DESIGN STUDY OF A MINIATURIZED MULTI-LAYERED
METAMATERIAL-INSPIRED DYNAMICALLY TUNABLE ANTENNA
By
Joshua C. Myers

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering – Master of Science
2014

ABSTRACT
DESIGN STUDY OF A MINIATURIZED MULTI-LAYERED
METAMATERIAL-INSPIRED DYNAMICALLY TUNABLE ANTENNA
By
Joshua C. Myers
A multi-layered metamaterial inspired minaturized antenna with pixel grid loading structures is introduced. The antenna consists of two conducting layers separated by a thin dielectric substrate. The first layer contains a folded monopole antenna surrounded by a metal
pixel based loading structure, while the second layer is envisioned to consist of a photo conductive pixel grid utilized to tune the antenna. The state of each pixel is controlled by a
binary genetic algorithm, which is implemented with a Matlab-HFSS interface. As a proof
of concept, the pixel grid on the second layer is initially made of a metal conductor. HFSS
simulations show that the second layer has a wide tuning ability with the appropriate state
formed through optimization. A wide range of other conductivities are also shown to provide
pixel combinations that meet the required antenna characteristics. The radiation efficiency
of the antenna with the second layer is also examined and optimized, and the theoretical
tuning range is investigated. Fabrication of multiple antenna configurations with the pixels
made of metal conductors are explored. Thin PET films are first investigated to be used as
simple loading elements that can be placed directly on the antenna. However, the airgap
and misalignment between the layers caused by this method is shown to be too large. A
novel multi-layer fabrication technique is then investigated which uses a SU-8 photoresist
as the dielectric layer. This layer can be spun directly onto the antenna, eliminating the
airgap. The alignment between the two layers using this method is also much better than
the previous method. Multiple antenna configurations corresponding to a wide frequency
range are constructed using this fabrication method. The measured reflection coefficients and
radiation patterns are shown to be in good agreement with HFSS simulations, successfully
demonstrating the ability to dramatically tune the antenna with a second pixel grid.

For my love Melissa, my beautiful Sunshine, and my comedian Shane.

iii

ACKNOWLEDGMENTS

I would like to first thank my advisor Dr. Premjeet Chahal. Since first meeting him
in ECE 405 as an undergraduate, I have learned more than I thought possible about electromagnetics. Although at times he may be a taskmaster, his creative and unique visions
have inspired me with numerous research ideas that neither I nor anyone else could have
envisioned. Without his research guidance and assistance I can truly say that I do not know
what I would be doing with my life. In addition to all of this, I would like to thank him
greatly for giving me the opportunity to complete my graduate degree and welcome me into
the Electromagnetics Research Group at Michigan State University.
Second, I would like to thank Dr. Edward Rothwell for all of the meaningful and insightful
conversations we have had not only about this research, but also about many other areas
in electromagnetics and life. While his sarcasm is sometimes thick, I have had nothing but
positive experiences working with him. Although he is not my advisor, he has treated me
with the same respect and helpfulness that he typically reserves only for his students. I
would also like to thank Dr. Nelson Sepulveda, not only for his excellent softball playing,
but also his assistance in my research and for serving on my committee.
I would like to also acknowledge all of my excellent co-workers and friends in the EM
group. In particular, I would like to thank Xianbo Yang not only for his friendship, but also
for his knowledge and helpfulness in the cleanroom. Without him, I highly doubt this thesis
would have been possible. In addition, I would like to sincerely thank Dr. Raoul Ouedraogo
for the use of his GA optimization code that is the basis of the optimizer used in this work.
Further, I would like to thank Benjamin Crowgey for his friendship and our frequent talks
about sports that successfully distracted me from completing this thesis at an earlier date.
I would also like to thank Junyan Tang for his help in soldering connectors, as well as his
many tang-isms. Last but not least, I would like to thank my former co-worker Dr. Jose
Hejase for not only his help with my research, but also his endearing friendship. Without

iv

Jose I would never have came to work with Dr. Chahal. I don’t think I could have ever
found a better mentor and friend.
Most importantly, I would like to thank my wife Melissa Myers for all her support and
love throughout this process. Without her I would have never known my beautiful daughter
Sunshine and my amazing son Shane. For this I will be eternally grateful to her. Additionally,
I would like to thank her parents and brothers for putting up with me all these years. I would
also like to thank my parents Barbara and Darryl Myers. Without them I would obviously
not be here. Their love and support has helped mold me into the person I am, and I am
extremely lucky to be their son.

v

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . .

xi

CHAPTER 1
Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Antenna Miniaturization . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Antenna Tuning and Reconfiguration . . . . . . . . . . . . . . . . . .
1.3 A Brief Overview of Metamaterials . . . . . . . . . . . . . . . . . . .
1.3.1 Applications of Metamaterials to Antenna Miniaturization and
1.4 Optimization Techniques in Electromagnetics . . . . . . . . . . . . .
1.4.1 Calculus-based Optimization Methods . . . . . . . . . . . . .
1.4.2 Enumerative Optimization Schemes . . . . . . . . . . . . . . .
1.4.3 Random Search Algorithms . . . . . . . . . . . . . . . . . . .
1.4.4 Genetic Algorithm Optimization . . . . . . . . . . . . . . . . .
1.4.4.1 Fitness Functions and Chromosomes . . . . . . . . .
1.4.4.2 Variable Selection . . . . . . . . . . . . . . . . . . .
1.4.4.3 Population . . . . . . . . . . . . . . . . . . . . . . .
1.4.4.4 Selection . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4.5 Mating . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4.6 Mutations . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4.7 Convergence and Future Generations . . . . . . . . .
1.4.5 Applications of Genetic Algorithms in Electromagnetics . . . .
1.5 HFSS-MATLAB Optimization Interface . . . . . . . . . . . . . . . . .
1.6 Research Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 2
Design and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Miniaturization of a Monopole Antenna Using a Distributed Pixel Grid . . .
2.1.1 GA optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Demonstration of Antenna Miniaturization . . . . . . . . . . . . . . .
2.1.3 Fabrication and Measured Results . . . . . . . . . . . . . . . . . . . .
2.2 Effects of a Second Tuning Layer on Antenna Characteristics . . . . . . . . .
2.3 Design of Second Tuning Layer . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Second Layer Configuration . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 GA Optimization of Tuning Layer . . . . . . . . . . . . . . . . . . . .
2.3.3 Simulated Reflection Coefficients of Selected Antenna Configurations
2.3.4 Investigation of Pixel Conductivities . . . . . . . . . . . . . . . . . .
2.4 Simultaneous Optimization of Radiation Efficiency and Reflection Coefficient

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CHAPTER 3
Fabrication and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1
3.2

3.3
3.4

Fabrication of Second Layer Using PET Films . . .
3.1.1 Effects of Misalignment and Air Gap . . . .
Fabrication of Second Layer Using SU-8 Photoresist
3.2.1 Fabrication Process . . . . . . . . . . . . . .
3.2.2 Fabricated Antenna Samples . . . . . . . . .
Measurement of Antenna Reflection Coefficients . .
Measurement of Antenna Radiation Parameters . .

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CHAPTER 4
Conclusions and Future Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

vii

LIST OF FIGURES

Figure 1.1

Geometrical miniaturization of a monopole antenna. . . . . . . . . .

3

Figure 1.2

Miniaturization of a monopole antenna with LC loading.

5

Figure 1.3

Simple configuration of a patch antenna loaded with a varactor diode.

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Figure 1.4

Summary of artificial material types. . . . . . . . . . . . . . . . . . .

11

Figure 1.5

Dipole antenna surrounded by a metamaterial shell. . . . . . . . . .

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Figure 1.6

Local maximum of function f(x,y,z).

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Figure 1.7

Extended domain of function f(x,y,z) showing multiple maximas. . .

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Figure 1.8

Sample discretized pixel grid. . . . . . . . . . . . . . . . . . . . . . .

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Figure 1.9

Mating process in the binary genetic algorithm. . . . . . . . . . . . .

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Figure 1.10

Summary of genetic algorithm optimization process.

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Figure 1.11

Genetic algorithm Matlab-HFSS interface flowchart . . . . . . . . .

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Figure 1.12

Optical excitation of second pixel grid. . . . . . . . . . . . . . . . .

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Figure 2.1

Geometry of miniaturized antenna. The selected pixel configuration
is shown in red, corresponding to a resonance of 2.5 GHz. . . . . . .

35

Reflection coefficient of monopole antenna with and without optimized pixel grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 2.3

Simulated radiation patterns in the XY and YZ planes. . . . . . . .

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Figure 2.4

Fabricated pixel edges. . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 2.5

Fabricated antenna compared with a US dime. . . . . . . . . . . . .

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Figure 2.6

Simulated and measured reflection coefficient of miniaturized antenna. 42

Figure 2.7

Miniaturized antenna with added dielectric layer and pixels. . . . . .

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Figure 2.8

Effect of multiple pixel configurations of second layer on reflection
coefficient of the antenna. . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 2.2

viii

. . . . . .

Figure 2.9

Comparison of the radiation patterns of the antenna with the added
dielectric layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 2.10

Antenna configuration with a second pixelized tuning grid. . . . . .

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Figure 2.11

Reflection coefficients of some sample antenna configurations within
the 3-4 GHz frequency range. . . . . . . . . . . . . . . . . . . . . . .

56

Reflection coefficients of some sample antenna configurations within
the 5-6 GHz frequency range. . . . . . . . . . . . . . . . . . . . . . .

57

Effect of changing the top layer pixel conductivity on the reflection
coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Reflection coefficient of optimized antenna configurations corresponding to very low top layer pixel conductivity. . . . . . . . . . . . . . .

60

Reflection coefficient versus radiation efficiency for the random search
and after 3 generations of GA optimization for 2.5 GHz. . . . . . . .

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Reflection coefficient versus radiation efficiency for the random search
and after 3 generations of GA optimization for 4 GHz. . . . . . . . .

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Reflection coefficient versus radiation efficiency for the random search
and after 3 generations of GA optimization for 6 GHz. . . . . . . . .

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Reflection coefficient versus radiation efficiency for the random search
and after 3 generations of GA optimization for 8 GHz. . . . . . . . .

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Best radiation efficiency corresponding to a reflection coefficient of at
least -10dB for a variety of frequencies. . . . . . . . . . . . . . . . .

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Figure 3.1

Antenna with PET layer setup.

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Figure 3.2

Fabricated PET layers. . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 3.3

Antenna with and without PET layers. . . . . . . . . . . . . . . . .

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Figure 3.4

Reflection coefficient of antenna with PET layer at 3.75GHz. . . . .

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Figure 3.5

Effects of miss-alignment between layers on the reflection coefficient
of the antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Effects of air-gap between layers on the reflection coefficient of the
antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Overview of fabrication process. . . . . . . . . . . . . . . . . . . . .

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Figure 2.12

Figure 2.13

Figure 2.14

Figure 2.15

Figure 2.16

Figure 2.17

Figure 2.18

Figure 2.19

Figure 3.6

Figure 3.7

ix

Figure 3.8

Close up of a fabricated antenna configuration. . . . . . . . . . . . .

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Figure 3.9

Alignment marks between SU-8 and bottom layer. . . . . . . . . . .

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Figure 3.10

Non-uniformity of SU-8 layer. . . . . . . . . . . . . . . . . . . . . .

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Figure 3.11

Excess copper on the second layer edges. . . . . . . . . . . . . . . .

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Figure 3.12

Example of one fabricated antenna configuration with a SMC connector attached. The simulated model of all pixel configurations are
also shown for comparison. . . . . . . . . . . . . . . . . . . . . . . .

85

Measured and simulated reflection coefficients of one antenna configuration at 2.5GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Measured and simulated reflection coefficients of one antenna configuration at at 4GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Measured and simulated reflection coefficients of one antenna configuration at at 5GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Measured and simulated reflection coefficients of one antenna configuration at at 8GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Measured and simulated reflection coefficients of all four antenna configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Figure 3.18

Antenna radiation pattern measurement setup.

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Figure 3.19

Measured and simulated radiation patterns in XY plane at 2.5 GHz.

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Figure 3.20

Measured and simulated radiation patterns in YZ plane at 2.5 GHz.

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Figure 3.21

Measured and simulated radiation patterns in XY plane at 4 GHz. .

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Figure 3.22

Measured and simulated radiation patterns in YZ plane at 4 GHz. .

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Figure 3.23

Measured and simulated radiation patterns in XY plane at 8 GHz. .

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Figure 3.24

Measured and simulated radiation patterns in YZ plane at 8 GHz. .

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Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 3.17

x

KEY TO SYMBOLS AND ABBREVIATIONS

❼ CPW: Coplanar Waveguide
❼ DNG: Double Negative Parameters
❼ DPS: Double Positive Parameters
❼ ENG: Epsilon Negative Parameters
❼ GA: Genetic Algorithm
❼ HFSS: High Frequency Structural Simulator
❼ MEMS: Micro-Electro-Mechanical Switches
❼ MNG: Mu Negative Parameters
❼ PET: Polyethylene terephthalate
❼ SRR: Split Ring Resonator
❼ SU-8: Novalac Epoxy Resin Photoresist

xi

CHAPTER 1
Introduction and Background
1.1

Antenna Miniaturization

The need for small and compact antennas has increased rapidly over recent years. This
increased interest has been primarily led by the desire to further shrink both military and
consumer devices in size. Meanwhile, the need for antennas in nearly all modern devices
has also increased exponentially as device connectivity has become crucial. While many of
the other components in these devices can easily be reduced in size, antenna miniaturization
presents a uniquely difficult problem. Since the operational band of an antenna is primarily
determined by the size of the antenna with respect to a wavelength, reducing the size of
antennas without losing the desired frequency of operation is somewhat counter-intuitive.
In addition, the desire for small antennas to maintain an acceptable level of radiation efficiency, bandwidth, and preserve a reasonable radiation pattern while being much less than
a wavelength in size further complicates the design process.
While the overall goal of antenna miniaturization is to reduce the geometric footprint
of the antenna, the process used to accomplish this reduction can be through electrical
or geometrical means. Perhaps the most intuitive means of antenna miniaturization is to
simply reduce the overall size impact of the antenna through geometry modification. A

1

simple example of size reduction through geometry modification can be displayed for a
simple monopole antenna. A monopole antenna is classically thought of as a single straight
wire, which produces a omni-directional radiation pattern in the azimuth plane, and a bidirectional pattern in any of the elevation planes [1]. These characteristics can be expected
of the monopole antenna when the length of the antenna is approximately equal to λ/4.
A monopole antenna can also be realized on a simple dielectric by creating a thin metal
strip λ/4 in length on one side of the dielectric, and a ground plane on the other. As long as
the overall length of the antenna is maintained, the shape of the monopole can be altered. A
common method of changing the shape of the antenna to reduce its size is by simply folding
the monopole back toward its excitation while maintaining the same overall length. The
overall process of this miniaturization is shown in Figure 1.1.
More novel methods of antenna miniaturization through geometry modification are also
commonly used. In [3]-[6], patch antennas are miniaturized using fractal structures. Size
reductions are achieved by using complex fractal structures, such as minkowski and sierpinski fractals. Fractals are space filling contours, meaning electrically large features can be
efficiently packed into small areas. A typical patch antenna is simply a rectangular radiating
patch placed above a ground plane, usually with a dielectric used for separation [2]. The
radiation achieved by the patch antenna is directly related to the current distribution of the
surface of the patch. However, the rectangular patch can also be replaced by a more complex
structure such as a fractal. These structures cause the current residing on the surface of the
antenna to travel along a longer path than a traditional patch. If the structure is more
compact than the patch, as in the case of a fractal, then less realization area is required for
the antenna.

2

Altering the Geometry of an Antenna:
“Folded” Monopole

Antenna Size Decreased

Figure 1.1 Geometrical miniaturization of a monopole antenna.

3

While modifying the antenna geometry can provide some antenna miniaturization, the
miniaturization cannot be electrically controlled. In addition, only a modest amount of
miniaturization can be achieved by modifying the geometry. Also, geometry modification
does not alter the resonant frequency of the antenna, providing only a limited frequency
tuning. Perhaps the simplest way of miniaturizing a patch antenna without modifying
its geometry is by using a high permittivity dielectric. Since the resonant frequency of
√
the patch is proportional to the relative permittivity of the dielectric by 1/ r , increasing
the permittivity will decrease the resonance frequency without altering the geometry [7][8]. Selecting a dielectric material with a large permittivity will dramatically decrease the
resonance of the antenna compared with traditional patch antenna substrate materials.
Antennas can also be miniaturized through the use of capacitive and inductive shorting
pins [11]-[14]. For a patch antenna, the miniaturization is achieved by altering the location
of the dominant patch mode from the center to the edge through the use of shorting pins
between the feed and edge of the patch. While a variety of size reductions are reported, a
miniaturization of nearly 50 % can be achieved for a rectangular patch antenna. Much larger
reductions have been investigated with more complex structures as shown in [15]-[16].
Another possible way to miniaturize an antenna is through the use of a variety of RLC
loading structures. Placing a loading structure near the antenna will drive down the resonant
frequency of the antenna, depending on the resonance of the loading structure. While the
antenna geometry remains the same, the resonant frequency of the antenna is reduced.
Thinking of the antenna size in terms of fractions of wavelength, if the antenna operating
frequency is reduced, than the effective size of the antenna is also reduced. This creates
a miniaturized antenna in terms of wavelength, even though the antenna dimensions are
unchanged. This loading concept applied to a simple monopole antenna is shown in Figure
1.2.

