflection coefficient of 0dB, while zero denotes
an average reflection coefficient that is below ?20dB.
The non-dominated
sorting method [40] is used to combine the two costs for each solution by means
of the Pareto ranking. This method assigns rank 1 to the non-dominated solutions
of the population. The term non-dominated solution means that there are no other
solutions that are superior to this solution in both objectives. Then the next nondominated solutions among the remaining solutions are assigned to the nexthighest rank. The process is iterated until all the solutions in the population are
ranked. Based on the rank, the same reproduction process described in Section 4.3
is performed to refine the population into the next generation. The set of rank 1
solutions is called the Pareto front. In order to avoid the solutions on the Pareto
front from converging to a single point in the cost space, a sharing scheme
described in [41] is performed. In this sharing process, the rank is modified by
penalizing those members on the front that are too close to each other in the cost
space. This is accomplished by multiplying a niche count (mi) to the assigned
rank. The niche count is calculated according to:
N
1 p
mi =
Sh(d ij )
( 4. 4 )
N p j =1
where the Np is the number of rank 1 members and the sharing function, Sh(dij), is
a function of the cost distance between solutions expressed as:
62
Sh(d ij ) =
2?
d ij
d share
1
if d ij < d share
(4.5)
if d ij > d share
and
d ij =
(Cost 1(i) ? Cost 1( j ) )2 + (Cost 2(i ) ? Cost 2( j ) )2
As can be seen, the sharing function increases linearly if the other members on the
front are closer than dshare from a chosen member i in the cost space.
Consequently, those members that have close-by neighbors in the cost space are
assigned lower ranks in the reproduction process.
4.4.2 Pareto GA Results
In this Pareto GA, the population size is chosen to be 100. A crossover
probability of 0.8, a mutation probability of 0.1 and a dshare distance of 1 are used.
Figs. 4.11(a)-(d) show the convergence of the solutions for this multi-objective
problem (reflection cost versus the height of the profile) as the number of
generations is increased. The period of the absorber, the material for the coating,
and the angle of incidence are the same as those used in Section 4.3. The height is
constrained to be less than 8mm. Fig. 4.11(a) is the plot of the initial population.
The majority of the solutions are located in the upper-right side of the cost
domain. Figs. 4.11(b)-(d) are plots of the population after 5, 20 and 200
generations, respectively. They show that as the number of generations increases,
63
the Pareto front spreads out and converges toward the lower-left region of the cost
space. Fig. 4.12 shows the final converged Pareto front and four optimized
coating shapes that are on the front. Inset shape (a) shows the lowest profile of the
four samples, but it has the highest reflection among the four designs. Inset shape
(d) has the highest profile and the lowest reflection. As expected, the absorbing
performance must be traded off against the profile height. If we look in detail at
the optimized shapes, it appears that as the height of the absorber decreases, the
top of the profile gets more flattened. However, the shapes maintain a rectangular
profile that is only slightly modified by the coating height. This is consistent with
the physical interpretation of the absorption process for the more dominant
horizontal polarization discussed in Section 4.3. Another observation from Fig.
4.12 is that the Pareto front is not smooth due to the quantization effect of the
coating height. If the height is discretized with more binary bits, the shape of the
Pareto front becomes smoother.
Next, the first cost is changed from coating height to coating weight while
keeping the reflection cost the same. The cost for the coating weight is normalized
to be 1 when all of the design area (period � maximum coating height) is filled by
the coating material while it is zero when no coating material exists. Fig. 4.13
shows the converged Pareto front for this problem. Also shown in insets (a) to (d)
are four optimized shapes with different coating weights.
64
0.45
0.45
0.4
0.4
0.35
0.3
0.25
0.2
A bsorbing P erformance
0.5
Reflection cost
Absorbing Performance
Reflection cost
0.5
0.3
0.25
0.2
0.15
0.1
0.35
0.15
4
4.5
5
5.5
6
6.5
7
7.5
0.1
8
4
4.5
Height of absorber (mm)
5
6.5
7
7.5
8
(b)
0.5
0.45
0.45
0.4
0.4
0.35
0.3
0.25
0.2
A bsorbing P erformance
0.5
Reflection cost
Absorbing Perform ance
Reflection cost
6
Height of absorber (mm)
(a)
0.35
0.3
0.25
0.2
0.15
0.15
0.1
5.5
0.1
4
4.5
5
5.5
6
6.5
7
7.5
8
Height of absorber (mm)
4.5
5
5.5
6
6.5
Heigh of the profile (mm)
7
7.5
Height of absorber (mm)
(c)
Fig. 4.11
4
(d)
Convergence of the Pareto front as a function of the number of generations
for absorbing performance versus absorber height. (a) Initial population. (b)
After 5 generations. (c) After 20 generations. (d) After 200 generations.
65
8
(a)
(b)
0.5
(c)
0.45
Reflection cost
0.4
(d)
0.35
0.3
0.25
0.2
0.15
0.1
Fig. 4.12
4
4.5
5
5.5
6
6.5
Height of profile (mm)
7
7.5
8
Final converged Pareto front of absorbing performance versus absorber
height. The insets show four sample designs on the Pareto front.
66
(a)
(b)
0.7
(c)
0.6
Reflection cost
0.5
0.4
(d)
0.3
0.2
0.1
0.15
0.2
0.25
0.3
Normalized Weight
0.35
0.4
Weight of the absorber
Fig. 4.13
Final converged Pareto front of absorbing performance versus absorber
weight. The insets show four sample designs on the Pareto front.
67
It should be noticed that inset (d) is very similar in shape to inset (d) in Fig. 4.12.
However, instead of trimming the top off in order to reduce the height, the weight
consideration results in designs that become progressively skinnier, as shown by
insets (c), (b) and (a). Nevertheless, the shapes still preserve the sharp sidewalls
as those presented in Fig. 4.12.
4.5 Summary
Optimized shapes for a corrugated absorber under near-grazing incidence
have been investigated using GA. First, GA was applied to design corrugated
coating depending on incident polarizations. The designed absorber shape for the
vertical polarization resembled a triangular profile, while that for the horizontal
polarization resembled a rectangular profile. The optimized shapes were
compared to canonical planar and triangular shaped designs, and were shown to
have better absorbing performance. The sensitivity of the designs to variations in
shape and incident angles were also tested, and the results showed reasonable
tolerance. A physical interpretation for the optimized shape was presented. It was
shown that the sharp sidewalls of the resulting shape effectively changed the
incident polarization from horizontal to the vertical case, thus facilitating wave
absorption.
The Pareto GA has also been applied to efficiently map out absorbing
performance versus absorber height. The non-dominated sorting method was used
68
to combine the two costs for each solution by means of the Pareto ranking. A
sharing scheme was implemented to avoid the solutions on the Pareto front from
converging to a single point in the cost space. The converged Pareto front showed
that better absorbing performance must be traded off against absorber height.
Similar conclusions were also found for the absorbing performance versus
absorber weight.
69
Chapter 5
Design of Electrically Small Wire Antennas Using a Pareto
Genetic Algorithm
5.1 Introduction
In Chapter 2 and 3, the basic GA for microstrip antenna design was
introduced. In Chapter 4, the more advanced Pareto GA was investigated to solve
the more challenging multi-objective problem. In this chapter, the developed
methodology is applied to the design of electrically small wire antennas.
As the size of wireless devices shrinks, the design of electrically small
antennas is an area of growing interest [42,43]. By the classical definition, an
electrically small antenna is one that can be enclosed in a volume of radius r
much less than a quarter of a wavelength. It is well known that the bandwidth of
an electrically small antenna decreases as the third power of the radius [44-46].
Much research has been carried out to increase the bandwidth of small antennas
using structures such as folded design, disk-loaded monopole, inverted-L or
inverted-F designs, multi-armed spiral and conical helix [47-50]. Fig. 5.1 shows
the bandwidth performance of these antennas in terms of the normalized antenna
size kr, where k=2?/? is the wave number. Recently, Altshuler reported on the
use of a GA in designing electrically small wire antennas [9]. Instead of using a
70
regular shape, he used GA to search for an arbitrary wire configuration in 3-D
space that results in maximum bandwidth for a given antenna size.
While much of the small antenna research has been focused on antenna
bandwidth, antenna miniaturization also impacts antenna efficiency. The objective
of this chapter is to apply GA in the design optimization of electrically small wire
antennas, taking into account of bandwidth, efficiency and antenna size. To
efficiently map out this multi-objective problem, the Pareto GA [2,4] is utilized.
3dB Bandwidth (%)
Fundamental Limit (L. J. Chu, 1948)
100
Goubau [50]
50
Dobbins & Rogers [49]
30
Foltz & McLean [48]
20
0.6
Fig. 5.1
0.8
1.2
Antenna size (kr)
1/4?
? monopole
1.6
Achievable bandwidth in terms of antenna size and some examples of small
antennas.
71
The concept of divided range multi-objective [51] is employed to
accelerate convergence in the GA process. In this GA approach, the multisegment wire structure similar to the one used in [9] is employed. The Numerical
Electromagnetics Code (NEC) [52] is used to predict the performance of each
wire structure. Then an optimal set of designs is generated by considering
bandwidth, efficiency and antenna size. To verify the GA results, several GA
designs are built, measured and compared to the simulation. Physical
interpretations of the GA-optimized structures, showing the different operating
principles depending on the antenna size are also provided. The performance
curve achieved by the GA approach is compared against the well-known
fundamental limit for small antennas [44-46]. To more easily assess the
performance of the antennas, the efficiency-bandwidth product is normalized by
the antenna size in order to represent the antenna performance as a single figureof-merit [54]. Finally, further improvement of the GA results is attempted by
exploring additional geometrical design freedoms to better approach the
fundamental limit.
This chapter is organized as follows. In Section 5.2, the details of the GA
implementation are described. Section 5.3 describes the GA designs and the
measurement verification of the results. In Section 5.4, the GA results are
compared to the fundamental limits. In Section 5.5, other design freedoms are
72
explored to further improve performance. Section 5.6 provides conclusions
gathered from this research.
5.2 Pareto GA Approach
The basic antenna configuration considered in this chapter is shown in Fig.
5.2. The antenna consists of M connected wire segments. Each segment of the
antenna is confined in a hemispheric design space with a radius r and an infinite
ground plane.
r
Fig. 5.2
Configuration of the multi-segment wire antenna used in the GA design.
The three design goals are: broad bandwidth, high efficiency and small
antenna size. The Pareto GA is employed to efficiently map out this multi-
73
objective problem. The advantage of using the Pareto GA over the conventional
GA is that a wide range of solutions corresponding to more than one objective can
be mapped by running the optimization only once.
Binary Chromosome
Antenna Shape
1
0
0
0
1
0
1
1
0
0
1
1
Fig. 5.3
0
1
0
1
1
0
0
1
1
0
1
0
1
0
0
1
1
1
1
0
1
1
1
0
1
1
1
0
1
1
0
1
1
1
0
0
1
0
1
1
1
1
1
0
0
1
1
0
1
1
0
0
1
1
0
1
1
0
1
1
Encoding of the wire configuration into a binary chromosome.
In this GA implementation, the hemispheric design space is evenly
n
discretized into 2 grid points, and the location of each joint of the antenna is
encoded into an n-bit binary string, as shown in Fig. 5.3. Thus, the total number of
bits in the chromosome is nM when M connected wire segments are used. The
three costs associated with these design goals are:
74
Cost 1 = 1 ?
Antenna Bandwidth
Theoretical Bandwidth Limit
(5.1)
Cost 2 = 1 ? Efficiency
Cost 3 = Antenna Size ( kr )
In
the
above
definition,
the
theoretical
bandwidth
limit
of
1/((1/kr)+(1/kr)3) derived in [46] is used. NEC [52] is used to predict the antenna
performance in order to compute the cost functions. Multiple Linux machines are
used in parallel to carry out this computation.
After evaluating the three cost functions of each sample structure using
NEC, all the samples of the population are ranked using the non-dominated
sorting method [40]. Here, the higher the rank (1 denotes the highest rank), the
better the solution. This method assigns rank1 to the non-dominated solutions of
the population. The term non-dominated solution means that there are no other
solutions that are superior to this solution with respect to all design objectives.
Then the next non-dominated solutions among the remaining solutions are
assigned rank2. The process is iterated until all the solutions in the population are
ranked. Based on the rank, a reproduction process is performed to refine the
population into the next generation. The set of rank1 solutions is termed the
Pareto front. By favoring the higher-ranked solutions in the reproduction process,
the Pareto GA tries to push the Pareto front as close to the optimum solution in
the cost space as possible.
