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

код для вставкиСкачатьCopyright by Hosung Choo 2003 The Dissertation Committee for Hosung Choo certifies that this is the approved version of the following dissertation: Application of Genetic Algorithms to the Design of Microstrip Antennas, Wire Antennas and Microwave Absorbers Committee: Hao Ling, Supervisor Mircea D. Driga Edward J. Powers Ross Baldick Robert Rogers Application of Genetic Algorithms to the Design of Microstrip Antennas, Wire Antennas and Microwave Absorbers by Hosung Choo, BS, MSE Dissertation Presented to the Faculty of the Graduate School of the University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin May 2003 UMI Number: 3110761 Copyright 2003 by Choo, Hosung All rights reserved. ________________________________________________________ UMI Microform 3110761 Copyright 2004 ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ____________________________________________________________ ProQuest Information and Learning Company 300 North Zeeb Road PO Box 1346 Ann Arbor, MI 48106-1346 To my wife Seongsin Acknowledgements First of all, I would like to especially thank my supervisor Professor Hao Ling, who greatly supported me during my five years of graduate study. He has done an outstanding job of supplying me with suggestions and encouragement during my studies at the University of Texas at Austin. He was always helpful and gave me plenty of new ideas. Without his support, this work would have been impossible to pursue. I would also like to extend my gratitude to Dr. Robert Rogers, who has been a valuable source of information in my small antenna research. His deep knowledge of electrically small antenna theory and practice enabled me to complete the work presented in this dissertation. I also appreciate Professor Mircea Driga for graciously giving his time to serve as the chair of my dissertation committee. I am grateful to Professor Edward Powers and Professor Ross Baldick for serving on my dissertation committee and kindly reviewing this work. I would also like to thank all the current and former group members: Dr. Hyeongdong Kim, Dr. Tao Su, Dr. Yuan Wang, Mr. Yong Zhou and Mr. Sungkyun Lim. Special thanks to Dr. Luiz C. Trintinalia and Adrian Hutani, my friends as well as my co-workers, for providing me with valuable numerical simulation code and for sharing their expertise in setting up the antenna measurement and fabrication systems. Finally, I would like to thank my wife and my parents for their endless support and love, which I will never forget. Application of Genetic Algorithms to the Design of Microstrip Antennas, Wire Antennas and Microwave Absorbers Publication No._____________ Hosung Choo, Ph.D. The University of Texas at Austin, 2003 Supervisor: Hao Ling This dissertation explores the general methodology for designing electromagnetic (EM) systems by combining genetic algorithms (GA) with computational electromagnetic (EM) simulations. The EM problems investigated are broadband and multi-band microstrip antennas, low-profile microwave absorbers and electrically small wire antennas. It is shown that optimized performance can be designed and realized using simple shape control. In addition, novel GA approaches are investigated for more challenging multiobjective problems. The developed methodology is also used to explore performance bounds in complex EM systems. First, the use of GA to design microstrip antenna shapes for broadband and multi-band applications is investigated. A full-wave electromagnetic solver is vii employed to predict the performance of microstrip antennas with arbitrary patch shapes. A GA with two-point crossover and geometrical filtering is implemented to optimize the patch shape. For broadband application, the optimized patch antenna achieves a four-fold improvement in bandwidth when compared to a standard square microstrip. For multi-band application, the optimized patches show that arbitrary frequency spacing ranging from 1:1.1 to 1:2 can be achieved. Tri-band and quad-band microstrip shapes are also generated and the resulting designs show good operations at the designated frequencies. Second, the use of GA for designing optimal shapes for corrugated coatings under near-grazing incidence is examined. Optimized coating shapes depending on different polarizations are generated. Physical interpretations for the optimized structure are discussed, and the resulting shape is compared to conventional planar and triangular shaped designs. This problem is also extended from the single to multi-objective optimization using the Pareto GA. Optimization results using two different objectives, the height (or weight) of the coating versus absorbing performance, are presented. Finally, this dissertation reports on the use of GA in the design optimization of electrically small wire antennas, taking into account of bandwidth, efficiency and antenna size. To efficiently map out this multi-objective problem, the Pareto GA is implemented with the concept of divided range optimization. An optimal set of designs, trading off bandwidth, efficiency and antenna size, are viii generated. Several GA designs are built, measured and compared to the simulation. Physical interpretations of the GA-optimized structures are provided, and the results are compared against the well-known fundamental limit for small antennas. Further improvements using other geometrical design freedoms are also discussed. ix Table of Contents List of Figures ..................................................................................................xii Chapter 1 Introduction .........................................................................................1 Chapter 2 Design of Broadband Microstrip Antennas on High-Dielectric Substrate Using a Genetic Algorithm...........................................................7 2.1 Introduction .........................................................................................7 2.2 GA Optimization ...................................................................................9 2.3 Broadband Microstrip Antenna Design .............................................. 15 2.4 Summary ........................................................................................... 24 Chapter 3 Design of Multi-Band Microstrip Antennas Using a Genetic Algorithm.................................................................................................. 25 3.1 Introduction ....................................................................................... 25 3.2 Miniaturized Dual-Band Microstrip Antenna Design ........................... 26 3.3 Multi-band Microstrip Without Slots-cut ............................................. 35 3.4 Summary............................................................................................. 43 Chapter 4 Shape Optimization of Corrugated Coatings Under Grazing Incidence Using a Genetic Algorithm ...................................................... 44 4.1 Introduction ....................................................................................... 44 4.2 Approach............................................................................................. 46 4.3 GA-Optimized Coating Shape ............................................................. 50 4.4 Multi-Objective Optimization.............................................................. 61 4.5 Summary............................................................................................. 68 Chapter 5 Design of Electrically Small Wire Antennas Using a Pareto Genetic Algorithm.................................................................................................. 70 5.1 Introduction ....................................................................................... 70 5.2 Pareto GA Approach ........................................................................... 73 5.3 GA-Optimized Results ........................................................................ 78 5.4 Comparison to Fundamental Limit ...................................................... 85 x 5.5 Further Improvement on GA Designs .................................................. 89 5.6 Summary............................................................................................. 90 Chapter 6 Conclusions ....................................................................................... 94 Appendix A ....................................................................................................... 98 Bibliography.................................................................................................... 112 Vita ............................................................................................... 118 xi List of Figures Figure 1.1: Flow chart of the genetic algorithm. Figure 2.1: 2-D encoding of patch shapes for broadband shape optimization. Figure 2.2: Calculation time of single shape depends on the chromosome size. Figure 2.3: Coarse-to-fine 2 dimensional chromosome. Figure 2.4: Median filtering. Figure 2.5: Two-point crossover with three chromosomes. Figure 2.6: Convergence rates of the conventional one-point crossover scheme and our two-point crossover scheme. A population of 30, a crossover rate of 0.8 and a mutation rate of 0.1 are used. In addition, geometrical filtering is applied. The results are averaged over 10 trials. Figure 2.7: Parallel processing for a GA application. Figure 2.8: (a) GA-optimized microstrip antenna using 40� resolution within a 72mm�mm area. The gray pixels are metal and the white dot shows the position of the probe feed. (b) A picture of the microstrip antenna built on FR-4 circuit board. (c) Return loss (dB) of the GAoptimized antenna from simulation (------) and measurement (? ??). Figure 2.9: Bandwidth enhancement is achieved by two-mode operation and ragged edges. Figure 2.10: (a) Current distribution on the surface of the GA-optimized patch at 1.98GHz. (b) Current distribution on the surface of the GAoptimized patch at 2.1GHz. Figure 2.11: (a) Measured radiation pattern at 1.98GHz (?=?45o plane). (b) Measured radiation pattern at 2.1GHz (?=45o plane). Figure 2.12: Bandwidth enhancement is achieved by two-mode operation and ragged edges. xii Figure 3.1: Encoding scheme for the dual-band design. Figure 3.2: (a) GA-optimized slot shape for dual-band operation. (b) Return loss (dB) of the GA-optimized antenna from simulation ( ) and measurement (? ??). Figure 3.3: Simulated total metal loss (dBm) of the GA-optimized dual-band microstrip (? ??), a square microstrip of 72mm�mm (------) and a square microstrip of 36mm�mm (-?-?-?-?). Figure 3.4: (a) Current distribution on the surface of the GA-optimized patch at 1 GHz. (b) Current distribution at 2 GHz. Figure 3.5: (a) Measured radiation patterns at 1 GHz (?=0 plane). (b) Measured radiation pattern at 2 GHz (?=0 plane). Figure 3.6: Shapes of three GA-optimized dual-band microstrip antennas, and the resulting return loss from simulation (------) and measurement (? ??). (a) Frequency ratio of 1:1.3 (1.8GHz and 2.34GHz). (b) Frequency ratio 1:1.6 (1.8GHz and 2.9GHz). (c) Frequency ratio of 1:1.9 (1.8GHz and 3.42GHz). Figure 3.7: Shapes of dual-band microstrip antennas. Frequency ratio from 1:1.1 to 1:2. Figure 3.8: Shape of the GA-optimized tri-band microstrip antenna that operates at 1.6GHz, 1.8GHz and 2.45GHz, and the resulting return loss of the antenna from simulation (------) and measurement (? ??). Figure 3.9: Shape of the GA-optimized quad-band microstrip antenna that operates at 0.9GHz, 1.6GHz, 1.8GHz and 2.45GHz, and the resulting return loss of the antenna from simulation (------) and measurement (? ??). Figure 4.1: Geometry of the corrugated absorber. Figure 4.2: Encoding of corrugated absorber into a binary chromosome. Figure 4.3: (a) Before geometrical filter. (b) After 7-point sliding window filter. (c) After the descending order filter. xiii Figure 4.4: (a) GA-optimized shape for the vertical polarization. (b) Reflection coefficient (dB) versus frequency at 30 degrees from grazing. Figure 4.5: (a) GA-optimized shape for the horizontal polarization. (b) Reflection coefficient (dB) versus frequency. Figure 4.6: Performance comparison of the planar absorber (------), triangular shaped absorber (? ??), and GA-optimized absorber (-?-?-?-?) for the horizontal polarization. Figure 4.7: Effect of cost definitions on the optimization results for the horizontal polarization. (a) Cost definition in (4.1). (b) Minimax cost definition. (c) Average power reflection coefficient cost definition. Figure 4.8: (a)Performance sensitivity to variations in the shape of the profile for the horizontal polarization. (i) Original GA-optimized design. (ii) GA-optimized shape with RMS error of 0.4 mm. (iii) Smoother shape after 6:1 undersampling. (b) Reflection coefficient (dB) versus frequency at 25o, 30 o and 35 o from grazing. Figure 4.9: (a) GA-optimized shape taking into account of both the vertical and horizontal polarizations. (b) Reflection coefficient (dB) versus frequency. Figure 4.10: (a) Planar absorber with horizontal polarized wave incidence. (b) Rectangular profile with horizontal polarized wave incidence. Figure 4.11: Convergence of the Pareto front as a function of the number of generations for absorbing performance versus absorber height. (a) Initial population. (b) After 5 generations. (c) After 20 generations. (d) After 200 generations. Figure 4.12: Final converged Pareto front of absorbing performance versus absorber height. The insets show 4 sample designs on the Pareto front. Figure 4.13: Final converged Pareto front of absorbing performance versus absorber weight. The insets show four sample designs on the Pareto front. xiv Figure 5.1: Achievable bandwidth in terms of antenna size and some examples of small antennas. Figure 5.2: Configuration of the multi-segment wire antenna used in the GA design. Figure 5.3: Encoding of the wire configuration into a binary chromosomes. Figure 5.4: Divided range multi-objective GA approach. Figure 5.5: Convergence of the Pareto front as a function of the number of generations in terms of bandwidth, efficiency and antenna size. (a) Initial generation. (b) After 200 generations. (c) After 1000 generations. Figure 5.6: Three samples from the Pareto front (a) kr=0.34, (b) kr=0.42 and (c) kr=0.50. Figure 5.7: (a) Photo for antenna B, which has an antenna size of kr=0.42. (b) Return loss and (c) efficiency versus frequency of antenna B. The efficiency measurement was done using the Wheeler cap method. Figure 5.8: Pareto front of the GA designs after convergence. The surface is generated using a least squares fitting to best fit the GA results shown as dots. Figure 5.9: (a) Projection of the Pareto front onto the size and efficiency plane. (b) Projection of the Pareto front onto the size and bandwidth plane. (c) Projection of the Pareto front onto the bandwidth and efficiency plane. (d) through (f) show the corresponding fundamental limit based on (5.4). Figure 5.10: Small antenna performance using the definition of ? = (Eff � BW) / Theoretical BW Limit. (a) Original 7-wire configuration. (b) Variable input impedance. (c) Multi-arm configuration. (d) Multiple wire radii. Figure A.1: Input resistance of a standard square-shaped (36mm�mm) microstrip built on FR-4 before (------) and after (? ??) using the Wheeler cap size of 10cmm�cmm�m. (a) Measured (b) ENEMBLE simulated. xv Figure A.2: Wheeler cap measured efficiency based on parallel circuit model (-----), Wheeler cap measured efficiency based on series circuit model, efficiency by Wheeler cap simulation and efficiency by gain simulation. (a) microstrip (36mm�mm) built on FR-4 substrate (b) microstrip (17.5mm�mm) build on Duroid. Figure A.3: Measured input resistance of a standard square-shaped (36mm x 36mm) microstrip built on Duroid before (------) and after (??) using (a) the Wheeler cap size of 17cmm x 17cmm x 8.5cm. (b) using the Wheeler cap size of 17cm x 17cm x 2cm. Figure A.4: Interior cavity modes spectrums for two Wheeler cap size. The upper spectrum is for the Wheeler cap size of 17cm x 17cm x 8.5cm.The lower spectrum is for the Wheeler cap size of 17cm x 17cm x 2.0cm. Figure A.5: (a) A photo of one of the sample miniaturized microstrip. Wheeler cap measured efficiencies for the microstrips (b) built on FR-4 substrate in terms of % from the regular size and (c) the microstrips built on Duroid. Figure A.6: EB Product against physical microstrip patch size for microstrip built on three different substrates. (a) FR-4 (? ??), Duroid (------)and Air (-?-?-?-?). xvi Chapter 1 Introduction The optimization of a real-world electromagnetic (EM) system is in general a challenging problem, as it typically involves a large number of degrees of freedom. In addition, the cost function of a complex EM problem often contains many local minima and may include non-differentiable regions. These factors make it difficult to find a global optimum using deterministic optimization methods such as the quasi-Newton method or the conjugate gradient method [1]. Genetic algorithm (GA) is a stochastic search method based on the principles of natural selection and evolution [2]. GA is classified as a global optimizer instead of a local optimizer, so the solutions are less dependent on the initial values. For this reason, GA has shown highly efficient optimization capability in various engineering applications. In particular, GA is more attractive for finding an approximate global optimum in a problem that has a very highdimensional space with many local minima. The idea of GA was developed by modeling the natural processes of evolution and adaptation. In 1975, Holland first applied a GA to the design of an artificial system [3]. Since then, GA has proven to be a powerful tool in solving various kinds of design and optimization problems ranging from building heating 1 systems and job scheduling to VLSI layout and cell planning for communication systems. The block diagram of a basic GA is shown in Fig. 1.1. The GA process can roughly be divided into three parts: initialization, evaluation, and reproduction. The algorithm starts with an initial population of possible solutions. The solutions are encoded as binary or real-valued chromosomes. In the evaluation phase, the performance of each solution in the population is predicted using a simulation tool. Initial Solutions Reproduction Crossover Performance Simulation Mutation Geometrical Filtering Evaluate Cost Optimized Solution Fig. 1.1 Flow chart of the genetic algorithm. 