4

Loading with LC Structures:
Loading Structure

f0

f1

f2 f0 > f2

Antenna Frequency Driven Downward

Figure 1.2 Miniaturization of a monopole antenna with LC loading.

5

Using loading structures for miniaturization increases the possible frequency tuning range
of the antenna as the resonant frequency can be driven down based on the loading structure.
Loading structures also offer a much greater amount of miniaturization compared with simply
changing the antenna geometry. In addition, loading structures can easily become frequency
tunable through the use of active devices and a variety of other methods. A simple example
of a loading structure that reduces the resonance of an antenna is discussed in [17]. This
work utilizes a series of loaded slits to miniaturize a slot antenna fed with a CPW line.
This method is shown to allow for a 42% reduction in resonant frequency, while maintaining
the radiation properties of the antenna. More complex cases of loading structures will be
explored in section 1.3.1 of this thesis.
While this is only a small subset of techniques used to miniaturize antennas, it represents
a relative overview of some of the simplest methods. However, in recent years a large amount
of research has been applied to the use of metamaterials in antenna miniaturization.

1.2

Antenna Tuning and Reconfiguration

Unfortunately, loading antennas to achieve miniaturization only provides a fixed frequency
change in resonance, based on the resonace of the loading material. Therefore, it is desirable
to also make miniaturized antennas frequency tunable. One common way of frequency tuning
antennas is through the use of varactor diodes. Varactor diodes are typically operated in
a reverse-bias state. No current flows through the diode, but since the thickness of the
depletion region in a diode is increased with applied voltage, the capacitance of the diode
can be made to vary [20]. As the capacitance of a diode is dependent on the depletion region
width, which is dependent on the square root of the applied voltage, a relation between the
capacitance and applied voltage can be determined from the following relation.

6

1

C∝

(1.1)

Vapplied
While all diodes exhibit this behavior to a certain extent, varactor diodes are manufactured specifically to exploit this effect and increase the capacitance. Essentially, varactor
diodes can be used as voltage controlled variable capacitors. These capacitors can often
be used as loading elements on many types of antennas to tune the antenna in frequency
[21]-[25]. These tuning elements also allow for a continuous tuning range as opposed to a
discrete range created by tuning switches or pins. In [26] The operating frequency of a patch
antenna is tuned using a varactor diode mounted onto one of the radiating edges, similar
to the configuration shown in Figure 1.3. The patch is shown to be tunable from 4.9 to 5.4
GHz, with a supply voltage from 0 to 20V. In this case, the diodes provide an increase tuning
range near the original resonant frequency of approximately 10%. Unfortunately, varactor
diodes only provide a moderate increase in frequency tuning range.
Multiple types of switching methods have also been demonstrated for use in antenna
frequency tuning systems. Simple switches can be placed between the patch antenna and
its ground plane [27]. The resonance frequency of the antenna can be changed based on the
location and switch position. Micro-Electro-Mechanical Switches (MEMS) devices have also
been used in antenna tuning [28]-[29], where they are used as ultra-fast capacitive switches.
In particular, [30] uses periodically placed MEMS on a co-planar waveguide feed to a patch
antenna to tune the resonant frequency. Approximate 400 MHz of additional bandwidth
is achieved in this work. However, to achieve adequate frequency tuning, usually a large
number of switches or MEMS devices are necessary.
There are many additional frequency tuning methods besides just varactor diodes and
switches. In [31] the reconfiguration of antenna operating frequency is achieved by pin diodes
through switching states. Frequency tuning is achieved by changing

7

Reverse Bias

Side View

Top View

Figure 1.3 Simple configuration of a patch antenna loaded with a varactor diode.

8

the length of the coupled conductors. Mechanically means are also sometimes used to tune
antennas in frequency [32]-[33]. As an example, in [34] a patch antenna with an air-glass
substrate is tuned by mechanically changing the air-gap height between the antenna and
ground plane. Frequency tuning of approximately 12% are reported, with a 400MHz resonance shift. A number of new methods have also been presented to both miniaturize and
tune antenna systems. However, one method that is becoming increasingly important is in
the use of metamaterials. An overview of these artificial materials, as well as selected applications concerning the miniaturization and tuning of antennas, is presented in the following
section.

1.3

A Brief Overview of Metamaterials

The response of a system to an excitation can largely be determined by the properties of the
materials involved. In the case of an electromagnetic response, the most important material
properties are the permittivity ( ) and permeability (µ). The permittivity of a material
describes the measure of the ability to support the formation of a electric field within itself,
while the permeability of a material is a similar response, but for the support of magnetic
fields.
Materials can then be classified depending on the values of ( ) and (µ). A material with
both permittivity and permeability greater than zero ( > 0, µ > 0) is generally referred to
as a double-positive (DPS) medium. Most natural materials fall under this category, such
as dielectrics. Materials with permittivity greater than zero, and permeability less than zero
( > 0, µ < 0) are referred to as mu-negative (MNG) mediums [36]. Gyrotripic materials
exhibit these properties naturally over certain frequency spectrums. Noble metals have been
shown to exhibit negative permittivities in the IR and visible frequency spectrum. A material
with permittivity below zero, and a permeability above zero ( < 0, µ > 0) is referred to
as a epsilon-negative material (ENG) [37]. Mediums with both negative permittivity and

9

permeability ( < 0, µ < 0) are known as double-negative (DNG) materials [38]. The
classifications of these mediums is shown in Figure 1.4.
Unlike the other mediums discussed above, DNG material properties have never been
observed in nature. Mediums exhibiting these characteristics have been demonstrated with
artificial constructs. Artificial materials have been constructed that also have DPS, ENG,
and MNG properties. One of the first attempts to explore the concept of artificial materials dates back to the early nineteenth century, when Jagadish Bose conducted a series RF
experiments on twisted structures, essentially artificial chiral elements [35]. Since then, artificially engineered materials have become the subject of research for many groups worldwide.
However, materials with DNG parameters are of primary interest in this work. Artificially
engineered materials that exhibit electromagnetic properties which are not naturally found
(DNG paramaters) are commonly referred to as metamaterials.
In recent years, there has been a growing interest in metamaterials. In [39], the plane
wave propagation in a material with DNG parameters is theoretically investigated. The
direction of the poynting vector is found to be anti-parallel to the direction of propagation,
as opposed to parallel for the case of plane wave propagation in conventional materials. The
index of refraction has also been shown to be theoretically negative for DNG materials in
several works, and also experimentally verified [40]-[42]. Metamaterial mediums have also
been investigated through numerical means as well. In [43]-[45], finite-difference time-domain
(FTDT) simulations are used to study the wave physics associated with DNG materials.
Lossy Drude polarization and magnetization models are used in these simulations.
Metamaterials are also used for a variety of applications. two-dimensional metamaterials
are readily used in many microstrip-based applications. In [46]-[47], two microstrip based

10

µ
ENG Material
(Ɛ < 0, µ > 0)

DPS Material
(Ɛ > 0, µ > 0)

Plasmas

Dielectrics

DNG Material
(Ɛ < 0, µ < 0)

MNG Material
(Ɛ > 0, µ < 0)

Not found in nature

Gyrotropic

Figure 1.4 Summary of artificial material types.

11

Ɛ

coupler lines are presented. The first is a broadband backward-wave coupled-line directional
coupler, while the second is a asymmetric phase coupler. Metamaterials also have a number
of applications that are relevant in antenna design. In [48], a backward leaky-wave antenna
is presented. The radiation is achieved by capacitive gaps which excite a TM radiating wave.
Other antenna applications based on metamaterials include a dual-band ring antenna [49],
complex artificial ground planes [50], and resonance cone antennas [51]-[52]. In addition,
metamaterials have a wide range of uses in the miniaturization and tuning of antennas,
which is further explored in the following sections.

1.3.1

Applications of Metamaterials to Antenna Miniaturization
and Tuning

Metamaterials also have many applications in the miniaturization of antennas. This miniaturization can be achieved through a variety of techniques, though most of which involve
some form of metamaterial loading. In [53] and [54], an electrically small dipole antenna is
matched to free space by surrounding the antenna with a homogeneous shell having ENG
properties. If the proper ENG parameters are selected for the shell, a match can be found
between the shell and for any size dipole. This leaves an effectively miniaturized antenna.
However, the large size of the required shell eliminates any advantage gained by miniaturizing
the dipole. Figure 1.5
A similar concept is presented in [55], where a loop antenna is surrounded by a MNG
layer. The MNG layer is created by an array of split-ring resonators (SRRs) surrounding the
loop antenna. The resonance of the loop antenna is then reduced to that of the SRR elements.
Since the resonance of a typical SRR is mcuh less than a wavelength, the antenna resonance
is driven downward by at least λ/10. An even more compact design was is presented in
[56], where the loop antenna is placed directly into a single SRR resonance. A microstrip
patch antenna is also loaded with a DNG metamaterial so that the resonant frequency of
the T M110 mode can be made as small as desired [57]. Outside of using SRRs directly as
12

ŝƉŽůĞ ŶƚĞŶŶĂ

DĞƚĂŵĂƚĞƌŝĂů ^ŚĞůů
Figure 1.5 Dipole antenna surrounded by a metamaterial shell.

13

loading elements, metamaterials have been used in the place of ground planes to miniaturize
antennas. A number of authors [58]-[60] have investigated etching SRRs on the ground plane
of an antenna. The authors show that it is possible to lower the resonance of the radiating
patch to the resonance frequency of the SRRs. Although easier to implement than the other
techniques involving SRRs, only a size reduction of 31% is achieved in [58]. Additionally,
altering the ground plane of the antenna can dramatically effect other aspects of the antenna,
such as the radiation pattern and efficiency.
Metamaterials also have many potential applications in the tuning of antennas. Antennas
that are miniaturized usually have relatively narrow bandwidth as discussed in the previous
section. Due to these bandwidth limitations, it is commonly desirable to combine both
antenna miniaturization and frequency tuning in the design of a metamaterial based antenna.
A simple way of frequency tuning a miniaturized metamaterial antenna is presented in [61].
Varactor diodes are used to load an antenna miniaturized with SRRs. The varactor diodes
are placed between the gaps in the SRR, changing the resonant frequency of the resonators.
A tuning range of approximately 20% is reported. Similar results are also presented in
[62]-[63], with more or less the same level of tuning.
Metamaterials themselves have also been used as antenna tuning elements. In [64], a
metamaterial line antenna is tuned using two inductor poles. Changing the inductance
allows for a shift of the antenna resonance without sacrificing the antenna performance.
A number of other methods for antennas using metamaterials are presented in [65]-[67].
However, the loading effect of metamaterials for antenna tuning and miniaturization has
inspired other techniques used for a similar purpose. In particular, the use of optimized LC
structures in place of traditional metamaterials has been extensively investigated [68]-[71].
In these particular cases, a genetic algorithm (GA) optimization technique is used for both
miniaturization and tuning of loop, patch, and monopole antennas. In the next section, the
GA optimization technique is reviewed and explained in detail, and some select applications
of GA optimization in electromagnetic systems are also presented.

14

1.4

Optimization Techniques in Electromagnetics

A number of problems in electromagnetics become difficult to solve with conventional analytic equations. Most of these problems can be solved using numerical means, however, a
number of problems exist that require very complex structures with many possible variables.
Optimization can be used to solve these types of problems, and is widely used in many area
of electromagnetics such as antenna design and material characterization, along with others. There are three conventional categories of optimization and search methods used in the
current literature. These three methods include calculus-based, exhaustive search, and random search methods [72]-[73]. All of these methods, as well as non-traditional optimization
method based on natural selection, are subsequently introduced in this section.

1.4.1

Calculus-based Optimization Methods

Calculus-based methods are widely studied throughout a variety of research areas. These
methods involve seeking local maxima and minima by solving the gradient of the desired objective function set equal to zero. Given a smooth objective function, finding a corresponding
peak involves calculating derivatives in a local area and finding where they converge to zero.
While this method is local in scope, for a function similar to the one shown in Figure 1.6 the
peak location and value of the objective function can be easily found. However, this method
is extremely dependent on the location of the initial guess for the search. Depending on
the guess, there may be no way to tell if the maximum of the objective function is a local
maxima, or a global maxima. This means that the solution the method converges to may not
be the optimal, or even best, solution available. For example, if the domain of the function
shown in Figure 1.6 is extended as shown in Figure 1.7. The current maxima is revealed
to not be the global maxima in the area, meaning a better solution could be found with a
different initial guess. In addition, these methods depend upon the existence of well defined
derivatives which may not always be available.
15

Figure 1.6 Local maximum of function f(x,y,z).

16

Figure 1.7 Extended domain of function f(x,y,z) showing multiple maximas.

17

1.4.2

Enumerative Optimization Schemes

The idea of an enumerative (also known as exhaustive) search is possibly the most straight
forward method. Within a finite search space, the algorithm will evaluate the desired objective function at all possible points within the space. The simplicity of this search is very
attractive, and for small solution spaces can be effective. However, for more complex problems, these methods lack the necessary efficiency. Searching a large sample space element
by element simply takes far to long.

1.4.3

Random Search Algorithms

Random search algorithms have been shown to be effective for many problems. These search
methods involve randomly selecting points along the desired solution space, as opposed to
an exhaustive search searching every point. These techniques also have some of the short
comings of the exhaustive searches. Once again, the efficiency of such a search can be far
too low for very complex problems. However, for many desired solution spaces, a random
search is able to find satisfactory results.

1.4.4

Genetic Algorithm Optimization

The genetic algorithm (GA) is an optimization and search technique based on the principles of natural selection. These algorithms are widely used in many different engineering
applications in a wide range of fields. A GA allows a population composed of many individuals to ”evolve” under specific rules in order to maximize a desired fitness, or cost,
function associated with the current problem. GA optimization has many advantages over
the optimization techniques described previously. For example, the GA can optimize with
continuous or discrete values, although in this work only the discrete binary GA is used. In
addition, this method does not require and derivative information. GAs can also simultaneously search a wide range of objective functions and deal with a large number of variables.
18

In addition, finding a local minimum or maximum as opposed to a global one is not an issue
with a GA, meaning that extremely complex objective functions can be evaluated. Finally,
GAs also work very well with numerically or experimentally generated data as opposed to
analytic functions. These advantages provide an interesting optimization approach that can
solve many problems the previous search and optimization methods cannot. A thorough
explanation of the processes of the GA optimization technique is presented in the section,
with each component explained separately. In addition, some current applications of GA
optimization in electromagnetic problems are presented.

1.4.4.1

Fitness Functions and Chromosomes

A fitness function (sometimes called a cost function) represent the desired solution space
that the GA will attempt to minimize (sometimes maximize). This function generates an
output from a set of input variables. These input variables in the GA are referred to as a
chromosome. The GA begins by defining a chromosome array to be optimized. If the length
of the chromosome is Nbits , and each element of the chromosome are given by PN bits , then
a chromosome can be written in the following way.

chromosome = [P1 , P2 , ......, PN bits ]
1.4.4.2

(1.2)

Variable Selection

As the variables of the GA are represented as binary, the actual meaning of the variables
must be translated into binary bits. To illustrate this concept, a sheet discretized into square
pixels with equal sides of a length is shown in Figure 1.8. Each pixel can be in two possible
states, on or off. In this case, a 1 can be thought of as an on state, and a 0 can be thought of as
an off state. The grid has 10 rows and 10 columns, and the overall geometry is representative
of a chromosome of 100 bit length. For example, the the chromosome representing the

19

K Ŷ ƐƚĂƚĞ

K ĨĨ ƐƚĂƚĞ

Ă
Ă

Figure 1.8 Sample discretized pixel grid.

20

geometry of the grid can be written as be represented as in equation 1.3.

chromosome = [1110101101....]

(1.3)

With this discretization, the binary bits are able to be represented in the geometry we wish
to optimize.

1.4.4.3

Population

The collection of all the chromosomes within the GA is known as the population. The
population has Npop chromosomes and each chromosome has Nbits so the population will
be an matrix of Npop by Nbits in size. In order to begin the GA optimization process,
an initial population must be generated. The initial population is found by performing a
random search and filling the population with random ones and zeros. This initial population
is then passed into the fitness function to be evaluated.

1.4.4.4

Selection

From the evaluation of the initial population, the best chromosomes are selected by the
GA. This is done by first ranking the chromosomes from best to worse according to the
desired fitness function. From here, only the best chromosomes are kept and the others are
discarded. The number of chromosomes to be kept are selected based on the selection rate,
Xrate , which represented the fraction of the Npop chromosomes that are kept. The number
of chromosomes that are kept are given by

Nkeep = Xrate Npop

(1.4)

Selection occurs at each iteration (known as a generation) of the algorithm. Deciding how
21

many chromosomes are kept depends on the application. However, keeping too few will limit
the availability of genes in the offspring. Keeping too many chromosomes could bring in too
many bad performers from the previous generation.
Two chromosomes are then selected from the remaining chromosomes Nkeep to produce
two new offspring until the discard chromosomes are replaced. While there are a number of
techniques used for the selection of the new chromosomes, a tournament selection method is
used throughout this thesis. Tournament selection involves randomly picking a small subset
of chromosomes from the remaining pool, and the chromosome within this selection with the
best fitness is chosen to become a parent. The tournament repeats until every parent needed
to regenerate the population of chromosomes are found. This method works especially well
for large populations because there is no extra sorting necessary.