75
For the crossover operation, a two-point crossover scheme involving three
chromosomes is used. For numerical stability, a geometrical check is applied to
prevent the wires from intersecting one another. In order to avoid the solutions
from converging to a single point, a sharing scheme as described in [41] is
performed to generate a well-dispersed population. In the sharing process, the
rank is modified by penalizing those members on the front that are too close to
each other in the cost space. This is accomplished by multiplying a niche count
(mi) to the assigned rank. The niche count is calculated according to:
1
mi =
Np
Np
j =1
Sh (dij )
(5.2)
where the Np is the number of rank-1 members and the sharing function,
Sh(dij), is a function of the cost distance between solutions expressed as:
Sh (dij ) =
2?
1
dij =
dij
dshare
if dij < dshare
(5.3)
if dij > dshare
( Cost 1(i) ? Cost 1( j) )2 + ( Cost 2 (i) ? Cost 2 ( j) )2 + ( Cost 3 (i) ? Cost 3 ( j) )2
The sharing function increases linearly if the other members on the front
are closer than dshare from a chosen member i in the cost space. Consequently,
76
those members that have close-by neighbors in the cost space are assigned lower
ranks in the reproduction process.
The standard Pareto GA did not always give satisfactory results in this
problem, since it is much harder for small-sized antennas to converge than for
large-sized antennas. Thus, large-sized antennas usually dominate the whole
population after several generations of the GA process.
Cost1 (Size)
Range 1
Range 2
Cost3 (Efficiency)
Range 3
Cost2 (Bandwidth)
Fig. 5.4
Divided range multi-objective GA approach.
77
As a result, the final Pareto front contains only antenna designs with sizes
kr?0.45. To avoid this bias, the concept of the divided range multi-objective GA
[51] is employed in this implementation. As shown in Fig. 5.4, the size is
partitioned into multiple ranges and carry out the Pareto GA on each range
individually. After each range converges to an optimal solution, the populations
from all ranges are merged and the combined population is optimized in the last
step of this process. This algorithm shows much improved performance for this
more difficult multi-objective problem. Using the scheme good results for smallsized antennas (kr<0.45) as well as large-sized ones (kr>0.45) can be achieved.
5.3 GA-Optimized Results
5.3.1 GA Optimized Designs
This section investigates the optimal antenna shapes that produce the best
efficiency-bandwidth (EB) product for a given antenna size. Seven wire segments
are used in the antenna configuration. The 3-D hemisphere with radius r is
discretized into 215 points and the locations of the 7 wire segments are encoded as
a binary chromosome of 7� bits. In addition, an extra three bits are added for
choosing 8 different wire conductivities varying from 1�6 (graphite) to 5.7�7
(copper). The population size is chosen to be 2000 and the population is divided
into
four
sub
ranges
(0.29?kr<0.38,
78
0.38?kr<0.46,
0.46?kr<0.54
and
0.54?kr<0.63). Each of four ranges has a population size of 500. A crossover
probability of 0.8, a mutation probability of 0.1 and a dshare distance of 1 are used.
The target design frequency is chosen at 400 MHz and an infinite ground plane is
assumed in the simulation. All antennas are designed to match to a 50 ?
impedance.
The total computational time is about 20 hours using four Pentium IV
1.7GHz machines running in parallel. Figs. 5.5 (a), (b) and (c) show the designs in
the population with a rank of 1 at respectively, the initial, 200 and 1000
generations of the Pareto GA process. The three axes are bandwidth, efficiency
and antenna size. Each dot represents a particular rank-1 design. In the initial
generation, only a few rank-1 solutions exist. After 200 generations, many more
rank1 solutions appear. After 1000 generations, a large portion of solutions (770
over 2000) is on the Pareto front. The solutions are relatively well spread out over
the Pareto front due to the sharing operation.
79
(a)
Size (kr)
0.6
0.55
0.5
0.45
0.4
0.35
0.3
2
1.8
1.5
1
1.6
1.4
Efficiency log10(%)
0.5
Bandwidth log10(%)
0
(c)
0.6
C
Size (kr)
0.55
0.5
B
0.45
0.4
A
0.35
0.3
(b)
2
1.8
1.5
1
1.6
Size (kr)
0.6
1.4
Efficiency log10(%)
0.55
0.5
0.5
0
Bandwidth log10(%)
0.45
0.4
0.35
0.3
2
1.8
1.5
1
1.6
1.4
Efficiency log10(%)
0.5
0
Bandwidth log10(%)
Fig. 5.5 Convergence of the Pareto front as a function of the number of generations in
terms of bandwidth, efficiency and antenna size. (a) Initial generation. (b) After
200 generations. (c) After 1000 generations.
80
5.3.2 Verification of the GA-Optimized Results
To verify the GA results, three GA-optimized designs are selected from
the Pareto front (at points A (kr=0.32), B (kr=0.42) and C (kr=0.5)) and are
shown in Fig. 5.6. The smallest sample, at point A, somewhat resembles a helix,
while the largest sample C resembles a complicated loop where the end of the
wire is connected to the ground plane. These three designs were built and their
performances were measured. Copper wire of radius 0.5mm is used, and a 1.6m �
1.6m conducting plate as the ground plane is used. Fig. 5.7(a) is a photo of design
B and Fig. 5.7(b) is the resulting return loss (dB) as a function of frequency by
simulation and measurement. Except for a slight (3%) shift in the resonant
frequency, the simulation and measurement results show nearly the same
bandwidth (about 5.3% based on |S11| ? -3dB). Fig. 5.7(c) is the resulting
efficiency of the antenna. The standard Wheeler cap method [55,56] is used to
measure the efficiency. The measured efficiency matches the simulation well at
the resonant frequency of 400 MHz, as indicated by the arrow in Fig. 5.7(c). At
other frequencies, the agreement is also good except in the neighborhood of
385MHz. The presence of the large efficiency dip based on the measured data is
due to an anti-resonance in the antenna, as the Wheeler cap method fails near this
anti-resonance.
81
(a)
kr=0.34
(c)
kr=0.50
(b)
kr=0.42
Fig. 5.6
Three samples from the Pareto front (a) kr=0.34, (b) kr=0.42 and (c)
kr=0.50.
82
(a)
(c)
0
100
-1
90
-2
80
-3
70
Efficiency (%)
S11 (dB)
(b)
-4
-5
-6
-7
60
50
30
-8
20
-9
10
-10
350
360
370
380
390
400
410
420
430
440
450
0
350
Simulation
360
370
380
390
400
410
420
Frequency (MHz)
Frequency (MHz)
Fig. 5.7
Measurement
40
(a) Photo for antenna B, which has an antenna size of kr=0.42. (b) Return
loss and (c) efficiency versus frequency of antenna B. The efficiency
measurement was done using the Wheeler cap method.
83
430
440
450
Similar good agreements were also found for antennas A (kr=0.34) and C
(kr=0.50). The results are summarized in Table 5.1. It is noted that both the
achievable bandwidth and efficiency drop as the antenna size is reduced.
Measured
BW (3dB)
Simulated
BW (3dB)
Measured
Eff (%)
Simulated
Eff (%)
Antenna A
2.1 %
2.5 %
84 %
88 %
Antenna B
5.3 %
5.5 %
92 %
94 %
Antenna C
8.5 %
9.8 %
94 %
97 %
Table 5.1
Bandwidth and efficiency for the sample antennas A, B and C by measurement
and simulation.
5.3.3 Physical Interpretation of GA-Optimized Design
By examining the GA-optimized antenna structure in Fig. 5.6(c), we see
that the end of the antenna is connected to the ground plane. Most of the antennas
with 0.45<kr<0.65 have this characteristic, which is similar to a folded monopole
antenna. Since a folded monopole has four times the input impedance of a
standard monopole [57], the GA-designed antennas use this basic structure to
boost up the impedance of the antenna to approach 50 ?.
As we examine at the GA-optimized structures for even smaller-sized
antennas (kr<0.45) such as those in Figs. 5.6(a) and (b), most of the antennas are
shorted to the ground plane at the joint between the first and second segments
84
from the feed. This turns the first segment into an inductive feed. Segments 2
through 7 become the radiating part of the antenna, carrying most of the current.
The strength of the inductive coupling depends on the distance between the first
and second segments. Since inductive coupling can greatly increase the input
impedance, the GA finds this as an optimized structure for very small-sized
antennas (kr<0.45), which need a large impedance step-up to get to 50 ?. This
concept is currently investigated in more detail in order to design very small
antennas [58].
5.4 Comparison to Fundamental Limit
In this section, the GA results are compared to the fundamental limits for
small antennas. Fig. 5.8 depicts all of the GA-optimized designs on the Pareto
front plotted in the 3-D bandwidth, efficiency and antenna size space. A least
squares fit is used a to create a surface that best fits the GA results. To more easily
interpret the results, the 3-D plot is projected onto three planes, and the results are
shown in Figs. 5.9(a), 5.9(b) and 5.9(c). Then these GA results are compared to
the well-known fundamental limit using a combination of equations in [45] and
[46]:
BW =
1
Eff 譗
Where Q =
(5.4)
1
1
+ 3
kr (kr)
85
Antenna size (kr)
C
B
A
Efficiency log10(%)
Fig. 5.8
Bandwidth log10(%)
Pareto front of the GA designs after convergence. The surface is generated
using a least squares fitting to best fit the GA results shown as dots.
The curves based on (5.4) are shown in Figs. 5.9(d)-5.9(f). Fig. 5.9(a)
shows the maximum bandwidth curve achievable by the GA designs as a function
of antenna size for different efficiencies. As expected, for a given efficiency, the
achievable bandwidth decreases as the antenna size is reduced. Also, the higher
the efficiency, the lower the achievable bandwidth. It is similar to the trend of the
86
fundamental limit in Fig. 5.9(d). However, the GA performance is lower than the
fundamental limit. Fig. 5.9(b) is the projection of GA designs on the antenna size
versus efficiency plane. For a given bandwidth, the achievable efficiency
decreases as the antenna size is reduced. This trend can also be observed in Fig.
5.9(e). Fig. 5.9(c) is the projection of the GA results on the bandwidth versus
efficiency plane. For a given antenna size, the tradeoff between antenna
bandwidth and antenna efficiency can be clearly seen. Again, the trend on this
graph is similar to the fundamental limit in Fig. 5.9(f).
To make the antenna performance easier to assess, the figure-of-merit ?
suggested in [54] is used:
?=
Eff � BW
2
Q
(5.5)
where Q is given in (5.4) and an extra factor of 2 is used to account for the loaded
Q. Using this expression, the fundamental limit on ? is always 1 for antennas of
arbitrary sizes. All of the GA-optimized designs are re-plotted using this figureof-merit in Fig. 5.10(a). As a reference, the disk-loaded monopole from [48] is
plotted on the same figure. As can seen in Fig. 5.10(a), most of the GA designs
are ?<0.5. The smaller the antennas, the worse the performance of the GA
designs. Next the way to further improvement of the GA designs is investigated.
87
(a)
50
40
50
40
30
30
Bandwidth (%)
Bandwidth (%)
(d)
20
10
Eff
Eff
Eff
Eff
1
0.3
0.35
0.4
0.45
0.5
=
=
=
=
0.55
40%
55%
70%
85%
20
10
1
0.3
0.6
Eff
Eff
Eff
Eff
0.35
Antenna size (kr)
0.4
0.45
0.5
=
=
=
=
0.55
40%
55%
70%
85%
0.6
Antenna size (kr)
(b)
(e)
Efficiency (%)
80
BW
BW
BW
BW
=
=
=
=
100
5%
10%
20%
30%
80
Efficiency (%)
100
60
40
20
0.3
0.35
0.4
0.45
0.5
0.55
60
40
20
0.3
0.6
BW
BW
BW
BW
0.35
Antenna size (kr)
0.4
0.45
0.5
=
=
=
=
0.55
5%
10%
20%
30%
0.6
Antenna size (kr)
(f)
100
100
80
80
Efficiency (%)
Efficiency (%)
(c)
60
40
20
kr
kr
kr
kr
1
=
=
=
=
0.34
0.42
0.50
0.59
10
20
30
40
20
40 50
Bandwidth (%)
Fig. 5.9
60
kr =
kr =
kr =
kr =
1
0.34
0.42
0.50
0.59
10
20
30
40 50
Bandwidth (%)
(a) Projection of the Pareto front onto the size and efficiency plane.
(b) Projection of the Pareto front onto the size and bandwidth plane.
(c) Projection of the Pareto front to the bandwidth and efficiency plane.
(d) through (f) show the corresponding fundamental limit based on (5.4).