2 Then the cost (or fitness) of each solution is evaluated using the proper cost function defined in terms of the design goals. According to the cost, the chromosomes are refined into the next generation through a reproduction process that includes crossover, mutation and geometrical filtering. This series of processes is repeated until the cost is minimized, meaning an optimum solution has been found. Recently, GA has been applied to a number of EM design and inverse problems [4]. For instance, Alatan et al. [5] employed a GA to design circularly polarized (CP) microstrip antennas. Johnson and Rahmat-Samii in [6] showed that it is possible to obtain microstrip shapes for broadband and dual-band applications by using a GA. Haupt reported on the use of GA in designing antenna arrays [7]. Altshuler and Linden examined the use of GA in designing wire antennas [8,9]. Michielssen et al. applied a GA to find optimal thicknesses for multi-layer planar coatings for radar cross section (RCS) reduction [10]. GA has also been applied to inverse scattering problems to reconstruct the shape of an unknown target from its scattered field data [11,12]. While the application of GA to EM problems has met with initial success, much more research is still needed to apply GA to real-world EM design problems. In addition, the adaptation of advanced GA techniques is also needed in order to address more challenging problems. In this dissertation, GAs are applied to the design of antennas and absorbers. The objectives of the dissertation are 3 threefold. The first objective is to formulate general design methodologies that combine GAs with state-of-the-art computational EM simulations to create novel EM designs. The second objective is to investigate more advanced GA techniques such as the Pareto GA for multi-objective problems. The third objective is to explore the use of GA for mapping the fundamental performance bounds of a given EM system. Three classes of EM design problems are investigated in this dissertation: microstrip antennas, shaped microwave absorbers and electrically small wire antennas. This dissertation is organized as follows. In Chapters 2 and 3, the design of microstrip antennas using GA for broadband and multi-band applications is addressed. Because of their low profile and ease of fabrication, microstrip antennas are a very popular choice for many antenna applications. However, a well-known drawback of the microstrip is that it is an intrinsically narrow-band device. The goal is to improve the frequency characteristics of microstrip antennas by exploring arbitrary patch shapes. Therefore a design methodology based on a GA in conjunction with a fast EM solver is developed to find unique patch shapes for achieving optimal performance. For the broadband application discussed in Chapter 2, the design methodology is applied to explore the maximum achievable bandwidth of microstrips. Basic antenna measures such as return loss, current distributions and far field radiation patterns are examined to provide a physical interpretation of the optimized design. In Chapter 3, the design 4 methodology is applied to search for patch shapes for dual-band, tri-band and quad-band operations. Chapter 4 discusses the design of shaped microwave absorbers using GA. Microwave absorbers are often used to reduce the RCS of a target. An effective absorber needs to suppress the reflection over a wide range of frequencies. In addition, the profile of the absorber needs to be small for practical constraints. Thus the goal is to design low-profile microwave absorbers with broadband frequency characteristics. These low-profile and broadband requirements are usually not compatible with each other. The Pareto GA is an attractive technique of efficiently mapping out this multi-objective problem. The Pareto GA is applied to examine the optimized absorber shapes in terms of the height (or weight) of the coating versus its absorbing performance. This chapter also discusses a physical interpretation of the optimized structure and compares the GA-designed shapes to conventional planar and triangular shaped designs. Chapter 5 examines the design of electrically small wire antennas using GA. Many antenna applications require physically small antennas. However, the size of an antenna usually needs to be on the order of a wavelength of the operating frequency in order for the antenna to operate efficiently. In 1948, Chu derived a formula showing that the fundamental limit of the bandwidth performance of an antenna is a function of the antenna?s electrical size [13]. The smaller the antenna size, the narrower the bandwidth. In addition to bandwidth, 5 antenna efficiency is also very important in the design of small antennas. The goal is to use GA to design electrically small wire antennas, taking into account bandwidth, efficiency and antenna size. Because smaller-sized solutions are particularly difficult to optimize, the Pareto GA is enhanced by the concept of divided range optimization to efficiently map out this challenging multi-objective problem. The GA generates an optimal set of designs, trading off bandwidth, efficiency and antenna size. The performance of the designs is analyzed by comparing it against the well-known fundamental limit for small antennas. This chapter also discusses further improvements to the design using other geometrical design freedoms. Finally, Chapter 6 concludes the dissertation and discusses directions for future research. Appendix A is included to provide some new insights on the Wheeler cap method for measuring antenna efficiency, which is a central issue considered in several chapters in the dissertation. 6 Chapter 2 Design of Broadband Microstrip Antennas on High-Dielectric Substrate Using a Genetic Algorithm 2.1 Introduction Microstrip antenna has many favorable characteristics such as low profile, conformal shape and ease of fabrication. In many antenna applications the use microstrip antennas is preferred over other types of antennas. However, a wellknown drawback of the microstrip is that it is an intrinsically narrow-band device. Increasingly, wireless devices are being asked to handle larger amounts of data at faster speed, which requires the antenna to be broadband. Much research has been carried out to improve the bandwidth of microstrips, and a number of different techniques have been published including adding parasite patches, adopting multilayer structures, using thick air substrate and adding shorting posts as reactive loading [14,15]. However, these techniques are usually accompanied by an increase in overall size and/or manufacturing cost. In the past several years, some researchers have applied GA to the design of microstrip antennas. Alatan et al. [5] employed a GA to design circularly polarized (CP) microstrip antennas by optimizing the corners of a square microstrip. Johnson and Rahmat-Samii in [6] showed that it is possible to obtain novel shapes for broadband and dual-band 7 applications by using a GA. They used air as substrate material by suspending a patch above the ground plane. The attractiveness of GA shape optimization is that improved performance can be achieved without increasing overall volume or manufacturing cost. In this chapter, the use of GA for broadband applications is examined. In contrast to the work of Johnson and Rahmat-Samii, standard FR-4 is employed as the substrate, since it is the most commonly used material in wireless devices. Microstrips built on high dielectric material substrate such as FR-4 (dielectric constant of 4.3) have narrower bandwidth. Consequently, it is more challenging to obtain the desired frequency characteristics without sufficient degrees of freedom in the design process. The approach is to employ a full-wave EM patch code to predict the performance of an arbitrary shaped microstrip. A GA is implemented to optimize the patch shape that is encoded into a 2-D chromosome. A two-point crossover scheme with three chromosomes is used as a crossover operator to achieve faster convergence. In addition, geometrical filtering is adopted to create more realizable shapes. This methodology is applied to broadband patch antenna design. This chapter is organized as follows. In Section 2.2, the EM simulation tool and details of the GA algorithm are described. In Section 2.3, the application of the GA algorithm to broadband microstrip antenna design is described. A summary of this chapter is given in Section 2.4. 8 2.2 GA Optimization To predict the performance of each patch shape, an EM simulation code adapted from a code for analyzing frequency selective surfaces is used [17,18]. The code uses the electric field integral equation (EFIE) formulation in conjunction with the periodic Green?s function for a layered medium. Rooftop functions are used as basis functions to represent the currents on the patch. To speed up the computation of the matrix elements, the fast Fourier transform (FFT) is employed. To further reduce the matrix fill time during GA, the matrix elements for all the possible basis positions are computed once and stored before the GA process. Consequently, the moment matrix for any patch shape can be easily assembled without any computation. To achieve broadband operations, GA is implemented as an optimizer for the microstrip patch shapes. The algorithm starts with an initial population of shapes that are encoded as chromosomes. After these chromosomes are evaluated by the EM simulation code, a cost function is computed. According to the cost function, the chromosomes are refined into the next generation through a reproduction process that includes crossover, mutation and geometrical filtering. This series of processes is repeated until the cost function is minimized. In this GA process, each patch shape is represented as a binary bitmap. In the resulting 2-D chromosome [19], ones represent the metallized areas and zeros represent the areas without metal (Fig. 2.1). 9 2-D Chromosome Shapes Substrate Metallic Patch 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 r =4.3 FR-4 Fig. 2.1 2-D encoding of patch shapes for broadband shape optimization. Additionally, the coarse-to-fine concept is adopted to save computation time. To get better design, it is necessary to use large size of chromosome. However, the use of large size chromosome causes a considerable computational cost. Fig.2.2 is calculation time depend on number of genes in chromosome. The calculation time for one shape scales O(N3) where N is the number of genes in the chromosome. In order to use large size of chromosome without that much computational burden, coarse-to-fine size variant chromosome is introduced in this GA. 10 -- Computation time by Pentium III 550Mhz Fig. 2.2 Calculation time of single shape depends on the chromosome size. The basic idea of the coarse-to-fine chromosome is to begin the initial generations with coarse resolution and increase the resolution forward the later generations. Fig. 2.3 describes how coarse-to-fine chromosome works. It starts with coarse resolution of chromosome. Without that much time, GA finds the result that has the lowest cost for this resolution. If the cost does not satisfy the design goal, then the resolution is increased to twice as much as before. In transition from coarse to fine, shapes are already partially optimized with the coarse resolution. Therefore, at higher resolutions, GA only needs to slightly tune the results. This variable resolution of chromosome from coarse to fine makes it possible to save much computation time as compared to starting with very fine resolution. Since it is more desirable to have a well-congregated patch shape in the final design, a 11 2-D median filter [20] is applied to the chromosomes at each generation of the GA process. YES Convergence Check Converge? Increase the resolution NO Stay in current resolution Fig. 2.3 Coarse-to-fine 2 dimensional chromosome. Fig. 2.4 shows a sample chromosome before and after the 2-D median filter operation. Before median filtering, the chromosome shows many isolated patches. After median filtering, most of the isolated patches in the chromosome are gone, and the overall shape of the chromosome becomes more gathered. For the crossover operation, a two-point crossover scheme using three chromosomes is used to boost the GA convergence rate. Fig. 2.5 depicts the crossover scheme. It starts by selecting three chromosomes as parents and divides each chromosome into three parts. 12 After Before Median Filtering Fig. 2.4 Median filtering. Then the next generation is made by intermingling the three parent chromosomes. This crossover scheme exhibits a more disruptive characteristic for regeneration than the conventional one-point or two-point crossover. It counteracts against the median filtering effect and is found to result in better convergence rate. A comparison of the results between the two-point crossover and the conventional one-point crossover is shown in Fig. 2.6. Each crossover scheme is applied to the broadband patch antenna design problem to compare the convergence rate. The two-point crossover scheme results in faster convergence. 13 Parent Children Fig. 2.5 two-point crossover with three chromosomes. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Fig. 2.6 10 20 30 40 50 Convergence rates of the one-point crossover scheme (------), and the two-point crossover scheme (? ??). A population of 30, a crossover rate of 0.8 and a mutation rate of 0.1 are used. In addition, geometrical filtering is applied. The results are averaged over 10 trials. 14 2.3 Broadband Microstrip Antenna Design 2.3.1 GA Optimization for Broadband Application In this section, the design goal is to broaden the bandwidth of a microstrip by exploring arbitrary patch shapes through GA. To achieve the design goal, a cost function is defined as follows: Cost = 1 N N n =1 Pn where Pn = S 11 (dB ) + 10 dB if S 11 (dB ) ? ? 10 dB 0 if S 11 (dB ) < ?10 dB (2.1) The cost function in (2.1) averages those S11 values (in dB) that exceed ? 10dB within the desired frequency range. In the GA process, the cost function is an important factor not only for the final design but also for the overall GA convergence time. To reduce the computational burden, an adaptive cost function is used. In the first stage of the GA process, a small frequency range compared to the eventual design goal is used. Then the frequency range is gradually increased whenever the cost converges to a minimum. This process is repeated until the eventual design goal is achieved. In general, the number of frequency calculations required in this scheme is much less than if the entire design frequency band is used throughout the GA iterations. In this GA, the size of the population is 30. A crossover probability of 80% is used, while the probability of mutation is set to 10%. The frequency range 15 is set between 1.9GHz and 2.1GHz. A square design area (72mm�mm) and FR-4 substrate with a thickness of 1.6mm are used. The design area is discretized into a 40 � 40 grid. Since lots of repeated EM computations are necessary at each generation, cost calculations for the population are distributed to multiple computers using the parallel processing concept as shown in Fig. 2.7. The total time of the design process is about 24 hours on ten Pentium 4 (1.7GHz) machines running in parallel. Fig. 2.8(a) is the resulting shape of the GA-optimized microstrip. The white dot represents the location of the probe feed. Note that the position of the feed is a part of the parameters to be optimized, since the patch has the freedom to be located anywhere within the total design area. Calculation results Chromosome Information Fig. 2.7 Parallel processing for a GA application. 16 For experimental verification, a prototype of the GA-optimized microstrip patch using aluminum tape is built (copper tape was also tried with nearly identical results). A photo of the tested patch is shown in Fig. 2.8(b). Fig. 2.8(c) shows the return loss comparison between the measurement (solid line) and the simulation (dashed line) results. Good agreement is observed. The bandwidth is found to be 8.5% by simulation and 8.1% by measurement. It should be noted that the same GA methodology using coarser 16� grids and 32� grids within the same design area have also been applied [16]. As the design resolution of the grid increases from 16� to 32�, the bandwidth was improved by about 30% as shown in Fig. 2.9. However, the higher resolution 40� grid shows only a slight bandwidth increase over that of the 32� grid. The gain of the GA-optimized microstrip relative to the reference square microstrip under the same construction was also measured. Nearly negligible gain loss is observed (0.9dB gain loss at 1.98GHz, and 1.0dB at 2.1GHz). 