1.4.4.5

Mating

Mating is the creation of offspring from the parents found in the selection process. The
most common form of mating involves two parents which create two offspring. A crossover
point between two parents is chosen randomly between the first and last bit of both parents
chromosomes. The first parent will pass its binary code to the left of the crossover point
to the first offspring. The second parent also passes its binary code to the second offspring
from the left of the same crossover point. The binary code to the right of the crossover point
of the first parent is then passed to the second offspring, while the binary bits to the right
of the crossover point for the second parent are passed to the first offspring. This leaves us
with two offspring with the genetic makeup of both parents, which are used to replace the
two parents selected for Nkeep . This process is summarized in Figure 1.9
1.4.4.6

Mutations

Random mutations are also used to alter the a percentage of the remaining population matrix.
Mutation introduces another element to the optimization process by introducing traits into

22

the population that were not initially present in the selected parents and offspring. In the
binary GA, mutation can be as simple as changing a single bit in a chromosome from a 1
to a 0, or vice versa. Mutation is important for the GA process as it introduces traits not
found in the selected parents and offspring to fill the population. Without mutation, the GA
may converge too quickly to an answer that does not include other possible combinations.
The number of mutations are given by equation 1.5, where µrate is the desired number of
mutations for the entire population.

mutations = µrate (Npop − 1)Nbits
1.4.4.7

(1.5)

Convergence and Future Generations

After the mutations takes place and the fitness functions associated with the offspring are
calculated, the results are compared with a stopping criteria. If the stopping criteria is not
met, the algorithm takes the current population and begins the selection process again in
a new generation. If the criteria is met, the algorithm will stop and the results will be
exported. The overall GA optimization process is shown in Figure 1.10.

23

WĂ ƌĞ Ŷ ƚ η ϭ

WĂ ƌĞ Ŷ ƚ η Ϯ

ϭϭϭϬϬ
ϬϭϬϭϭ
ϭϭϬϬϭ
ϬϬϭϭϬ

ϭϭϬϬϭϭϬ
ϬϬϬϭϬϭϬ
ϭϭϬϭϬϭϬ
ϬϭϬϭϬϭϭ

ϭϭϭϬϬ ϬϭϬϭϬϭϭ
ϬϬϭϭϬ ϭϭϬϬϭϭϬ

Figure 1.9 Mating process in the binary genetic algorithm.

24

K ĨĨƐƉ ƌŝŶ Ő η ϭ
K ĨĨƐƉ ƌŝŶ Ő η Ϯ

G en erate In itial
P o p u latio n
E v alu ate F itn ess

S to p

M eet R eq u irem en ts

Yes

No
F ill N ew G en eratio n
P erfo rm S electio n
P erfo rm M u tatio n

P erfo rm C ro sso v er

Figure 1.10 Summary of genetic algorithm optimization process.

25

1.4.5

Applications of Genetic Algorithms in Electromagnetics

Genetic algorithms are used as an optimization tool in a wide variety of Electromagnetics
problems. In [75], genetic algorithms are used to synthesize light-weight, broadband microwave absorbers. The absorber is intended to be as light-weight as possible, while being
thin and multi-layered, backed with a PEC conductor. Each layer of the antenna is optimized with a frequency dependent material, while the permittivity and permeability are
varied. The reflectivity of the array is optimized while changing the materials in the arraystack, while keeping the stack as thin as possible. A wide range of frequency absorbers are
designed, with reflection coefficients ranging from .2-8GHz.
These algorithms have also been used in the design of frequency selective surface (FSS)
structures [76]. In this work, the FSS structures are desired to be used as waveguide filters.
As the filter characteristics depend upon the shape and size of the individual elements, as
well as the number of element layers, both are optimized with the GA. Very strong bandpass
and bandstop filters are shown in this work, at frequencies as high as 25GHz.
GAs are also widely used in antenna array design. In [77], the sidelobe levels of an
antenna array are decreased by thinning or removing the elements that makeup an array.
This is done by representing the elements with a binary number, and turning them to either
a 1 or 0 based on if the element is on or off. Extremely low sidelobe levels, as low as -22dB,
are reported. Refrence [78] uses the genetic algorithm to optimize a linear antenna array
to shape the mainbeam of the radiation pattern. In this case, the goal is to select a set
of amplitude and phase coefficients for the antenna to achieve a narrow beam at a specific
location.
Optimization has also been done extensively on single antenna elements. In [79], a patch
antenna is optimized so that the bandwidth of the antenna is significantly wider than a
traditional patch. The GA is coupled with a method of moments so that a binary string
could represent the presence or absence of a subsection of metal in the patch. With this

26

method, the bandwidth of one patch was able to be increased by approximately 20%. The
GA optimization method has also been recently used to miniaturize antennas.
In [69], a patch antenna is miniaturized by placing SRRs between the top layer of the
antenna and the ground plane. The geometry of the SRRs are optimized with a GA, while
the fitness function is designed to minimize the reflection coefficient of the antenna. A good
impedance match and radiation characteristics are maintained with the optimization, and
one state is shown to achieve a miniaturization of approximately 1/16th the size of the
original patch. [68] presents a loop antenna that is miniaturized using a binary GA. The
loop antenna is printed on a planar dielectric, with a pixelized sheet directly above it. Each
pixel is controlled with the binary GA and the fitness function is chosen to miniaturize the
reflection coefficient at a desired frequency.
Another miniaturization method is presented in [71]. In this work, a monopole antenna
is surrounded by a metallic pixelized grid controlled by a binary GA. The pixel grid is used
to miniaturize the antenna through control of the GA. The radiation characteristics of the
monopole are well maintained throughout the miniaturization, and an overall miniaturization
of λ0 /26 is achieved. All three of these methods are implemented using a HFSS-matlab
interface, which will be discussed in detail in the following section.

1.5

HFSS-MATLAB Optimization Interface

In order to implement the GA optimization successfully with an electromagnetics problem, a
numerical solution method must usually be implemented. While some simple problems can
be solved through analytic equations, most require approximate solutions. There are many
numerical solution techniques that can be used, however, perhaps the most popular method
is in the use of commercial electromagnetics solvers. One such commercial solver is Ansys
High Frequency Structural Simulator (HFSS), which is based on a finite element method.
This solver has a CAD like interface which can be easily used and can implement a number

27

of solution methods in both the frequency and time domain.
Implementing a GA to solve a problem iteratively is unfortunately not available in HFSS.
However, HFSS commands are able to be executed through simple macro scripts that can
be imported and run in real time. These macros can be executed through MATLAB, and
compiled into a script which can be imported to HFSS. This allows for the automated creation
of geometry, boundary conditions, and excitations in HFSS. This allows the geometry to be
altered after every iteration of a parameter, meaning a GA can be used to optimize structures
with an interface between HFSS and matlab.
An overall flow of the optimization process is as follows. First, matlab is launched and
the GA parameters are initialized. The geometry is then created for the first iteration of
the GA. The script is then assembled and HFSS is launched. The script is passed to HFSS,
and the geometry is drawn in real time. The structure is then simulated, and the relevant
results are exported back to matlab for post processing. If these results meet the desired
requirements, then the GA is stopped. However, if the results do not meet the requirements
desired, the overall process is restarted with a new geometry and the next iteration of the
GA is performed. The overall flowchart for this process is shown in Figure 1.11.

28

L au n ch M AT L A B

No

Yes

C reate G eo m etry &
G A P aram eters

M eet R eq u irem en ts

C reate H F S S S crip t

P o st P ro cessin g

L au n ch H F S S

E x p o rt P aram eters

D raw G eo m etry

S im u late S tru ctu re

Figure 1.11 Genetic algorithm Matlab-HFSS interface flowchart

29

S to p

1.6

Research Overview

The work in this thesis introduces a new approach to the tuning of a miniaturized antenna.
A distributed pixelized grid structure is placed on a thin substrate directly on top of the
miniaturized antenna introduced in [71]. The pixel grid is envisioned to be composed of a
photo conductive or phase change material layer, allowing each discrete pixel to be turned
on or off through photo-illumination. This second grid layer is optimized via a binary GA
in a similar manner to the bottom layer, where a 1 corresponds to a high conductivity
(illuminated pixel) and a 0 corresponds to no pixel (no illumination). Once a suitable state
for a desired frequency is found with the GA, each pixel corresponding to that state may be
illuminated in real-time, tuning the antenna to the desired frequency. This overall process
is shown in Figure 1.12.
Chapter 2 introduces the specifics regarding the original miniaturized antenna presented
in [71]. A proof of concept of adding the second layer of the antenna is first presented.
In this case, the reflection coefficients and radiation patterns of the antenna are studied as
the layer and pixels are added to determine the level of effect on the monopole antenna
characteristics. Next, the design of the second layer and numerous simulated results are
presented. The radiation efficiency and reflection coefficients are optimized in unison using
the HFSS-matlab GA interface. The theoretical tuning range of the second layer is also
investigated. These simulations are first carried out using copper as the pixel material,
however, a variety of conductivities are also tested showing a wide range of pixel materials
are possible.
Chapter 3 describes the fabrication and measurement of the antennas with a copper
tuning layer. Multiple states are fabricated that cover a wide range of frequencies to show
the potential of the second layer as a tuning element. Thin Polyethylene terephthalate
(PET) layers are first investigated to be used as a simple insert that can be placed on the
antenna as a loading structure. The airgap and misalignment between the PET layer and

30

the antenna are investigated and prove to be to large of a obstacle to overcome. However,
a multi-layer fabrication method is then presented which eliminates airgaps between the
layers and minimizes misalignment. Measured reflection coefficients and radiation patterns
are investigated of multiple states corresponding to a very wide frequency range. The final
chapter discusses any concluding results and suggestions for future work.

31

Optimize bottom layer

Select desired tuning
frequency of antenna

Repeat for additional
tuning frequencies

Optimize top layer pixels
for tuning frequency

Save pixel configuration

Illuminate desired
pixels within grid

Illuminated Pixel

Optical Excitation

Pixel grid made of
photo-sensitive material
Figure 1.12 Optical excitation of second pixel grid.

32

CHAPTER 2
Design and Simulations
In this chapter, the miniaturized monopole antenna of [71]is briefly introduced. The bandwidth of this antenna is shown to be relatively small, leading to a need for a reliable frequency
tuning technique. To tune the antenna in frequency, the pixelized grid is extended directly
on top of the antenna. First, a brief proof of concept is presented, showing that the radiation
and reflection characteristics of the antenna are largely unaltered by adding a second grid.
The design of the second layer tuning grid is then introduced, and the GA used to optimize
the structure is explained in detail. Next, the tuning layer is optimized over a wide range
of frequencies, and a multitude of antenna configurations are able to be found with very
good reflection coefficients over a wide frequency range. The pixel conductivities are also
examined, showing that nearly any metal or conductive material can be used as the material
for the pixel grid. Finally, the optimization is extended to not only minimize the reflection
coefficient of the antenna, but also increase the radiation efficiency. This optimization is used
to not only produce a more efficient antenna, but also demonstrate the theoretical tuning
range of the second layer.

33

2.1

Miniaturization of a Monopole Antenna Using a
Distributed Pixel Grid

The initial miniaturized antenna presented in [71] is briefly summarized in this section.
A folded monopole antenna is fed by a co-planar waveguide (CPW) and surrounded by a
metallic grid structure. The metallic grid structure is intended to be used as a parasitic
loading element to alter the electrical characteristics of the patch. The monopole antenna
and pixelized patch are placed on a .5mm thick Rogers RO4003 substrate, along with the
co-planar ground plane. The Rogers RO4003 substrate has a dielectric constant of r = 3.55
and a loss tangent of δ = .0027. The structure is shaped into a half-disk of radius 5.2mm to
further miniaturize the overall size. the grid is discretized into 134 square pixels 500µm in
size. The total number of possible combinations is then 2134 or 2.1778x1040 possible pixel
states.

2.1.1

GA optimization

The large number of pixel configurations suggests the use of a binary optimizer capable
of quickly finding pixel configurations that produce an acceptable performance. A brief
summary of the GA optimization applied to the miniaturized antenna is presented here, but
a more in depth explanation is presented in section 2.3.2 Each pixel is encoded with a binary
1 or 0, correspond to metal either being in place or absent. The geometry of this antenna
is displayed in Figure 2.1, with one possible configuration shown in red. The state of each
pixel is optimized with a binary GA, and the fitness function is given by equation 2.1.

f itness = 20log10 (|S11 |)

34

(2.1)

Substrate

Pixelized Patch

Selected Pixel Configuration
Z

Monopole

Ground Plane
Y
X

Figure 2.1 Geometry of miniaturized antenna. The selected pixel configuration is shown in
red, corresponding to a resonance of 2.5 GHz.

35

The objective of the GA is to minimize this fitness function at a frequency of interest.
The GA uses a tournament selection method with an initial population of 200 and a 10%
selection rate. Each chromosome is then 134 bits long, with a population matrix with 134 by
200 entries. The mutation rate is also set at 20%, with a 2-point crossover with an evolving
single bit mutation. The GA is allowed to run for a maximum of three generations or until
the desired results are met. The GA is built in matalb, and interface with HFSS to allow
for automation of the antenna geometry and solutions as described in section 1.5.

2.1.2

Demonstration of Antenna Miniaturization

A particular antenna configuration is selected which corresponds to a resonant frequency of
2.5 GHz. The pixels selected that compose this configuration are shown in red in Figure
2.1. To determine the amount of antenna miniaturization, the reflection coefficient before
and after the added grid are studied. Both reflection coefficients are shown Figure 2.2.
The resonance of the antenna without any of the pixels added is around 5GHz, with a
magnitude of under -20dB. However, once the optimized pixels are added, the resonance
shifts to approximately 2.5GHz. At this frequency the antenna is miniaturized by roughly
50%, and the overall size of the antenna is approximately λ0 /24, reduced from λ0 /4 . The
simulated radiation patterns are also investigated to determine if typical monopole antenna
radiation characteristics are maintained. Figure 2.3 shows the radiation pattern in the
azimuth (XY) and elevation (YZ) planes. Both patterns exhibit what we would expect
from a monopole antenna, uni-directional in azimuth, and a bi-directional pattern in the
elevation planes. From these results it can be seen that the antenna exhibits very high
miniaturization in simulation, but the antenna must also be built and tested to determine
its real world performance.

36

0
Pixel Grid
No Grid

−10

11

|S | (dB)

−5

−15

−20

−25
2

2.5

3

3.5
4
Frequency (GHz)

4.5

5

5.5

Figure 2.2 Reflection coefficient of monopole antenna with and without optimized pixel grid.

37

Figure 2.3 Simulated radiation patterns in the XY and YZ planes.

38

2.1.3

Fabrication and Measured Results

The antenna is fabricated using conventional micro-fabrication in a clean room environment.
Care is taken so that the edges of the pixels are overlapping. If the pixels edges are simply
left the same as in the simulations, undercut will cause the pixels to no longer be electrically
connected. A small shorting pin between the pixels is added to the structure so that the
electrical connection is maintained. Figure 2.4 shows the pixel structure under magnification,
where the edges of the pixels can be seen. Without the shorting structure, it is clear that
the pixels would no be electrically connected, although this method may slightly effect the
measured results, it is a necessary step. A fabricated antenna designed for operation at 2.5
GHz is shown in Figure 2.5.
To properly characterize the antenna miniaturization, the reflection coefficient was measured using a Agilent N5227A network analyzer. The measured reflection coefficient of an
antenna configuration at 2.5 GHz compared with the simulated result is shown in Figure
2.6. The results show very good agreement between measurement and simulations. The
magnitude of the reflection coefficient matches very well, while the location is only shifted
by approximately 100MHz. These slight discrepancies are most likely caused by the added
structure between pixels to prevent undercut. Overall, this antenna presents many desired
traits, such as a high amount of miniaturization, a strongly resonant reflection coefficient,
and good agreement between simulations and measurement. However, as discussed in chapter 1, miniaturized antennas of this kind have very narrow bandwidth, requiring frequency
tuning structures.

39

Figure 2.4 Fabricated pixel edges.

40

Figure 2.5 Fabricated antenna compared with a US dime.

41

Figure 2.6 Simulated and measured reflection coefficient of miniaturized antenna.