88
5.5 Further Improvement on GA Designs
To bring the figure-of-merit of the designs even closer to the fundamental
limit, additional design freedoms are explored to the original 7-wire
configuration. First, the number of segments is increased up to 16 wires.
However, these results show almost no improvement compared to the original 7wire ones. Next, the selection of characteristic impedance is permitted to vary
from 1 ? to 300 ?, instead of requiring a fixed 50 ? for the input port. This is the
assumption used in the work of Altshuler [9], who assumes that a perfect
impedance transformer is available. When allowed this freedom, the GA produced
the results plotted in Fig. 5.10(b). As the graph shows, using variable
characteristic impedance only gives a slight improvement in performance over the
original design plotted in Fig. 5.10(a).
Then a multi-arm configuration is examined to improve antenna
performance. Two arms are used for the antennas structure and the resulting
figure-of-merits are shown in Fig. 5.10(c). It shows good improvement for largesized antennas (kr>0.45). Using the multi-arm configuration, the antenna
efficiency is increased while the antenna bandwidth is preserved. This is achieved
by spreading the current on multiple branches and lowering the power dissipation.
However, the multi-arm design does not show much improvement for
small-sized antennas (kr<0.45). The number of arms is further increased to four,
but adding more number of arms does not show more improvement over the two89
arm design. It is due to the difficulty in packing a multi-arm structure in a limited
design space.
Finally, two different wire radii are allowed in each design as an
additional degree of design freedom. The resulting antenna has one wire radius
for the lower portion of the antenna and another radius for the upper portion.
Stepping the wire radius has the effect of increasing the input impedance, similar
to the way a radius step up is used in a folded monopole design.
Since it is
known that the NEC version 2 used in the simulation does not accurately model
the wire radius change, the radius change is limited to less than two.
Fig. 5.10(d) shows the result of the design.
It shows improved
performance for both small-sized antennas and large-sized antennas. Based on
these preliminary results, it appears that the use of targeted design freedoms can
further improve the performance of the GA-optimized designs toward the
fundamental limit.
5.6 Summary
The Pareto GA has been applied to design electrically small wire antennas
by considering antenna bandwidth, efficiency and size. Wire structures
comprising of multiple segments were considered. The key advantage of using
the Pareto GA method is that a whole series of optimal designs of varying size,
bandwidth and efficiency can be generated efficiently in a single GA run.
90
�- 7-segment wire
(a)
1.2
Fundamental Limit (2/Q)
1
Figure-of-merit (?)
[48]
0.8
0.6
C
B
A
0.4
0.2
0
0.3
0.35
0.4
0.45
0.5
Antenna size (kr)
0.55
0.6
0.65
�- Variable characteristic impedance
(b)
1.2
Fundamental Limit (2/Q)
Figure-of-merit (?)
1
[48]
0.8
0.6
0.4
0.2
0
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Antenna size (kr)
Fig. 5.10
Small antenna performance using the definition of ? = (Eff � BW) /
Theoretical BW Limit. (a) Original 7-wire configuration. (b) Variable input
impedance.
91
�- Multi-arm
(c)
1.2
Fundamental Limit (2/Q)
1
[48]
Figure-of-merit (?)
0.8
0.6
0.4
0.2
0
0.3
0.35
0.4
0.45
0.5
Antenna size (kr)
0.55
0.6
0.65
�- Radius step up
(d)
1.2
Fundamental Limit (2/Q)
1
Figure-of-merit (?)
[48]
0.8
0.6
0.4
0.2
0
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Antenna size (kr)
Fig. 5.10 (Cont'
d)
Small antenna performance using the definition of ? = (Eff � BW) /
Theoretical BW Limit. (c) Multi-arm configuration. (d) Multiple wire radii.
92
By incorporating the concept of the divided range GA, a well-formed Pareto front
in terms of the three objectives was achieved. To verify the GA results, several
antennas based on the GA designs were built and their bandwidth and efficiency
were measured. Both the bandwidth and efficiency measurements agreed well
with the simulation for all the sample antennas.
Based on the Pareto front, it was observed that for a given antenna size,
broader antenna bandwidth must be traded off against lower antenna efficiency.
The results also showed that the maximum achievable bandwidth and efficiency
both decrease as the antenna size is reduced. The performance achieved by the
GA designs was also compared against the well-known fundamental limit for
small antennas. The resulting GA designs followed the trend of the fundamental
limit, but were about a factor of two below the limit.
To further improve the performance of the GA-designed antennas, other
design freedoms were explored such as variable characteristic impedance, multiarm wires and multiple wire radii.
Results showed that the use of targeted
design freedoms could further improve the optimization performance toward the
fundamental limit.
93
Chapter 6
Conclusions
In this dissertation, GA has been applied to three classes of EM design
problems: microstrip antennas, low-profile microwave absorbers and electrically
small wire antennas.
In Chapter 2, GA-optimized patch shapes for broadband microstrip
antennas on thin FR-4 substrate was investigated. The optimized shape showed a
four-fold improvement in bandwidth compared to the standard square microstrip.
This result was verified by laboratory measurement. The basic operating principle
of the optimized shape can be explained in terms of a combination of two-mode
operation and ragged edge shape.
Chapter 3 examined GA-optimized patch shapes for multi-band microstrip
antennas. It was shown that for dual-band operation, a frequency ratio ranging
from 1:1.1 to 1:2 between the two bands can be achieved using the GA
methodology. Tri-band and quad-band microstrip shapes were also generated, and
the resulting antennas performed well at the design frequencies. All results were
verified by laboratory measurements.
In Chapter 4, GA-optimized shapes for corrugated microwave absorbers
under near-grazing incidence were investigated. Corrugated coatings were studied
94
for different incident polarizations, and a physical interpretation for the optimized
shapes was presented. The Pareto GA was applied to efficiently map out the
absorbing performance versus absorber height. The converged Pareto front
showed that better absorbing performance must be traded off against absorber
height.
Chapter 5 described the use of the Pareto GA to design electrically small
wire antennas by considering the bandwidth, efficiency and antenna size. Using
the concept of divided-range GA, a well-formed Pareto front in terms of all three
objectives was achieved. The performance achieved by the GA designs was
compared against the well-known fundamental limit for small antennas. The
resulting GA designs followed the trend of the fundamental limit but were below
the limit by approximately a factor of two. Further improvement of the GAdesigned antennas was tried by incorporating additional design freedoms.
In this dissertation, it has been demonstrated that the GA methodology is
very effective in solving real-world EM design problems. It has also been shown
that GA is highly adaptable to a variety of design problems. However, a detailed
understanding of the actual physics of a particular problem is essential for the
success of GA. The physics of the problem should be incorporated into such GA
process as chromosome encoding and geometrical filtering to achieve fast
convergence and satisfactory solutions. Another consideration is that the total
computation time needed for the GA is usually much longer than deterministic
95
algorithms. To solve the time issue, parallel computation was implemented in this
dissertation. Other ways to boost the convergence rate should also be further
explored.
Advanced GA techniques such as the Pareto optimization and the divided
range optimization have been employed in this dissertation to solve multiobjective EM problems. These advanced techniques showed highly efficient
capability to generate an entire set of optimal solutions with a single optimization
run. Consequently, it became possible to utilize such methodology to map out
the performance bounds associated with a given problem. The resulting
performance bounds provided valuable information on the design of very complex
EM systems.
In future research, both microstrip and wire type antenna structures should
be further investigated for new applications. In particular, the study of electrically
small wire antennas is currently being extended to HF communications, where the
physical size of antennas is usually very large. Some other candidate EM
problems are listed below.
Design of antennas for ultra-wideband (UWB) Applications: The recent
emergence of UWB communication systems requires new types of antennas that
can operate over a frequency bandwidth of more than 100%. Furthermore, UWB
systems require antennas to have not only flat amplitude but also linear phase
response over the broad bandwidth. Additional design concerns include low
96
cross-polarization and high efficiency. These criteria for UWB systems are very
challenging and have not been fully addressed.
Design of multiple-input multiple-output (MIMO) communications
systems: Multiple antennas have recently been investigated to increase channel
capacity and service quality in mobile communication systems. Some preliminary
research has been done to design polarization and pattern diversity antennas for
mobile handset terminals in MIMO systems. One challenging problem yet to be
addressed is the design of diversity antennas with a small antenna form factor.
Design of on-glass antennas for vehicular applications: On-glass antenna
design for automobiles is a costly and time-consuming engineering process. The
design methodology developed in this dissertation can be applied to generate
optimized printed antenna designs for vehicle rear windows (on sedans) or vehicle
side windows (on SUVs and minivans).
Design of microwave absorbers: Chapter 4 in this dissertation examined
microwave absorber shapes to reduce the RCS of military targets. A possible
commercial application could be the design of absorbers to minimize EM
interference in RF systems.
97
Appendix A
Wheeler Cap Method for Measuring the Efficiency of Microstrip
Antennas
In this Appendix, some work into the Wheeler cap method for measuring
the efficiency of microstrip antennas is described.
Since antenna efficiency
characterization is an important issue in several of the chapters in this dissertation,
this material is included here for completeness.
A.1 Introduction
The Wheeler cap method is a simple and well-known technique for
measuring antenna efficiency [55,56]. The method involves making only two
input impedance measurements of the antenna under test: one with a conducting
cap enclosing the antenna and one without.
The antenna efficiency is then
estimated based on either a parallel or a series RLC circuit model for the antenna.
Pozar and Kaufman reported on the use of this method for measuring the
efficiency of microstrips [59]. Even though it is generally believed that a
microstrip antenna is more appropriately modeled as a parallel circuit [14,15,60],
their measurement results did not support the parallel RLC model, and they
concluded that the loss mechanism in the microstrip is similar to that of a series
98
RLC circuit. Even recently microstrip antenna efficiency has been measured using
a series RLC circuit model [61].
In this Appendix, the Wheeler cap method for measuring the efficiency of
microstrip antennas is revisited. The main interest stems from the need to
characterize the efficiency of a class of miniaturized microstrip antennas that was
designed using a GA [62].
Here three main findings are presented. First, it is
shown that the parallel RLC model is indeed a more appropriate model to use than
the series model for microstrips. The results are corroborated by numerical
simulations using the commercial software ENSEMBLE.
Second, the role of
interior cap modes is investigated and an optimal shape of the Wheeler cap for
microstrips is proposed. Finally, the Wheeler cap method is applied to investigate
the efficiency of the GA-designed miniaturized microstrip antennas.
A.2 Antenna Circuit Model
In the Wheeler cap method, two input resistance measurements are needed
to obtain the efficiency of the antenna. One is the input resistance before using the
cap, Rbefore, and the other is the input resistance R after putting the cap on, Rafter.
Since this method is valid only for antennas with a simple dominant loss
mechanism, we should know whether a given antenna follows the series circuit
model or the parallel circuit model. If the test antenna works more like the series
circuit model, then R decreases after applying the cap, and the efficiency is
calculated by the following expression:
99
Eff =
Rbefore ? Rafter
PR
RR
=
=
PR + PL RR + RL
Rbefore
( A.1)
where PR is the total radiated power, PL is the power loss, RR is the radiation
resistance and RL is the loss resistance.
If the test antenna works more like the
parallel circuit model, R should increase after applying the cap, and the efficiency
can be calculated using this expression:
Eff =
Rafter ? Rbefore
PR
RL
=
=
PR + PL RR + RL
Rafter
( A.2)
As a test, this Wheeler cap method is applied to measure a standard
square-shaped microstrip antenna built on lossy FR-4 substrate with thickness of
1.6mm. The test antenna?s dimensions are 36mm � 36mm, and it has a resonant
frequency of 2 GHz. For the Wheeler cap, a conducting rectangular cap (10cm �
5cm � 10cm) is used to completely enclose the test microstrip. Then an HP 8753C
network analyzer is used to gather the input impedance of the test antenna. As is
stated in [56,59], a perfect contact between the cap and the ground plane is critical
for an accurate measurement. Therefore, aluminum tape is used to shield the
slight gap between the cap and the ground plane. Fig. A.1(a) shows the resulting
input resistance before and after applying cap, marked as a dashed line and a solid
line, respectively. At the resonant frequency, it shows Rafter of 113 ohms and
Rbefore of 65 ohms. Since the test antenna is represented by a parallel RLC circuit
100
model the cap should increases the input resistance. Based on the parallel RLC
model, the Wheeler cap measurement shows an efficiency of 34%.
(b)
(a)
Measurement (FR-4)
Input Resistance (?)