17 72mm 40� (b) (a) -5 S11 (dB) -10 -15 -20 1.90 1.96 2.02 2.08 2.14 2.20 Frequency (GHz) (c) Fig. 2.8 (a) GA-optimized microstrip antenna using 40� resolution within a 72mm�mm area. The gray pixels are metal, and the white dot shows the position of the probe feed. (b) A picture of the microstrip antenna built on FR-4 circuit board. (c) Return loss (dB) of the GA-optimized antenna from simulation (------) and measurement (? ??). 18 72mm 72mm � 1.3 . 16x16 32x32 (a) (b) 72mm � 1.05 40x40 (c) Fig. 2.9 Bandwidth enhancement is achieved by two-mode operation and ragged edges. 19 2.3.2 Operating Principle of the GA Design Next, an interpretation on the operating principle of the GA-designed antenna is provided. The two frequency dips in Fig. 2.8(c) show that the antenna supports two operating modes very close to each other in frequency. To verify the two operating modes, the current distributions on the patch around these frequencies are analyzed. Figs. 2.10(a) and 2.10(b) show the current plots at 1.98GHz and 2.1GHz respectively. At 1.98GHz, current flows predominantly in the lower-right direction, while at 2.1GHz, current flows predominantly in the upper-right direction. The currents corresponding to the two modes are nearly perpendicular to each other. The two-mode operation can be seen even more clearly in the measured radiation pattern plots. Figs. 2.11(a) and 2.11(b) show the radiation patterns at the two frequencies along the two dashed cuts. It is observed that at 1.98GHz the dominant polarization is in the ?=?45o plane while at 2.1GHz the dominant polarization is in the ?=45o plane, as expected from the current plots. This two-mode operation improves the bandwidth of a square microstrip by a factor of about three. Note that the enhancement of bandwidth in this antenna comes at the price of polarization purity, which was not a constraint in the GA process. For some applications where the depolarization from the propagation channel is dominant, polarization purity might not be a strong design consideration. 20 (a) (b) Fig. 2.10 (a) Current distribution on the surface of the GA-optimized patch at 1.98GHz. (b) Current distribution on the surface of the GA-optimized patch at 2.1GHz. 21 90 120 -30 60 -20 150 30 -10 180 0 o ?=-45 210 330 E E 240 300 270 (a) 90 120 60 -30 ?=45o 150 30 -20 -10 180 0 210 330 240 300 E E 270 (b) Fig. 2.11 (a) Measured radiation pattern at 1.98GHz (?=-45o plane). (b) Measured radiation pattern at 2.1GHz (?=45o plane). 22 An additional important bandwidth enhancement effect is also observed through the ragged edge shape. When the patch is restricted to single-mode operation by symmetry constraints, the ragged edges in the GA-optimized shape enhance the bandwidth by about 30% compared to the reference square microstrip. 72mm 72mm � 1.3 (a) (b) � 4.0 72mm � 3.0 (c) Fig. 2.12 Bandwidth enhancement is achieved by two-mode operation and ragged edges. 23 This is shown in Figs. 2.12(a) and 2.12(b). The ragged edges cause the broadening of the resonant frequency by introducing multiple resonant lengths between the two radiating edges on the two sides of the patch. Therefore, this ragged edge shape in conjunction with the two-mode operation in the GAoptimized design in Fig. 2.12(c) results in the broadest bandwidth possible (�0 compared to Fig. 2.12(a)). 2.4 Summary Optimized patch shapes for broadband microstrip antennas on thin FR-4 substrate have been investigated using the genetic algorithm. The optimized shape showed a four-fold improvement in bandwidth compared to a standard square microstrip. This result has been verified by laboratory measurement. The basic operating principle of the optimized shape can be explained in terms of a combination of two-mode operation and ragged edge shape. The number of iterations needed for GA to converge is in general quite large in this implementation. This results in long computation time during the design. The convergence rate could potentially be reduced by hybridizing GA with other optimization algorithms such as local search [23,24] or Tabu search [25]. 24 Chapter 3 Design of Multi-Band Microstrip Antennas Using a Genetic Algorithm 3.1 Introduction In the previous chapter, a design method for broadband microstrip antenna was developed. Arbitrary microstrip patch shapes with the broadest bandwidth were found using a GA. The designed antenna operates at the frequency band around 2GHz. However, with the growing demand in wireless applications, microstrip antennas that can operate at more than one frequency band have become an area of great research interest. This chapter examines the use of GA to design optimal shapes for microstrip antennas to achieve multi-band operation. Various multi-band methods for microstrip antennas have been proposed to date. For example, multi-layered structures, parasitic patches and shorting posts are some of the well-known techniques for achieving multi-band operation [14,15]. However, these techniques usually lead to an increase in antenna size or manufacturing cost. The design of dual-band microstrip antennas using GA was first addressed by Johnson and Rahmat-Samii [6]. The attractiveness of the GA design over the aforementioned methods is its ability to achieve the desired performance using a 25 single, unique patch shape. In their study, they used air as the substrate martial. Just as in the broadband design in Chapter 2, we focus on FR-4 as the substrate material here, as it is the most commercially viable material in wireless devices. First, dual-band microstrips are designed by optimizing the size of vertical slots on the metallic patch. Unfortunately, these antenna designs exhibit poor antenna efficiency. To solve the low efficiency problem, dual-band microstrips are designed without slots cut, but with an emphasis on optimizing the overall shape. The optimized patches can achieve arbitrary frequency spacing ranging from 1:1.1 to 1:2. Furthermore, tri-band and quad-band microstrip shapes are also generated and the resulting designs show good operation at the designated frequencies. All results are verified by laboratory measurements on FR-4 substrate. 3.2 Miniaturized Dual-Band Microstrip Using Slots-Cut 3.2.1 GA and Cost Function for Dual-Band Design GA is applied to the design of a dual-band microstrip using slots cut on the metallic patch. This dual-band design uses the same substrate material and substrate thickness as used in the broadband design. The design goal is to produce a good impedance match at the frequencies of 1 and 2GHz. In addition, the size of the patch is constrained to 42.5mm�mm. This patch size is 40% smaller than that of a standard square microstrip working at the frequency of 1GHz. It is 26 known that slots in microstrips have the effect of lowering the resonance frequency [21]. Therefore, a slot-encoding scheme as shown in Fig. 3.1 is adopted to encode each shape into a chromosome. Substrate Metal Feed Position 0 0 1 Fig. 3.1 0 1 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 Encoding scheme for the dual-band design. 27 1 1 0 0 1 1 0 1 1 0 1 1 Five equally spaced vertical lines are chosen in the patch area. These pre-selected lines are encoded into a binary chromosome where ones denote metal, and zeros denote non-metal. In addition, three more bits are added to the chromosome to represent the position of the feed along the dashed line on the patch. Since significant power dissipation is expected due to the introduction of the slots, metal loss is taken into account by adding a second term to the cost function. The first part of the cost function accounts for the impedance mismatch, in the same way as in (2.1). The second part of the cost function accounts for the metal loss generated by the current flowing on the patch. The conductivity of aluminum (?=3.82�7 S/m) is used. Cost = 1 N N ( Pn n =1 where Pn = Qn = + Qn ) (3.1) S11 (dB) + 10dB if S11 (dB) ? ? 10dB if S11 (dB) < ?10dB 0 PJ (dB) + 70dB if PJ (dB) ? ? 30dB if PJ (dB) < ? 30dB 0 Rs 2 J s ds (dB) 2 s w� Rs = 2? PJ = 28 3.2.2 Simulation and Measurement Results Fig. 3.2(a) shows the GA-optimized microstrip for dual-band operation. The white dot shows the position of the feed, and it resides 12.5mm from the left edge. Fig. 3.2(b) shows the measurement and the simulation results of the return loss. Other than a slight shift in the resonant frequencies, the graph shows good agreement between the measurement and the simulation. The bandwidths at the two frequencies of 1 and 2GHz are 1.2% and 1.37%, respectively. The measured relative gain of this microstrip shows a gain loss of ?7 dB at 1GHz and ?7.5 dB at 2GHz when compared to square microstrips operating at those two frequencies. To explain the gain loss, Fig. 3.3 shows the calculated metal loss (dBm) of the microstrips. The solid line is the metal loss of the GA-optimized microstrip, and the dashed lines are the losses of two reference square microstrips. At the low-band frequency (1GHz), the GA-optimized microstrip shows a metal loss 5 dB higher than the square microstrip of size 72mm�mm, while at the high-band it shows a metal loss 6 dB higher than the square microstrip of size 36mm�mm. These values show that the main cause for the gain loss comes from metal loss. It should be noted that a dual-band microstrip without any constraint on the metal loss has been also optimized, and the resulting design shows an even larger metal loss. Thus the GA design achieves better radiation efficiency by taking the metal loss on the patch into consideration. 29 40mm y x 42.5mm (a) S11 (dB) -5 -10 -15 -20 0.80 1.08 1.36 1.64 1.92 Frequency (GHz) 2.20 (b) Fig. 3.2 (a) GA-optimized slot shape for dual-band operation. (b) Return loss (dB) of the GA-optimized antenna from simulation ( ) and measurement (? ??). 30 30 25 Loss (dBm) 20 15 10 5 0 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Frequency (GHz) Fig. 3.3 Simulated total metal loss (dBm) of the GA-optimized dual-band microstrip (? ??), a square microstrip of 72mm�mm (------) and a square microstrip of 36mm�mm (-?-?-?-?). 3.2.3 Operating Principle of the GA Design To explain the miniaturization principle of the GA design, it should be noted that a microstrip of size 42.5mm�mm without any slots has a resonant frequency of 1.7GHz for the TM10 mode, and a resonant frequency of 3.4GHz for the TM20 mode. Fig. 3.4 shows the current plots for the GA-optimized microstrip with slots at the resonant frequencies of 1 and 2GHz. At the low band, the surface current distribution resembles the TM10 mode. 31 (a) (b) Fig. 3.4 (a) Current distribution on the surface of the GA-optimized patch at 1 GHz. (b) Current distribution at 2 GHz. 32 The excited surface current is forced through the narrow channel near the feed. Since the current path has to meander around the slots, the total length of the current path is increased. Therefore, the resonant frequency is reduced by a factor of about 1.7 compared to the microstrip without slots. At the high frequency band, the patch current resembles the TM20 mode. The currents meander around the two large vertical slots near the two vertical radiating edges on the patch. In contrast to the slots near the feed, these slots lower the resonant frequency for the high band. These two effects have been verified through numerical simulation. If the length of the large slots near the two radiating edges is slightly reduced, the resonant frequency of the high band is increased. If the length of the slots near the feed is reduced, the resonant frequency of the low band is increased. These results clearly show that the miniaturization of the GA-optimized design is achieved by the frequency lowering effect of the slots to both the low and high frequency bands of the microstrip. Figs. 3.5(a) and (b) illustrate the measured radiation patterns at the frequency of 1GHz and 2GHz respectively. The radiation plots exhibit broadside radiation and show less than ?20dB cross polarization level at both frequencies. In addition, the radiation patterns are almost identical at the two frequencies. 33 90 120 -10 60 -20 150 30 -30 180 0 210 330 240 300 E E 270 (a) 90 120 60 -10 -20 150 30 -30 180 0 210 330 240 300 270 E E (b) Fig. 3.5 (a) Measured radiation patterns at 1 GHz (?=0 plane). (b) Measured radiation pattern at 2 GHz (?=0 plane). 34 3.3 Multi-Band Microstrip Without Slots-cut In Section 3.2, a GA-designed dual-band microstrip on high-permittivity substrate based on slots cut into the patch was attempted. However, metal loss was found to be significant due to the use of the slots. Also only a particular dual-band frequency spacing ratio of 1:2 (1GHz and 2GHz) was investigated. This section examines the design of optimal patch shapes for multi-band operation. To reduce the metal loss, the patch shape without slots cut is investigated. Unlike the fixed spacing of dual-band operation, as reported in Section 3.2, the general set of frequency spacing between the two frequency bands is investigated in this section. The optimized patches achieve arbitrary frequency spacing ranging from 1:1.1 to 1:2. In addition to dual-band operation, tri-band and quad-band microstrip shapes are also generated. 3.3.1 GA and Cost Function for Multi-Band Design The GA methodology similar to one reported earlier for broadband application in Chapter 2 is implemented to optimize the patch shape. In this GA implementation, a 2-D chromosome is used to encode each patch shape into a binary map [19]. The metallic sub-patches are represented by ones and the nonmetallic areas are represented by zeros. Since it is more desirable to obtain optimized patch shapes that are well connected from the antenna efficiency and 35 manufacturing point of view, a 2-D median filter [20] is applied to the chromosomes to create a more realizable population at each generation of the GA. To evaluate the performance of each patch shape, a full-wave periodic patch code [17,18] is used. The design goal is to maximize antenna bandwidth at multiple frequency bands by changing the patch shape. To achieve the design goal, the cost function in (3.2) is defined as the average of those S11 values that exceed ?10dB (i.e., VSWR=2:1) within the frequency bands of interest. Cost = 1 N N n =1 where Pn = ( Pn ) (3.2) S11 (dB) + 10dB if S11 (dB) ? ? 10dB if S11 (dB) < ?10dB 0 Based on the cost function, the next generation is created by a reproduction process that uses crossover, mutation, and 2-D median filtering. A two-point crossover scheme involving three chromosomes is used. 3.3.2 GA and Cost Function for Dual-Band Design First, the dual-band design is carried out to achieve different frequency ratios between the low and the high bands. For each microstrip, the low frequency band is fixed at 1.8GHz, and the high frequency band is varied. The insets in Figs. 3.6(a), (b) and (c) are the GA-optimized designs for the frequency ratios of 1:1.3, 1:1.6 and 1:1.9, respectively. A 72mm � 72mm square design area on which the metallic patch can reside is discretized into a 32 � 32 grid for the chromosome 36 definition. The thickness of the FR-4 substrate (dielectric constant of about 4.3) is 1.6 mm. The white dot shows the position of the probe feed. In the same figure, the calculated return loss (S11 in dB) of the resulting microstrip is plotted as a dashed line. It shows good dual-band operation at the designed frequencies. To verify the GA design experimentally, the microstrip patches described above are built and measured. Copper tape is used to construct the metallic patches and the dimension of the ground plane is 15.3cm � 15.3cm. The measurements were taken on an HP8753C network analyzer. The solid lines in Fig. 3.6 are the measured return losses versus frequency. Good agreement can be observed between the measurements and simulations. The radiation patterns for these microstrips are also measured. All three microstrips show broadside radiation patterns at both operating frequencies, with linear polarizations that are nearly orthogonal to each other. The measured realized gains for these three antennas range from ?1.3dB to 2dB in the broadside direction. Due to the high loss tangent of FR-4, the dissipation in the antenna (and thus the radiation efficiency) is mainly dominated by dielectric loss. 37 S11 (dB) -5 -10 72mm -15 -20 1.65 2.03 2.41 2.79 3.17 3.55 Frequency (GHz) (a) S11 (dB) -5 -10 72mm -15 -20 1.65 2.03 2.41 2.79 3.17 3.55 Frequency (GHz) (b) Fig. 3.6 Shapes of three GA-optimized dual-band microstrip antennas, and the resulting return loss from simulation (------) and measurement (??). (a) Frequency ratio of 1:1.3 (1.8GHz and 2.34GHz). (b) Frequency ratio 1:1.6 (1.8GHz and 2.9GHz). (c) Frequency ratio of 1:1.9 (1.8GHz and 3.42GHz). 38 -10 -15 72mm S11 (dB) -5 -20 1.65 2.03 2.41 2.79 3.17 3.55 Frequency (GHz) (c) Fig. 3.6 (Cont' d) (c) Frequency ratio of 1:1.9 (1.8GHz and 3.42GHz). This was verified by running the simulation with and without dielectric loss, and found that dielectric loss causes a 4 to 8 dB loss in gain. In addition to the three frequency spacings presented in Fig. 3.6, designs for other frequency ratios ranging from 1:1.1 to 1:2 were also realized using GA. Figs. 3.7(a) to (g) are optimized microstrip shapes for other frequency spacings. 39 (f) Ratio 1:1.8 72mm (e) Ratio 1:1.7 72mm 72mm (d) Ratio 1:1.5 (c) Ratio 1:1.4 72mm (b) Ratio 1:1.2 72mm 72mm (a) Ratio 1:1.1 72mm (g) Ratio 1:2 Fig. 3.7 Shapes of dual-band microstrip antennas. Frequency ratio from 1:1.1 to 1:2. 