42

2.2

Effects of a Second Tuning Layer on Antenna Characteristics

To use a second metamaterial structure as a tuning element for the antenna first requires
a study of the effects this layer will have on the original antenna’s characteristics. The
effects of a second layer are first tested using a thin 25m piece of the same rogers material
that the antenna was originally fabricated with. The effect of the layer itself, as well as
multiple sample pixel combinations, are studied. The overall geometry and pixel setup is
shown in Figure 2.7. The layer is covered in two pixel groups, labeled as pixels A and B. The
simulated reflection coefficient with each combination is shown in Figure 2.8. The reflection
coefficients show relatively little change between each combination in terms of the magnitude
of the resonance, however, the location of the resonance is slightly shifted.
Two primary observations can be made from these results. The first is that little change
is shown in the reflection coefficients between the adding of the pixels and the second layer
and the original antenna. This is a desired results because if the resonance was greatly
affected with just a few pixels than adding an entire pixel grid may completely change the
characteristics of the antenna. Although it is more applicable to the radiation characteristics
of the antenna, these observations are also applicable to the reflection coefficients as well.
The second is that some tuning is shown between each pixel combination, meaning that
pixels added on top of the layer have potential to be used as a tuning element.
The radiation patterns of the antenna are also studied with and without the dielectric
layer. The patterns are studied in two primary planes, the azimuth and elevation plane.
In this case, the azimuth plane refers to the XY plane, and the elevation plane refers to
the YZ plane. In the azimuth plane, a monopole antenna is expected to exhibit a simple
uni-directional antenna pattern (similar to a circle). Both the antenna with and without the
layer added have very little differences in the radiation patterns in this plane. Similarly, in
the elevation plane, the expected pattern for a monopole antenna is bi-directional in the
43

Test Pixels on Top Layer

Bottom Layer
Pixels (unchanged)

A

Thin Dielectric Layer

B
Z
Y

X
Figure 2.7 Miniaturized antenna with added dielectric layer and pixels.

44

−4

−8
−10

|S

11

| (dB)

−6

−12
−14
−16
2.3

Original
Thin Layer
A Pixels
B Pixels
2.4

2.5

2.6
2.7
2.8
Frequency (GHz)

2.9

3

Figure 2.8 Effect of multiple pixel configurations of second layer on reflection coefficient of
the antenna.

45

shape of a figure eight. In this plane, the results also match very well for both with and
without the dielectric layer. These results are very encouraging as the radiation pattern is
maintained in both planes despite the addition of the layer. This suggests that the pixelization, as well as the added dielectric layer, have little effect on the radiation characteristics
of the monopole antenna.

46

Figure 2.9 Comparison of the radiation patterns of the antenna with the added dielectric
layer.

47

2.3
2.3.1

Design of Second Tuning Layer
Second Layer Configuration

As discussed in the previous section, the addition of a second layer to the antenna does not
have negative impact on the radiation characteristics of the monopole. However, the location
of the reflection coefficient is shown to shift with different pixel combinations. In order to
exploit this shift in resonance, another pixelized patch is placed directly above the dielectric
layer. In this thesis, this added layer will be commonly referred to as the top layer of the
antenna, and the original miniaturized antenna will be referred to as the bottom layer of the
antenna. The pixel size for the second layer is designed to be the same as the first layer,
but the grid is no longer limited to the geometry surrounding the antenna. Thus, the pixel
grid is extended to envelope the entire antenna, utilizing a 178 pixel grid, with 2178 possible
combinations. The geometry of the antenna with the second pixel grid is shown in Figure
2.10.
To utilize the top layer as a tuning element, each pixel is made of a suitable material
whose conductive properties can be changed in real time. This layer could be composed of an
optically controlled semiconductor, or a phase change material. In the case of a semiconducting layer, each pixel can be configured to an ”on” or ”off” state through optical illumination
to produce a configuration that is found from the GA optimization for a desired resonant
frequency. To initially test the tuning layer, the pixels are modeled with the conductivity of
copper.
The dielectric layer thickness between the two layers is chosen to be 25µm, and is again
modeled with the same Rogers material as the bottom layer. The bottom layer of the antenna
is maintained fixed throughout this work at the configuration shown in section 2.1 designed
for a resonance at 2.5GHz. Changing the bottom layer of the antenna may change the
possible tuning range of the second layer, as well as other characteristics of the antenna, but

48

is not explored in this work. While the bottom layer pixel configuration remains constant,
the top layer of the antenna is optimized through a similar GA techniques used on the bottom
layer.

49

Bottom Layer
Pixels (unchanged)

Top Layer Pixels
(Controlled with
GA)

Thin Substrate
Z

Y

X

Antenna & Coplanar Waveguide
Figure 2.10 Antenna configuration with a second pixelized tuning grid.

50

2.3.2

GA Optimization of Tuning Layer

The GA optimization for the second layer is similar to the first layer, however, the number
of pixel combinations are now higher due to a decrease in geometrical constraints on the new
layer. The GA uses a tournament selection method with an initial population of 300 and a
10% selection rate. The initial population is increased for the second layer as a result of the
increase in pixel combinations. Each chromosome is then 178 bits long, with a population
matrix with 178 by 300 entries. The mutation rate is also set at 20%, with a 2-point crossover
with an evolving single bit mutation. The GA is allowed to run for a maximum of three
generations or until the desired results are met. The initial fitness function of the GA is
the same as in equation 2.1, and the objective is to minimize the reflection coefficient at a
desired frequency.
To feed the GA, a random search is first implemented where the chromosomes of the
population matrix are randomly selected. This random selection is accomplished with the
following matlab code.

1

for ip=1:popsize;

2

for jp = 1:ChromLength
if rand() > 0.5

3

Pop(ip,jp) = 1;

4

else

5

Pop(ip,jp) =0;

6

end

7
8

end

The corresponding pixel arrangements on the second layer are then created one by one
and passed to HFSS. The matlab code used to create each pixel and pass the geometry to
HFSS is shown in the following code.

51

1

for i=1:m; NN=0;

2

for j=1:n
if Meta(j,i)==1;

3

NN=1+ NN;

4
5

end

6

if NN>0
if Meta(j,i)==0

7
8

N=N+1;

9

xp=(i)*Pw;

10

yp=(j)*Pw;

11

hfssRectangle(fid, sprintf('RecB%d',N), 'Y' , ...
[−xp+SRR LX/2, −.025, yp−(1+NN)*Pw], Pw*NN, Pw,'mm');
hfssAssignFiniteCond(fid, sprintf('RecBPEC%d',N), 0, ...

12

'um', {sprintf('RecB%d',N)})
NN=0;

13

end

14

if j==n

15

if Meta(j,i)==1;

16
17

N=N+1;

18

xp=(i)*Pw;

19

yp=(j)*Pw;

20

hfssRectangle(fid, sprintf('RecB%d',N), 'Y', ...
[−xp+SRR LX/2, −.025, yp−NN*Pw], Pw*NN, Pw,'mm');
hfssAssignFiniteCond(fid, sprintf('RecBPEC%d',N), 0, ...

21

'um', {sprintf('RecB%d',N)})
NN=0;

22

end

23

end

24

end

25

end

26
27

end

52

These random chromosomes are then evaluated with equation 2.1, and ranked from lowest
to highest. During each iteration of both the random search and the GA, every antenna
configuration is evaluated in HFSS and saved in a local directory. This allows for easy
access to any configuration after the GA is finished, as opposed to only saving the pixel
combinations and having to re-simulate each desired configuration. Once the random search
is finished, the re-organized fitness function is passed to the GA portion of the code. First,
the chromosomes to be kept are selected, which is accoumplished in the following way.

1

% Selection of chromozomes to be kept

2

indx=find(fitness≤mean(fitness));

3

% Determines the chroms with fitness < mean fitness

4

keep=ceil(selection*popsize); %Number of chrom to keep

5

fitkeep=fitness(indx);% fitnesses to keep

6

Popkeep=Pop(indx,:);%Chroms to keep

7

M1=ceil((popsize−keep)*.6);

8
9
10

% Chromosomes to be replaced via Xover and Mutation for case 1
M2=ceil((popsize−keep)*.4);
% Chromosomes to be replaced via Xover and Mutation for case 2

Next, a random selection of possible parents is performed.

1

for ic2=1:2:M2

2

%Random Selection of Parents

3

%first Parent

4

ma=ceil(keep*rand());% indicies of first parent

5

%Second parent

6

pa=ceil(keep*rand());% indicies of second parent

From here, a 2-point crossover is performed between parents to create two offspring.

53

1

%2pt Crossover

2

Xpt1=ceil(rand()*Nt/2);

3

Xpt2=Xpt1 + ceil(rand()*(Nt−1)/2);

4

Pop(keep+M1+ic2−1,:)=[Pop(ma,1:Xpt1) Pop(pa,Xpt1+1:Xpt2) ...
Pop(ma,Xpt2+1:Nt)]; % Offspring 1

5

Pop(keep+M1+ic2,:)=[Pop(pa,1:Xpt1) Pop(ma,Xpt1+1:Xpt2) ...
Pop(pa,Xpt2+1:Nt)];%Offspring 2

6

end

Then the best two individuals are selected and 2-point crossover is performed again.

1

%selection the best of 2 individual

2

%first Parent

3

rc=ceil(keep*rand(1,Ntourn)); %Randomly select 2 chroms among those kept

4

[c,ci]=min(fitness(rc)); % Select the chrom with lowest fitness

5

ma=rc(ci);

6

%Second parent

7

rc=ceil(keep*rand(1,Ntourn));%%Randomly select 2 chroms among those kept

8

[c,ci]=min(fitness(rc)); % Select the chrom with lowest fitness

9

pa=rc(ci);

% indicies of first parent

% indicies of second parent

Finally, single-bit mutations are performed based on the probability of mutation.

1

%mutation: keep best two and mutate a single bit on the other

2

%chromosomes based on mutating probability

3

for iMu=keep:popsize

4

if Muprob≥rand()

5

Mupt=ceil(rand()*Nt);% bit to be mutated

6

Pop(iMu,Mupt)=not(Pop(iMu,Mupt));

7
8

end
end

54

After the population matrix is refilled, the chromosomes are evaluated in the same way as
the random search. Once all the possible pixel combinations are evaluated, the population
matrix is passed back to the beginning of the GA and the process if repeated in a new
generation. In the case of this work, the GA is allowed to optimize over 3 generations,
corresponding to the simulation of 1200 different pixel configurations. At approximately 15
minutes per simulation, the optimization can take over a week to complete for a desired
frequency. However, the results are also displayed in real-time, so if a configuration that
meets the desired criteria is found, the GA can be stopped early. The entire code for the
matlab-HFSS GA can be found in appendix A.

2.3.3

Simulated Reflection Coefficients of Selected Antenna Configurations

The second layer tuning element is first tested over a variety of possible frequency ranges.
Figure 2.11 and 2.12 show some selected reflection coefficients in the 3-4GHz frequency
range as well as the 5-6GHz frequency range. These antennas show just some of the possible
configurations with reflection coefficients of less than -10dB. The gaps between reflection
coefficients in these regions are only a result of computational time, as antennas with well
defined resonances can also be found for those frequencies.
At the upper end of the 6GHz frequency range, the antenna is tuned in frequency by
nearly 240 %.Compared with the methods introduced in sections 1.2 and 1.3.1, this is an
extremely drastic tuning range. In addition, the magnitude of many of these antenna reflection coefficients are very large, and in some cases even better than the original antenna.
From these results, the potential for a dynamically tunable antenna is clear, however, the
conductivities of the pixels in this case are very high, correspond to copper. In the next section, similar antenan configurations will be investigated but with much lower conductivities
for the second layer pixel grid.

55

−5
−10
|S11| (dB)

−15
−20
−25
−30
−35
−40

3

3.2

3.4
3.6
Frequency (GHz)

3.8

4

Figure 2.11 Reflection coefficients of some sample antenna configurations within the 3-4 GHz
frequency range.

56

|S11| (dB)

−10

−20

−30

−40
5

5.2

5.4
5.6
Frequency (GHz)

5.8

6

Figure 2.12 Reflection coefficients of some sample antenna configurations within the 5-6 GHz
frequency range.

57

2.3.4

Investigation of Pixel Conductivities

While the previous analysis only takes into account highly conductive metals such as copper,
the top layer pixels of the antenna are envisioned to be made of materials that may have
a conductivity much less than typical metals. To investigate the effect of change in the
conductivity of the pixels on the top layer, an antenna with a resonance at 3.8 GHz is
chosen. The conductivity is varied from a value corresponding to a typical metal 5.9x107 S/m
(copper), to an order of magnitude less than most metals. As the conductivity is decreased
from the original conductivity the pixel states were optimized for, the reflection coefficient
of the antenna grows, until at one point the antenna is no longer highly resonant. This
trend is shown is shown for a variety of conductivities in Figure 2.13. While changing the
conductivity of an already optimized pixel state increases the magnitude of the reflection
coefficient of the antenna to a state where it is unusable, the conductivity can also be change
prior to the original optimization of the second layer to dissipate this effect.
Two frequencies corresponding to antennas with pixel combinations made from much
lower conductivities are optimized with the same GA parameters as previously shown. Figure
2.14 shows a pair of the best possible antennas found for each of the individual conductivities
within the 3-4GHz band. One antenna corresponds to a pixel with conductivity of 2x105 S/m,
while the other corresponds to 2x104 S/m. Clearly, antenna configurations can be found with
conductivities much below common metals that are still highly resonant.

58

0

|S11| (dB)

−5
−10
−15
−20
−25
3

1x106 S/m
1x107 S/m
2x107 S/m
3x107 S/m
5x107 S/m
5.9x107 S/m
3.2

3.4
3.6
Frequency (GHz)

3.8

4

Figure 2.13 Effect of changing the top layer pixel conductivity on the reflection coefficient.

59

0

|S11| (dB)

−10
−20
−30
2x105 S/m
2x104 S/m

−40
−50
3

3.2

3.4
3.6
Frequency (GHz)

3.8

4

Figure 2.14 Reflection coefficient of optimized antenna configurations corresponding to very
low top layer pixel conductivity.

60

2.4

Simultaneous Optimization of Radiation Efficiency
and Reflection Coefficient

While these antenna configurations correspond to very good reflection coefficients at a wide
range of possible frequencies, the radiation characteristics of the antenna are also important.
However, the antenna with the highest reflection coefficient will not necessarily correspond
to the best radiator. While some of these antennas could maintain an acceptable radiation
pattern and efficiency, others could simply be acting as strongly resonant structures with
little radiation. In order to properly ensure the device is acting as an antenna, the fitness
function of the GA is altered to also weigh the radiation efficiency of the antenna. This is
achieved by using the radiation efficiency(η) as a weighting function of the magnitude of the
reflection coefficient which was initially minimized, i.e.

f itness = η ∗ 20log10 (|S11 |)

(2.2)

In order to determine the antenna configuration that will correspond to the best fit for
our desired results, the radiation efficiency and reflection coefficient are monitored for every
optimized antenna state. From the trend between the two, a antenna configuration which
has a tradeoff between a high radiation efficiency and a good reflection coefficient can be
selected. In this case, any antenna with a reflection coefficient lower than -10dB can be
thought of as a desirable antenna, so the highest radiation efficiency for any state meeting
this reflection coefficient will be selected. Figures 2.15-2.18 show the radiation efficiency
versus reflection coefficient for four discrete frequency antenna designs. Each graph shows
the antenna configurations corresponding to the initial random search, as well as after the
third generation of the GA. It can be seen that the antennas found after three generations
have much better characteristics than the initial random search. This confirms that the GA
61

0
−5
|S11| (dB)

−10
−15
−20
−25
−30
0

Random Search
GA (3 Generations)
10

20

30
40
50
Radiation Efficiency (%)

60

70

80

Figure 2.15 Reflection coefficient versus radiation efficiency for the random search and after
3 generations of GA optimization for 2.5 GHz.

62

0

|S11| (dB)

−10

−20

−30
Random Search
GA (3 Generations)
−40
0

10

20

30
40
50
Radiation Efficiency (%)

60

70

80

Figure 2.16 Reflection coefficient versus radiation efficiency for the random search and after
3 generations of GA optimization for 4 GHz.

63

0

−10
−15

|S

11

| (dB)

−5

−20
−25
−30
0

Random Search
GA (3 Generations)
10

20

30
40
50
Radiation Efficiency (%)

60

70

80

Figure 2.17 Reflection coefficient versus radiation efficiency for the random search and after
3 generations of GA optimization for 6 GHz.

64

0

| (dB)

−5

|S

11

−10

−15
Random Search
GA (3 Generations)
−20
0

10

20

30
40
50
Radiation Efficiency (%)

60

70

80

Figure 2.18 Reflection coefficient versus radiation efficiency for the random search and after
3 generations of GA optimization for 8 GHz.

65

can create much better results than a simple random search. From these results the theoretical tuning range of the antenna can begin to be discussed. At frequencies lower or higher than
the tuning range of the antenna, some configurations may have a low reflection coefficient,
but the radiation efficiency of the antenna will be near zero. To illustrate this, radiation
efficiency and reflection coefficients are calculated over three GA generations for a very wide
frequency range. This range goes from 1.5GHz to 18 GHz. The antenna corresponding to
the best radiation efficiency and a reflection coefficient of -10dB at each frequency are then
recorded. From these results, it can be observed that at 1.5 GHz there is essentially no radiation. As the frequency increases towards the resonance of the bottom layer, the efficiency
increases and peaks locally at 2.5 GHz. From this point onward, the efficiency oscillates
but does not drive downward towards zero. These results are shown in Figure 2.19, where
the dotted line corresponds to the resonance of the bottom layer. The upper portion of the
tuning is not clearly found in these results, however, as the frequency is increased, the size
of the antenna structure becomes much larger with respect to the wavelength. While the
antenna will still radiate efficiently at these frequencies, the radiation pattern becomes much
more complex and the original pattern of the monopole can no longer be maintained.