ENSEMBLE Simulation (FR-4)
cap
140
120
140
------ Before
?? After
Rafter
------ Before
?? After
120
100
100
Rbefore
80
80
60
60
40
40
20
20
0
0
-20
1.8
Fig. A.1
1.85
1.9
1.95
2
2.05
2.1
2.15
Frequency (GHz)
2.2
2.25
2.3
-20
1.8
1.85
1.9
1.95
2
2.05
2.1
2.15
Frequency (GHz)
2.2
2.25
2.3
Input resistance of a standard square-shaped (36mm�mm) microstrip built on
FR-4 before (------) and after (?
??) using the Wheeler cap size of
10cmm�cmm�m. (a) Measured (b) ENEMBLE simulated.
Next, the measurement result is verified with two different numerical
simulations using the full wave EM simulator ENSEMBLE [63]. In the first
simulation, the Wheeler cap is modeled to rigorously predict Rafter and Rbefore. To
model the conducting rectangular cap, we generate a cavity on top of the
microstrip. The side of the cavity that faces the microstrip is opened. The
simulated input resistance is shown in Fig. A.1(b). Similar to Fig. A.1(a), the
result shows that the input resistance is actually increased after the cap is placed.
This again shows that the loss mechanism in the microstrip more closely
101
resembles a parallel circuit model. Based on the parallel circuit model, the
efficiency computed by this Wheeler cap simulation method is 41%. In the second
simulation, the gain of the test antenna is computed with and without dielectric
and metal loss. Note that metal loss occurs not only on the patch but also on the
ground plane, and this should be taken into account when modeling the
microstrips. The efficiency is then calculated using eq. (A.3), which gives us a
value of 32%.
Eff =
Gwith loss
( A.3)
Gwithout loss
The comparisons of two simulation and measurements in terms of
frequency are shown in Fig. A.2(a). These two simulation results are reasonably
close to the measurement results over the frequency range of interest. The
efficiency using the series RLC model is plotted as thin solid line. This shows that
the parallel RLC circuit model is more appropriate to measure the efficiency of
microstrip. This Wheeler cap method is also applied to a microstrip which is built
on a low loss substrate, RT Duroid 5880 (loss tangent of around 0.001). This test
microstrip has dimensions of 17.5mm�mm and operates at near 5.3GHz. This
measurement shows a 92% efficiency, and this compares favorably to
corresponding simulation results (95% in the gain simulation, and 98% in the
Wheeler cap simulation) as shown in Fig. A.2(b). The small deep at the frequency
of 5.12GHz is caused by an excitation of a cavity mode (TM111) by a Wheeler
102
cap. These results confirm the validity of our Wheeler cap measurements. In [59],
Pozar and Kauffman reported Rafter < Rbefore in their measurement of the simple
microstrip, and they concluded that the loss mechanism in the microstrip is
similar to the a series circuit model. It should be noted that this is probably due to
the influence of interior cap modes when using a large-sized Wheeler cap. In the
next section, cap modes and their dependence on the size of the cap will be
discussed.
(b)
(a)
100
100
Measured
Measured (Series)
Wheeler Cap Simulation
Gain Simulation
90
90
80
70
Efficiency (%)
Efficiency
(%)
Efficiency (% )
80
-10dB bandwidth
60
50
40
30
60
50
40
30
20
20
10
10
0
1.96
1.98
2
2.02
2.04
2.06
0
5.1
2.08
Frequency (GHz)
Fig. A.2
-10dB bandwidth
70
Measured
Measured (Series)
Wheeler Cap Simulation
Gain Simulation
5.15
5.2
5.25
5.3
5.35
5.4
Frequency (GHz)
Wheeler cap measured efficiency based on parallel circuit model (??),
Wheeler cap measured efficiency based on series circuit model (??Purple),
efficiency by Wheeler cap simulation (------) and efficiency by gain
simulation (------). (a) microstrip built on FR-4 substrate (b) microstrip build
on Duroid.
103
5.45
A.3 Effect of Cap Dimensions
Wheeler recommended that the cap radius be around 1/6 of a wavelength
to cause no change in the current distribution on the antenna [55,56]. However,
for microstrip antennas, a larger size Wheeler cap may have to be used to enclose
an extended substrate or to enclose a microstrip array.
Cap Height: 8.5cm
------ Before
?? After
700
600
17�5�cm
(2.83�41�83?)
500
400
------ Before
?? After
800
Input Resistance (?)
800
Input Resistance (?)
Cap Height: 2.0cm
900
900
300
200
700
600
17�0�cm
(2.83�33�83?)
500
400
300
200
100
100
0
0
4.6
4.8
5
5.2
5.4
5.6
4.6
5.8
Frequency (GHz)
Fig. A.3
4.8
5
5.2
5.4
Frequency (GHz)
5.6
5.8
Measured input resistance of a standard square-shaped (36mm x 36mm)
microstrip built on Duroid before (------) and after (??) using (a) the
Wheeler cap size of 17cmm x 17cmm x 8.5cm. (b) using the Wheeler cap size
of 17cm x 17cm x 2cm.
Fig. A.3(a) is an example of the effect using a larger size
(17cm�5cm�cm) Wheeler cap for the same microstrip (17.5mm�mm) built
on low loss substrate. In a small Wheeler cap, interior cap modes exist, but at such
high frequencies that they do not significantly interfere with the resonant
104
frequency of the microstrip as shown in Fig. A.1(a) and (b). On the other hand, a
larger Wheeler cap, shown in Fig. A.3(a), creates interior cap modes near the
resonant frequency of the microstrip, which causes a deviation in the input
resistance value. This can cause an inaccurate efficiency measurement when using
the Wheeler cap method.
Thus, the interior cavity modes should be as sparse as possible. This may
be achieved by using a smaller cap size. However, the finite size of the microstrip
substrate restricts the minimum size of the cap. If we take a detailed look at the
interior cavity modes, only TM modes are dominant in the Wheeler cap since the
microstrip works similarly to a horizontal magnetic current parallel to the ground
plane.
f
TM
MNP
1
2? �
M?
a
2
N?
+
b
2
P?
+
c
2
( A.4)
Fig. A.4(a) shows the interior cavity mode-spectrums for a Wheeler cap
sizes of 17cm x 8.5cm x 17cm. The solid line is the measured input resistance and
the dashed line is mode-spectrum calculated using eq. (A.4) [64]. The measured
cavity modes compare well with the ones by calculation. The index N is
associated with the cap height while the other indexes M and P are associated
with the length and the width of the cap, respectively.
105
(a)
Cavity Mode Spectrum of 8.5cm Cap Height
2
1
0.5
(b)
3
3.5
4
4.5
5
5.5
5
5.5
TM114
TM313
TM214
2.5
TM212
TM113
2
TM111
0
1.5
TM112
Log10 [Rin (?)]
1.5
Cavity Mode Spectrum of 2.0cm Cap Height
Log10 [Rin (?)]
2
1.5
1
0.5
0
1.5
Fig. A.4
2
2.5
3
3.5
4
Frequency (GHz)
4.5
Interior cavity modes spectrums for two Wheeler cap size. The upper spectrum is for
the Wheeler cap size of 17cm x 17cm x 8.5cm.The lower spectrum is for the Wheeler
cap size of 17cm x 17cm x 2.0cm.
106
Then the cavity mode-spectrum is observed by reducing only the height of the
Wheeler cap since the planar profile of the microstrip prevents us from decreasing
the other two dimensions. As shown in Fig. A.4(b), the mode spectrum of the 2cm
height Wheeler cap (17cm x 2cm x 17cm) is sparser than the mode spectrum of
the 8.5cm one. This is due to the fact that a cap height reduction makes the mode
spectrum sparser by pushing the interior modes with index N=1 to much higher
frequencies. Sparser mode spectrums provide more space in which to make
Wheeler cap measurements. This theory is applied to the test by measuring the
input resistance of the same microstrip shown in Fig. A.3(a) but using a reduced
cap height of only 2.0cm (0.33?). The results shown in Fig. A.3(b), indeed show
the sparser cavity mode-spectrum as expected. However, the resonant frequency
with cap is about 91MHz lower than the resonant frequency without cap. One
solution is to shift the capped resonant frequency up to 91MHz to compensate the
frequency shift by cap effect [65,66]. After the frequency compensation, an
efficiency value of 94% with this reduced height (17cm x 2.0cm x 17cm) Wheeler
cap is obtained, which is closer to the simulation value than the efficiency of
72.5% by using the size of Wheeler cap. This shows that reduced height wheeler
caps are helpful in obtaining more accurate efficiency values without sacrificing
the length and width of the cap.
107
A.4 Results for Efficiency of Miniaturized Microstrips
In this section, the Wheeler cap method described above is applied to
measure the efficiency of our miniaturized microstrips. A GA is previously
applied to minimize the size of a microstrip patch while keeping its bandwidth as
broad as possible [62].
(a)
(b)
100
50
Efficiency(%)
Efficiency (%)
40
35
30
25
20
15
90
Standard Square
Microstrips
80
Efficiency
(%)
Efficiency(%)
-Measurement
-Simulation
45
GA Microstrips
70
60
50
40
30
10
20
5
10
0
40
50
60
70
80
90
0
40
100
% from regular size
Fig. A.5
GA Microstrips
-Measurement
-Simulation
50
60
70
80
90
100
% from regular size
(a) built on FR-4 substrate in terms of % from the regular size and (b) the
microstrips built on Duroid.
The insets in Fig. A.5(a) are the samples of GA-miniaturized microstrips.
The achievable bandwidth of these miniaturized antenna drops as the size of the
antennas is reduced from 8% to 1.3%. It also shows that even when the size of the
patch is reduced to 40% of the regular size, it still maintains a bandwidth of
around 1.3%, which is good compared to the microstrip?s small size. However,
we thought that investigating the efficiency of these microstrips is crucial due to
108
the high loss in FR-4 substrate. Thus, the efficiencies of these microstrip are
observed and the results are shown in Fig. A.5 as the solid line with the
efficiencies using Wheeler cap measurements and the dashed line is using gain
simulations. The simulated and measured efficiencies are close, showing that our
measurements are accurate, even if the results show that our microstrips have very
low efficiencies. This is somewhat to be expected due to the high loss on the FR-4
substrate, which has a loss tangent of around 0.025. For this reason, our study is
extended to look at how low-loss substrate materials such as Duroid and air
increase the efficiency of these miniaturized microstrips. Fig. A.5(b) shows the
measured and simulated efficiencies of microstrips using the Duroid substrate
with loss tangent of about 0.001. An improved efficiency of more than 65% for all
samples can be observed. This gain in efficiency is not without a trade-off. The
achievable bandwidth using the Duroid substrate is reduced compared to the
bandwidth using the FR-4 substrate. This forces us to evaluate each antenna in
terms of both its efficiency and its bandwidth using the antenna?s EB product
[66]. Fig. A.6 plots the EB product against the physical antenna size for antennas
built on three different substrates: FR-4, Duroid and air substrate, marked as solid,
dashed and solid-dotted lines, respectively. Using Fig. A.6, It could be determined
which substrate material provides the highest EB product for a given antenna size.
109
7
Air (3.25mm)
5
Bandwidth * Efficiency
Efficiency x Bandwidth (EB)
6
Duroid (3.2mm)
4
3
2
FR-4 (1.6mm)
1
0
10
20
30
40
50
60
70
Physical Antenna Size (mm)
Fig. A.6 EB Product against physical microstrip patch size for microstrip built on three
different substrates. (a) FR-4 (?
??), Duroid (------)and Air (-?-?-?-?).
A.5 Summary
The Wheeler cap method for measuring microstrip efficiency was
revisited, and it was shown that the parallel circuit model is appropriate for the
microstrip loss mechanism. The measured efficiency values were verified using a
numerical simulation code. Then interior cap modes were investigated, and a way
to diminish them using a reduced height Wheeler cap was found. Finally, the
reduced-height Wheeler cap method was applied to investigate the efficiency of
miniaturized microstrip antennas on various substrate materials. However, this
110
Wheeler cap method is only valid for a limited frequency near a resonance. Also,
the method is valid for an antenna that has only one dominant loss mechanism.
111
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Vita
Hosung Choo was born in Seoul, Korea on April 10, 1972, the
son of Hyungsam Choo, and Kabrae Cho. After completing his work
at Dankuk High School, Seoul, Korea in 1991, he entered Hanyang
University in Seoul, Korea. He received the degree of Bachelor of
Science from Hanyang University in February 1998. In September
1998, he entered the Graduate School at The University of Texas at
Austin. He received the degree of Master of Science from The
University of Texas at Austin in August 2000.
Permanent Address:
2501 Lake Austin Blvd. APT J205
Austin, TX, 78703
This dissertation was typed by the author.
118
entional GA.