40 S11 (dB) -5 -10 72mm -15 -20 1.40 1.68 1.96 2.10 2.52 2.80 Frequency (GHz) Fig. 3.8 Shape of the GA-optimized tri-band microstrip antenna that operates at 1.6GHz, 1.8GHz and 2.45GHz, and the resulting return loss of the antenna from simulation (------) and measurement (? ??). All designed shapes showed good dual-band operation at the two design frequencies. It was also numerically verified that these shapes could be scaled in size to different operating frequencies of interest or to other substrate materials with only minor modifications. Next, tri-band designs having three operating frequencies at 1.6GHz (GPS/L1), 1.8GHz (DCS) and 2.45GHz (ISM/Bluetooth) is attempted. Fig. 3.8 shows the optimized shape using the GA technique and the corresponding return 41 loss. It shows excellent tri-band operation at the design frequencies. The measured result again shows close agreement with the simulation result. The bandwidths obtained at these frequency bands are respectively 2.36%, 2.54% and 1.22% from simulation and 1.81%, 2.16% and 1.42% from measurement. Finally, quad-band designs having operating frequencies at 0.9GHz (GSM900), 1.6GHz, 1.8GHz and 2.45GHz are tried. Reasonably good quad-band operation is demonstrated in Fig. 3.9. Simulation shows a return loss of less than ?10dB (the design goal) at all four bands. S11 (dB) -5 -10 144mm -15 -20 0.80 Fig. 3.9 1.16 1.52 1.88 Frequency (GHz) 2.24 2.60 Shape of the GA-optimized quad-band microstrip antenna that operates at 0.9GHz, 1.6GHz, 1.8GHz and 2.45GHz, and the resulting return loss of the antenna from simulation (------) and measurement (? ??). 42 The measured result shows a return loss of less than ?10dB at the first, third and fourth band, while the second band has a slightly higher (?9.4dB) return loss. The results demonstrate that it is possible to use this GA approach in designing a multi-band microstrip that requires very specific frequency bands of operation. 3.4 Summary Optimized patch shapes for multi-band microstrip antennas have been investigated using GA. For the dual-band design with slots cut, the optimized shape showed good operation at both frequencies with a 40% reduction in size compared to a standard microstrip. The measurement result matched well with the numerical prediction. The operating principle of the optimized dual-band microstrip can be explained by the frequency lowering effect from the narrow slots for the different modes operating at the two frequency bands. Gain loss due to the presence of the slots, however, is intrinsic to the miniaturized design. It is only partially alleviated by the GA process. For the multi-band design without slots cut, it has been shown that frequency ratios between the two bands ranging from 1:1.1 to 1:2 can be achieved using the GA methodology. Tri-band and quad-band microstrip shapes have also been generated, and the resulting antennas showed good operation at the design frequencies. All results have been verified by laboratory measurements on FR-4 substrate. 43 Chapter 4 Shape Optimization of Corrugated Coatings Under Grazing Incidence Using a Genetic Algorithm 4.1 Introduction In this chapter, GA is utilized to design corrugated coatings to reduce the forward scattering under grazing incidence. Lossy material coatings are commonly used to reduce scattering from conducting bodies. In general, design of coatings should meet multiple criteria including low reflection, small volume and light weight. These design goals often conflict with one another. Multi-layer planar coatings have been studied extensively for their wideband absorbing characteristics [27,28]. GA have been applied with success in finding optimal thicknesses for multi-layer coatings in either planar or cylindrical configurations [29-30]. Corrugated coatings with non-planar profiles offer additional degrees of freedom and have been studied in [31-33]. In particular, it was shown in [34,35] that single-material corrugated coating can be exploited to alleviate polarization dependence and improve the absorption performance over a wide range of frequencies at near-grazing incidence. However, only a few simple shapes were considered. In this chapter, GA is used to explore more arbitrary coating shapes in an attempt to achieve better absorber performance. With more degrees of freedom 44 in the design, arbitrarily shaped coatings may give rise to better absorbing performance. At the same time, finding an optimal shape is more challenging as the design parameter space is much larger. In this approach, a full-wave EM simulation code is used to evaluate the absorbing performance of each shape. GA is implemented to optimize the shape of the coating, which is encoded into a binary chromosome. A two-point crossover scheme involving three chromosomes is used to achieve fast convergence. In addition, geometrical filtering is adopted to create more realizable shapes. This study focuses on the near-grazing incidence case. A single-layer MAGRAM material [36] is used for the coating. This method is first applied to achieve optimal shapes under various polarization constraints. The physical interpretation of the optimized structures is discussed and their performance is compared to the baseline results obtained from conventional planar and triangular shaped designs. Next, the Pareto GA [2,4] is employed to map the more general set of optimal solutions trading off coating thickness (or weight) versus absorbing characteristics. This chapter is organized as follows. Section 4.2 describes the EM simulation code utilized and other details in this GA implementation. Section 4.3 describes the application of the GA to the design of optimal coating shapes under various geometrical and polarization constraints. In Section 4.4, multi-objective optimization is applied to the design of coating, taking both absorbing 45 performance and coating thickness into account. Finally, Section 4.5 summarizes this chapter. 4.2 Approach 4.2.1 EM Simulation Code The geometry considered in this chapter is shown in Fig. 4.1. The shaped grooves in the coating have a period of p along the x direction and extend to infinity along the z direction. The bottom of the coating is backed by a conducting ground plane. A plane wave is obliquely incident upon the infinite grating with ?el and ?grating. y ?grating ?el x z Fig. 4.1 Geometry of the corrugated absorber. 46 To evaluate the performance of each shape for the coating, a full-wave EM code based on a boundary-integral equation formulation [35] is used. The formulation entails dividing one cell of the grating into different homogeneous regions according to the material layers. A homogeneous Green?s function is first used to calculate the moment matrix. Boundary integral equations are then obtained for each region. Field continuity at region interfaces and periodic boundary conditions at cell boundaries are then enforced. The fields in the top half-space are expanded into a sum of Floquet harmonics and are matched to the fields in the lower region so that the reflection coefficients can be found. The code has previously been validated by comparing the simulation results to measurement data in [34, 35]. 4.2.2 GA Optimization GA is employed to optimize the shape of the coating profile. In this GA implementation, each possible absorber shape is encoded into a binary chromosome, as shown in Fig. 4.2. The period of the absorber is divided into M points. The height of the coating at each point along x is represented as a binary number. A symmetry constraint is applied in the x-direction so that only the right half of the absorber is encoded into the chromosome. In order to obtain coating shapes that are not too complicated from the manufacturing point of view and to speed up convergence of the GA, a geometrical filter is applied to the chromosomes at each generation of the GA. 47 8mm 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 Period = 2.032mm Fig. 4.2 Encoding of corrugated absorber into a binary chromosome. Two different geometrical filters were tried: a 1-D sliding window filter [38] and a descending order filter. Figs. 4.3(a) and (b) show the shapes before and after the 1-D 7-point sliding window filter, which is a low-pass, moving-average filter. As expected, the surface shape after the filtering looks smoother without any sharp peaks. From the results of using this filter, it is found that the GA-optimized profiles consistently had shapes that monotonically decreased from a central peak. Therefore, an alternative ?descending order filter? was also tried. This filter simply rearranges the heights of the absorber at each of the M points so that the highest point is at the center and all other points are placed in descending order. 48 Sliding Window Filter (b) Descending Order Filter (a) (c) Fig. 4.3 (a) Before any geometrical filters. (b) After the 7-point sliding window filter. (c) After the descending order filter. 49 Fig. 4.3(c) shows the shape after the descending order filter. Note that this filter preserves the sharp edges in the design while making the shape less oscillatory. It is found that optimized shapes from the descending order filter gave better performance than that from the 1-D sliding window filter. Therefore, all the results presented in this chapter are generated by using the descending order filter. After these chromosomes are evaluated by the EM simulation code, a cost function related to the absorbing performance is computed. Based on the cost function, chromosomes are refined into the next generation by a reproduction process that involves crossover, mutation and geometrical filtering. For the crossover operation, a two-point crossover scheme involving three chromosomes is used. This series of processes is iterated until the cost function is minimized. 4.3 GA-Optimized Coating Shapes This section investigates coating profiles that produce the best absorbing characteristic for a given coating height. The design frequency band is chosen to be from 8 GHz to 18 GHz, and the maximum height of the coating is restricted to 8mm. To avoid higher order diffraction, the period of the coating is set to 2.032mm. A MAGRAM material is used for the coating (the detailed absorption characteristics can be found in Fig. 7 of [35]). The incident angle is ?el=30� and ?grating=0�. 50 To encode each possible shape of the coating into a binary string, the period of the coating is discretized into 30 points. The height of the groove at each point is described by a 6-bit number (i.e., in 64 steps) that ranges between 0 and 8mm. When the GA process converges to an optimal value, the discretization for the period and the height is increased to 60 points and 8 bits, respectively, to achieve a more refined coating shape. Associated with the design goal, the cost function is defined as the average of those reflection coefficient values, ?(dB), that exceed ?20dB within the frequency band of interest: Cost = 1 N N n =1 where Pn = ( Pn ) ?(dB) + 20 dB 0 (4.1) if ?(dB) ? ? 20 dB if ?(dB) < ?20 dB In the GA, the population size is 30. A crossover probability of 0.8 is used, and the probability of mutation is set to 0.1. The computational time is about 8 hours on a Pentium IV 1.7GHz machine for a typical design. First, only the reflection coefficient for the vertical polarization is used in the cost function definition. Fig. 4.4(a) shows the resulting GA-optimized shape, which closely resembles a triangular profile. Fig. 4.4(b) is a plot of the simulated reflection coefficient (in dB) versus frequency for the optimized shape. The reflection coefficient of the vertical polarization nearly meets the ?20 dB design 51 goal over the entire frequency band from 8GHz to 18GHz. The horizontal polarization is not optimized and shows a much higher reflection coefficient. Next, the reverse situation when only the horizontally polarized reflection coefficient is used in the cost function. Fig. 4.5(a) shows the resulting GAoptimized shape. The optimal shape of the corrugated coating resembles a rectangular profile. Fig. 4.5(b) shows the associated reflection coefficient (in dB) versus frequency for the optimized shape. In this case, the reflection coefficient of the horizontal polarization meets the ?20 dB design goal for all the frequencies above 10 GHz while the vertical polarization is higher. Further improvement in the low-frequency performance will likely require a thicker coating. Then this optimized shape is compared to the conventional planar and the triangular shaped coatings that are also optimized using GA. The maximum height of all three coatings is limited to the same 8mm thickness. Fig. 4.6 shows the horizontally polarized reflection coefficients for the three coatings. The dashed, solid, and dash-dotted lines are the respective reflection coefficients for the GA-optimized planar, triangular, and arbitrarily shaped coatings. The planar shaped coating (thickness of 1.19mm) shows a reflection of about ?5 dB within the frequency range of interest. 52 8 7 Height (mm) 6 E 5 4 3 2 MAGRAM 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Period = 2.032 mm (a) 0 H | ? | (dB) -10 -20 V -30 10 12 14 Frequency (GHz) 16 18 (b) Fig. 4.4 (a) GA-optimized shape for the vertical polarization. (b) Reflection coefficient (dB) versus frequency at 30 degrees from grazing. 53 8 7 Height (mm) 6 E 5 4 3 2 MAGRAM 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Period = 2.032 mm (a) 00 -5 | ? | (dB) -10-10 V -15 -20-20 H -25 -30-30 -35 10 12 14 Frequency (GHz) 16 18 (b) Fig. 4.5 (a) GA-optimized shape for the horizontal polarization. (b) Reflection coefficient (dB) versus frequency. 54 0 0 | ? | (dB) -10 -20 -30 10 Fig. 4.6 12 14 Frequency (GHz) 16 18 Performance comparison of the planar absorber (------), triangular shaped absorber (? ??), and GA-optimized absorber (-?-?-?-?) for the horizontal polarization. Using the triangular shaped profile (base thickness of 0.03mm and triangular height of 6.12mm) the reflection coefficient can be reduced to less than ?10dB. The GA-optimized arbitrarily shaped coating shows better absorbing performance in terms of the cost definition in (4.1) than either of the conventional designs. 55 8 (a) 7 0 (b) 6 5 4 3 2 -10 1 | ? | (dB) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (b) (c) (c) -20 (a) -30 0 8 Fig. 4.7 9 10 10 11 12 12 13 14 14 Frequency (GHz) 15 16 16 17 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 18 18 Effect of cost definitions on the optimization results for the horizontal polarization. (a) Cost definition in (4.1). (b) Minimax cost definition. (c) Average power reflection coefficient cost definition. Next, the sensitivity of the GA design is tested. To see the effects of different cost definitions, the designs are optimized using two other cost functions. The first alternative cost is the maximum reflection value across the whole frequency range of interest (typically called the Minimax cost function). The resulting performance is indicated by the dashed line in Fig. 4.7. The second alternative cost function is the averaged power reflection coefficient (on a linear scale) across the frequency band. This performance of design is indicated by the 56 dash-dotted line. Some difference in the overall performance is noted. However, it is observed that the optimized shapes retain the overall feature of the original design based on the cost definition in (4.1). To test the sensitivity of the GAoptimized shape to manufacturing tolerances, random RMS deviations of 0.4 mm are introduced into the profile height. The resulting performance is shown by the dashed line in Fig. 4.8(a). GA description of the profile is also intentionally undersampled as the 60-point by a factor of 6, resulting in a more smoothed-out profile. The performance is shown by the dash-dotted line. This shows that the performance is not too sensitive to the deviation to the optimized profile. Fig. 4.8(b) shows the performance of the optimized coating for close-by incident angles of 25o and 35o. The results indicate some degradation toward the smaller grazing angles. Optimizing the coating shape for both polarizations is also tried using the average of the reflection coefficients from the horizontal and vertical polarizations in the cost function. The resulting shape is shown in Fig. 4.9(a). As can be seen from the previous examples, the design for the horizontal polarization is more difficult than that for the vertical polarization. Therefore, in this case, the cost is dominated by the horizontal polarization and the resulting GA-optimized shape is not that different from that for the horizontal polarization shown in Fig. 4.5(a). 57 8 (ii) (i) 7 6 0 5 4 3 2 -10 1 | ? | (dB) 0 -20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 (i) 8 9 0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (iii) (iii) -30 0.2 (ii) 10 11 10 12 13 12 14 15 14 16 17 16 18 18 0 0.2 0.4 1.6 1.8 2 Frequency (GHz) (a) 0 | ? | (dB) -10 25o 30o -20 35o -30 8 9 10 10 11 12 13 14 15 12 14 Frequency (GHz) 16 16 17 18 18 (b) Fig. 4.8 (a) Performance sensitivity to variations in the shape of the profile for the horizontal polarization. (i) Original GA-optimized design. (ii) GAoptimized shape with RMS error of 0.4 mm. (iii) Smoother shape after 6:1 undersampling. (b) Reflection coefficient (dB) versus frequency at 25o, 30o and 35o from grazing. 58 8 7 Height (mm) 6 5 4 3 2 MAGRAM 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Period = 2.032 mm (a) 0 (V+H)/2 | ? | (dB) -10 V -20 H -30 8 9 10 10 11 12 12 13 14 14 15 16 16 17 18 18 Frequency (GHz) (b) Fig. 4.9 (a) GA-optimized shape taking into account of both the vertical and horizontal polarizations. (b) Reflection coefficient (dB) versus frequency. 