66

Radiation Efficiency (%)

80

60

40

20

0
1.5

4

6

8
10
12
Frequency (GHz)

14

16

18

Figure 2.19 Best radiation efficiency corresponding to a reflection coefficient of at least -10dB
for a variety of frequencies.

67

CHAPTER 3
Fabrication and Experimental Results
In this chapter, the fabrication and measurement of the antenna with the second tuning
layer is presented. The second layer in this work is made of a metallic pixel grid in order
to test the concept of the layer as a tuning element. First, the layer is constructed using
thin PET films with the pixel grid etched on. The PET film is envisioned to be used
as a hot swappable loading element that can be placed directly on the bottom layer of
the antenna. The problem with this method is that the misalignment and airgap between
the layers has a largely negative impact on the performance of the second layer. To solve
these problems, a multi-layer fabrication process is developed which uses a simple photo
resist as a dielectric layer. In this case, the photo resist used is a 25µm SU-8 layer. The
SU-8 can be directly spun onto the bottom layer of the antenna, eliminating airgaps and
reducing the misalignment. With this method, multiple antennas are fabricated over a wide
frequency range. The reflection coefficient and radiation patterns of each of the antennas
are measured, and each antenna configuration shows relatively good agreement with their
simulated counterpart.

68

3.1

Fabrication of Second Layer Using PET Films

The first method implemented in the fabrication of the second layer is using PET thin films.
The configuration of the antenna with the PET film layer is shown in Figure 3.1. The PET
layer is envisioned to be placed directly on the bottom layer of the antenna. Each pixel
configuration found through simulation can then be fabricated directly onto the PET film,
and then placed directly on the antenna to act as the second layer. The PET film in this
case is 25µm thick, and a thin layer of aluminum is deposited on the film. The desired pixel
state is then etched onto the PET film, and the film is placed onto the bottom layer of the
antenna. A number of pixel configurations corresponding to different resonant frequencies
are fabricated using the PET layers. Figure 3.2 shows some a sample of some of the PET
thin film states.
With the second layer fabricated, each pixel configuration is placed directly on the bottom
layer to achieve frequency tuning. The antenna with and without the PET layer is shown in
Figure 3.3. It is easy to observe that the alignment between the layers in this case is difficult
to control. In addition, the layers do not adhere directly to the substrate, making airgap
between the layers a possible problem. Each antenna is fed with a thin-pin SMA connector
from Lighthouse technologies (part # LTI-SASF55ZGT-P2-4). These SMA connectors have
a very thin center pin, making them ideal for this application, as the connector needs to
be soldered directly on the monopole antenna. To characterize the antenna, the reflection
coefficient of the antenna is measured with a Agilent N5227A network analyzer.

69

Figure 3.1 Antenna with PET layer setup.

70

Figure 3.2 Fabricated PET layers.

71

Figure 3.3 Antenna with and without PET layers.

72

Figure 3.4 shows the reflection coefficient for a antenna configuration chosen for a resonance of 3.75GHz. From these results we can see that the measured and simulated reflection
coefficients have poor matching. The cause of this discrepancy is mostly attributed to the
possible airgap between the layers, and the misalignment in the lateral directions of the antenna. The next section examines the effects of adding an airgap and misalignment between
the top and bottom antenna layers.

3.1.1

Effects of Misalignment and Air Gap

First, the misalignment between the layers is studied. The top layer for a particular pixel
configuration that produces a antenna resonance at 1.8GHz is chosen. The layer is moved in
four directions along the lateral direction of the antenna. These correspond to shifts in the
x and y directions of 250m and 500m each. Figure 3.5 shows the effect of misalignment on
the reflection coefficient of the antenna. These misalignments correspond to both shifts in
the location of the resonant frequency and alter the magnitude of the reflection coefficient.
While there is not a direct relation between a shift in alignment and the effect it has on the
reflection coefficient, it is noticeable that shifts in the y direction seem to alter the location
of the resonance much more than in the x direction.
It is clear that the location of the resonance is shown to dramatically shift even with only
slight misalignment. Although the exact level of misalignment in the fabricated antennas was
not measured, levels similar to the above could easily be expected. While the misalignment
between layers greatly alters the reflection coefficient, the amount of disagreement between
the measured and simulated results presented in the previous section are not fully explained
by just the misalignment.

73

0

|S11| (dB)

−5
−10
−15
−20
−25
3

Simulated
Measured
3.5

4
Frequency (GHz)

4.5

Figure 3.4 Reflection coefficient of antenna with PET layer at 3.75GHz.

74

5

Figure 3.5 Effects of miss-alignment between layers on the reflection coefficient of the antenna.

75

The airgap between the layers can also be a major cause of disagreement in the measured
results. Since the PET layers do not physically adhere to the bottom layer, some air can
easily be trapped between the PET film and the antenna. To determine the effect of airgap
between the tuning layer of the antenna, the antenna is simulated with a 10µm airgap
between the PET film and the bottom layer of the antenna. Figure 3.6 shows the effect
of this airgap in the reflection coefficient of the antenna. Obviously, the effect here is very
dramatic and essentially destroys the performance of the antenna. However, as the exact
airgap between the layers is unknown, this would most likely be the most extreme possibility.
Both of these effects hamper the performance of the antenna greatly. While the attractiveness of easily hot swappable PET films could provide many advantages, a more precise
and permanent fabrication method must be used to ensure the antenna results match well
with the simulated expectations.

3.2

Fabrication of Second Layer Using SU-8 Photoresist

In this section, a fabrication process is developed that eliminates the problems found in the
attempt to use PET films for the second layer tuning element. In this case, a thin SU8 photoresist is used as the dielectric layer between the bottom layer of the antenna and
the tuning layer. This layer can be directly spin coated onto the antenna, eliminating any
airgaps between the layers. Additionally, the thickness of the layer can be controlled within
a few microns of the desired results. This method also treats the dielectric layer and second
metallic layer with a mask aligner which allows the second tuning layer to be precisely aligned
with the bottom layer of the antenna.

76

0
−5

|S11| (dB)

−10
−15
−20
−25
−30

Original Reasonance
With Air Gap

−35
3

3.2

3.4
3.6
Frequency (GHz)

3.8

Figure 3.6 Effects of air-gap between layers on the reflection coefficient of the antenna.

77

4

3.2.1

Fabrication Process

To properly realize the antenna, a precise multi-layer fabrication process must be utilized.
Slight miss-alignment between the top tuning layer and the bottom antenna can have dramatic effects due to the high sensitivity of the antenna. Additionally, small air gaps between
the top layer and the antenna can negatively effect the antenna resonance. To ensure both
of these problems are minimized as much as possible, the dielectric layer which hosts the
tuning pixels is developed directly onto the antenna. In this case, a 25µm layer of SU-8
photo resist is used as the dielectric layer. The SU-8 layer has the properties of a relatively
low loss material within the lower GHz spectrum, with r = 3.28 and tanδ = .015 according
to the Microchem data sheet.
The overall fabrication process is summarized in Figure 3.7. First, the bottom layer of
the antenna is chemically etched onto a Rogers RO4003 substrate using a sodium persulfate
etchant. After the patterning of the bottom layer is finished, the SU-8 is spun onto the
antenna. In this case, the SU-8 used is developed by Microchem, and the product number is
SU-8 25. For a 25µm thick layer, the SU-8 is spun onto the bottom layer with a Laurel 6”
Photoresist Spinner at 2000 rpm. Before and after development, the SU-8 also requires a prebake and post-bake of 65 and 95◦ C, respectively. The SU-8 is aligned with the bottom layer
using a Karl Suss MJB3 contact mask aligner. Alignment marks on the bottom layer made
of copper are aligned with small SU-8 squares in the photomask to ensure proper alignment
between the SU-8 and the bottom layer. A thin layer of titanium is then deposited on the
SU-8 layer using a sputtering system to act as an adhesive layer for the copper. Next, a layer
of copper (≈ 1.5µm) is sputtered onto the titanium layer using the same system. The copper
is then patterned to the desired pixel configuration using the same wet-etching technique as
the bottom antenna layer. Finally, the sample is then quickly dipped in hydrofluoric acid to
remove the excess titanium.

78

Figure 3.7 Overview of fabrication process.

79

3.2.2

Fabricated Antenna Samples

Using the previously described fabrication method, four antennas were fabricated and measured. The four antennas correspond to resonant frequencies of 2.5GHz, 4GHz, 5GHz, and
8GHz. A close up view of one of the fabricated antenna configurations is shown in Figure
3.8. In this case, the SU-8 layer is the semi-transparent layer shown between each metal.
The metal on the top of the antenna is represented in black, while the bottom layer is the
traditional copper color. Overall, the dimensions of the second layer and the antenna match
well with the desired specifications. In addition, to the naked eye the layer appear to be
very well aligned. Figure 3.9 shows the alignment marks between the SU-8 and the bottom
copper layer. The SU-8 is represented by the semi-transparent layer which has a box shape
that is placed around the metallic cross alignment mark for the bottom layer. From this
figure, it is clear that the alignment between the layers is very good as the alignment marks
match well.
However, the SU-8 is very difficult to spin evenly over the bottom layer of the antenna.
Figure 3.10 shows the SU-8 one the antenna under high levels of magnification. Due to the
uneven spread of the SU-8, ripples become prevalent along the surface of the antenna. These
ripples essentially represent a difference in the thickness of the SU-8 at certain locations. In
addition, along the edges of some spots of the second layer, excess copper can develop as the
photoresist may have not fully developed. Figure 3.11 shows this excess copper for one pixel
configuration along the edge of the SU-8 layer. This copper can be scraped off, but either
way some will remain and may have a small effect on the antenna performance.
Each antenna is fed with a small SMC connector that is machined so that only the pin is
exposed to prevent any further interference with the antenna. The ground connections are
made with small wires from the sides of the SMC connector to the CPW feed. A picture
of one of the final fabricated antennas with the SMC connector attached is shown in Figure
3.12.

80

Figure 3.8 Close up of a fabricated antenna configuration.

81

Figure 3.9 Alignment marks between SU-8 and bottom layer.

82

Figure 3.10 Non-uniformity of SU-8 layer.

83

Figure 3.11 Excess copper on the second layer edges.

84

Figure 3.12 Example of one fabricated antenna configuration with a SMC connector attached.
The simulated model of all pixel configurations are also shown for comparison.

85

3.3

Measurement of Antenna Reflection Coefficients

The reflection coefficients of the antennas are measured using a Agilent N5227A network
analyzer. The reflection coefficients of antenna configurations corresponding to a resonance
at 2.5, 4, 5, and 8GHz are measured. The measured reflection coefficient of each antenna
as well as the simulated result are shown in Figures 3.13-3.16. The results show relatively
good agreement with the simulations. However, the locations of the resonances are slightly
shifted.
While a number of factors can contribute to this disagreement, the most likely cause
is misalignment between the two antenna layers. Referencing Figure 3.5, resonance shifts
similar to the ones displayed when misalignment is introduced are similar to the difference to
the measured and simulated results shown here. Although the exact degree of misalignment
in the fabricated antennas is unknown, levels similar to the above could easily be expected.
Overall, the antenna resonance is within a reasonable range of the simulated values, especially
considering the sensitivity of the antenna to misalignment. Figure 3.17 shows all four antenna
configurations on the same frequency axis. From this figure it is clear that the tuning range
of the antenna with the second layer is extremely dramatic, with a tuning range from 2.5GHz
to 8GHz, or a tuning range of 320%.

86

0

|S11| (dB)

−5
−10
−15
−20
−25
2

Measured
Simulated
2.2

2.4

2.6

2.8
3
3.2
Frequency (GHz)

3.4

3.6

3.8

Figure 3.13 Measured and simulated reflection coefficients of one antenna configuration at
2.5GHz.

87

0
−5

|S11| (dB)

−10
−15
−20
−25
−30
−35
2.8

3

3.2

3.4

3.6
3.8
Frequency (GHz)

4

Measured
Simulated
4.2
4.4

Figure 3.14 Measured and simulated reflection coefficients of one antenna configuration at
at 4GHz.

88

0

| (dB)

−5

11

−10

|S

−15
−20
−25
4

Measured
Simulated
4.5

5
Frequency (GHz)

5.5

6

Figure 3.15 Measured and simulated reflection coefficients of one antenna configuration at
at 5GHz.

89

0
−5
|S11| (dB)

−10
−15
−20
Measured
Simulated

−25
−30

6.5

7

7.5
8
Frequency (GHz)

8.5

9

Figure 3.16 Measured and simulated reflection coefficients of one antenna configuration at
at 8GHz.

90

0

−10
|S11| (dB)

Measured 2.5 GHz
Simulated 2.5 GHz
Measured 3.5 GHz
Simulated 3.5 GHz
Measured 5 GHz
Simulated 5 GHz
Measured 8 GHz
Simulated 8 GHz

−20

−30

−40
2

3

4

5
6
Frequency (GHz)

7

8

9

Figure 3.17 Measured and simulated reflection coefficients of all four antenna configurations.

91

3.4

Measurement of Antenna Radiation Parameters

The radiation characteristics of the antenna are also measured for three selected antenna
configurations. The 2.5, 4, and 8GHz cases are measured using a SATIMO near-field measurement system. The radiation patterns are captured for both the azimuth (XY) and elevation (YZ) planes of the antenna. The antenna measurment setup is shown in Figure 3.18.
The measured and simulated radiation patterns in both planes for the 2.5 GHz antenna are
shown in Figures 3.19 and 3.20. The measured radiation patterns in both the azimuth and
elevation planes match relatively well with the simulated results. Yet, in the elevation plane
there is some disagreement in the back lobe of the pattern. Both patterns do however match
well in terms of their maximum gain (directivity). In addition, the expected characteristics
of the monopole are maintained, i.e. omnidirectional in the azimuth and bidirectional in the
elevation plane.
Similar results are displayed for the patterns for the 4GHz plane, shown in Figures 3.21
and 3.22. However, in the azimuth plane, the pattern is further distorted than in the 2.5 GHz
antenna, as there is an additional null in the pattern. Unfortunately, the measured results
for the pattern of the 8GHz antenna do not match as well as the previous antennas. These
results are shown in Figures 3.23 and 3.24. In the azimuth plane, many lobes are displayed
in the measured results, as opposed to the omnidirectional pattern that is expected. The
elevation plane matches slightly better than the azimuth, but neither show particularly good
agreement with the simulated results.
These discrepancies are primary thought to be caused by the antenna becoming unbalanced at such a high frequency. In addition, at 8GHz the antenna is now much greater
in wavelength than originally intended as the size is unchanged from the 2.5 GHz antenna
configuration. While the radiation of the characteristics of the antenna do not match as well
for the higher frequencies, most of the patterns match very well with the simulated results.

92

Figure 3.18 Antenna radiation pattern measurement setup.

93

Figure 3.19 Measured and simulated radiation patterns in XY plane at 2.5 GHz.

94

Figure 3.20 Measured and simulated radiation patterns in YZ plane at 2.5 GHz.

95

Figure 3.21 Measured and simulated radiation patterns in XY plane at 4 GHz.

96

Figure 3.22 Measured and simulated radiation patterns in YZ plane at 4 GHz.

97

Figure 3.23 Measured and simulated radiation patterns in XY plane at 8 GHz.

98

Figure 3.24 Measured and simulated radiation patterns in YZ plane at 8 GHz.

99

CHAPTER 4
Conclusions and Future Studies
A multi-layered metamaterial inspired antenna capable of dynamic tunability through a
photoconductive pixel grid is introduced in this thesis. A GA optimization technique is used
to tune the antenna with a second pixel grid that is placed directly on the miniaturized
antenna layer. The GA design process is exlpained at length, and the simulated results
show that many pixel combinations can be found which produce a large range of tuning.
In addition, the radiation patterns of the antenna are shown to not be greatly affected by
the second pixel tuning layer, and the radiation efficiency is optimized to produce the most
efficient antenna possible. Further, a variety of conductivities are investigated to ensure the
tuning layer will be useable with non-traditional conductors. Antenna states are fabricated
with a metallic pixel grid to test the tuning ability of the antenna. Initially, PET thin
films are investigated to be used as hot swappable loading elements, but the airgap and
misalignment between layers are shown to be too large for this method to produce accurate
results. A multi-layer fabrication method is then investigated which uses an SU-8 layer
that can be precisely fabricated to eliminate the airgap and minimize the misalignment
between layers. A large number of states are then fabricated using this method, and states
corresponding to resonant frequencies of 2.5, 4, 5, and 8GHz are presented. The reflection
coefficient for each antenna is measured and are shown to match extremely well with the

100

simulated results. Finally, the radiation patterns of each antenna were measured, which in the
lower frequency spectrum show a good matching with the simulated results. These antennas
show that a very large range of tuning can be achieved with this method. In comparison with
the traditional tuning methods described in chapter one, this method provides at least 10
times the tuning capability achieved over the conventional approaches. If only the measured
results are used as a measure of the possible tuning range of the antenna, a tuning range of
approximately 320% is presented in this work. However, higher frequency antennas could
easily be fabricated and this range could be extended much farther. Although in this case
a metallic pixel grid is explored through fabrication, our expectations are that this second
metal layer can be replaced by a photoconductor or phase change material to achieve real
time tuning capability.