Pareto GA [2,4] is a useful tool for this problem. In the Pareto GA, a wide range
of solutions corresponding to more than one objective can be mapped by running
the optimization only once. In this GA implementation, two cost functions are
defined:
Cost 1 = Normalized coating height
Cost 2 = Normalized value of
where Pn =
1
N
N
n =1
(4.2)
( Pn )
(?? + ?| | ) / 2 (dB) + 20 dB
if (?? + ?|| ) / 2 (dB) ? ? 20 dB
0
if (?? + ?|| ) / 2 (dB) < ?20 dB
(4.3)
Cost 1 is determined by the coating height and Cost 2 is associated with the
reflection cost. Both costs are normalized to a value between zero and one. For
61
Cost 2, one denotes an average reflection coefficient of 0dB, while zero denotes
an average reflection coefficient that is below ?20dB.
The non-dominated
sorting method [40] is used to combine the two costs for each solution by means
of the Pareto ranking. This method assigns rank 1 to the non-dominated solutions
of the population. The term non-dominated solution means that there are no other
solutions that are superior to this solution in both objectives. Then the next nondominated solutions among the remaining solutions are assigned to the nexthighest rank. The process is iterated until all the solutions in the population are
ranked. Based on the rank, the same reproduction process described in Section 4.3
is performed to refine the population into the next generation. The set of rank 1
solutions is called the Pareto front. In order to avoid the solutions on the Pareto
front from converging to a single point in the cost space, a sharing scheme
described in [41] is performed. In this sharing process, the rank is modified by
penalizing those members on the front that are too close to each other in the cost
space. This is accomplished by multiplying a niche count (mi) to the assigned
rank. The niche count is calculated according to:
N
1 p
mi =
Sh(d ij )
( 4. 4 )
N p j =1
where the Np is the number of rank 1 members and the sharing function, Sh(dij), is
a function of the cost distance between solutions expressed as:
62
Sh(d ij ) =
2?
d ij
d share
1
if d ij < d share
(4.5)
if d ij > d share
and
d ij =
(Cost 1(i) ? Cost 1( j ) )2 + (Cost 2(i ) ? Cost 2( j ) )2
As can be seen, the sharing function increases linearly if the other members on the
front are closer than dshare from a chosen member i in the cost space.
Consequently, those members that have close-by neighbors in the cost space are
assigned lower ranks in the reproduction process.
4.4.2 Pareto GA Results
In this Pareto GA, the population size is chosen to be 100. A crossover
probability of 0.8, a mutation probability of 0.1 and a dshare distance of 1 are used.
Figs. 4.11(a)-(d) show the convergence of the solutions for this multi-objective
problem (reflection cost versus the height of the profile) as the number of
generations is increased. The period of the absorber, the material for the coating,
and the angle of incidence are the same as those used in Section 4.3. The height is
constrained to be less than 8mm. Fig. 4.11(a) is the plot of the initial population.
The majority of the solutions are located in the upper-right side of the cost
domain. Figs. 4.11(b)-(d) are plots of the population after 5, 20 and 200
generations, respectively. They show that as the number of generations increases,
63
the Pareto front spreads out and converges toward the lower-left region of the cost
space. Fig. 4.12 shows the final converged Pareto front and four optimized
coating shapes that are on the front. Inset shape (a) shows the lowest profile of the
four samples, but it has the highest reflection among the four designs. Inset shape
(d) has the highest profile and the lowest reflection. As expected, the absorbing
performance must be traded off against the profile height. If we look in detail at
the optimized shapes, it appears that as the height of the absorber decreases, the
top of the profile gets more flattened. However, the shapes maintain a rectangular
profile that is only slightly modified by the coating height. This is consistent with
the physical interpretation of the absorption process for the more dominant
horizontal polarization discussed in Section 4.3. Another observation from Fig.
4.12 is that the Pareto front is not smooth due to the quantization effect of the
coating height. If the height is discretized with more binary bits, the shape of the
Pareto front becomes smoother.
Next, the first cost is changed from coating height to coating weight while
keeping the reflection cost the same. The cost for the coating weight is normalized
to be 1 when all of the design area (period � maximum coating height) is filled by
the coating material while it is zero when no coating material exists. Fig. 4.13
shows the converged Pareto front for this problem. Also shown in insets (a) to (d)
are four optimized shapes with different coating weights.
64
0.45
0.45
0.4
0.4
0.35
0.3
0.25
0.2
A bsorbing P erformance
0.5
Reflection cost
Absorbing Performance
Reflection cost
0.5
0.3
0.25
0.2
0.15
0.1
0.35
0.15
4
4.5
5
5.5
6
6.5
7
7.5
0.1
8
4
4.5
Height of absorber (mm)
5
6.5
7
7.5
8
(b)
0.5
0.45
0.45
0.4
0.4
0.35
0.3
0.25
0.2
A bsorbing P erformance
0.5
Reflection cost
Absorbing Perform ance
Reflection cost
6
Height of absorber (mm)
(a)
0.35
0.3
0.25
0.2
0.15
0.15
0.1
5.5
0.1
4
4.5
5
5.5
6
6.5
7
7.5
8
Height of absorber (mm)
4.5
5
5.5
6
6.5
Heigh of the profile (mm)
7
7.5
Height of absorber (mm)
(c)
Fig. 4.11
4
(d)
Convergence of the Pareto front as a function of the number of generations
for absorbing performance versus absorber height. (a) Initial population. (b)
After 5 generations. (c) After 20 generations. (d) After 200 generations.
65
8
(a)
(b)
0.5
(c)
0.45
Reflection cost
0.4
(d)
0.35
0.3
0.25
0.2
0.15
0.1
Fig. 4.12
4
4.5
5
5.5
6
6.5
Height of profile (mm)
7
7.5
8
Final converged Pareto front of absorbing performance versus absorber
height. The insets show four sample designs on the Pareto front.
66
(a)
(b)
0.7
(c)
0.6
Reflection cost
0.5
0.4
(d)
0.3
0.2
0.1
0.15
0.2
0.25
0.3
Normalized Weight
0.35
0.4
Weight of the absorber
Fig. 4.13
Final converged Pareto front of absorbing performance versus absorber
weight. The insets show four sample designs on the Pareto front.
67
It should be noticed that inset (d) is very similar in shape to inset (d) in Fig. 4.12.
However, instead of trimming the top off in order to reduce the height, the weight
consideration results in designs that become progressively skinnier, as shown by
insets (c), (b) and (a). Nevertheless, the shapes still preserve the sharp sidewalls
as those presented in Fig. 4.12.
4.5 Summary
Optimized shapes for a corrugated absorber under near-grazing incidence
have been investigated using GA. First, GA was applied to design corrugated
coating depending on incident polarizations. The designed absorber shape for the
vertical polarization resembled a triangular profile, while that for the horizontal
polarization resembled a rectangular profile. The optimized shapes were
compared to canonical planar and triangular shaped designs, and were shown to
have better absorbing performance. The sensitivity of the designs to variations in
shape and incident angles were also tested, and the results showed reasonable
tolerance. A physical interpretation for the optimized shape was presented. It was
shown that the sharp sidewalls of the resulting shape effectively changed the
incident polarization from horizontal to the vertical case, thus facilitating wave
absorption.
The Pareto GA has also been applied to efficiently map out absorbing
performance versus absorber height. The non-dominated sorting method was used
68
to combine the two costs for each solution by means of the Pareto ranking. A
sharing scheme was implemented to avoid the solutions on the Pareto front from
converging to a single point in the cost space. The converged Pareto front showed
that better absorbing performance must be traded off against absorber height.
Similar conclusions were also found for the absorbing performance versus
absorber weight.
69
Chapter 5
Design of Electrically Small Wire Antennas Using a Pareto
Genetic Algorithm
5.1 Introduction
In Chapter 2 and 3, the basic GA for microstrip antenna design was
introduced. In Chapter 4, the more advanced Pareto GA was investigated to solve
the more challenging multi-objective problem. In this chapter, the developed
methodology is applied to the design of electrically small wire antennas.
As the size of wireless devices shrinks, the design of electrically small
antennas is an area of growing interest [42,43]. By the classical definition, an
electrically small antenna is one that can be enclosed in a volume of radius r
much less than a quarter of a wavelength. It is well known that the bandwidth of
an electrically small antenna decreases as the third power of the radius [44-46].
Much research has been carried out to increase the bandwidth of small antennas
using structures such as folded design, disk-loaded monopole, inverted-L or
inverted-F designs, multi-armed spiral and conical helix [47-50]. Fig. 5.1 shows
the bandwidth performance of these antennas in terms of the normalized antenna
size kr, where k=2?/? is the wave number. Recently, Altshuler reported on the
use of a GA in designing electrically small wire antennas [9]. Instead of using a
70
regular shape, he used GA to search for an arbitrary wire configuration in 3-D
space that results in maximum bandwidth for a given antenna size.
While much of the small antenna research has been focused on antenna
bandwidth, antenna miniaturization also impacts antenna efficiency. The objective
of this chapter is to apply GA in the design optimization of electrically small wire
antennas, taking into account of bandwidth, efficiency and antenna size. To
efficiently map out this multi-objective problem, the Pareto GA [2,4] is utilized.
3dB Bandwidth (%)
Fundamental Limit (L. J. Chu, 1948)
100
Goubau [50]
50
Dobbins & Rogers [49]
30
Foltz & McLean [48]
20
0.6
Fig. 5.1
0.8
1.2
Antenna size (kr)
1/4?
? monopole
1.6
Achievable bandwidth in terms of antenna size and some examples of small
antennas.
71
The concept of divided range multi-objective [51] is employed to
accelerate convergence in the GA process. In this GA approach, the multisegment wire structure similar to the one used in [9] is employed. The Numerical
Electromagnetics Code (NEC) [52] is used to predict the performance of each
wire structure. Then an optimal set of designs is generated by considering
bandwidth, efficiency and antenna size. To verify the GA results, several GA
designs are built, measured and compared to the simulation. Physical
interpretations of the GA-optimized structures, showing the different operating
principles depending on the antenna size are also provided. The performance
curve achieved by the GA approach is compared against the well-known
fundamental limit for small antennas [44-46]. To more easily assess the
performance of the antennas, the efficiency-bandwidth product is normalized by
the antenna size in order to represent the antenna performance as a single figureof-merit [54]. Finally, further improvement of the GA results is attempted by
exploring additional geometrical design freedoms to better approach the
fundamental limit.
This chapter is organized as follows. In Section 5.2, the details of the GA
implementation are described. Section 5.3 describes the GA designs and the
measurement verification of the results. In Section 5.4, the GA results are
compared to the fundamental limits. In Section 5.5, other design freedoms are
72
explored to further improve performance. Section 5.6 provides conclusions
gathered from this research.
5.2 Pareto GA Approach
The basic antenna configuration considered in this chapter is shown in Fig.
5.2. The antenna consists of M connected wire segments. Each segment of the
antenna is confined in a hemispheric design space with a radius r and an infinite
ground plane.
r
Fig. 5.2
Configuration of the multi-segment wire antenna used in the GA design.
The three design goals are: broad bandwidth, high efficiency and small
antenna size. The Pareto GA is employed to efficiently map out this multi-
73
objective problem. The advantage of using the Pareto GA over the conventional
GA is that a wide range of solutions corresponding to more than one objective can
be mapped by running the optimization only once.
Binary Chromosome
Antenna Shape
1
0
0
0
1
0
1
1
0
0
1
1
Fig. 5.3
0
1
0
1
1
0
0
1
1
0
1
0
1
0
0
1
1
1
1
0
1
1
1
0
1
1
1
0
1
1
0
1
1
1
0
0
1
0
1
1
1
1
1
0
0
1
1
0
1
1
0
0
1
1
0
1
1
0
1
1
Encoding of the wire configuration into a binary chromosome.
In this GA implementation, the hemispheric design space is evenly
n
discretized into 2 grid points, and the location of each joint of the antenna is
encoded into an n-bit binary string, as shown in Fig. 5.3. Thus, the total number of
bits in the chromosome is nM when M connected wire segments are used. The
three costs associated with these design goals are:
74
Cost 1 = 1 ?
Antenna Bandwidth
Theoretical Bandwidth Limit
(5.1)
Cost 2 = 1 ? Efficiency
Cost 3 = Antenna Size ( kr )
In
the
above
definition,
the
theoretical
bandwidth
limit
of
1/((1/kr)+(1/kr)3) derived in [46] is used. NEC [52] is used to predict the antenna
performance in order to compute the cost functions. Multiple Linux machines are
used in parallel to carry out this computation.