59 Finally, an interpretation on the operating principle of the GA optimized shape is attempted. The left side of Fig. 4.10 is a planar absorber. Typically it is more difficult for the horizontal polarization to infiltrate an absorber near grazing than the vertical polarization. However, if we look at the incident electric field on the near-vertical sidewalls of the profile shown on the right side of Fig. 4.10, it behaves more like the vertical polarization. Thus, the absorbing performance is improved by effectively changing the horizontal polarization into the vertical polarization. This explains why the optimized design for the horizontal polarization resembles a rectangular profile. Other researchers have discussed the difference in coupling into corrugated profiles depending on polarization [39]. E E (a) (b) Fig. 4.10 (a) Planar absorber with horizontal polarized wave incidence. (b) Rectangular profile with horizontal polarized wave incidence. 60 4.4 Multi-Objective Optimization 4.4.1 Pareto GA and Cost Function Definition In addition to the absorbing performance of the coating, another design criterion of interest is the coating volume, which is measured by the coating height. An investigation of the absorbing performance versus the coating height is studied. This can be done by repeatedly using the same methodology described in Section 4.3 for various heights. However, it is much more efficient to cast this problem into a multi-objective problem rather than using the conventional GA. Pareto GA [2,4] is a useful tool for this problem. In the Pareto GA, a wide range of solutions corresponding to more than one objective can be mapped by running the optimization only once. In this GA implementation, two cost functions are defined: Cost 1 = Normalized coating height Cost 2 = Normalized value of where Pn = 1 N N n =1 (4.2) ( Pn ) (?? + ?| | ) / 2 (dB) + 20 dB if (?? + ?|| ) / 2 (dB) ? ? 20 dB 0 if (?? + ?|| ) / 2 (dB) < ?20 dB (4.3) Cost 1 is determined by the coating height and Cost 2 is associated with the reflection cost. Both costs are normalized to a value between zero and one. For 61 Cost 2, one denotes an average reflection coefficient of 0dB, while zero denotes an average reflection coefficient that is below ?20dB. The non-dominated sorting method [40] is used to combine the two costs for each solution by means of the Pareto ranking. This method assigns rank 1 to the non-dominated solutions of the population. The term non-dominated solution means that there are no other solutions that are superior to this solution in both objectives. Then the next nondominated solutions among the remaining solutions are assigned to the nexthighest rank. The process is iterated until all the solutions in the population are ranked. Based on the rank, the same reproduction process described in Section 4.3 is performed to refine the population into the next generation. The set of rank 1 solutions is called the Pareto front. In order to avoid the solutions on the Pareto front from converging to a single point in the cost space, a sharing scheme described in [41] is performed. In this sharing process, the rank is modified by penalizing those members on the front that are too close to each other in the cost space. This is accomplished by multiplying a niche count (mi) to the assigned rank. The niche count is calculated according to: N 1 p mi = Sh(d ij ) ( 4. 4 ) N p j =1 where the Np is the number of rank 1 members and the sharing function, Sh(dij), is a function of the cost distance between solutions expressed as: 62 Sh(d ij ) = 2? d ij d share 1 if d ij < d share (4.5) if d ij > d share and d ij = (Cost 1(i) ? Cost 1( j ) )2 + (Cost 2(i ) ? Cost 2( j ) )2 As can be seen, the sharing function increases linearly if the other members on the front are closer than dshare from a chosen member i in the cost space. Consequently, those members that have close-by neighbors in the cost space are assigned lower ranks in the reproduction process. 4.4.2 Pareto GA Results In this Pareto GA, the population size is chosen to be 100. A crossover probability of 0.8, a mutation probability of 0.1 and a dshare distance of 1 are used. Figs. 4.11(a)-(d) show the convergence of the solutions for this multi-objective problem (reflection cost versus the height of the profile) as the number of generations is increased. The period of the absorber, the material for the coating, and the angle of incidence are the same as those used in Section 4.3. The height is constrained to be less than 8mm. Fig. 4.11(a) is the plot of the initial population. The majority of the solutions are located in the upper-right side of the cost domain. Figs. 4.11(b)-(d) are plots of the population after 5, 20 and 200 generations, respectively. They show that as the number of generations increases, 63 the Pareto front spreads out and converges toward the lower-left region of the cost space. Fig. 4.12 shows the final converged Pareto front and four optimized coating shapes that are on the front. Inset shape (a) shows the lowest profile of the four samples, but it has the highest reflection among the four designs. Inset shape (d) has the highest profile and the lowest reflection. As expected, the absorbing performance must be traded off against the profile height. If we look in detail at the optimized shapes, it appears that as the height of the absorber decreases, the top of the profile gets more flattened. However, the shapes maintain a rectangular profile that is only slightly modified by the coating height. This is consistent with the physical interpretation of the absorption process for the more dominant horizontal polarization discussed in Section 4.3. Another observation from Fig. 4.12 is that the Pareto front is not smooth due to the quantization effect of the coating height. If the height is discretized with more binary bits, the shape of the Pareto front becomes smoother. Next, the first cost is changed from coating height to coating weight while keeping the reflection cost the same. The cost for the coating weight is normalized to be 1 when all of the design area (period � maximum coating height) is filled by the coating material while it is zero when no coating material exists. Fig. 4.13 shows the converged Pareto front for this problem. Also shown in insets (a) to (d) are four optimized shapes with different coating weights. 64 0.45 0.45 0.4 0.4 0.35 0.3 0.25 0.2 A bsorbing P erformance 0.5 Reflection cost Absorbing Performance Reflection cost 0.5 0.3 0.25 0.2 0.15 0.1 0.35 0.15 4 4.5 5 5.5 6 6.5 7 7.5 0.1 8 4 4.5 Height of absorber (mm) 5 6.5 7 7.5 8 (b) 0.5 0.45 0.45 0.4 0.4 0.35 0.3 0.25 0.2 A bsorbing P erformance 0.5 Reflection cost Absorbing Perform ance Reflection cost 6 Height of absorber (mm) (a) 0.35 0.3 0.25 0.2 0.15 0.15 0.1 5.5 0.1 4 4.5 5 5.5 6 6.5 7 7.5 8 Height of absorber (mm) 4.5 5 5.5 6 6.5 Heigh of the profile (mm) 7 7.5 Height of absorber (mm) (c) Fig. 4.11 4 (d) Convergence of the Pareto front as a function of the number of generations for absorbing performance versus absorber height. (a) Initial population. (b) After 5 generations. (c) After 20 generations. (d) After 200 generations. 65 8 (a) (b) 0.5 (c) 0.45 Reflection cost 0.4 (d) 0.35 0.3 0.25 0.2 0.15 0.1 Fig. 4.12 4 4.5 5 5.5 6 6.5 Height of profile (mm) 7 7.5 8 Final converged Pareto front of absorbing performance versus absorber height. The insets show four sample designs on the Pareto front. 66 (a) (b) 0.7 (c) 0.6 Reflection cost 0.5 0.4 (d) 0.3 0.2 0.1 0.15 0.2 0.25 0.3 Normalized Weight 0.35 0.4 Weight of the absorber Fig. 4.13 Final converged Pareto front of absorbing performance versus absorber weight. The insets show four sample designs on the Pareto front. 67 It should be noticed that inset (d) is very similar in shape to inset (d) in Fig. 4.12. However, instead of trimming the top off in order to reduce the height, the weight consideration results in designs that become progressively skinnier, as shown by insets (c), (b) and (a). Nevertheless, the shapes still preserve the sharp sidewalls as those presented in Fig. 4.12. 4.5 Summary Optimized shapes for a corrugated absorber under near-grazing incidence have been investigated using GA. First, GA was applied to design corrugated coating depending on incident polarizations. The designed absorber shape for the vertical polarization resembled a triangular profile, while that for the horizontal polarization resembled a rectangular profile. The optimized shapes were compared to canonical planar and triangular shaped designs, and were shown to have better absorbing performance. The sensitivity of the designs to variations in shape and incident angles were also tested, and the results showed reasonable tolerance. A physical interpretation for the optimized shape was presented. It was shown that the sharp sidewalls of the resulting shape effectively changed the incident polarization from horizontal to the vertical case, thus facilitating wave absorption. The Pareto GA has also been applied to efficiently map out absorbing performance versus absorber height. The non-dominated sorting method was used 68 to combine the two costs for each solution by means of the Pareto ranking. A sharing scheme was implemented to avoid the solutions on the Pareto front from converging to a single point in the cost space. The converged Pareto front showed that better absorbing performance must be traded off against absorber height. Similar conclusions were also found for the absorbing performance versus absorber weight. 69 Chapter 5 Design of Electrically Small Wire Antennas Using a Pareto Genetic Algorithm 5.1 Introduction In Chapter 2 and 3, the basic GA for microstrip antenna design was introduced. In Chapter 4, the more advanced Pareto GA was investigated to solve the more challenging multi-objective problem. In this chapter, the developed methodology is applied to the design of electrically small wire antennas. As the size of wireless devices shrinks, the design of electrically small antennas is an area of growing interest [42,43]. By the classical definition, an electrically small antenna is one that can be enclosed in a volume of radius r much less than a quarter of a wavelength. It is well known that the bandwidth of an electrically small antenna decreases as the third power of the radius [44-46]. Much research has been carried out to increase the bandwidth of small antennas using structures such as folded design, disk-loaded monopole, inverted-L or inverted-F designs, multi-armed spiral and conical helix [47-50]. Fig. 5.1 shows the bandwidth performance of these antennas in terms of the normalized antenna size kr, where k=2?/? is the wave number. Recently, Altshuler reported on the use of a GA in designing electrically small wire antennas [9]. Instead of using a 70 regular shape, he used GA to search for an arbitrary wire configuration in 3-D space that results in maximum bandwidth for a given antenna size. While much of the small antenna research has been focused on antenna bandwidth, antenna miniaturization also impacts antenna efficiency. The objective of this chapter is to apply GA in the design optimization of electrically small wire antennas, taking into account of bandwidth, efficiency and antenna size. To efficiently map out this multi-objective problem, the Pareto GA [2,4] is utilized. 3dB Bandwidth (%) Fundamental Limit (L. J. Chu, 1948) 100 Goubau [50] 50 Dobbins & Rogers [49] 30 Foltz & McLean [48] 20 0.6 Fig. 5.1 0.8 1.2 Antenna size (kr) 1/4? ? monopole 1.6 Achievable bandwidth in terms of antenna size and some examples of small antennas. 71 The concept of divided range multi-objective [51] is employed to accelerate convergence in the GA process. In this GA approach, the multisegment wire structure similar to the one used in [9] is employed. The Numerical Electromagnetics Code (NEC) [52] is used to predict the performance of each wire structure. Then an optimal set of designs is generated by considering bandwidth, efficiency and antenna size. To verify the GA results, several GA designs are built, measured and compared to the simulation. Physical interpretations of the GA-optimized structures, showing the different operating principles depending on the antenna size are also provided. The performance curve achieved by the GA approach is compared against the well-known fundamental limit for small antennas [44-46]. To more easily assess the performance of the antennas, the efficiency-bandwidth product is normalized by the antenna size in order to represent the antenna performance as a single figureof-merit [54]. Finally, further improvement of the GA results is attempted by exploring additional geometrical design freedoms to better approach the fundamental limit. This chapter is organized as follows. In Section 5.2, the details of the GA implementation are described. Section 5.3 describes the GA designs and the measurement verification of the results. In Section 5.4, the GA results are compared to the fundamental limits. In Section 5.5, other design freedoms are 72 explored to further improve performance. Section 5.6 provides conclusions gathered from this research. 5.2 Pareto GA Approach The basic antenna configuration considered in this chapter is shown in Fig. 5.2. The antenna consists of M connected wire segments. Each segment of the antenna is confined in a hemispheric design space with a radius r and an infinite ground plane. r Fig. 5.2 Configuration of the multi-segment wire antenna used in the GA design. The three design goals are: broad bandwidth, high efficiency and small antenna size. The Pareto GA is employed to efficiently map out this multi- 73 objective problem. The advantage of using the Pareto GA over the conventional GA is that a wide range of solutions corresponding to more than one objective can be mapped by running the optimization only once. Binary Chromosome Antenna Shape 1 0 0 0 1 0 1 1 0 0 1 1 Fig. 5.3 0 1 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 Encoding of the wire configuration into a binary chromosome. In this GA implementation, the hemispheric design space is evenly n discretized into 2 grid points, and the location of each joint of the antenna is encoded into an n-bit binary string, as shown in Fig. 5.3. Thus, the total number of bits in the chromosome is nM when M connected wire segments are used. The three costs associated with these design goals are: 74 Cost 1 = 1 ? Antenna Bandwidth Theoretical Bandwidth Limit (5.1) Cost 2 = 1 ? Efficiency Cost 3 = Antenna Size ( kr ) In the above definition, the theoretical bandwidth limit of 1/((1/kr)+(1/kr)3) derived in [46] is used. NEC [52] is used to predict the antenna performance in order to compute the cost functions. Multiple Linux machines are used in parallel to carry out this computation. After evaluating the three cost functions of each sample structure using NEC, all the samples of the population are ranked using the non-dominated sorting method [40]. Here, the higher the rank (1 denotes the highest rank), the better the solution. This method assigns rank1 to the non-dominated solutions of the population. The term non-dominated solution means that there are no other solutions that are superior to this solution with respect to all design objectives. Then the next non-dominated solutions among the remaining solutions are assigned rank2. The process is iterated until all the solutions in the population are ranked. Based on the rank, a reproduction process is performed to refine the population into the next generation. The set of rank1 solutions is termed the Pareto front. By favoring the higher-ranked solutions in the reproduction process, the Pareto GA tries to push the Pareto front as close to the optimum solution in the cost space as possible. 75 For the crossover operation, a two-point crossover scheme involving three chromosomes is used. For numerical stability, a geometrical check is applied to prevent the wires from intersecting one another. In order to avoid the solutions from converging to a single point, a sharing scheme as described in [41] is performed to generate a well-dispersed population. In the sharing process, the rank is modified by penalizing those members on the front that are too close to each other in the cost space. This is accomplished by multiplying a niche count (mi) to the assigned rank. The niche count is calculated according to: 1 mi = Np Np j =1 Sh (dij ) (5.2) where the Np is the number of rank-1 members and the sharing function, Sh(dij), is a function of the cost distance between solutions expressed as: Sh (dij ) = 2? 1 dij = dij dshare if dij < dshare (5.3) if dij > dshare ( Cost 1(i) ? Cost 1( j) )2 + ( Cost 2 (i) ? Cost 2 ( j) )2 + ( Cost 3 (i) ? Cost 3 ( j) )2 The sharing function increases linearly if the other members on the front are closer than dshare from a chosen member i in the cost space. Consequently, 76 those members that have close-by neighbors in the cost space are assigned lower ranks in the reproduction process. The standard Pareto GA did not always give satisfactory results in this problem, since it is much harder for small-sized antennas to converge than for large-sized antennas. Thus, large-sized antennas usually dominate the whole population after several generations of the GA process. Cost1 (Size) Range 1 Range 2 Cost3 (Efficiency) Range 3 Cost2 (Bandwidth) Fig. 5.4 Divided range multi-objective GA approach. 