101

APPENDIX

102

Appendix: Genetic Algorithm
Optimization Code
This particular GA is meant for use at a resonant frequency of 4 GHz. However, this frequency can easily be changed. To implement this code successfully, all of the local directories
and paths need to be changed to fit the users desired setup.

1
2

clc

3

clear all;

4

format long

5

%START GA

6

%

7

maxit=3;

8

mincost=−10000;

9

popsize=300;

III. GA parameters
% max number of iterations
% minimum cost
% set population size

10

Muprob=.2;

% set mutation rate

11

selection=0.1;

12

m=20;

%Number of bits per column

13

n=10 ;

%Number of bits per row

14

Nt=m*n;

15

ChromLength=Nt;

16

keep=floor(selection*popsize);

17

iga=1;

% fraction of population kept

% number of bits in each chromosome

% #population members that survive

%INITIALIZE GENERATION COUNTER

103

18

bestfit=10000;

19

%

20

%

Create the initial population for each individual

21
22
23

% Make folders for each generation and the random search hfss simulations

24

mkdir('Gen1')

25

mkdir('Gen2')

26

mkdir('Gen3')

27

mkdir('RandomSearch')

28
29
30

% add paths to the required m−files.

31

addpath('C:\AFRL MAT\Efficiency4GHz\Files\');

32

FarFieldpath='C:\AFRL MAT\Efficiency4GHz\farFieldEff.m' ;

33

% Frequency sweep parameters

34

eps o=8.8541878176e−12;

35

mu o= 4*pi*1e−7;

36

Co=1/sqrt(eps o * mu o); %Speed of light in free space

37

fLow =6e9;

38

fHigh =10e9;

39

nPoints = 1;

40

np = 1;

41

fC =4e9;

42

Wv = (Co/fC)*1000;

%Epsilon
%Mu

%Simulation start point
%Simulation end point
%frequency range discretization

% Frequency of Interest.
% Wavelength.

43
44

Pw=.5;

% Pixel Width

45

XL=m;

46

ZL=n; %Nuber of pixels in Y direction

47

SRR LX=(Pw*XL); %Unit cells length X−direction

48

SRR LZ=(Pw*ZL);

% Nuber of pixels in X direction

%Unit cell height Y direction

49
50

%Ground Plane

104

51

Gnd X=Wv/2;

52

Gnd Y=Wv/2;

53

Gnd Z=3*Pw+3*Pw;

54
55

X sp=Pw/3;

56

Z sp=X sp/2;

57
58

%Antenna Parameter

59

PwL=Pw;

60

LoopW=5*PwL;

61

P X=1*PwL;

62

P Z=3.5*PwL;

63

P w=PwL*1.4;

64

P sx=P X−P w;

65

P sz=P Z−0.5* P w;

66

sub P=P Z/3−PwL;

67

Gndgap=P w+.3;

68
69

%Substrate Dimensions

70

Sub X= SRR LX+2*Pw;

71

Sub Y=−.5 ;%Substrate width

72

Sub2 Y=−.5;

73

Sub Z= SRR LZ;%SRR LZ %54.6100;%Substrate height

74

Cir Sub Y=−.3;

75

%Feed parameters

76

R oc=2.6/2;

77

Th oc=R oc−.1;

78

Hi oc=30;

79

R feed=.635/2;

80

Hi feed=Hi oc;

81

R tf=Th oc; %R oc−.1;

82

Th tf=R feed;

83

Feed=(−P X+P w/2)/2;

%Substrate length

105

84

Deembed=Hi feed;

85

FX=−2.5*Pw−.2;

86

CapZ=1;

87

% Airbox Dimensions

88

G R =Wv/3+SRR LX *2 ;

89

G H = Wv/2+SRR LX+Hi oc;

%Waveguide length
%Waveguide height

90
91

Cop Th=.017;

92
93

% HFSS Executable Path.

94

hfssExePath = 'C:\"Program Files (x86)"\Ansoft\HFSS13.0\hfss.exe';

95
96
97

for ip=1:popsize;
tic

98
99
100

for jp = 1:ChromLength
if rand() > 0.5

101

Pop(ip,jp) = 1;

102

else

103

Pop(ip,jp) =0;

104

end

105
106

end

107
108
109
110
111

Meta=(reshape(Pop(ip,:),n,m));
tt=0;
for ix=1:6

112

for iy=1:5

113

Meta(n−iy+tt,ix)=0;

114

end

115

tt=tt+1;

116

end

106

117
118

tt=0;

119

for ix=1:6

120

for iy=1:5

121

Meta(n−iy+tt,m−ix+1)=0;

122

end

123

tt=tt+1;

124
125

end
% Temp Files.

126
127
128

tmpPrjFile = 'C:\AFRL MAT\Efficiency4GHz\GALoop.hfss';

129

tmpDataFile = 'C:\AFRL MAT\Efficiency4GHz\GALoopData.m';

130

tmpScriptFile = 'C:\AFRL MAT\Efficiency4GHz\GALoopScript.vbs';

131
132
133

% Create a new temporary HFSS script file.

134

fid = fopen(tmpScriptFile, 'wt');

135
136

% Create a new HFSS Project and insert a new design.

137

hfssNewProject(fid);

138

hfssInsertDesign(fid, 'GA Loop');

139
140
141
142

%Draw AirBox
hfssCylinder(fid, 'Air Box', 'Z', [0, 0, −(Hi oc+Gnd Z+CapZ+10)], G R, ...
G H, 'mm');

143

hfssAssignRadiation(fid, 'Box Rad', 'Air Box');

144

hfssSetTransparency(fid, {'Air Box'}, 0.95);

145

%Draw Feed (Outer Conductor)

146

hfssHollowCylinder(fid, 'Outer C', 'Z', [FX, 0, −(Hi oc+Gnd Z)], ...
Th oc, R oc, (Hi oc), 'mm');

147

hfssAssignPE(fid, 'PEC oc',

{'Outer C'});

107

148

%

Teflon
hfssHollowCylinder(fid, 'Tef', 'Z', [FX, 0, −(Hi oc+Gnd Z)], Th tf, ...

149

R tf, (Hi oc), 'mm');
hfssAssignMaterial(fid, 'Tef', 'Neltec NX9294 (tm)');

150
151
152

%

Inner Conductor
hfssCylinder(fid, 'Inner C', 'Z', [FX, 0, −(Hi oc+Gnd Z)], R feed, ...
(Hi oc), 'mm');

153
154
155

hfssAssignPE(fid, 'PEC Ic',

{'Inner C'});

%Draw waveport
hfssCircularPort(fid, 'LPort', 'port1', 'Z', [FX, 0, ...
−(Hi oc+Gnd Z)], R tf,Deembed, 'mm');

156

hfssCylinder(fid, 'PortCup1', 'Z', [FX, 0, −(Hi oc+Gnd Z)], R oc, ...
−CapZ, 'mm');

157

hfssAssignMaterial(fid, 'PortCup1', 'pec');

158
159
160

%Draw Ground plane

161

%Ground 1

162

hfssRectangle(fid, 'Gnd', 'Y', [−Sub X/2+Pw, 0, ...
−Gnd Z],Gnd Z,Sub X−2*Pw, 'mm');

163

hfssRectangle(fid, 'Gnd Sub', 'Y', [−Gndgap/2+FX, 0, −Gnd Z], Gnd Z, ...
Gndgap, 'mm');

164

hfssSubtract(fid, {'Gnd'}, {'Gnd Sub'});

165

hfssAssignFiniteCond(fid, 'PEC Gnd', 0, 'um', {'Gnd'});

166
167
168

%Circuitry

169

hfssRectangle(fid, 'Circuit1', 'Y', [−Sub X/2, −Sub Y, ...
−Gnd Z+1.62],Gnd Z−1.62,Sub X, 'mm');

170

hfssCircle(fid, 'Circuit2', 'Y', [0, −Sub Y, 0],Sub X/2, 'mm');

171

hfssRectangle(fid, 'Circuit2Sub', 'Y', [−Sub X/2, −Sub Y, ...
0],−Sub X/2,Sub X, 'mm');

172

hfssSubtract(fid, {'Circuit2'}, {'Circuit2Sub'});

108

173

hfssUnite(fid, 'Circuit1','Circuit2');

174

hfssAssignFiniteCond(fid, 'PEC Circuit1', 0, 'um', {'Circuit1'});

175
176
177

%Draw Substrate

178

hfssBox(fid, 'Substrate', [−Sub X/2, 0, −Gnd Z], [Sub X, −Sub Y, ...
Gnd Z], 'mm');

179

hfssAssignMaterial(fid, 'Substrate', 'Rogers RO4003 (tm)');

180

hfssSetColor(fid, 'Substrate', [0, 128, 0]);

181

hfssSetTransparency(fid, {'Substrate'}, 0.8);

182

hfssCylinder(fid, 'Sub2', 'Y', [0, 0, 0], Sub X/2, −Sub Y, 'mm');

183

hfssBox(fid, 'Sub2sub', [−Sub X/2, 0, −Sub X/2], [Sub X, −Sub Y, ...
Sub X/2], 'mm');

184

hfssSubtract(fid, {'Sub2'}, {'Sub2sub'});

185

hfssUnite(fid, 'Substrate','Sub2');

186
187

%Draw Extra Layer Substrate

188

hfssBox(fid, 'Layer', [−Sub X/2, 0, −Gnd Z], [Sub X, .025, Gnd Z], 'mm');

189

hfssAssignMaterial(fid, 'Layer', 'GIL MC5 (tm)');

190

hfssSetColor(fid, 'Layer', [0, 900, 0]);

191

hfssSetTransparency(fid, {'Layer'}, 0.8);

192

hfssCylinder(fid, 'Sub0', 'Y', [0, 0, 0], Sub X/2, .025, 'mm');

193

hfssBox(fid, 'Sub2sub0', [−Sub X/2, 0, −Sub X/2], [Sub X, .025, ...
Sub X/2], 'mm');

194

hfssSubtract(fid, {'Sub0'}, {'Sub2sub0'});

195

hfssUnite(fid, 'Layer','Sub0');

196

hfssMove(fid, {'Layer'}, [0, −.025, 0], 'mm');

197
198

%Draw Folded Monopole

199

hfssRectangle(fid, 'mono', 'Y', [−P w/2+FX, 0, −Gnd Z], Gnd Z, P w,'mm');

200

hfssRectangle(fid, 'Loop', 'Y', [−P w/2+FX, 0, 0], P Z, LoopW,'mm');

201

hfssRectangle(fid, 'Loopsub', 'Y', [FX+P w/2, 0, 0], P Z−P w, ...
LoopW−2*P w,'mm');

109

202

hfssRectangle(fid, 'Loopsub2', 'Y', [FX+LoopW−3*P w/2, 0, 0], P w *1, ...
P w,'mm');

203

hfssAssignFiniteCond(fid, 'PEC Loop', 0, 'um', {'mono'});

204

hfssSubtract(fid, {'Loop'}, {'Loopsub'});

205
206

hfssSubtract(fid, {'Loop'}, {'Loopsub2'});
hfssUnite(fid, 'mono','Loop');

207
208

% Draw Bottom Substrate pads

209

hfssRectangle(fid, 'Pad1', 'Y' , [−2.5,0,0], .5, .5,'mm');

210

hfssAssignFiniteCond(fid, 'Pad1', 0, 'um', {'Pad1'})

211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229

hfssRectangle(fid, 'Pad2', 'Y' , [−5,0,.5], 1.5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad2', 0, 'um', {'Pad2'})
hfssRectangle(fid, 'Pad3', 'Y' , [−4.5,0,.5], 1, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad3', 0, 'um', {'Pad3'})
hfssRectangle(fid, 'Pad4', 'Y' , [−4,0,1.5], 1.5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad4', 0, 'um', {'Pad4'})
hfssRectangle(fid, 'Pad5', 'Y' , [−3.5,0,1], 1.5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad5', 0, 'um', {'Pad5'})
hfssRectangle(fid, 'Pad6', 'Y' , [−3,0,1.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad6', 0, 'um', {'Pad6'})
hfssRectangle(fid, 'Pad7', 'Y' , [−2.5,0,1], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad7', 0, 'um', {'Pad7'})
hfssRectangle(fid, 'Pad8', 'Y' , [−2.5,0,2], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad8', 0, 'um', {'Pad8'})
hfssRectangle(fid, 'Pad9', 'Y' , [−3.5,0,3], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad9', 0, 'um', {'Pad9'})
hfssRectangle(fid, 'Pad10', 'Y' , [−3,0,3], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad10', 0, 'um', {'Pad10'})
hfssRectangle(fid, 'Pad11', 'Y' , [−2.5,0,3], 1.5, .5,'mm');

230

hfssAssignFiniteCond(fid, 'Pad11', 0, 'um', {'Pad11'})

231

hfssRectangle(fid, 'Pad12', 'Y' , [−2,0,2], 2, .5,'mm');

232

hfssAssignFiniteCond(fid, 'Pad12', 0, 'um', {'Pad12'})

233

hfssRectangle(fid, 'Pad13', 'Y' , [−2,0,4.5], .5, .5,'mm');

110

234
235

hfssAssignFiniteCond(fid, 'Pad13', 0, 'um', {'Pad13'})
hfssRectangle(fid, 'Pad14', 'Y' , [−1.5,0,2], 1, .5,'mm');

236

hfssAssignFiniteCond(fid, 'Pad14', 0, 'um', {'Pad14'});

237

hfssRectangle(fid, 'Pad15', 'Y' , [−1.5,0,3.5], .5, .5,'mm');

238

hfssAssignFiniteCond(fid, 'Pad15', 0, 'um', {'Pad15'});

239

hfssRectangle(fid, 'Pad16', 'Y' , [−1.5,0,4.5], .5, .5,'mm');

240

hfssAssignFiniteCond(fid, 'Pad16', 0, 'um', {'Pad16'});

241

hfssRectangle(fid, 'Pad17', 'Y' , [−1,0,2], 1.5, .5,'mm');

242

hfssAssignFiniteCond(fid, 'Pad17', 0, 'um', {'Pad17'});

243

hfssRectangle(fid, 'Pad18', 'Y' , [−1,0,4.5], .5, .5,'mm');

244

hfssAssignFiniteCond(fid, 'Pad18', 0, 'um', {'Pad18'});

245

hfssRectangle(fid, 'Pad19', 'Y' , [−.5,0,3.5], .5, .5,'mm');

246

hfssAssignFiniteCond(fid, 'Pad19', 0, 'um', {'Pad19'});

247

hfssRectangle(fid, 'Pad20', 'Y' , [−.5,0,4.5], .5, .5,'mm');

248

hfssAssignFiniteCond(fid, 'Pad20', 0, 'um', {'Pad20'});

249

hfssRectangle(fid, 'Pad21', 'Y' , [0,0,2.5], .5, .5,'mm');

250

hfssAssignFiniteCond(fid, 'Pad21', 0, 'um', {'Pad21'});

251

hfssRectangle(fid, 'Pad22', 'Y' , [0,0,4], .5, .5,'mm');

252

hfssAssignFiniteCond(fid, 'Pad22', 0, 'um', {'Pad22'});

253

hfssRectangle(fid, 'Pad23', 'Y' , [.5,0,3], .5, .5,'mm');

254

hfssAssignFiniteCond(fid, 'Pad23', 0, 'um', {'Pad23'});

255

hfssRectangle(fid, 'Pad24', 'Y' , [.5,0,4.5], .5, .5,'mm');

256

hfssAssignFiniteCond(fid, 'Pad24', 0, 'um', {'Pad24'});

257

hfssRectangle(fid, 'Pad25', 'Y' , [1,0,3], .5, .5,'mm');

258

hfssAssignFiniteCond(fid, 'Pad25', 0, 'um', {'Pad25'});

259

hfssRectangle(fid, 'Pad26', 'Y' , [1,0,4], .5, .5,'mm');

260

hfssAssignFiniteCond(fid, 'Pad26', 0, 'um', {'Pad26'});

261

hfssRectangle(fid, 'Pad27', 'Y' , [1.5,0,3], .5, .5,'mm');

262

hfssAssignFiniteCond(fid, 'Pad27', 0, 'um', {'Pad27'});

263

hfssRectangle(fid, 'Pad28', 'Y' , [1.5,0,4], 1, .5,'mm');

264

hfssAssignFiniteCond(fid, 'Pad28', 0, 'um', {'Pad28'});

265

hfssRectangle(fid, 'Pad29', 'Y' , [2,0,3.5], 1, .5,'mm');

266

hfssAssignFiniteCond(fid, 'Pad29', 0, 'um', {'Pad29'});

111

267
268
269
270
271

hfssRectangle(fid, 'Pad30', 'Y' , [1,0,.5], 1.5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad30', 0, 'um', {'Pad30'});
hfssRectangle(fid, 'Pad31', 'Y' , [1.5,0,1], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad31', 0, 'um', {'Pad31'});
hfssRectangle(fid, 'Pad32', 'Y' , [1.5,0,2], .5, .5,'mm');