After evaluating the three cost functions of each sample structure using
NEC, all the samples of the population are ranked using the non-dominated
sorting method [40]. Here, the higher the rank (1 denotes the highest rank), the
better the solution. This method assigns rank1 to the non-dominated solutions of
the population. The term non-dominated solution means that there are no other
solutions that are superior to this solution with respect to all design objectives.
Then the next non-dominated solutions among the remaining solutions are
assigned rank2. The process is iterated until all the solutions in the population are
ranked. Based on the rank, a reproduction process is performed to refine the
population into the next generation. The set of rank1 solutions is termed the
Pareto front. By favoring the higher-ranked solutions in the reproduction process,
the Pareto GA tries to push the Pareto front as close to the optimum solution in
the cost space as possible.
75
For the crossover operation, a two-point crossover scheme involving three
chromosomes is used. For numerical stability, a geometrical check is applied to
prevent the wires from intersecting one another. In order to avoid the solutions
from converging to a single point, a sharing scheme as described in [41] is
performed to generate a well-dispersed population. In the sharing process, the
rank is modified by penalizing those members on the front that are too close to
each other in the cost space. This is accomplished by multiplying a niche count
(mi) to the assigned rank. The niche count is calculated according to:
1
mi =
Np
Np
j =1
Sh (dij )
(5.2)
where the Np is the number of rank-1 members and the sharing function,
Sh(dij), is a function of the cost distance between solutions expressed as:
Sh (dij ) =
2?
1
dij =
dij
dshare
if dij < dshare
(5.3)
if dij > dshare
( Cost 1(i) ? Cost 1( j) )2 + ( Cost 2 (i) ? Cost 2 ( j) )2 + ( Cost 3 (i) ? Cost 3 ( j) )2
The sharing function increases linearly if the other members on the front
are closer than dshare from a chosen member i in the cost space. Consequently,
76
those members that have close-by neighbors in the cost space are assigned lower
ranks in the reproduction process.
The standard Pareto GA did not always give satisfactory results in this
problem, since it is much harder for small-sized antennas to converge than for
large-sized antennas. Thus, large-sized antennas usually dominate the whole
population after several generations of the GA process.
Cost1 (Size)
Range 1
Range 2
Cost3 (Efficiency)
Range 3
Cost2 (Bandwidth)
Fig. 5.4
Divided range multi-objective GA approach.
77
As a result, the final Pareto front contains only antenna designs with sizes
kr?0.45. To avoid this bias, the concept of the divided range multi-objective GA
[51] is employed in this implementation. As shown in Fig. 5.4, the size is
partitioned into multiple ranges and carry out the Pareto GA on each range
individually. After each range converges to an optimal solution, the populations
from all ranges are merged and the combined population is optimized in the last
step of this process. This algorithm shows much improved performance for this
more difficult multi-objective problem. Using the scheme good results for smallsized antennas (kr<0.45) as well as large-sized ones (kr>0.45) can be achieved.
5.3 GA-Optimized Results
5.3.1 GA Optimized Designs
This section investigates the optimal antenna shapes that produce the best
efficiency-bandwidth (EB) product for a given antenna size. Seven wire segments
are used in the antenna configuration. The 3-D hemisphere with radius r is
discretized into 215 points and the locations of the 7 wire segments are encoded as
a binary chromosome of 7� bits. In addition, an extra three bits are added for
choosing 8 different wire conductivities varying from 1�6 (graphite) to 5.7�7
(copper). The population size is chosen to be 2000 and the population is divided
into
four
sub
ranges
(0.29?kr<0.38,
78
0.38?kr<0.46,
0.46?kr<0.54
and
0.54?kr<0.63). Each of four ranges has a population size of 500. A crossover
probability of 0.8, a mutation probability of 0.1 and a dshare distance of 1 are used.
The target design frequency is chosen at 400 MHz and an infinite ground plane is
assumed in the simulation. All antennas are designed to match to a 50 ?
impedance.
The total computational time is about 20 hours using four Pentium IV
1.7GHz machines running in parallel. Figs. 5.5 (a), (b) and (c) show the designs in
the population with a rank of 1 at respectively, the initial, 200 and 1000
generations of the Pareto GA process. The three axes are bandwidth, efficiency
and antenna size. Each dot represents a particular rank-1 design. In the initial
generation, only a few rank-1 solutions exist. After 200 generations, many more
rank1 solutions appear. After 1000 generations, a large portion of solutions (770
over 2000) is on the Pareto front. The solutions are relatively well spread out over
the Pareto front due to the sharing operation.
79
(a)
Size (kr)
0.6
0.55
0.5
0.45
0.4
0.35
0.3
2
1.8
1.5
1
1.6
1.4
Efficiency log10(%)
0.5
Bandwidth log10(%)
0
(c)
0.6
C
Size (kr)
0.55
0.5
B
0.45
0.4
A
0.35
0.3
(b)
2
1.8
1.5
1
1.6
Size (kr)
0.6
1.4
Efficiency log10(%)
0.55
0.5
0.5
0
Bandwidth log10(%)
0.45
0.4
0.35
0.3
2
1.8
1.5
1
1.6
1.4
Efficiency log10(%)
0.5
0
Bandwidth log10(%)
Fig. 5.5 Convergence of the Pareto front as a function of the number of generations in
terms of bandwidth, efficiency and antenna size. (a) Initial generation. (b) After
200 generations. (c) After 1000 generations.
80
5.3.2 Verification of the GA-Optimized Results
To verify the GA results, three GA-optimized designs are selected from
the Pareto front (at points A (kr=0.32), B (kr=0.42) and C (kr=0.5)) and are
shown in Fig. 5.6. The smallest sample, at point A, somewhat resembles a helix,
while the largest sample C resembles a complicated loop where the end of the
wire is connected to the ground plane. These three designs were built and their
performances were measured. Copper wire of radius 0.5mm is used, and a 1.6m �
1.6m conducting plate as the ground plane is used. Fig. 5.7(a) is a photo of design
B and Fig. 5.7(b) is the resulting return loss (dB) as a function of frequency by
simulation and measurement. Except for a slight (3%) shift in the resonant
frequency, the simulation and measurement results show nearly the same
bandwidth (about 5.3% based on |S11| ? -3dB). Fig. 5.7(c) is the resulting
efficiency of the antenna. The standard Wheeler cap method [55,56] is used to
measure the efficiency. The measured efficiency matches the simulation well at
the resonant frequency of 400 MHz, as indicated by the arrow in Fig. 5.7(c). At
other frequencies, the agreement is also good except in the neighborhood of
385MHz. The presence of the large efficiency dip based on the measured data is
due to an anti-resonance in the antenna, as the Wheeler cap method fails near this
anti-resonance.
81
(a)
kr=0.34
(c)
kr=0.50
(b)
kr=0.42
Fig. 5.6
Three samples from the Pareto front (a) kr=0.34, (b) kr=0.42 and (c)
kr=0.50.
82
(a)
(c)
0
100
-1
90
-2
80
-3
70
Efficiency (%)
S11 (dB)
(b)
-4
-5
-6
-7
60
50
30
-8
20
-9
10
-10
350
360
370
380
390
400
410
420
430
440
450
0
350
Simulation
360
370
380
390
400
410
420
Frequency (MHz)
Frequency (MHz)
Fig. 5.7
Measurement
40
(a) Photo for antenna B, which has an antenna size of kr=0.42. (b) Return
loss and (c) efficiency versus frequency of antenna B. The efficiency
measurement was done using the Wheeler cap method.
83
430
440
450
Similar good agreements were also found for antennas A (kr=0.34) and C
(kr=0.50). The results are summarized in Table 5.1. It is noted that both the
achievable bandwidth and efficiency drop as the antenna size is reduced.
Measured
BW (3dB)
Simulated
BW (3dB)
Measured
Eff (%)
Simulated
Eff (%)
Antenna A
2.1 %
2.5 %
84 %
88 %
Antenna B
5.3 %
5.5 %
92 %
94 %
Antenna C
8.5 %
9.8 %
94 %
97 %
Table 5.1
Bandwidth and efficiency for the sample antennas A, B and C by measurement
and simulation.
5.3.3 Physical Interpretation of GA-Optimized Design
By examining the GA-optimized antenna structure in Fig. 5.6(c), we see
that the end of the antenna is connected to the ground plane. Most of the antennas
with 0.45<kr<0.65 have this characteristic, which is similar to a folded monopole
antenna. Since a folded monopole has four times the input impedance of a
standard monopole [57], the GA-designed antennas use this basic structure to
boost up the impedance of the antenna to approach 50 ?.
As we examine at the GA-optimized structures for even smaller-sized
antennas (kr<0.45) such as those in Figs. 5.6(a) and (b), most of the antennas are
shorted to the ground plane at the joint between the first and second segments
84
from the feed. This turns the first segment into an inductive feed. Segments 2
through 7 become the radiating part of the antenna, carrying most of the current.
The strength of the inductive coupling depends on the distance between the first
and second segments. Since inductive coupling can greatly increase the input
impedance, the GA finds this as an optimized structure for very small-sized
antennas (kr<0.45), which need a large impedance step-up to get to 50 ?. This
concept is currently investigated in more detail in order to design very small
antennas [58].
5.4 Comparison to Fundamental Limit
In this section, the GA results are compared to the fundamental limits for
small antennas. Fig. 5.8 depicts all of the GA-optimized designs on the Pareto
front plotted in the 3-D bandwidth, efficiency and antenna size space. A least
squares fit is used a to create a surface that best fits the GA results. To more easily
interpret the results, the 3-D plot is projected onto three planes, and the results are
shown in Figs. 5.9(a), 5.9(b) and 5.9(c). Then these GA results are compared to
the well-known fundamental limit using a combination of equations in [45] and
[46]:
BW =
1
Eff 譗
Where Q =
(5.4)
1
1
+ 3
kr (kr)
85
Antenna size (kr)
C
B
A
Efficiency log10(%)
Fig. 5.8
Bandwidth log10(%)
Pareto front of the GA designs after convergence. The surface is generated
using a least squares fitting to best fit the GA results shown as dots.
The curves based on (5.4) are shown in Figs. 5.9(d)-5.9(f). Fig. 5.9(a)
shows the maximum bandwidth curve achievable by the GA designs as a function
of antenna size for different efficiencies. As expected, for a given efficiency, the
achievable bandwidth decreases as the antenna size is reduced. Also, the higher
the efficiency, the lower the achievable bandwidth. It is similar to the trend of the
86
fundamental limit in Fig. 5.9(d). However, the GA performance is lower than the
fundamental limit. Fig. 5.9(b) is the projection of GA designs on the antenna size
versus efficiency plane. For a given bandwidth, the achievable efficiency
decreases as the antenna size is reduced. This trend can also be observed in Fig.
5.9(e). Fig. 5.9(c) is the projection of the GA results on the bandwidth versus
efficiency plane. For a given antenna size, the tradeoff between antenna
bandwidth and antenna efficiency can be clearly seen. Again, the trend on this
graph is similar to the fundamental limit in Fig. 5.9(f).
To make the antenna performance easier to assess, the figure-of-merit ?
suggested in [54] is used:
?=
Eff � BW
2
Q
(5.5)
where Q is given in (5.4) and an extra factor of 2 is used to account for the loaded
Q. Using this expression, the fundamental limit on ? is always 1 for antennas of
arbitrary sizes. All of the GA-optimized designs are re-plotted using this figureof-merit in Fig. 5.10(a). As a reference, the disk-loaded monopole from [48] is
plotted on the same figure. As can seen in Fig. 5.10(a), most of the GA designs
are ?<0.5. The smaller the antennas, the worse the performance of the GA
designs. Next the way to further improvement of the GA designs is investigated.
87
(a)
50
40
50
40
30
30
Bandwidth (%)
Bandwidth (%)
(d)
20
10
Eff
Eff
Eff
Eff
1
0.3
0.35
0.4
0.45
0.5
=
=
=
=
0.55
40%
55%
70%
85%
20
10
1
0.3
0.6
Eff
Eff
Eff
Eff
0.35
Antenna size (kr)
0.4
0.45
0.5
=
=
=
=
0.55
40%
55%
70%
85%
0.6
Antenna size (kr)
(b)
(e)
Efficiency (%)
80
BW
BW
BW
BW
=
=
=
=
100
5%
10%
20%
30%
80
Efficiency (%)
100
60
40
20
0.3
0.35
0.4
0.45
0.5
0.55
60
40
20
0.3
0.6
BW
BW
BW
BW
0.35
Antenna size (kr)
0.4
0.45
0.5
=
=
=
=
0.55
5%
10%
20%
30%
0.6
Antenna size (kr)
(f)
100
100
80
80
Efficiency (%)
Efficiency (%)
(c)
60
40
20
kr
kr
kr
kr
1
=
=
=
=
0.34
0.42
0.50
0.59
10
20
30
40
20
40 50
Bandwidth (%)
Fig. 5.9
60
kr =
kr =
kr =
kr =
1
0.34
0.42
0.50
0.59
10
20
30
40 50
Bandwidth (%)
(a) Projection of the Pareto front onto the size and efficiency plane.