77 As a result, the final Pareto front contains only antenna designs with sizes kr?0.45. To avoid this bias, the concept of the divided range multi-objective GA [51] is employed in this implementation. As shown in Fig. 5.4, the size is partitioned into multiple ranges and carry out the Pareto GA on each range individually. After each range converges to an optimal solution, the populations from all ranges are merged and the combined population is optimized in the last step of this process. This algorithm shows much improved performance for this more difficult multi-objective problem. Using the scheme good results for smallsized antennas (kr<0.45) as well as large-sized ones (kr>0.45) can be achieved. 5.3 GA-Optimized Results 5.3.1 GA Optimized Designs This section investigates the optimal antenna shapes that produce the best efficiency-bandwidth (EB) product for a given antenna size. Seven wire segments are used in the antenna configuration. The 3-D hemisphere with radius r is discretized into 215 points and the locations of the 7 wire segments are encoded as a binary chromosome of 7� bits. In addition, an extra three bits are added for choosing 8 different wire conductivities varying from 1�6 (graphite) to 5.7�7 (copper). The population size is chosen to be 2000 and the population is divided into four sub ranges (0.29?kr<0.38, 78 0.38?kr<0.46, 0.46?kr<0.54 and 0.54?kr<0.63). Each of four ranges has a population size of 500. A crossover probability of 0.8, a mutation probability of 0.1 and a dshare distance of 1 are used. The target design frequency is chosen at 400 MHz and an infinite ground plane is assumed in the simulation. All antennas are designed to match to a 50 ? impedance. The total computational time is about 20 hours using four Pentium IV 1.7GHz machines running in parallel. Figs. 5.5 (a), (b) and (c) show the designs in the population with a rank of 1 at respectively, the initial, 200 and 1000 generations of the Pareto GA process. The three axes are bandwidth, efficiency and antenna size. Each dot represents a particular rank-1 design. In the initial generation, only a few rank-1 solutions exist. After 200 generations, many more rank1 solutions appear. After 1000 generations, a large portion of solutions (770 over 2000) is on the Pareto front. The solutions are relatively well spread out over the Pareto front due to the sharing operation. 79 (a) Size (kr) 0.6 0.55 0.5 0.45 0.4 0.35 0.3 2 1.8 1.5 1 1.6 1.4 Efficiency log10(%) 0.5 Bandwidth log10(%) 0 (c) 0.6 C Size (kr) 0.55 0.5 B 0.45 0.4 A 0.35 0.3 (b) 2 1.8 1.5 1 1.6 Size (kr) 0.6 1.4 Efficiency log10(%) 0.55 0.5 0.5 0 Bandwidth log10(%) 0.45 0.4 0.35 0.3 2 1.8 1.5 1 1.6 1.4 Efficiency log10(%) 0.5 0 Bandwidth log10(%) Fig. 5.5 Convergence of the Pareto front as a function of the number of generations in terms of bandwidth, efficiency and antenna size. (a) Initial generation. (b) After 200 generations. (c) After 1000 generations. 80 5.3.2 Verification of the GA-Optimized Results To verify the GA results, three GA-optimized designs are selected from the Pareto front (at points A (kr=0.32), B (kr=0.42) and C (kr=0.5)) and are shown in Fig. 5.6. The smallest sample, at point A, somewhat resembles a helix, while the largest sample C resembles a complicated loop where the end of the wire is connected to the ground plane. These three designs were built and their performances were measured. Copper wire of radius 0.5mm is used, and a 1.6m � 1.6m conducting plate as the ground plane is used. Fig. 5.7(a) is a photo of design B and Fig. 5.7(b) is the resulting return loss (dB) as a function of frequency by simulation and measurement. Except for a slight (3%) shift in the resonant frequency, the simulation and measurement results show nearly the same bandwidth (about 5.3% based on |S11| ? -3dB). Fig. 5.7(c) is the resulting efficiency of the antenna. The standard Wheeler cap method [55,56] is used to measure the efficiency. The measured efficiency matches the simulation well at the resonant frequency of 400 MHz, as indicated by the arrow in Fig. 5.7(c). At other frequencies, the agreement is also good except in the neighborhood of 385MHz. The presence of the large efficiency dip based on the measured data is due to an anti-resonance in the antenna, as the Wheeler cap method fails near this anti-resonance. 81 (a) kr=0.34 (c) kr=0.50 (b) kr=0.42 Fig. 5.6 Three samples from the Pareto front (a) kr=0.34, (b) kr=0.42 and (c) kr=0.50. 82 (a) (c) 0 100 -1 90 -2 80 -3 70 Efficiency (%) S11 (dB) (b) -4 -5 -6 -7 60 50 30 -8 20 -9 10 -10 350 360 370 380 390 400 410 420 430 440 450 0 350 Simulation 360 370 380 390 400 410 420 Frequency (MHz) Frequency (MHz) Fig. 5.7 Measurement 40 (a) Photo for antenna B, which has an antenna size of kr=0.42. (b) Return loss and (c) efficiency versus frequency of antenna B. The efficiency measurement was done using the Wheeler cap method. 83 430 440 450 Similar good agreements were also found for antennas A (kr=0.34) and C (kr=0.50). The results are summarized in Table 5.1. It is noted that both the achievable bandwidth and efficiency drop as the antenna size is reduced. Measured BW (3dB) Simulated BW (3dB) Measured Eff (%) Simulated Eff (%) Antenna A 2.1 % 2.5 % 84 % 88 % Antenna B 5.3 % 5.5 % 92 % 94 % Antenna C 8.5 % 9.8 % 94 % 97 % Table 5.1 Bandwidth and efficiency for the sample antennas A, B and C by measurement and simulation. 5.3.3 Physical Interpretation of GA-Optimized Design By examining the GA-optimized antenna structure in Fig. 5.6(c), we see that the end of the antenna is connected to the ground plane. Most of the antennas with 0.45<kr<0.65 have this characteristic, which is similar to a folded monopole antenna. Since a folded monopole has four times the input impedance of a standard monopole [57], the GA-designed antennas use this basic structure to boost up the impedance of the antenna to approach 50 ?. As we examine at the GA-optimized structures for even smaller-sized antennas (kr<0.45) such as those in Figs. 5.6(a) and (b), most of the antennas are shorted to the ground plane at the joint between the first and second segments 84 from the feed. This turns the first segment into an inductive feed. Segments 2 through 7 become the radiating part of the antenna, carrying most of the current. The strength of the inductive coupling depends on the distance between the first and second segments. Since inductive coupling can greatly increase the input impedance, the GA finds this as an optimized structure for very small-sized antennas (kr<0.45), which need a large impedance step-up to get to 50 ?. This concept is currently investigated in more detail in order to design very small antennas [58]. 5.4 Comparison to Fundamental Limit In this section, the GA results are compared to the fundamental limits for small antennas. Fig. 5.8 depicts all of the GA-optimized designs on the Pareto front plotted in the 3-D bandwidth, efficiency and antenna size space. A least squares fit is used a to create a surface that best fits the GA results. To more easily interpret the results, the 3-D plot is projected onto three planes, and the results are shown in Figs. 5.9(a), 5.9(b) and 5.9(c). Then these GA results are compared to the well-known fundamental limit using a combination of equations in [45] and [46]: BW = 1 Eff 譗 Where Q = (5.4) 1 1 + 3 kr (kr) 85 Antenna size (kr) C B A Efficiency log10(%) Fig. 5.8 Bandwidth log10(%) Pareto front of the GA designs after convergence. The surface is generated using a least squares fitting to best fit the GA results shown as dots. The curves based on (5.4) are shown in Figs. 5.9(d)-5.9(f). Fig. 5.9(a) shows the maximum bandwidth curve achievable by the GA designs as a function of antenna size for different efficiencies. As expected, for a given efficiency, the achievable bandwidth decreases as the antenna size is reduced. Also, the higher the efficiency, the lower the achievable bandwidth. It is similar to the trend of the 86 fundamental limit in Fig. 5.9(d). However, the GA performance is lower than the fundamental limit. Fig. 5.9(b) is the projection of GA designs on the antenna size versus efficiency plane. For a given bandwidth, the achievable efficiency decreases as the antenna size is reduced. This trend can also be observed in Fig. 5.9(e). Fig. 5.9(c) is the projection of the GA results on the bandwidth versus efficiency plane. For a given antenna size, the tradeoff between antenna bandwidth and antenna efficiency can be clearly seen. Again, the trend on this graph is similar to the fundamental limit in Fig. 5.9(f). To make the antenna performance easier to assess, the figure-of-merit ? suggested in [54] is used: ?= Eff � BW 2 Q (5.5) where Q is given in (5.4) and an extra factor of 2 is used to account for the loaded Q. Using this expression, the fundamental limit on ? is always 1 for antennas of arbitrary sizes. All of the GA-optimized designs are re-plotted using this figureof-merit in Fig. 5.10(a). As a reference, the disk-loaded monopole from [48] is plotted on the same figure. As can seen in Fig. 5.10(a), most of the GA designs are ?<0.5. The smaller the antennas, the worse the performance of the GA designs. Next the way to further improvement of the GA designs is investigated. 87 (a) 50 40 50 40 30 30 Bandwidth (%) Bandwidth (%) (d) 20 10 Eff Eff Eff Eff 1 0.3 0.35 0.4 0.45 0.5 = = = = 0.55 40% 55% 70% 85% 20 10 1 0.3 0.6 Eff Eff Eff Eff 0.35 Antenna size (kr) 0.4 0.45 0.5 = = = = 0.55 40% 55% 70% 85% 0.6 Antenna size (kr) (b) (e) Efficiency (%) 80 BW BW BW BW = = = = 100 5% 10% 20% 30% 80 Efficiency (%) 100 60 40 20 0.3 0.35 0.4 0.45 0.5 0.55 60 40 20 0.3 0.6 BW BW BW BW 0.35 Antenna size (kr) 0.4 0.45 0.5 = = = = 0.55 5% 10% 20% 30% 0.6 Antenna size (kr) (f) 100 100 80 80 Efficiency (%) Efficiency (%) (c) 60 40 20 kr kr kr kr 1 = = = = 0.34 0.42 0.50 0.59 10 20 30 40 20 40 50 Bandwidth (%) Fig. 5.9 60 kr = kr = kr = kr = 1 0.34 0.42 0.50 0.59 10 20 30 40 50 Bandwidth (%) (a) Projection of the Pareto front onto the size and efficiency plane. (b) Projection of the Pareto front onto the size and bandwidth plane. (c) Projection of the Pareto front to the bandwidth and efficiency plane. (d) through (f) show the corresponding fundamental limit based on (5.4). 88 5.5 Further Improvement on GA Designs To bring the figure-of-merit of the designs even closer to the fundamental limit, additional design freedoms are explored to the original 7-wire configuration. First, the number of segments is increased up to 16 wires. However, these results show almost no improvement compared to the original 7wire ones. Next, the selection of characteristic impedance is permitted to vary from 1 ? to 300 ?, instead of requiring a fixed 50 ? for the input port. This is the assumption used in the work of Altshuler [9], who assumes that a perfect impedance transformer is available. When allowed this freedom, the GA produced the results plotted in Fig. 5.10(b). As the graph shows, using variable characteristic impedance only gives a slight improvement in performance over the original design plotted in Fig. 5.10(a). Then a multi-arm configuration is examined to improve antenna performance. Two arms are used for the antennas structure and the resulting figure-of-merits are shown in Fig. 5.10(c). It shows good improvement for largesized antennas (kr>0.45). Using the multi-arm configuration, the antenna efficiency is increased while the antenna bandwidth is preserved. This is achieved by spreading the current on multiple branches and lowering the power dissipation. However, the multi-arm design does not show much improvement for small-sized antennas (kr<0.45). The number of arms is further increased to four, but adding more number of arms does not show more improvement over the two89 arm design. It is due to the difficulty in packing a multi-arm structure in a limited design space. Finally, two different wire radii are allowed in each design as an additional degree of design freedom. The resulting antenna has one wire radius for the lower portion of the antenna and another radius for the upper portion. Stepping the wire radius has the effect of increasing the input impedance, similar to the way a radius step up is used in a folded monopole design. Since it is known that the NEC version 2 used in the simulation does not accurately model the wire radius change, the radius change is limited to less than two. Fig. 5.10(d) shows the result of the design. It shows improved performance for both small-sized antennas and large-sized antennas. Based on these preliminary results, it appears that the use of targeted design freedoms can further improve the performance of the GA-optimized designs toward the fundamental limit. 5.6 Summary The Pareto GA has been applied to design electrically small wire antennas by considering antenna bandwidth, efficiency and size. Wire structures comprising of multiple segments were considered. The key advantage of using the Pareto GA method is that a whole series of optimal designs of varying size, bandwidth and efficiency can be generated efficiently in a single GA run. 90 �- 7-segment wire (a) 1.2 Fundamental Limit (2/Q) 1 Figure-of-merit (?) [48] 0.8 0.6 C B A 0.4 0.2 0 0.3 0.35 0.4 0.45 0.5 Antenna size (kr) 0.55 0.6 0.65 �- Variable characteristic impedance (b) 1.2 Fundamental Limit (2/Q) Figure-of-merit (?) 1 [48] 0.8 0.6 0.4 0.2 0 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Antenna size (kr) Fig. 5.10 Small antenna performance using the definition of ? = (Eff � BW) / Theoretical BW Limit. (a) Original 7-wire configuration. (b) Variable input impedance. 91 �- Multi-arm (c) 1.2 Fundamental Limit (2/Q) 1 [48] Figure-of-merit (?) 0.8 0.6 0.4 0.2 0 0.3 0.35 0.4 0.45 0.5 Antenna size (kr) 0.55 0.6 0.65 �- Radius step up (d) 1.2 Fundamental Limit (2/Q) 1 Figure-of-merit (?) [48] 0.8 0.6 0.4 0.2 0 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Antenna size (kr) Fig. 5.10 (Cont' d) Small antenna performance using the definition of ? = (Eff � BW) / Theoretical BW Limit. (c) Multi-arm configuration. (d) Multiple wire radii. 92 By incorporating the concept of the divided range GA, a well-formed Pareto front in terms of the three objectives was achieved. To verify the GA results, several antennas based on the GA designs were built and their bandwidth and efficiency were measured. Both the bandwidth and efficiency measurements agreed well with the simulation for all the sample antennas. Based on the Pareto front, it was observed that for a given antenna size, broader antenna bandwidth must be traded off against lower antenna efficiency. The results also showed that the maximum achievable bandwidth and efficiency both decrease as the antenna size is reduced. The performance achieved by the GA designs was also compared against the well-known fundamental limit for small antennas. The resulting GA designs followed the trend of the fundamental limit, but were about a factor of two below the limit. To further improve the performance of the GA-designed antennas, other design freedoms were explored such as variable characteristic impedance, multiarm wires and multiple wire radii. Results showed that the use of targeted design freedoms could further improve the optimization performance toward the fundamental limit. 