272

hfssAssignFiniteCond(fid, 'Pad32', 0, 'um', {'Pad32'});

273

hfssRectangle(fid, 'Pad33', 'Y' , [2,0,.5], 1, .5,'mm');

274

hfssAssignFiniteCond(fid, 'Pad33', 0, 'um', {'Pad33'});

275
276
277
278
279

hfssRectangle(fid, 'Pad34', 'Y' , [2,0,2.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad34', 0, 'um', {'Pad34'});
hfssRectangle(fid, 'Pad35', 'Y' , [2.5,0,1], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad35', 0, 'um', {'Pad35'});
hfssRectangle(fid, 'Pad36', 'Y' , [2.5,0,2], 1.5, .5,'mm');

280

hfssAssignFiniteCond(fid, 'Pad36', 0, 'um', {'Pad36'});

281

hfssRectangle(fid, 'Pad37', 'Y' , [3,0,2], 1.5, .5,'mm');

282

hfssAssignFiniteCond(fid, 'Pad37', 0, 'um', {'Pad37'});

283
284
285
286
287
288
289

hfssRectangle(fid, 'Pad38', 'Y' , [3.5,0,3], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad38', 0, 'um', {'Pad38'});
hfssRectangle(fid, 'Pad39', 'Y' , [3.5,0,1.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad39', 0, 'um', {'Pad39'});
hfssRectangle(fid, 'Pad40', 'Y' , [3.5,0,2.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad40', 0, 'um', {'Pad40'});
hfssRectangle(fid, 'Pad41', 'Y' , [4,0,.5], 1,.5,'mm');

290

hfssAssignFiniteCond(fid, 'Pad41', 0, 'um', {'Pad41'});

291

hfssRectangle(fid, 'Pad42', 'Y' , [4,0,2], .5, .5,'mm');

292

hfssAssignFiniteCond(fid, 'Pad42', 0, 'um', {'Pad42'});

293
294
295
296

hfssRectangle(fid, 'Pad43', 'Y' , [4.5,0,0], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad43', 0, 'um', {'Pad43'});
hfssRectangle(fid, 'Pad44', 'Y' , [4.5,0,1], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad44', 0, 'um', {'Pad44'});

297
298
299

N=1;
for i=1:m

112

300

NN=0;

301

for j=1:n
if Meta(j,i)==1;

302

NN=1+ NN;

303
304

end

305

if NN>0
if Meta(j,i)==0

306
307

N=N+1;

308

xp=(i)*Pw;

309

yp=(j)*Pw;

310

hfssRectangle(fid, sprintf('RecB%d',N), 'Y' , ...
[−xp+SRR LX/2, −.025, yp−(1+NN)*Pw], Pw*NN, Pw,'mm');
hfssAssignFiniteCond(fid, sprintf('RecBPEC%d',N), 0, ...

311

'um', {sprintf('RecB%d',N)})
NN=0;

312

end

313

if j==n

314

if Meta(j,i)==1;

315
316

N=N+1;

317

xp=(i)*Pw;

318

yp=(j)*Pw;

319

hfssRectangle(fid, sprintf('RecB%d',N), 'Y', ...
[−xp+SRR LX/2, −.025, yp−NN*Pw], Pw*NN, Pw,'mm');
hfssAssignFiniteCond(fid, sprintf('RecBPEC%d',N), 0, ...

320

'um', {sprintf('RecB%d',N)})
321

NN=0;

322

end
end

323

end

324

end

325
326

end

327

% Add a Solution Setup.

328

hfssInsertSolution(fid, 'Setup', fC/1e9, 0.02,20);

113

329

% Save the project to a temporary file and solve it.

330

hfssSaveProject(fid, tmpPrjFile, true);

331

hfssSolveSetup(fid, 'Setup');

332

hfssInsertFarField(fid, 'Infinite Sphere1',0,0,0,0,0,0,'deg')

333

% Export the Network data as an m−file.

334

hfssExportNetworkData(fid, tmpDataFile, 'Setup', 'LastAdaptive');

335

FFc=fC*(1e−9);

336

hfssCreateReportDualComp(fid,FarFieldpath, FFc, 'GHz');

337

hfssCreateReport(fid, FFc, 'GHz');

338
339

% Close the HFSS Script File.

340

fclose(fid);

341

% Execute the Script by starting HFSS.

342

disp('Solving using HFSS ..');

343

hfssExecuteScript(hfssExePath, tmpScriptFile);

344
345

% Load the data by running the exported matlab file.

346
347

run(tmpDataFile);

348
349

[heffi2, RadEff] = hdrload('farFieldEff.m');

350

radF=RadEff(:,1);

351

RE=RadEff(:,2);

352
353
354

for np=1:nPoints

355

S1r(np)=20*log10(abs(S(np,:,:)));

356

end

357
358
359
360
361

fitness(ip)= RE*S1r;

114

362

Sfc(ip) = S1r;

363

Rfc(ip) = RE;

364

% min(S1r) and rad efficiency

365
366
367
368

Name = ...
sprintf('C:\\AFRL MAT\\Efficiency4GHz\\RandomSearch\\Name%d.hfss',ip);
movefile('GALoop.hfss',Name)

369
370
371

if ip == popsize

372

figure

373

scatter(Rfc,Sfc)

374

axis([ 0 1 −40 0])

375

set(gca,'FontSize',17);

376

xlabel('Efficiency')

377

ylabel(' | S11 | (dB)')

378

title('Random Search 4GHZ')

379

savefile = 'Sfc.mat';

380

save(savefile, 'Sfc');

381

savefile = 'Rfc.mat';

382

save(savefile, 'Rfc');

383

end

384
385
386
387

end

388
389
390
391

[fitness,ind]=sort(fitness) ;

%Sort fitness such that the lowest is ...

at ind 1
392

Pop=Pop(ind,:);

% sorts population with lowest fitness placed at ind 1

115

393

Sfc = Sfc(:,ind);

394

Rfc = Rfc(:,ind);

395

minc(1)=min(fitness);% find min of the population

396

max(1)=max(fitness);

397

meanc(1)=mean(fitness); %find mean of the population

% find max of the population

398
399

Nt=ChromLength;

400

h = 1;

401
402

% %Start Iterating through generations
while iga≤maxit

403
404

tic

405

NewPop=Pop;

406

Muprob=.2;

407

%store the previous population for reference

selection=0.2;

% set mutation rate
% fraction of population kept

408
409

%

%I) Selection of chromozomes to be kept

410

indx=find(fitness≤mean(fitness));% Determines the chroms with fitness < ...
mean fitness

411

keep=ceil(selection*popsize); %Number of chrom to keep

412

fitkeep=fitness(indx);% fitnesses to keep

413

Popkeep=Pop(indx,:);%Chroms to keep

414

M1=ceil((popsize−keep)*.6); % Chromosomes to be replaced via Xover and ...
Mutation for case 1

415

M2=ceil((popsize−keep)*.4); % Chromosomes to be replaced via Xover and ...
Mutation for case 2

416
417

%II) Pairing Chromozomes for Xover/Mutation

418

%Case I: random selection of 2 individual

419
420
421
422

Ntourn=2;
for ic2=1:2:M2
%Random Selection of Parents

116

423

%first Parent

424

ma=ceil(keep*rand());

425

%Second parent

426

pa=ceil(keep*rand());

% indicies of first parent

% indicies of second parent

427
428

if ic2≤ceil(M2/3)

429

%1pt Crossover

430

ma=1;

431

Xpt=ceil(rand()*(Nt−1));

432

Pop(keep+M1+ic2−1,:)=[Pop(ma,1:Xpt) Pop(pa,Xpt+1:Nt)]; % Offspring 1

433

Pop(keep+M1+ic2,:)=[Pop(pa,1:Xpt) Pop(ma,Xpt+1:Nt)];%Offspring 2

434

end

435

if ic2≥ceil(M2/3) && ic2≤ceil(2*M2/3)

436

%2pt Crossover

437

Xpt1=ceil(rand()*Nt/2);

438

Xpt2=Xpt1 + ceil(rand()*(Nt−1)/2);

439

Pop(keep+M1+ic2−1,:)=[Pop(ma,1:Xpt1) Pop(pa,Xpt1+1:Xpt2) ...
Pop(ma,Xpt2+1:Nt)]; % Offspring 1

440

Pop(keep+M1+ic2,:)=[Pop(pa,1:Xpt1) Pop(ma,Xpt1+1:Xpt2) ...
Pop(pa,Xpt2+1:Nt)];%Offspring 2

441

end

442

if ic2≥ceil(M2/3)

443

%3pt Crossover

444

Xpt1=ceil(rand()*Nt/3);

445

Xpt2=Xpt1 + ceil(rand()*(Nt)/3);

446

Xpt3=Xpt2 + ceil(rand()*(Nt−1)/3);

447

Pop(keep+M1+ic2−1,:)=[Pop(ma,1:Xpt1) Pop(pa,Xpt1+1:Xpt2) ...
Pop(ma,Xpt2+1:Xpt3) Pop(pa,Xpt3+1:Nt)]; % Offspring 1

448

Pop(keep+M1+ic2,:)=[Pop(pa,1:Xpt1) Pop(ma,Xpt1+1:Xpt2) ...
Pop(pa,Xpt2+1:Xpt3) Pop(ma,Xpt3+1:Nt)];%Offspring 2

449
450

end
end

451

117

452

%Case I:

selection the best of 2 individual

453

for ic=1:2:M1

454

%first Parent

455

rc=ceil(keep*rand(1,Ntourn)); %Randomly select 2 chroms among those kept

456

[c,ci]=min(fitness(rc)); % Select the chrom with lowest fitness

457

ma=rc(ci);

458

%Second parent

459

rc=ceil(keep*rand(1,Ntourn));%%Randomly select 2 chroms among those kept

460

[c,ci]=min(fitness(rc)); % Select the chrom with lowest fitness

461

pa=rc(ci);

462

if ic≤ceil(M1/5)

463

%1pt Crossover

464

Xpt=ceil(rand()*(Nt−1));

465

Pop(keep+ic−1,:)=[Pop(ma,1:Xpt) Pop(pa,Xpt+1:Nt)]; % Offspring 1

466

Pop(keep+ic,:)=[Pop(pa,1:Xpt) Pop(ma,Xpt+1:Nt)];%Offspring 2

467

end

468

if ic≥ceil(M1/5) && ic≤ceil(4*M1/5)

469

%2pt Crossover

470

Xpt1=ceil(rand()*Nt/2);

471

Xpt2=Xpt1 + ceil(rand()*(Nt−1)/2);

472

Pop(keep+ic−1,:)=[Pop(ma,1:Xpt1) Pop(pa,Xpt1+1:Xpt2) ...

% indicies of first parent

% indicies of second parent

Pop(ma,Xpt2+1:Nt)]; % Offspring 1
473

Pop(keep+ic,:)=[Pop(pa,1:Xpt1) Pop(ma,Xpt1+1:Xpt2) ...
Pop(pa,Xpt2+1:Nt)];%Offspring 2

474

end

475

if ic≥ceil(4*M1/5)

476

%3pt Crossover

477

Xpt1=ceil(rand()*Nt/3);

478

Xpt2=Xpt1 + ceil(rand()*(Nt)/3);

479

Xpt3=Xpt2 + ceil(rand()*(Nt−1)/3);

480

Pop(keep+ic−1,:)=[Pop(ma,1:Xpt1) Pop(pa,Xpt1+1:Xpt2) ...
Pop(ma,Xpt2+1:Xpt3) Pop(pa,Xpt3+1:Nt)]; % Offspring 1

481

Pop(keep+ic,:)=[Pop(pa,1:Xpt1) Pop(ma,Xpt1+1:Xpt2) Pop(pa,Xpt2+1:Xpt3) ...

118

Pop(ma,Xpt3+1:Nt)];%Offspring 2
482
483

end
end

484
485
486

%III) mutation: keep best two and mutate a single bit on the other

487

%chromosomes based on mutating probability

488
489

for iMu=keep:popsize

490

if Muprob≥rand()

491

Mupt=ceil(rand()*Nt);% bit to be mutated

492

Pop(iMu,Mupt)=not(Pop(iMu,Mupt));

493

end

494

end

495
496
497
498

CurrentChrom=keep;

499

for ip = keep:popsize

500
501

%Added so that repeated structures are not analyzed

502

for ipp=1:popsize
if Pop(ip,:)==NewPop(ipp,:)

503

for ipp1=1:Nt

504

if Muprob≥rand()

505

Pop(ip,ipp1)=not(Pop(ip,ipp1));

506

end

507

end

508

end

509
510

end

511
512

Meta=(reshape(Pop(ip,:),n,m));

513

119

514
515

tt=0;
for ix=1:6

516

for iy=1:5

517

Meta(n−iy+tt,ix)=0;

518

end

519

tt=tt+1;

520

end

521
522

tt=0;

523

for ix=1:6

524

for iy=1:5

525

Meta(n−iy+tt,m−ix+1)=0;

526

end

527

tt=tt+1;

528

end

529
530

% Temp Files.

531

tmpPrjFile = 'C:\AFRL MAT\Efficiency4GHz\GALoopGA.hfss';

532

tmpDataFile = 'C:\AFRL MAT\Efficiency4GHz\GALoopData.m';

533

tmpScriptFile = 'C:\AFRL MAT\Efficiency4GHz\GALoopScript.vbs';

534
535

% Create a new temporary HFSS script file.

536

fid = fopen(tmpScriptFile, 'wt');

537
538

% Create a new HFSS Project and insert a new design.

539

hfssNewProject(fid);

540

hfssInsertDesign(fid, 'GA Loop');

541
542
543
544

%Draw AirBox
hfssCylinder(fid, 'Air Box', 'Z', [0, 0, −(Hi oc+Gnd Z+CapZ+10)], G R, ...
G H, 'mm');

545

hfssAssignRadiation(fid, 'Box Rad', 'Air Box');

120

546

hfssSetTransparency(fid, {'Air Box'}, 0.95);

547

%Draw Feed (Outer Conductor)

548

hfssHollowCylinder(fid, 'Outer C', 'Z', [FX, 0, −(Hi oc+Gnd Z)], ...
Th oc, R oc, (Hi oc), 'mm');
hfssAssignPE(fid, 'PEC oc',

549
550

%

{'Outer C'});

Teflon
hfssHollowCylinder(fid, 'Tef', 'Z', [FX, 0, −(Hi oc+Gnd Z)], Th tf, ...

551

R tf, (Hi oc), 'mm');
hfssAssignMaterial(fid, 'Tef', 'Neltec NX9294 (tm)');

552
553
554

%

Inner Conductor
hfssCylinder(fid, 'Inner C', 'Z', [FX, 0, −(Hi oc+Gnd Z)], R feed, ...
(Hi oc), 'mm');

555
556
557

hfssAssignPE(fid, 'PEC Ic',

{'Inner C'});

%Draw waveport
hfssCircularPort(fid, 'LPort', 'port1', 'Z', [FX, 0, ...
−(Hi oc+Gnd Z)], R tf,Deembed, 'mm');

558

hfssCylinder(fid, 'PortCup1', 'Z', [FX, 0, −(Hi oc+Gnd Z)], R oc, ...
−CapZ, 'mm');

559

hfssAssignMaterial(fid, 'PortCup1', 'pec');

560
561
562

%Draw Ground plane

563

%Ground 1

564

hfssRectangle(fid, 'Gnd', 'Y', [−Sub X/2+Pw, 0, ...
−Gnd Z],Gnd Z,Sub X−2*Pw, 'mm');

565

hfssRectangle(fid, 'Gnd Sub', 'Y', [−Gndgap/2+FX, 0, −Gnd Z], Gnd Z, ...
Gndgap, 'mm');

566

hfssSubtract(fid, {'Gnd'}, {'Gnd Sub'});

567

hfssAssignFiniteCond(fid, 'PEC Gnd', 0, 'um', {'Gnd'});

568
569
570

%Circuitry

571

hfssRectangle(fid, 'Circuit1', 'Y', [−Sub X/2, −Sub Y, ...