(b) Projection of the Pareto front onto the size and bandwidth plane.
(c) Projection of the Pareto front to the bandwidth and efficiency plane.
(d) through (f) show the corresponding fundamental limit based on (5.4).
88
5.5 Further Improvement on GA Designs
To bring the figure-of-merit of the designs even closer to the fundamental
limit, additional design freedoms are explored to the original 7-wire
configuration. First, the number of segments is increased up to 16 wires.
However, these results show almost no improvement compared to the original 7wire ones. Next, the selection of characteristic impedance is permitted to vary
from 1 ? to 300 ?, instead of requiring a fixed 50 ? for the input port. This is the
assumption used in the work of Altshuler [9], who assumes that a perfect
impedance transformer is available. When allowed this freedom, the GA produced
the results plotted in Fig. 5.10(b). As the graph shows, using variable
characteristic impedance only gives a slight improvement in performance over the
original design plotted in Fig. 5.10(a).
Then a multi-arm configuration is examined to improve antenna
performance. Two arms are used for the antennas structure and the resulting
figure-of-merits are shown in Fig. 5.10(c). It shows good improvement for largesized antennas (kr>0.45). Using the multi-arm configuration, the antenna
efficiency is increased while the antenna bandwidth is preserved. This is achieved
by spreading the current on multiple branches and lowering the power dissipation.
However, the multi-arm design does not show much improvement for
small-sized antennas (kr<0.45). The number of arms is further increased to four,
but adding more number of arms does not show more improvement over the two89
arm design. It is due to the difficulty in packing a multi-arm structure in a limited
design space.
Finally, two different wire radii are allowed in each design as an
additional degree of design freedom. The resulting antenna has one wire radius
for the lower portion of the antenna and another radius for the upper portion.
Stepping the wire radius has the effect of increasing the input impedance, similar
to the way a radius step up is used in a folded monopole design.
Since it is
known that the NEC version 2 used in the simulation does not accurately model
the wire radius change, the radius change is limited to less than two.
Fig. 5.10(d) shows the result of the design.
It shows improved
performance for both small-sized antennas and large-sized antennas. Based on
these preliminary results, it appears that the use of targeted design freedoms can
further improve the performance of the GA-optimized designs toward the
fundamental limit.
5.6 Summary
The Pareto GA has been applied to design electrically small wire antennas
by considering antenna bandwidth, efficiency and size. Wire structures
comprising of multiple segments were considered. The key advantage of using
the Pareto GA method is that a whole series of optimal designs of varying size,
bandwidth and efficiency can be generated efficiently in a single GA run.
90
�- 7-segment wire
(a)
1.2
Fundamental Limit (2/Q)
1
Figure-of-merit (?)
[48]
0.8
0.6
C
B
A
0.4
0.2
0
0.3
0.35
0.4
0.45
0.5
Antenna size (kr)
0.55
0.6
0.65
�- Variable characteristic impedance
(b)
1.2
Fundamental Limit (2/Q)
Figure-of-merit (?)
1
[48]
0.8
0.6
0.4
0.2
0
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Antenna size (kr)
Fig. 5.10
Small antenna performance using the definition of ? = (Eff � BW) /
Theoretical BW Limit. (a) Original 7-wire configuration. (b) Variable input
impedance.
91
�- Multi-arm
(c)
1.2
Fundamental Limit (2/Q)
1
[48]
Figure-of-merit (?)
0.8
0.6
0.4
0.2
0
0.3
0.35
0.4
0.45
0.5
Antenna size (kr)
0.55
0.6
0.65
�- Radius step up
(d)
1.2
Fundamental Limit (2/Q)
1
Figure-of-merit (?)
[48]
0.8
0.6
0.4
0.2
0
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Antenna size (kr)
Fig. 5.10 (Cont'
d)
Small antenna performance using the definition of ? = (Eff � BW) /
Theoretical BW Limit. (c) Multi-arm configuration. (d) Multiple wire radii.
92
By incorporating the concept of the divided range GA, a well-formed Pareto front
in terms of the three objectives was achieved. To verify the GA results, several
antennas based on the GA designs were built and their bandwidth and efficiency
were measured. Both the bandwidth and efficiency measurements agreed well
with the simulation for all the sample antennas.
Based on the Pareto front, it was observed that for a given antenna size,
broader antenna bandwidth must be traded off against lower antenna efficiency.
The results also showed that the maximum achievable bandwidth and efficiency
both decrease as the antenna size is reduced. The performance achieved by the
GA designs was also compared against the well-known fundamental limit for
small antennas. The resulting GA designs followed the trend of the fundamental
limit, but were about a factor of two below the limit.
To further improve the performance of the GA-designed antennas, other
design freedoms were explored such as variable characteristic impedance, multiarm wires and multiple wire radii.
Results showed that the use of targeted
design freedoms could further improve the optimization performance toward the
fundamental limit.
93
Chapter 6
Conclusions
In this dissertation, GA has been applied to three classes of EM design
problems: microstrip antennas, low-profile microwave absorbers and electrically
small wire antennas.
In Chapter 2, GA-optimized patch shapes for broadband microstrip
antennas on thin FR-4 substrate was investigated. The optimized shape showed a
four-fold improvement in bandwidth compared to the standard square microstrip.
This result was verified by laboratory measurement. The basic operating principle
of the optimized shape can be explained in terms of a combination of two-mode
operation and ragged edge shape.
Chapter 3 examined GA-optimized patch shapes for multi-band microstrip
antennas. It was shown that for dual-band operation, a frequency ratio ranging
from 1:1.1 to 1:2 between the two bands can be achieved using the GA
methodology. Tri-band and quad-band microstrip shapes were also generated, and
the resulting antennas performed well at the design frequencies. All results were
verified by laboratory measurements.
In Chapter 4, GA-optimized shapes for corrugated microwave absorbers
under near-grazing incidence were investigated. Corrugated coatings were studied
94
for different incident polarizations, and a physical interpretation for the optimized
shapes was presented. The Pareto GA was applied to efficiently map out the
absorbing performance versus absorber height. The converged Pareto front
showed that better absorbing performance must be traded off against absorber
height.
Chapter 5 described the use of the Pareto GA to design electrically small
wire antennas by considering the bandwidth, efficiency and antenna size. Using
the concept of divided-range GA, a well-formed Pareto front in terms of all three
objectives was achieved. The performance achieved by the GA designs was
compared against the well-known fundamental limit for small antennas. The
resulting GA designs followed the trend of the fundamental limit but were below
the limit by approximately a factor of two. Further improvement of the GAdesigned antennas was tried by incorporating additional design freedoms.
In this dissertation, it has been demonstrated that the GA methodology is
very effective in solving real-world EM design problems. It has also been shown
that GA is highly adaptable to a variety of design problems. However, a detailed
understanding of the actual physics of a particular problem is essential for the
success of GA. The physics of the problem should be incorporated into such GA
process as chromosome encoding and geometrical filtering to achieve fast
convergence and satisfactory solutions. Another consideration is that the total
computation time needed for the GA is usually much longer than deterministic
95
algorithms. To solve the time issue, parallel computation was implemented in this
dissertation. Other ways to boost the convergence rate should also be further
explored.
Advanced GA techniques such as the Pareto optimization and the divided
range optimization have been employed in this dissertation to solve multiobjective EM problems. These advanced techniques showed highly efficient
capability to generate an entire set of optimal solutions with a single optimization
run. Consequently, it became possible to utilize such methodology to map out
the performance bounds associated with a given problem. The resulting
performance bounds provided valuable information on the design of very complex
EM systems.
In future research, both microstrip and wire type antenna structures should
be further investigated for new applications. In particular, the study of electrically
small wire antennas is currently being extended to HF communications, where the
physical size of antennas is usually very large. Some other candidate EM
problems are listed below.
Design of antennas for ultra-wideband (UWB) Applications: The recent
emergence of UWB communication systems requires new types of antennas that
can operate over a frequency bandwidth of more than 100%. Furthermore, UWB
systems require antennas to have not only flat amplitude but also linear phase
response over the broad bandwidth. Additional design concerns include low
96
cross-polarization and high efficiency. These criteria for UWB systems are very
challenging and have not been fully addressed.
Design of multiple-input multiple-output (MIMO) communications
systems: Multiple antennas have recently been investigated to increase channel
capacity and service quality in mobile communication systems. Some preliminary
research has been done to design polarization and pattern diversity antennas for
mobile handset terminals in MIMO systems. One challenging problem yet to be
addressed is the design of diversity antennas with a small antenna form factor.
Design of on-glass antennas for vehicular applications: On-glass antenna
design for automobiles is a costly and time-consuming engineering process. The
design methodology developed in this dissertation can be applied to generate
optimized printed antenna designs for vehicle rear windows (on sedans) or vehicle
side windows (on SUVs and minivans).
Design of microwave absorbers: Chapter 4 in this dissertation examined
microwave absorber shapes to reduce the RCS of military targets. A possible
commercial application could be the design of absorbers to minimize EM
interference in RF systems.
97
Appendix A
Wheeler Cap Method for Measuring the Efficiency of Microstrip
Antennas
In this Appendix, some work into the Wheeler cap method for measuring
the efficiency of microstrip antennas is described.
Since antenna efficiency
characterization is an important issue in several of the chapters in this dissertation,
this material is included here for completeness.
A.1 Introduction
The Wheeler cap method is a simple and well-known technique for
measuring antenna efficiency [55,56]. The method involves making only two
input impedance measurements of the antenna under test: one with a conducting
cap enclosing the antenna and one without.
The antenna efficiency is then
estimated based on either a parallel or a series RLC circuit model for the antenna.
Pozar and Kaufman reported on the use of this method for measuring the
efficiency of microstrips [59]. Even though it is generally believed that a
microstrip antenna is more appropriately modeled as a parallel circuit [14,15,60],
their measurement results did not support the parallel RLC model, and they
concluded that the loss mechanism in the microstrip is similar to that of a series
98
RLC circuit. Even recently microstrip antenna efficiency has been measured using
a series RLC circuit model [61].
In this Appendix, the Wheeler cap method for measuring the efficiency of
microstrip antennas is revisited. The main interest stems from the need to
characterize the efficiency of a class of miniaturized microstrip antennas that was
designed using a GA [62].
Here three main findings are presented. First, it is
shown that the parallel RLC model is indeed a more appropriate model to use than
the series model for microstrips. The results are corroborated by numerical
simulations using the commercial software ENSEMBLE.
Second, the role of
interior cap modes is investigated and an optimal shape of the Wheeler cap for
microstrips is proposed. Finally, the Wheeler cap method is applied to investigate
the efficiency of the GA-designed miniaturized microstrip antennas.
A.2 Antenna Circuit Model
In the Wheeler cap method, two input resistance measurements are needed
to obtain the efficiency of the antenna. One is the input resistance before using the
cap, Rbefore, and the other is the input resistance R after putting the cap on, Rafter.
Since this method is valid only for antennas with a simple dominant loss
mechanism, we should know whether a given antenna follows the series circuit
model or the parallel circuit model. If the test antenna works more like the series
circuit model, then R decreases after applying the cap, and the efficiency is
calculated by the following expression:
99
Eff =
Rbefore ? Rafter
PR
RR
=
=
PR + PL RR + RL
Rbefore
( A.1)
where PR is the total radiated power, PL is the power loss, RR is the radiation
resistance and RL is the loss resistance.
If the test antenna works more like the
parallel circuit model, R should increase after applying the cap, and the efficiency
can be calculated using this expression:
Eff =
Rafter ? Rbefore
PR
RL
=
=
PR + PL RR + RL
Rafter
( A.2)
As a test, this Wheeler cap method is applied to measure a standard
square-shaped microstrip antenna built on lossy FR-4 substrate with thickness of
1.6mm. The test antenna?s dimensions are 36mm � 36mm, and it has a resonant
frequency of 2 GHz. For the Wheeler cap, a conducting rectangular cap (10cm �
5cm � 10cm) is used to completely enclose the test microstrip. Then an HP 8753C
network analyzer is used to gather the input impedance of the test antenna. As is
stated in [56,59], a perfect contact between the cap and the ground plane is critical
for an accurate measurement. Therefore, aluminum tape is used to shield the
slight gap between the cap and the ground plane. Fig. A.1(a) shows the resulting
input resistance before and after applying cap, marked as a dashed line and a solid
line, respectively. At the resonant frequency, it shows Rafter of 113 ohms and
Rbefore of 65 ohms. Since the test antenna is represented by a parallel RLC circuit
100
model the cap should increases the input resistance. Based on the parallel RLC
model, the Wheeler cap measurement shows an efficiency of 34%.