93 Chapter 6 Conclusions In this dissertation, GA has been applied to three classes of EM design problems: microstrip antennas, low-profile microwave absorbers and electrically small wire antennas. In Chapter 2, GA-optimized patch shapes for broadband microstrip antennas on thin FR-4 substrate was investigated. The optimized shape showed a four-fold improvement in bandwidth compared to the standard square microstrip. This result was verified by laboratory measurement. The basic operating principle of the optimized shape can be explained in terms of a combination of two-mode operation and ragged edge shape. Chapter 3 examined GA-optimized patch shapes for multi-band microstrip antennas. It was shown that for dual-band operation, a frequency ratio ranging from 1:1.1 to 1:2 between the two bands can be achieved using the GA methodology. Tri-band and quad-band microstrip shapes were also generated, and the resulting antennas performed well at the design frequencies. All results were verified by laboratory measurements. In Chapter 4, GA-optimized shapes for corrugated microwave absorbers under near-grazing incidence were investigated. Corrugated coatings were studied 94 for different incident polarizations, and a physical interpretation for the optimized shapes was presented. The Pareto GA was applied to efficiently map out the absorbing performance versus absorber height. The converged Pareto front showed that better absorbing performance must be traded off against absorber height. Chapter 5 described the use of the Pareto GA to design electrically small wire antennas by considering the bandwidth, efficiency and antenna size. Using the concept of divided-range GA, a well-formed Pareto front in terms of all three objectives was achieved. The performance achieved by the GA designs was compared against the well-known fundamental limit for small antennas. The resulting GA designs followed the trend of the fundamental limit but were below the limit by approximately a factor of two. Further improvement of the GAdesigned antennas was tried by incorporating additional design freedoms. In this dissertation, it has been demonstrated that the GA methodology is very effective in solving real-world EM design problems. It has also been shown that GA is highly adaptable to a variety of design problems. However, a detailed understanding of the actual physics of a particular problem is essential for the success of GA. The physics of the problem should be incorporated into such GA process as chromosome encoding and geometrical filtering to achieve fast convergence and satisfactory solutions. Another consideration is that the total computation time needed for the GA is usually much longer than deterministic 95 algorithms. To solve the time issue, parallel computation was implemented in this dissertation. Other ways to boost the convergence rate should also be further explored. Advanced GA techniques such as the Pareto optimization and the divided range optimization have been employed in this dissertation to solve multiobjective EM problems. These advanced techniques showed highly efficient capability to generate an entire set of optimal solutions with a single optimization run. Consequently, it became possible to utilize such methodology to map out the performance bounds associated with a given problem. The resulting performance bounds provided valuable information on the design of very complex EM systems. In future research, both microstrip and wire type antenna structures should be further investigated for new applications. In particular, the study of electrically small wire antennas is currently being extended to HF communications, where the physical size of antennas is usually very large. Some other candidate EM problems are listed below. Design of antennas for ultra-wideband (UWB) Applications: The recent emergence of UWB communication systems requires new types of antennas that can operate over a frequency bandwidth of more than 100%. Furthermore, UWB systems require antennas to have not only flat amplitude but also linear phase response over the broad bandwidth. Additional design concerns include low 96 cross-polarization and high efficiency. These criteria for UWB systems are very challenging and have not been fully addressed. Design of multiple-input multiple-output (MIMO) communications systems: Multiple antennas have recently been investigated to increase channel capacity and service quality in mobile communication systems. Some preliminary research has been done to design polarization and pattern diversity antennas for mobile handset terminals in MIMO systems. One challenging problem yet to be addressed is the design of diversity antennas with a small antenna form factor. Design of on-glass antennas for vehicular applications: On-glass antenna design for automobiles is a costly and time-consuming engineering process. The design methodology developed in this dissertation can be applied to generate optimized printed antenna designs for vehicle rear windows (on sedans) or vehicle side windows (on SUVs and minivans). Design of microwave absorbers: Chapter 4 in this dissertation examined microwave absorber shapes to reduce the RCS of military targets. A possible commercial application could be the design of absorbers to minimize EM interference in RF systems. 97 Appendix A Wheeler Cap Method for Measuring the Efficiency of Microstrip Antennas In this Appendix, some work into the Wheeler cap method for measuring the efficiency of microstrip antennas is described. Since antenna efficiency characterization is an important issue in several of the chapters in this dissertation, this material is included here for completeness. A.1 Introduction The Wheeler cap method is a simple and well-known technique for measuring antenna efficiency [55,56]. The method involves making only two input impedance measurements of the antenna under test: one with a conducting cap enclosing the antenna and one without. The antenna efficiency is then estimated based on either a parallel or a series RLC circuit model for the antenna. Pozar and Kaufman reported on the use of this method for measuring the efficiency of microstrips [59]. Even though it is generally believed that a microstrip antenna is more appropriately modeled as a parallel circuit [14,15,60], their measurement results did not support the parallel RLC model, and they concluded that the loss mechanism in the microstrip is similar to that of a series 98 RLC circuit. Even recently microstrip antenna efficiency has been measured using a series RLC circuit model [61]. In this Appendix, the Wheeler cap method for measuring the efficiency of microstrip antennas is revisited. The main interest stems from the need to characterize the efficiency of a class of miniaturized microstrip antennas that was designed using a GA [62]. Here three main findings are presented. First, it is shown that the parallel RLC model is indeed a more appropriate model to use than the series model for microstrips. The results are corroborated by numerical simulations using the commercial software ENSEMBLE. Second, the role of interior cap modes is investigated and an optimal shape of the Wheeler cap for microstrips is proposed. Finally, the Wheeler cap method is applied to investigate the efficiency of the GA-designed miniaturized microstrip antennas. A.2 Antenna Circuit Model In the Wheeler cap method, two input resistance measurements are needed to obtain the efficiency of the antenna. One is the input resistance before using the cap, Rbefore, and the other is the input resistance R after putting the cap on, Rafter. Since this method is valid only for antennas with a simple dominant loss mechanism, we should know whether a given antenna follows the series circuit model or the parallel circuit model. If the test antenna works more like the series circuit model, then R decreases after applying the cap, and the efficiency is calculated by the following expression: 99 Eff = Rbefore ? Rafter PR RR = = PR + PL RR + RL Rbefore ( A.1) where PR is the total radiated power, PL is the power loss, RR is the radiation resistance and RL is the loss resistance. If the test antenna works more like the parallel circuit model, R should increase after applying the cap, and the efficiency can be calculated using this expression: Eff = Rafter ? Rbefore PR RL = = PR + PL RR + RL Rafter ( A.2) As a test, this Wheeler cap method is applied to measure a standard square-shaped microstrip antenna built on lossy FR-4 substrate with thickness of 1.6mm. The test antenna?s dimensions are 36mm � 36mm, and it has a resonant frequency of 2 GHz. For the Wheeler cap, a conducting rectangular cap (10cm � 5cm � 10cm) is used to completely enclose the test microstrip. Then an HP 8753C network analyzer is used to gather the input impedance of the test antenna. As is stated in [56,59], a perfect contact between the cap and the ground plane is critical for an accurate measurement. Therefore, aluminum tape is used to shield the slight gap between the cap and the ground plane. Fig. A.1(a) shows the resulting input resistance before and after applying cap, marked as a dashed line and a solid line, respectively. At the resonant frequency, it shows Rafter of 113 ohms and Rbefore of 65 ohms. Since the test antenna is represented by a parallel RLC circuit 100 model the cap should increases the input resistance. Based on the parallel RLC model, the Wheeler cap measurement shows an efficiency of 34%. (b) (a) Measurement (FR-4) Input Resistance (?) ENSEMBLE Simulation (FR-4) cap 140 120 140 ------ Before ?? After Rafter ------ Before ?? After 120 100 100 Rbefore 80 80 60 60 40 40 20 20 0 0 -20 1.8 Fig. A.1 1.85 1.9 1.95 2 2.05 2.1 2.15 Frequency (GHz) 2.2 2.25 2.3 -20 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 Frequency (GHz) 2.2 2.25 2.3 Input resistance of a standard square-shaped (36mm�mm) microstrip built on FR-4 before (------) and after (? ??) using the Wheeler cap size of 10cmm�cmm�m. (a) Measured (b) ENEMBLE simulated. Next, the measurement result is verified with two different numerical simulations using the full wave EM simulator ENSEMBLE [63]. In the first simulation, the Wheeler cap is modeled to rigorously predict Rafter and Rbefore. To model the conducting rectangular cap, we generate a cavity on top of the microstrip. The side of the cavity that faces the microstrip is opened. The simulated input resistance is shown in Fig. A.1(b). Similar to Fig. A.1(a), the result shows that the input resistance is actually increased after the cap is placed. This again shows that the loss mechanism in the microstrip more closely 101 resembles a parallel circuit model. Based on the parallel circuit model, the efficiency computed by this Wheeler cap simulation method is 41%. In the second simulation, the gain of the test antenna is computed with and without dielectric and metal loss. Note that metal loss occurs not only on the patch but also on the ground plane, and this should be taken into account when modeling the microstrips. The efficiency is then calculated using eq. (A.3), which gives us a value of 32%. Eff = Gwith loss ( A.3) Gwithout loss The comparisons of two simulation and measurements in terms of frequency are shown in Fig. A.2(a). These two simulation results are reasonably close to the measurement results over the frequency range of interest. The efficiency using the series RLC model is plotted as thin solid line. This shows that the parallel RLC circuit model is more appropriate to measure the efficiency of microstrip. This Wheeler cap method is also applied to a microstrip which is built on a low loss substrate, RT Duroid 5880 (loss tangent of around 0.001). This test microstrip has dimensions of 17.5mm�mm and operates at near 5.3GHz. This measurement shows a 92% efficiency, and this compares favorably to corresponding simulation results (95% in the gain simulation, and 98% in the Wheeler cap simulation) as shown in Fig. A.2(b). The small deep at the frequency of 5.12GHz is caused by an excitation of a cavity mode (TM111) by a Wheeler 102 cap. These results confirm the validity of our Wheeler cap measurements. In [59], Pozar and Kauffman reported Rafter < Rbefore in their measurement of the simple microstrip, and they concluded that the loss mechanism in the microstrip is similar to the a series circuit model. It should be noted that this is probably due to the influence of interior cap modes when using a large-sized Wheeler cap. In the next section, cap modes and their dependence on the size of the cap will be discussed. (b) (a) 100 100 Measured Measured (Series) Wheeler Cap Simulation Gain Simulation 90 90 80 70 Efficiency (%) Efficiency (%) Efficiency (% ) 80 -10dB bandwidth 60 50 40 30 60 50 40 30 20 20 10 10 0 1.96 1.98 2 2.02 2.04 2.06 0 5.1 2.08 Frequency (GHz) Fig. A.2 -10dB bandwidth 70 Measured Measured (Series) Wheeler Cap Simulation Gain Simulation 5.15 5.2 5.25 5.3 5.35 5.4 Frequency (GHz) Wheeler cap measured efficiency based on parallel circuit model (??), Wheeler cap measured efficiency based on series circuit model (??Purple), efficiency by Wheeler cap simulation (------) and efficiency by gain simulation (------). (a) microstrip built on FR-4 substrate (b) microstrip build on Duroid. 103 5.45 A.3 Effect of Cap Dimensions Wheeler recommended that the cap radius be around 1/6 of a wavelength to cause no change in the current distribution on the antenna [55,56]. However, for microstrip antennas, a larger size Wheeler cap may have to be used to enclose an extended substrate or to enclose a microstrip array. Cap Height: 8.5cm ------ Before ?? After 700 600 17�5�cm (2.83�41�83?) 500 400 ------ Before ?? After 800 Input Resistance (?) 800 Input Resistance (?) Cap Height: 2.0cm 900 900 300 200 700 600 17�0�cm (2.83�33�83?) 500 400 300 200 100 100 0 0 4.6 4.8 5 5.2 5.4 5.6 4.6 5.8 Frequency (GHz) Fig. A.3 4.8 5 5.2 5.4 Frequency (GHz) 5.6 5.8 Measured input resistance of a standard square-shaped (36mm x 36mm) microstrip built on Duroid before (------) and after (??) using (a) the Wheeler cap size of 17cmm x 17cmm x 8.5cm. (b) using the Wheeler cap size of 17cm x 17cm x 2cm. Fig. A.3(a) is an example of the effect using a larger size (17cm�5cm�cm) Wheeler cap for the same microstrip (17.5mm�mm) built on low loss substrate. In a small Wheeler cap, interior cap modes exist, but at such high frequencies that they do not significantly interfere with the resonant 104 frequency of the microstrip as shown in Fig. A.1(a) and (b). On the other hand, a larger Wheeler cap, shown in Fig. A.3(a), creates interior cap modes near the resonant frequency of the microstrip, which causes a deviation in the input resistance value. This can cause an inaccurate efficiency measurement when using the Wheeler cap method. Thus, the interior cavity modes should be as sparse as possible. This may be achieved by using a smaller cap size. However, the finite size of the microstrip substrate restricts the minimum size of the cap. If we take a detailed look at the interior cavity modes, only TM modes are dominant in the Wheeler cap since the microstrip works similarly to a horizontal magnetic current parallel to the ground plane. f TM MNP 1 2? � M? a 2 N? + b 2 P? + c 2 ( A.4) Fig. A.4(a) shows the interior cavity mode-spectrums for a Wheeler cap sizes of 17cm x 8.5cm x 17cm. The solid line is the measured input resistance and the dashed line is mode-spectrum calculated using eq. (A.4) [64]. The measured cavity modes compare well with the ones by calculation. The index N is associated with the cap height while the other indexes M and P are associated with the length and the width of the cap, respectively. 105 (a) Cavity Mode Spectrum of 8.5cm Cap Height 2 1 0.5 (b) 3 3.5 4 4.5 5 5.5 5 5.5 TM114 TM313 TM214 2.5 TM212 TM113 2 TM111 0 1.5 TM112 Log10 [Rin (?)] 1.5 Cavity Mode Spectrum of 2.0cm Cap Height Log10 [Rin (?)] 2 1.5 1 0.5 0 1.5 Fig. A.4 2 2.5 3 3.5 4 Frequency (GHz) 4.5 Interior cavity modes spectrums for two Wheeler cap size. The upper spectrum is for the Wheeler cap size of 17cm x 17cm x 8.5cm.The lower spectrum is for the Wheeler cap size of 17cm x 17cm x 2.0cm. 106 Then the cavity mode-spectrum is observed by reducing only the height of the Wheeler cap since the planar profile of the microstrip prevents us from decreasing the other two dimensions. As shown in Fig. A.4(b), the mode spectrum of the 2cm height Wheeler cap (17cm x 2cm x 17cm) is sparser than the mode spectrum of the 8.5cm one. This is due to the fact that a cap height reduction makes the mode spectrum sparser by pushing the interior modes with index N=1 to much higher frequencies. Sparser mode spectrums provide more space in which to make Wheeler cap measurements. This theory is applied to the test by measuring the input resistance of the same microstrip shown in Fig. A.3(a) but using a reduced cap height of only 2.0cm (0.33?). The results shown in Fig. A.3(b), indeed show the sparser cavity mode-spectrum as expected. However, the resonant frequency with cap is about 91MHz lower than the resonant frequency without cap. One solution is to shift the capped resonant frequency up to 91MHz to compensate the frequency shift by cap effect [65,66]. After the frequency compensation, an efficiency value of 94% with this reduced height (17cm x 2.0cm x 17cm) Wheeler cap is obtained, which is closer to the simulation value than the efficiency of 72.5% by using the size of Wheeler cap. This shows that reduced height wheeler caps are helpful in obtaining more accurate efficiency values without sacrificing the length and width of the cap. 107 A.4 Results for Efficiency of Miniaturized Microstrips In this section, the Wheeler cap method described above is applied to measure the efficiency of our miniaturized microstrips. A GA is previously applied to minimize the size of a microstrip patch while keeping its bandwidth as broad as possible [62]. (a) (b) 100 50 Efficiency(%) Efficiency (%) 40 35 30 25 20 15 90 Standard Square Microstrips 80 Efficiency (%) Efficiency(%) -Measurement -Simulation 45 GA Microstrips 70 60 50 40 30 10 20 5 10 0 40 50 60 70 80 90 0 40 100 % from regular size Fig. A.5 GA Microstrips -Measurement -Simulation 50 60 70 80 90 100 % from regular size (a) built on FR-4 substrate in terms of % from the regular size and (b) the microstrips built on Duroid. The insets in Fig. A.5(a) are the samples of GA-miniaturized microstrips. The achievable bandwidth of these miniaturized antenna drops as the size of the antennas is reduced from 8% to 1.3%. It also shows that even when the size of the patch is reduced to 40% of the regular size, it still maintains a bandwidth of around 1.3%, which is good compared to the microstrip?s small size. However, we thought that investigating the efficiency of these microstrips is crucial due to 108 the high loss in FR-4 substrate. Thus, the efficiencies of these microstrip are observed and the results are shown in Fig. A.5 as the solid line with the efficiencies using Wheeler cap measurements and the dashed line is using gain simulations. The simulated and measured efficiencies are close, showing that our measurements are accurate, even if the results show that our microstrips have very low efficiencies. This is somewhat to be expected due to the high loss on the FR-4 substrate, which has a loss tangent of around 0.025. For this reason, our study is extended to look at how low-loss substrate materials such as Duroid and air increase the efficiency of these miniaturized microstrips. Fig. A.5(b) shows the measured and simulated efficiencies of microstrips using the Duroid substrate with loss tangent of about 0.001. An improved efficiency of more than 65% for all samples can be observed. This gain in efficiency is not without a trade-off. The achievable bandwidth using the Duroid substrate is reduced compared to the bandwidth using the FR-4 substrate. This forces us to evaluate each antenna in terms of both its efficiency and its bandwidth using the antenna?s EB product [66]. Fig. A.6 plots the EB product against the physical antenna size for antennas built on three different substrates: FR-4, Duroid and air substrate, marked as solid, dashed and solid-dotted lines, respectively. Using Fig. A.6, It could be determined which substrate material provides the highest EB product for a given antenna size. 109 7 Air (3.25mm) 5 Bandwidth * Efficiency Efficiency x Bandwidth (EB) 6 Duroid (3.2mm) 4 3 2 FR-4 (1.6mm) 1 0 10 20 30 40 50 60 70 Physical Antenna Size (mm) Fig. A.6 EB Product against physical microstrip patch size for microstrip built on three different substrates. (a) FR-4 (? ??), Duroid (------)and Air (-?