121

−Gnd Z+1.62],Gnd Z−1.62,Sub X, 'mm');
572

hfssCircle(fid, 'Circuit2', 'Y', [0, −Sub Y, 0],Sub X/2, 'mm');

573

hfssRectangle(fid, 'Circuit2Sub', 'Y', [−Sub X/2, −Sub Y, ...
0],−Sub X/2,Sub X, 'mm');

574

hfssSubtract(fid, {'Circuit2'}, {'Circuit2Sub'});

575

hfssUnite(fid, 'Circuit1','Circuit2');

576

hfssAssignFiniteCond(fid, 'PEC Circuit1', 0, 'um', {'Circuit1'});

577
578
579

%Draw Substrate

580

hfssBox(fid, 'Substrate', [−Sub X/2, 0, −Gnd Z], [Sub X, −Sub Y, ...
Gnd Z], 'mm');

581

hfssAssignMaterial(fid, 'Substrate', 'Rogers RO4003 (tm)');

582

hfssSetColor(fid, 'Substrate', [0, 128, 0]);

583

hfssSetTransparency(fid, {'Substrate'}, 0.8);

584

hfssCylinder(fid, 'Sub2', 'Y', [0, 0, 0], Sub X/2, −Sub Y, 'mm');

585

hfssBox(fid, 'Sub2sub', [−Sub X/2, 0, −Sub X/2], [Sub X, −Sub Y, ...
Sub X/2], 'mm');

586

hfssSubtract(fid, {'Sub2'}, {'Sub2sub'});

587

hfssUnite(fid, 'Substrate','Sub2');

588
589

%Draw Extra Layer Substrate

590

hfssBox(fid, 'Layer', [−Sub X/2, 0, −Gnd Z], [Sub X, .025, Gnd Z], 'mm');

591

hfssAssignMaterial(fid, 'Layer', 'GIL MC5 (tm)');

592

hfssSetColor(fid, 'Layer', [0, 900, 0]);

593

hfssSetTransparency(fid, {'Layer'}, 0.8);

594

hfssCylinder(fid, 'Sub0', 'Y', [0, 0, 0], Sub X/2, .025, 'mm');

595

hfssBox(fid, 'Sub2sub0', [−Sub X/2, 0, −Sub X/2], [Sub X, .025, ...
Sub X/2], 'mm');

596

hfssSubtract(fid, {'Sub0'}, {'Sub2sub0'});

597

hfssUnite(fid, 'Layer','Sub0');

598

hfssMove(fid, {'Layer'}, [0, −.025, 0], 'mm');

599

122

600

%Draw Folded Monopole

601

hfssRectangle(fid, 'mono', 'Y', [−P w/2+FX, 0, −Gnd Z], Gnd Z, P w,'mm');

602

hfssRectangle(fid, 'Loop', 'Y', [−P w/2+FX, 0, 0], P Z, LoopW,'mm');

603

hfssRectangle(fid, 'Loopsub', 'Y', [FX+P w/2, 0, 0], P Z−P w, ...
LoopW−2*P w,'mm');

604

hfssRectangle(fid, 'Loopsub2', 'Y', [FX+LoopW−3*P w/2, 0, 0], P w *1, ...
P w,'mm');

605

hfssAssignFiniteCond(fid, 'PEC Loop', 0, 'um', {'mono'});

606

hfssSubtract(fid, {'Loop'}, {'Loopsub'});

607
608

hfssSubtract(fid, {'Loop'}, {'Loopsub2'});
hfssUnite(fid, 'mono','Loop');

609
610

% Draw Bottom Substrate pads

611

hfssRectangle(fid, 'Pad1', 'Y' , [−2.5,0,0], .5, .5,'mm');

612

hfssAssignFiniteCond(fid, 'Pad1', 0, 'um', {'Pad1'})

613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630

hfssRectangle(fid, 'Pad2', 'Y' , [−5,0,.5], 1.5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad2', 0, 'um', {'Pad2'})
hfssRectangle(fid, 'Pad3', 'Y' , [−4.5,0,.5], 1, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad3', 0, 'um', {'Pad3'})
hfssRectangle(fid, 'Pad4', 'Y' , [−4,0,1.5], 1.5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad4', 0, 'um', {'Pad4'})
hfssRectangle(fid, 'Pad5', 'Y' , [−3.5,0,1], 1.5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad5', 0, 'um', {'Pad5'})
hfssRectangle(fid, 'Pad6', 'Y' , [−3,0,1.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad6', 0, 'um', {'Pad6'})
hfssRectangle(fid, 'Pad7', 'Y' , [−2.5,0,1], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad7', 0, 'um', {'Pad7'})
hfssRectangle(fid, 'Pad8', 'Y' , [−2.5,0,2], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad8', 0, 'um', {'Pad8'})
hfssRectangle(fid, 'Pad9', 'Y' , [−3.5,0,3], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad9', 0, 'um', {'Pad9'})
hfssRectangle(fid, 'Pad10', 'Y' , [−3,0,3], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad10', 0, 'um', {'Pad10'})

123

631

hfssRectangle(fid, 'Pad11', 'Y' , [−2.5,0,3], 1.5, .5,'mm');

632

hfssAssignFiniteCond(fid, 'Pad11', 0, 'um', {'Pad11'})

633

hfssRectangle(fid, 'Pad12', 'Y' , [−2,0,2], 2, .5,'mm');

634

hfssAssignFiniteCond(fid, 'Pad12', 0, 'um', {'Pad12'})

635
636
637

hfssRectangle(fid, 'Pad13', 'Y' , [−2,0,4.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad13', 0, 'um', {'Pad13'})
hfssRectangle(fid, 'Pad14', 'Y' , [−1.5,0,2], 1, .5,'mm');

638

hfssAssignFiniteCond(fid, 'Pad14', 0, 'um', {'Pad14'});

639

hfssRectangle(fid, 'Pad15', 'Y' , [−1.5,0,3.5], .5, .5,'mm');

640

hfssAssignFiniteCond(fid, 'Pad15', 0, 'um', {'Pad15'});

641

hfssRectangle(fid, 'Pad16', 'Y' , [−1.5,0,4.5], .5, .5,'mm');

642

hfssAssignFiniteCond(fid, 'Pad16', 0, 'um', {'Pad16'});

643

hfssRectangle(fid, 'Pad17', 'Y' , [−1,0,2], 1.5, .5,'mm');

644

hfssAssignFiniteCond(fid, 'Pad17', 0, 'um', {'Pad17'});

645

hfssRectangle(fid, 'Pad18', 'Y' , [−1,0,4.5], .5, .5,'mm');

646

hfssAssignFiniteCond(fid, 'Pad18', 0, 'um', {'Pad18'});

647

hfssRectangle(fid, 'Pad19', 'Y' , [−.5,0,3.5], .5, .5,'mm');

648

hfssAssignFiniteCond(fid, 'Pad19', 0, 'um', {'Pad19'});

649

hfssRectangle(fid, 'Pad20', 'Y' , [−.5,0,4.5], .5, .5,'mm');

650

hfssAssignFiniteCond(fid, 'Pad20', 0, 'um', {'Pad20'});

651

hfssRectangle(fid, 'Pad21', 'Y' , [0,0,2.5], .5, .5,'mm');

652

hfssAssignFiniteCond(fid, 'Pad21', 0, 'um', {'Pad21'});

653

hfssRectangle(fid, 'Pad22', 'Y' , [0,0,4], .5, .5,'mm');

654

hfssAssignFiniteCond(fid, 'Pad22', 0, 'um', {'Pad22'});

655

hfssRectangle(fid, 'Pad23', 'Y' , [.5,0,3], .5, .5,'mm');

656

hfssAssignFiniteCond(fid, 'Pad23', 0, 'um', {'Pad23'});

657

hfssRectangle(fid, 'Pad24', 'Y' , [.5,0,4.5], .5, .5,'mm');

658

hfssAssignFiniteCond(fid, 'Pad24', 0, 'um', {'Pad24'});

659

hfssRectangle(fid, 'Pad25', 'Y' , [1,0,3], .5, .5,'mm');

660

hfssAssignFiniteCond(fid, 'Pad25', 0, 'um', {'Pad25'});

661

hfssRectangle(fid, 'Pad26', 'Y' , [1,0,4], .5, .5,'mm');

662

hfssAssignFiniteCond(fid, 'Pad26', 0, 'um', {'Pad26'});

663

hfssRectangle(fid, 'Pad27', 'Y' , [1.5,0,3], .5, .5,'mm');

124

664

hfssAssignFiniteCond(fid, 'Pad27', 0, 'um', {'Pad27'});

665

hfssRectangle(fid, 'Pad28', 'Y' , [1.5,0,4], 1, .5,'mm');

666

hfssAssignFiniteCond(fid, 'Pad28', 0, 'um', {'Pad28'});

667

hfssRectangle(fid, 'Pad29', 'Y' , [2,0,3.5], 1, .5,'mm');

668

hfssAssignFiniteCond(fid, 'Pad29', 0, 'um', {'Pad29'});

669

hfssRectangle(fid, 'Pad30', 'Y' , [1,0,.5], 1.5, .5,'mm');

670
671
672
673

hfssAssignFiniteCond(fid, 'Pad30', 0, 'um', {'Pad30'});
hfssRectangle(fid, 'Pad31', 'Y' , [1.5,0,1], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad31', 0, 'um', {'Pad31'});
hfssRectangle(fid, 'Pad32', 'Y' , [1.5,0,2], .5, .5,'mm');

674

hfssAssignFiniteCond(fid, 'Pad32', 0, 'um', {'Pad32'});

675

hfssRectangle(fid, 'Pad33', 'Y' , [2,0,.5], 1, .5,'mm');

676

hfssAssignFiniteCond(fid, 'Pad33', 0, 'um', {'Pad33'});

677
678
679
680
681

hfssRectangle(fid, 'Pad34', 'Y' , [2,0,2.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad34', 0, 'um', {'Pad34'});
hfssRectangle(fid, 'Pad35', 'Y' , [2.5,0,1], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad35', 0, 'um', {'Pad35'});
hfssRectangle(fid, 'Pad36', 'Y' , [2.5,0,2], 1.5, .5,'mm');

682

hfssAssignFiniteCond(fid, 'Pad36', 0, 'um', {'Pad36'});

683

hfssRectangle(fid, 'Pad37', 'Y' , [3,0,2], 1.5, .5,'mm');

684

hfssAssignFiniteCond(fid, 'Pad37', 0, 'um', {'Pad37'});

685
686
687
688
689
690
691

hfssRectangle(fid, 'Pad38', 'Y' , [3.5,0,3], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad38', 0, 'um', {'Pad38'});
hfssRectangle(fid, 'Pad39', 'Y' , [3.5,0,1.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad39', 0, 'um', {'Pad39'});
hfssRectangle(fid, 'Pad40', 'Y' , [3.5,0,2.5], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad40', 0, 'um', {'Pad40'});
hfssRectangle(fid, 'Pad41', 'Y' , [4,0,.5], 1,.5,'mm');

692

hfssAssignFiniteCond(fid, 'Pad41', 0, 'um', {'Pad41'});

693

hfssRectangle(fid, 'Pad42', 'Y' , [4,0,2], .5, .5,'mm');

694

hfssAssignFiniteCond(fid, 'Pad42', 0, 'um', {'Pad42'});

695
696

hfssRectangle(fid, 'Pad43', 'Y' , [4.5,0,0], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad43', 0, 'um', {'Pad43'});

125

697
698

hfssRectangle(fid, 'Pad44', 'Y' , [4.5,0,1], .5, .5,'mm');
hfssAssignFiniteCond(fid, 'Pad44', 0, 'um', {'Pad44'});

699
700
701

N=1;
for i=1:m

702

NN=0;

703

for j=1:n

704

if Meta(j,i)==1;
NN=1+ NN;

705
706

end

707

if NN>0

708

if Meta(j,i)==0

709

N=N+1;

710

xp=(i)*Pw;

711

yp=(j)*Pw;

712

hfssRectangle(fid, sprintf('RecB%d',N), 'Y' , ...
[−xp+SRR LX/2, −.025, yp−(1+NN)*Pw], Pw*NN, Pw,'mm');
hfssAssignFiniteCond(fid, sprintf('RecBPEC%d',N), 0, ...

713

'um', {sprintf('RecB%d',N)})
NN=0;

714
715
716

end
if j==n
if Meta(j,i)==1;

717
718

N=N+1;

719

xp=(i)*Pw;

720

yp=(j)*Pw;

721

hfssRectangle(fid, sprintf('RecB%d',N), 'Y', ...
[−xp+SRR LX/2, −.025, yp−NN*Pw], Pw*NN, Pw,'mm');
hfssAssignFiniteCond(fid, sprintf('RecBPEC%d',N), 0, ...

722

'um', {sprintf('RecB%d',N)})
723

NN=0;

724

end

725

end

126

end

726

end

727
728

end

729

% Add a Solution Setup.

730

hfssInsertSolution(fid, 'Setup', fC/1e9, 0.02,15);

731

% Save the project to a temporary file and solve it.

732

hfssSaveProject(fid, tmpPrjFile, true);

733

hfssSolveSetup(fid, 'Setup');

734

hfssInsertFarField(fid, 'Infinite Sphere1',0,0,0,0,0,0,'deg')

735

% Export the Network data as an m−file.

736

hfssExportNetworkData(fid, tmpDataFile, 'Setup', 'LastAdaptive');

737

FFc=fC*(1e−9);

738

hfssCreateReportDualComp(fid,FarFieldpath, FFc, 'GHz');

739

hfssCreateReport(fid, FFc, 'GHz');

740
741

% Close the HFSS Script File.

742

fclose(fid);

743

% Execute the Script by starting HFSS.

744

disp('Solving using HFSS ..');

745

hfssExecuteScript(hfssExePath, tmpScriptFile);

746
747
748
749
750
751
752

% Load the data by running the exported matlab file.

753
754

run(tmpDataFile);

755

[heffi2, RadEff] = hdrload('farFieldEff.m');

756

radF=RadEff(:,1);

757

RE=RadEff(:,2);

758

Rfc(ip) = RE;

127

759
760

Sfc(ip)=20*log10(abs(S(np,:,:)));

761

fitness(ip)= Rfc(ip)*Sfc(ip);

762
763

% min(S1r) and rad efficiency

764
765

if iga == 1

766
767

Name = ...

768

sprintf('C:\\AFRL MAT\\Efficiency4GHz\\ ...

769

Gen%d\\Gen%d %d.hfss',iga,iga,ip);

770

movefile('GALoopGA.hfss',Name)

771
772
773

if ip == popsize

774

Sfc1 = Sfc;

775

Rfc1 = Rfc;

776

ind1 = ind;

777

keep1 = keep;

778

figure

779

scatter(Rfc1,Sfc1)

780

axis([ 0 1 −40 0])

781

set(gca,'FontSize',17);

782

xlabel('Efficiency')

783

ylabel(' | S11 | (dB)')

784

title('First Generation 4 GHZ')

785
786

savefile = 'Sfc1.mat';

787

save(savefile, 'Sfc1');

788

savefile = 'Rfc1.mat';

789

save(savefile, 'Rfc1');

790

savefile = 'ind1.mat';

791

save(savefile, 'ind1')

128

792

savefile = 'keep1.mat';

793

save(savefile,'keep1')
end

794
795
796

h = h+1
end

797
798
799

if iga ==2
Name = ...

800

sprintf('C:\\AFRL MAT\\Efficiency4GHz\\ ...

801

Gen%d\\Name%d %d.hfss',iga,iga,ip);

802

movefile('GALoopGA.hfss',Name)

803
804

if ip == popsize

805

Sfc2 = Sfc;

806

Rfc2 = Rfc;

807

ind2 = ind;

808

keep2 = keep;

809

figure

810

scatter(Rfc2,Sfc2)

811

axis([ 0 1 −40 0])

812

set(gca,'FontSize',17);

813

xlabel('Efficiency')

814

ylabel(' | S11 | (dB)')

815

title('Second Generation 4 GHZ')

816
817

savefile = 'Sfc2.mat';

818

save(savefile, 'Sfc2');

819

savefile = 'Rfc2.mat';

820

save(savefile, 'Rfc2');

821

savefile = 'ind2.mat';

822

save(savefile, 'ind2')

823

savefile = 'keep2.mat';

824

save(savefile,'keep2')

129

825

end

826

h = h+1

827

end

828
829
830

if iga ==3
Name = ...

831

sprintf('C:\\AFRL MAT\\Efficiency4GHz\\ ...

832

Gen%d\\Name%d %d.hfss',iga,iga,ip);

833

movefile('GALoopGA.hfss',Name)

834
835
836

if ip == popsize

837

Sfc3 = Sfc;

838

Rfc3 = Rfc;

839

ind3 = ind;

840

keep3 = keep;

841

figure

842

scatter(Rfc3,Sfc3)

843

axis([ 0 1 −40 0])

844

set(gca,'FontSize',17);

845

xlabel('Efficiency')

846

ylabel(' | S11 | (dB)')

847

title('Third Generation 4 GHZ')

848
849

savefile = 'Sfc3.mat';

850

save(savefile, 'Sfc3');

851

savefile = 'Rfc3.mat';

852

save(savefile, 'Rfc3');

853

savefile = 'ind3.mat';

854

save(savefile, 'ind3')

855

savefile = 'keep3.mat';

856

save(savefile,'keep3')

857

end

130

858

h = h+1

859

end

860
861
862
863
864

CurrentChrom=CurrentChrom+1
end %of the evaluation of the current population Muprob= Muprob−Muprob/10
% % % %

865
866

%Do statistics for new generation

867
868

[fitness,ind]=sort(fitness) ;

%Sort fitness such that the lowest is ...

at ind 1
869

Pop=Pop(ind,:);

% sorts population with lowest fitness placed at ind 1

870

Sfc = Sfc(:,ind);

871

Rfc = Rfc(:,ind);

872

minc(iga+1)=min(fitness);% minc contains min of population

873

meanc(iga+1)=mean(fitness);

874
875
876
877
878

iga=iga+1

879
880
881
882
883

end

884

% remove all the added paths.

885

rmpath('C:\MatlabHfss\Files\')

131

BIBLIOGRAPHY

132

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