(b)
(a)
Measurement (FR-4)
Input Resistance (?)
ENSEMBLE Simulation (FR-4)
cap
140
120
140
------ Before
?? After
Rafter
------ Before
?? After
120
100
100
Rbefore
80
80
60
60
40
40
20
20
0
0
-20
1.8
Fig. A.1
1.85
1.9
1.95
2
2.05
2.1
2.15
Frequency (GHz)
2.2
2.25
2.3
-20
1.8
1.85
1.9
1.95
2
2.05
2.1
2.15
Frequency (GHz)
2.2
2.25
2.3
Input resistance of a standard square-shaped (36mm�mm) microstrip built on
FR-4 before (------) and after (?
??) using the Wheeler cap size of
10cmm�cmm�m. (a) Measured (b) ENEMBLE simulated.
Next, the measurement result is verified with two different numerical
simulations using the full wave EM simulator ENSEMBLE [63]. In the first
simulation, the Wheeler cap is modeled to rigorously predict Rafter and Rbefore. To
model the conducting rectangular cap, we generate a cavity on top of the
microstrip. The side of the cavity that faces the microstrip is opened. The
simulated input resistance is shown in Fig. A.1(b). Similar to Fig. A.1(a), the
result shows that the input resistance is actually increased after the cap is placed.
This again shows that the loss mechanism in the microstrip more closely
101
resembles a parallel circuit model. Based on the parallel circuit model, the
efficiency computed by this Wheeler cap simulation method is 41%. In the second
simulation, the gain of the test antenna is computed with and without dielectric
and metal loss. Note that metal loss occurs not only on the patch but also on the
ground plane, and this should be taken into account when modeling the
microstrips. The efficiency is then calculated using eq. (A.3), which gives us a
value of 32%.
Eff =
Gwith loss
( A.3)
Gwithout loss
The comparisons of two simulation and measurements in terms of
frequency are shown in Fig. A.2(a). These two simulation results are reasonably
close to the measurement results over the frequency range of interest. The
efficiency using the series RLC model is plotted as thin solid line. This shows that
the parallel RLC circuit model is more appropriate to measure the efficiency of
microstrip. This Wheeler cap method is also applied to a microstrip which is built
on a low loss substrate, RT Duroid 5880 (loss tangent of around 0.001). This test
microstrip has dimensions of 17.5mm�mm and operates at near 5.3GHz. This
measurement shows a 92% efficiency, and this compares favorably to
corresponding simulation results (95% in the gain simulation, and 98% in the
Wheeler cap simulation) as shown in Fig. A.2(b). The small deep at the frequency
of 5.12GHz is caused by an excitation of a cavity mode (TM111) by a Wheeler
102
cap. These results confirm the validity of our Wheeler cap measurements. In [59],
Pozar and Kauffman reported Rafter < Rbefore in their measurement of the simple
microstrip, and they concluded that the loss mechanism in the microstrip is
similar to the a series circuit model. It should be noted that this is probably due to
the influence of interior cap modes when using a large-sized Wheeler cap. In the
next section, cap modes and their dependence on the size of the cap will be
discussed.
(b)
(a)
100
100
Measured
Measured (Series)
Wheeler Cap Simulation
Gain Simulation
90
90
80
70
Efficiency (%)
Efficiency
(%)
Efficiency (% )
80
-10dB bandwidth
60
50
40
30
60
50
40
30
20
20
10
10
0
1.96
1.98
2
2.02
2.04
2.06
0
5.1
2.08
Frequency (GHz)
Fig. A.2
-10dB bandwidth
70
Measured
Measured (Series)
Wheeler Cap Simulation
Gain Simulation
5.15
5.2
5.25
5.3
5.35
5.4
Frequency (GHz)
Wheeler cap measured efficiency based on parallel circuit model (??),
Wheeler cap measured efficiency based on series circuit model (??Purple),
efficiency by Wheeler cap simulation (------) and efficiency by gain
simulation (------). (a) microstrip built on FR-4 substrate (b) microstrip build
on Duroid.
103
5.45
A.3 Effect of Cap Dimensions
Wheeler recommended that the cap radius be around 1/6 of a wavelength
to cause no change in the current distribution on the antenna [55,56]. However,
for microstrip antennas, a larger size Wheeler cap may have to be used to enclose
an extended substrate or to enclose a microstrip array.
Cap Height: 8.5cm
------ Before
?? After
700
600
17�5�cm
(2.83�41�83?)
500
400
------ Before
?? After
800
Input Resistance (?)
800
Input Resistance (?)
Cap Height: 2.0cm
900
900
300
200
700
600
17�0�cm
(2.83�33�83?)
500
400
300
200
100
100
0
0
4.6
4.8
5
5.2
5.4
5.6
4.6
5.8
Frequency (GHz)
Fig. A.3
4.8
5
5.2
5.4
Frequency (GHz)
5.6
5.8
Measured input resistance of a standard square-shaped (36mm x 36mm)
microstrip built on Duroid before (------) and after (??) using (a) the
Wheeler cap size of 17cmm x 17cmm x 8.5cm. (b) using the Wheeler cap size
of 17cm x 17cm x 2cm.
Fig. A.3(a) is an example of the effect using a larger size
(17cm�5cm�cm) Wheeler cap for the same microstrip (17.5mm�mm) built
on low loss substrate. In a small Wheeler cap, interior cap modes exist, but at such
high frequencies that they do not significantly interfere with the resonant
104
frequency of the microstrip as shown in Fig. A.1(a) and (b). On the other hand, a
larger Wheeler cap, shown in Fig. A.3(a), creates interior cap modes near the
resonant frequency of the microstrip, which causes a deviation in the input
resistance value. This can cause an inaccurate efficiency measurement when using
the Wheeler cap method.
Thus, the interior cavity modes should be as sparse as possible. This may
be achieved by using a smaller cap size. However, the finite size of the microstrip
substrate restricts the minimum size of the cap. If we take a detailed look at the
interior cavity modes, only TM modes are dominant in the Wheeler cap since the
microstrip works similarly to a horizontal magnetic current parallel to the ground
plane.
f
TM
MNP
1
2? �
M?
a
2
N?
+
b
2
P?
+
c
2
( A.4)
Fig. A.4(a) shows the interior cavity mode-spectrums for a Wheeler cap
sizes of 17cm x 8.5cm x 17cm. The solid line is the measured input resistance and
the dashed line is mode-spectrum calculated using eq. (A.4) [64]. The measured
cavity modes compare well with the ones by calculation. The index N is
associated with the cap height while the other indexes M and P are associated
with the length and the width of the cap, respectively.
105
(a)
Cavity Mode Spectrum of 8.5cm Cap Height
2
1
0.5
(b)
3
3.5
4
4.5
5
5.5
5
5.5
TM114
TM313
TM214
2.5
TM212
TM113
2
TM111
0
1.5
TM112
Log10 [Rin (?)]
1.5
Cavity Mode Spectrum of 2.0cm Cap Height
Log10 [Rin (?)]
2
1.5
1
0.5
0
1.5
Fig. A.4
2
2.5
3
3.5
4
Frequency (GHz)
4.5
Interior cavity modes spectrums for two Wheeler cap size. The upper spectrum is for
the Wheeler cap size of 17cm x 17cm x 8.5cm.The lower spectrum is for the Wheeler
cap size of 17cm x 17cm x 2.0cm.
106
Then the cavity mode-spectrum is observed by reducing only the height of the
Wheeler cap since the planar profile of the microstrip prevents us from decreasing
the other two dimensions. As shown in Fig. A.4(b), the mode spectrum of the 2cm
height Wheeler cap (17cm x 2cm x 17cm) is sparser than the mode spectrum of
the 8.5cm one. This is due to the fact that a cap height reduction makes the mode
spectrum sparser by pushing the interior modes with index N=1 to much higher
frequencies. Sparser mode spectrums provide more space in which to make
Wheeler cap measurements. This theory is applied to the test by measuring the
input resistance of the same microstrip shown in Fig. A.3(a) but using a reduced
cap height of only 2.0cm (0.33?). The results shown in Fig. A.3(b), indeed show
the sparser cavity mode-spectrum as expected. However, the resonant frequency
with cap is about 91MHz lower than the resonant frequency without cap. One
solution is to shift the capped resonant frequency up to 91MHz to compensate the
frequency shift by cap effect [65,66]. After the frequency compensation, an
efficiency value of 94% with this reduced height (17cm x 2.0cm x 17cm) Wheeler
cap is obtained, which is closer to the simulation value than the efficiency of
72.5% by using the size of Wheeler cap. This shows that reduced height wheeler
caps are helpful in obtaining more accurate efficiency values without sacrificing
the length and width of the cap.
107
A.4 Results for Efficiency of Miniaturized Microstrips
In this section, the Wheeler cap method described above is applied to
measure the efficiency of our miniaturized microstrips. A GA is previously
applied to minimize the size of a microstrip patch while keeping its bandwidth as
broad as possible [62].
(a)
(b)
100
50
Efficiency(%)
Efficiency (%)
40
35
30
25
20
15
90
Standard Square
Microstrips
80
Efficiency
(%)
Efficiency(%)
-Measurement
-Simulation
45
GA Microstrips
70
60
50
40
30
10
20
5
10
0
40
50
60
70
80
90
0
40
100
% from regular size
Fig. A.5
GA Microstrips
-Measurement
-Simulation
50
60
70
80
90
100
% from regular size
(a) built on FR-4 substrate in terms of % from the regular size and (b) the
microstrips built on Duroid.
The insets in Fig. A.5(a) are the samples of GA-miniaturized microstrips.
The achievable bandwidth of these miniaturized antenna drops as the size of the
antennas is reduced from 8% to 1.3%. It also shows that even when the size of the
patch is reduced to 40% of the regular size, it still maintains a bandwidth of
around 1.3%, which is good compared to the microstrip?s small size. However,
we thought that investigating the efficiency of these microstrips is crucial due to
108
the high loss in FR-4 substrate. Thus, the efficiencies of these microstrip are
observed and the results are shown in Fig. A.5 as the solid line with the
efficiencies using Wheeler cap measurements and the dashed line is using gain
simulations. The simulated and measured efficiencies are close, showing that our
measurements are accurate, even if the results show that our microstrips have very
low efficiencies. This is somewhat to be expected due to the high loss on the FR-4
substrate, which has a loss tangent of around 0.025. For this reason, our study is
extended to look at how low-loss substrate materials such as Duroid and air
increase the efficiency of these miniaturized microstrips. Fig. A.5(b) shows the
measured and simulated efficiencies of microstrips using the Duroid substrate
with loss tangent of about 0.001. An improved efficiency of more than 65% for all
samples can be observed. This gain in efficiency is not without a trade-off. The
achievable bandwidth using the Duroid substrate is reduced compared to the
bandwidth using the FR-4 substrate. This forces us to evaluate each antenna in
terms of both its efficiency and its bandwidth using the antenna?s EB product
[66]. Fig. A.6 plots the EB product against the physical antenna size for antennas
built on three different substrates: FR-4, Duroid and air substrate, marked as solid,
dashed and solid-dotted lines, respectively. Using Fig. A.6, It could be determined
which substrate material provides the highest EB product for a given antenna size.
109
7
Air (3.25mm)
5
Bandwidth * Efficiency
Efficiency x Bandwidth (EB)
6
Duroid (3.2mm)
4
3
2
FR-4 (1.6mm)
1
0
10
20
30
40
50
60
70
Physical Antenna Size (mm)
Fig. A.6 EB Product against physical microstrip patch size for microstrip built on three
different substrates. (a) FR-4 (?
??), Duroid (------)and Air (-?-?-?-?).
A.5 Summary
The Wheeler cap method for measuring microstrip efficiency was
revisited, and it was shown that the parallel circuit model is appropriate for the
microstrip loss mechanism. The measured efficiency values were verified using a
numerical simulation code. Then interior cap modes were investigated, and a way
to diminish them using a reduced height Wheeler cap was found. Finally, the
reduced-height Wheeler cap method was applied to investigate the efficiency of
miniaturized microstrip antennas on various substrate materials. However, this
110
Wheeler cap method is only valid for a limited frequency near a resonance. Also,
the method is valid for an antenna that has only one dominant loss mechanism.
111
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[3]
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Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by
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