-?-?-?). A.5 Summary The Wheeler cap method for measuring microstrip efficiency was revisited, and it was shown that the parallel circuit model is appropriate for the microstrip loss mechanism. The measured efficiency values were verified using a numerical simulation code. Then interior cap modes were investigated, and a way to diminish them using a reduced height Wheeler cap was found. Finally, the reduced-height Wheeler cap method was applied to investigate the efficiency of miniaturized microstrip antennas on various substrate materials. However, this 110 Wheeler cap method is only valid for a limited frequency near a resonance. Also, the method is valid for an antenna that has only one dominant loss mechanism. 111 Bibliography [1] D. G. Luenberger, Introduction to Linear and Nonlinear Programming. Reading, MA: Addison-Wesley, 1973. [2] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. Reading, MA: Addison-Wesley, 1989. [3] J. H. Holland, Adaptation in Natural and Artificial Systems. Ann Arbor: University of Michigan Press, 1975. [4] Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms. New York: John Wiley & Sons, 1999. [5] L. Alatan, M. I. Aksun, K. Leblebicioglu and M. T. Birand, ?Use of computationally efficient method of moments in the optimization of printed antennas,? IEEE Trans. Antennas Propagat., vol. 47, pp.725-732, Apr. 1999. [6] J. M. Johnson, and Y. Rahmat-Samii, ?Genetic algorithms and method of moments (GA/MOM) for the design of integrated antennas,? IEEE Trans. Antennas Propagat., vol. 47, pp. 1606-1614, Oct. 1999. [7] R. L. Haupt, ?Optimum quantised low sidelobe phase tapers for arrays,? Elec. Lett., vol. 31, pp. 1117-1118, July 1995. [8] E. E. Altshuler and D. S. Linden, ?Wire-antenna designs using genetic algorithm,? IEEE Antennas and Propagat. Mag., vol. 39, pp. 33-43, Apr. 1997. [9] E. E. Altshuler, ?Electrically small self-resonant wire antennas optimized using a genetic algorithm,? IEEE Trans. Antennas Propagat., vol. 50, pp. 297-300, Mar. 2002. [10] E. Michielssen, J. M. Sajer, S. Ranjithan and R. Mittra, ?Design of lightweight, broad-band microwave absorbers using genetic algorithm,? IEEE Trans. Microwave Theory Tech., vol. 41, pp. 1024-1031, June-July 1993. 112 [11] M. Pastorino, A. Massa and S. Caorsi, ?A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,? IEEE Trans. Instrumentation Measurement, vol. 49, pp. 573-578, June 2000. [12] Y. Zhou and H. Ling, ?Electromagnetic inversion of Ipswich objects with the use of the genetic algorithm,? Microwave Optical Tech. Lett., vol. 33, pp. 457-459, June 2002. [13] L. J. Chu, ?Physical limitations of omnidirectional antennas,? J. Appl. Phys., vol. 19, pp. 1163-1175, Dec. 1948. [14] J. R. James and P. S. Hall, Handbook of Microstrip Antennas. London: Peter Peregrinus, 1989. [15] D. Pozar and D. H. Schaubert, Microstrip Antennas. New York: The Institute of Electrical and Electronics Engineers, 1995. [16] H. Choo, A. Hutani, L. C. Trintinalia and H. Ling, ?Shape optimisation of broadband microstrip antennas using genetic algorithm,? Elec. Lett., vol. 36, pp. 2057-2058, Dec. 2000. [17] L. C. Trintinalia, ?Electromagnetic scattering from frequency selective surfaces,? M.S. Thesis, Escola Polit閏nica da Univ. de S鉶 Paulo, Brazil, 1992 [18] T. Cwik and R. Mittra, ?Scattering from a periodic array of free-standing arbitrarily shaped perfectly conducting or resistive patches?, IEEE Trans. Antennas Propagat., vol. 35, pp.1226-1234, Nov. 1987. [19] M. Villegas and O. Picon, ?Creation of new shapes for resonant microstrip structures by means of genetic algorithms?, Elec. Lett., vol. 33, pp. 15091510, Aug. 1997. [20] A. Rosenfeld and A. C. Kak, Digital Picture Processing. Academic Press, 1982, 2nd ed. [21] Y. X. Guo, K. M. Luk and K. F. Lee, ?Dual-band slot-loaded shortcircuited patch antenna,? Elec. Lett., vol. 36, pp. 289-291, Feb. 2000. [22] H. Choo, and H. Ling, ?Design of dual-band microstrip antennas using the genetic algorithm,? in Proc. 17th Annu. Rev. Progress Appl. Computat. Electromagn., Monterey, CA, Mar. 2001, pp. 600-605. 113 London: [23] J. Renders, and S. P. Flasse, ?Hybrid methods using genetic algorithm for global optimization,? IEEE Trans. Syst. Man Cybern., vol. 26, pp.243-258 , Apr. 1996. [24] Y. Zhou, J. Li and H. Ling, ?Shape inversion of metallic cavities using a hybrid genetic algorithm combined with a tabu list,? Elect. Lett., vol. 39, pp. 280-281, Feb. 2003. [25] F. Gloven and M. Laguna, Tabu Search. Publishers, 1997. [26] H. Choo and H. Ling, ?Design of broadband and dual-band microstrip antennas on high-dielectric substrate using the genetic algorithm,? IEE Proc. Inst. Elect. Eng. Microwaves, Antennas Propagat., to be published in 2003. [27] E. F. Knott, J. F. Shaeffer and M. T. Tuley, Radar Cross Section. Dedham, MA: Artech House, 1985. [28] J. J. Pesque, D. P. Bouche and R. Mittra, ?Optimization of multilayered antireflection coatings using an optimal control method,? IEEE Trans. Antennas Propagat., vol. 40, pp. 1789-1796, Sept. 1992. [29] D. Weile, E. Michielssen and D. E. Goldberg, ?Genetic algorithm design of pareto optimal broadband microwave absorbers,? IEEE Trans. Electromagnetic Compatibility, vol. 38, pp.518-525, Aug. 1996. [30] H. Mosallaei and Y. Rahmat-Samii, ?RCS reduction of canonical targets using genetic algorithm synthesized RAM,? IEEE Trans. Antennas Propagat., vol. 48, pp. 1594-1606, Oct. 2000. [31] C. Yang and W. D. Burnside, ?A periodic moment method solution for TM scattering from lossy dielectric bodies with application to wedge absorber,? IEEE Trans. Antennas Propagat., vol. 40, pp. 652-660, June 1992. [32] R. Janaswamy, ?Oblique scattering from lossy periodic surfaces with application to anechoic chamber absorbers,? IEEE Trans. Antennas Propagat., vol. 40, pp. 162-169, Feb. 1992. [33] C. L. Holloway and E. F. Kuester, ?A low-frequency model for wedge or pyramid absorber arrays-II: Computed and measured results,? IEEE Trans. Electromagnetic Compatibility, vol. 36, pp. 307-313, Nov. 1994. 114 Boston: Kluwer Academic [34] J. Moore, H. Ling and C. S. Liang, ?The scattering and absorption characteristics of material-coated periodic grating under oblique incidence,? IEEE Trans. Antennas Propagat., vol. 41, pp. 1281-1288, Sept. 1993. [35] J. Moore, H. Ling, C. Liang and H. Carter, ?Oblique scattering from coated periodic surfaces,? in Proc. 1994 Have Forum Symposium, Colorado Springs, CO, Oct. 1994, pp. 283-292. [36] Dielectric Property of MAGRAM, http://www.arc-tech.com/mag.html. [37] H. Choo, H. Ling and C. S. Liang, ?Design of corrugated absorbers for oblique incidence using genetic algorithm,? in Proc. IEEE Antennas Propagat. Soc. Int. Symp., Boston, MA, July 2001, pp. 708-711. [38] R. A. Haddad and T. W. Parsons, Digital Signal Processing. New York: Computer Science Press, 1991. [39] C. L. Holloway and E. F. Kuester, ?Power loss associated with conducting and superconducting rough interfaces,? IEEE Trans. Microwave Theory Tech., vol. 48, pp. 1601-1610, Oct. 2000. [40] N. Srinivas and K. Deb, ?Multiobjective optimization using nondominated sorting in genetic algorithm,? J. Evol. Comput., vol. 2, pp. 221-248, 1995. [41] J. Horn, N. Nafpliotis and D. E. Goldberg, ?A niched pareto genetic algorithm for multiobjective optimization,? in Proc. IEEE Conf. Evolutionary Computat., Orlando, FL, June 1994, pp. 82-87. [42] A. K. Skrivervik, J. F. Zurcher, O. Staub and J. R. Mosig, ?PCS antenna design: the challenge of miniaturization,? IEEE Antennas Propagat. Mag., vol. 43, pp. 12-27, Aug. 2001. [43] J. P. Gianvittorio and Y. Rahmat-Samii, ?Fractal antennas: a novel antenna miniaturization technique, and applications,? IEEE Antennas Propagat. Mag., vol. 44, pp. 20?36, Feb. 2002. [44] H. A. Wheeler, ?Fundamental limitations of small antennas,? Proc. IRE, vol. 35, pp. 1479-1484, Dec. 1947. [45] R. C. Hansen, ?Fundamental limitations in antennas,? IEEE Trans. Antennas Propagat., vol. 69, pp. 170-182, Feb. 1981. 115 retrieved Aug. 2002, from: [46] J. S. McLean, ?A re-examination of the fundamental limits on the radiation Q of electrically small antennas,? IEEE Trans. Antennas Propagat., vol. 44, pp. 672-676, May 1996. [47] C. H. Friedman, ?Wide-band matching of a small disk-loaded monopoles,? IEEE Trans. Antennas Propagat., vol. 33, pp. 1142-1148, Oct. 1985. [48] H. D. Foltz, J. S. McLean and G. Crook, ?Disk-loaded monopoles with parallel strip elements,? IEEE Trans. Antennas and Propagat., vol. 46, pp. 1894-1896, Dec. 1998. [49] J. A. Dobbins and R. L. Rogers, ?Folded conical helix antenna,? IEEE Trans. Antennas Propagat., vol. 49, pp. 1777-1781, Dec. 2001. [50] G. Goubau, ?Multi-element monopole antennas,? in Proc. Workshop Electrically Small Antennas, ECOM, Ft. Monmouth, NJ, May 1976, pp. 63-67. [51] T. Hiroyasu, M. Miki and S. Watanabe, ?The new model of parallel genetic algorithm in multi-objective optimization problems - divided range multi-objective genetic algorithm?, in Proc. 2000 Congress on Evolutionary Computat., vol. 1, pp. 333-340, 2000. [52] G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC)Method of Moments. Lawrence Livermore Laboratory, 1981. [53] H. Choo, H. Ling and R. L. Rogers, ?Design of electrically small wire antenna using genetic algorithm taking into consideration of both bandwidth and efficiency,? in Proc. IEEE Antennas Propagat. Soc. Int. Symp., San Antonio, TX, June 2002, pp. 330-333. [54] R. L. Rogers, D. P. Buhl, H. Choo and H. Ling, ?Size reduction of a folded conical helix antenna,? in Proc. IEEE Antennas Propagat. Soc. Int. Symp., San Antonio, TX, June 2002, pp. 34-37. 2002. [55] H. A. Wheeler, ?The Radiansphere Around a Small Antenna,? Proc. IRE, pp. 1325-1331 Aug. 1959. [56] E. Newman, P. Hohley, and C. H. Walter, ?Two methods for the measurement of antenna efficiency,? IEEE Trans. Antennas Propagat., vol. 23, pp. 457-461, July. 1975. 116 [57] C. A. Balanis, Antenna Theory Analysis and Design. New York: John Wiley & Sons, 1997. [58] H. Choo and H. Ling, ?Design of planar, electrically small antennas with inductively coupled feed using a genetic algorithm,? in Proc. IEEE Antennas Propagat. Soc. Int. Symp., Columbus, OH, June 2003, to be published. [59] D. M. Pozar and B. Kaufman, ?Comparison of three methods for the measurement of printed antenna efficiency,? IEEE Trans. Antennas Propagat, vol. 36, pp. 136-139, Jan. 1988. [60] K. R. Carver and J. W. Mink, ?Microstrip antenna technology," IEEE Trans. Antennas Propagat, vol. 29, pp. 2-24, Jan. 1981. [61] R. Chair, K. M. Luk and K. F. Lee, ?Radiation efficiency analysis on small antenna by Wheeler cap method,? Microwave Optical Tech. Lett., vol. 33, pp. 112-113, Apr. 2002. [62] H. Choo, A. Hutani and H. Ling, ?Shape optimization of microstrip antennas using genetic algorithm,? in Proc. USNC/URSI Radio Sci. Meet., Boston, MA, July 2001, p. 144. [63] ENSEMBLE� Version 8.0, Ansoft Corporation, Pittbug. PA, 15219. [64] C. A. Balnis, Advanced Engineering Electromagnetics. New York: John Wiley & Jons, 1989. [65] W. E. McKinzie, ?A modified wheeler cap method for measuring antenna efficiency,? in Proc. IEEE Antennas Propagat. Soc. Int. Symp., Montreal, Que, Canada, July 1997, pp. 542-545. [66] M. S. Smith, ?Properties of dielectrically loaded antennas,? Proc. IEE, vol. 124, pp. 837-839, 1977. 117 Vita Hosung Choo was born in Seoul, Korea on April 10, 1972, the son of Hyungsam Choo, and Kabrae Cho. After completing his work at Dankuk High School, Seoul, Korea in 1991, he entered Hanyang University in Seoul, Korea. He received the degree of Bachelor of Science from Hanyang University in February 1998. In September 1998, he entered the Graduate School at The University of Texas at Austin. He received the degree of Master of Science from The University of Texas at Austin in August 2000. Permanent Address: 2501 Lake Austin Blvd. APT J205 Austin, TX, 78703 This dissertation was typed by the author. 118 entional GA. Pareto GA [2,4] is a useful tool for this problem. In the Pareto GA, a wide range of solutions corresponding to more than one objective can be mapped by running the optimization only once. In this GA implementation, two cost functions are defined: Cost 1 = Normalized coating height Cost 2 = Normalized value of where Pn = 1 N N n =1 (4.2) ( Pn ) (?? + ?| | ) / 2 (dB) + 20 dB if (?? + ?|| ) / 2 (dB) ? ? 20 dB 0 if (?? + ?|| ) / 2 (dB) < ?20 dB (4.3) Cost 1 is determined by the coating height and Cost 2 is associated with the reflection cost. Both costs are normalized to a value between zero and one. For 61 Cost 2, one denotes an average reflection coefficient of 0dB, while zero denotes an average reflection coefficient that is below ?20dB. The non-dominated sorting method [40] is used to combine the two costs for each solution by means of the Pareto ranking. This method assigns rank 1 to the non-dominated solutions of the population. The term non-dominated solution means that there are no other solutions that are superior to this solution in both objectives. Then the next nondominated solutions among the remaining solutions are assigned to the nexthighest rank. The process is iterated until all the solutions in the population are ranked. Based on the rank, the same reproduction process described in Section 4.3 is performed to refine the population into the next generation. The set of rank 1 solutions is called the Pareto front. In order to avoid the solutions on the Pareto front from converging to a single point in the cost space, a sharing scheme described in [41] is performed. In this sharing process, the rank is modified by penalizing those members on the front that are too close to each other in the cost space. This is accomplished by multiplying a niche count (mi) to the assigned rank. The niche count is calculated according to: N 1 p mi = Sh(d ij ) ( 4. 4 ) N p j =1 where the Np is the number of rank 1 members and the sharing function, Sh(dij), is a function of the cost distance between solutions expressed as: 62 Sh(d ij ) = 2? d ij d share 1 if d ij < d share (4.5) if d ij > d share and d ij = (Cost 1(i) ? Cost 1( j ) )2 + (Cost 2(i ) ? Cost 2( j ) )2 As can be seen, the sharing function increases linearly if the other members on the front are closer than dshare from a chosen member i in the cost space. Consequently, those members that have close-by neighbors in the cost space are assigned lower ranks in the reproduction process. 4.4.2 Pareto GA Results In this Pareto GA, the population size is chosen to be 100. A crossover probability of 0.8, a mutation probability of 0.1 and a dshare distance of 1 are used. Figs. 4.11(a)-(d) show the convergence of the solutions for this multi-objective problem (reflection cost versus the height of the profile) as the number of generations is increased. The period of the absorber, the material for the coating, and the angle of incidence are the same as those used in Section 4.3. The height is constrained to be less than 8mm. Fig. 4.11(a) is the plot of the initial population. The majority of the solutions are located in the upper-right side of the cost domain. Figs. 4.11(b)-(d) are plots of the population after 5, 20 and 200 generations, respectively. They show that as the number of generations increases, 63 the Pareto front spreads out and converges toward the lower-left region of the cost space. Fig. 4.12 shows the final converged Pareto front and four optimized coating shapes that are on the front. Inset shape (a) shows the lowest profile of the four samples, but it has the highest reflection among the four designs. Inset shape (d) has the highest profile and the lowest reflection. As expected, the absorbing performance must be traded off against the profile height. If we look in detail at the optimized shapes, it appears that as the height of the absorber decreases, the top of the profile gets more flattened. However, the shapes maintain a rectangular profile that is only slightly modified by the coating height. This is consistent with the physical interpretation of the absorption process for the more dominant horizontal polarization discussed in Section 4.3. Another observation from Fig. 4.12 is that the Pareto front is not smooth due to the quantization effect of the coating height. If the height is discretized with more binary bits, the shape of the Pareto front becomes smoother. Next, the first cost is changed from coating height to coating weight while keeping the reflection cost the same. The cost for the coating weight is normalized to be 1 when all of the design area (period � maximum coating height) is filled by the coating material while it is zero when no coating material exists. Fig. 4.13 shows the converged Pareto front for this problem. Also shown in insets (a) to (d) are four optimized shapes with different coating weights. 64 0.45 0.45 0.4 0.4 0.35 0.3 0.25 0.2 A bsorbing P erformance 0.5 Reflection cost Absorbing Performance Reflection cost 0.5 0.3 0.25 0.2 0.15 0.1 0.35 0.15 4 4.5 5 5.5 6 6.5 7 7.5 0.1 8 4 4.5 Height of absorber (mm) 5 6.5 7 7.5 8 (b) 0.5 0.45 0.45 0.4 0.4 0.35 0.3 0.25 0.2 A bsorbing P erformance 0.5 Reflection cost Absorbing Perform ance Reflection cost 6 Height of absorber (mm) (a) 0.35 0.3 0.25 0.2 0.15 0.15 0.1 5.5 0.1 4 4.5 5 5.5 6 6.5 7 7.5 8 Height of absorber (mm) 4.5 5 5.5 6 6.5 Heigh of the profile (mm) 7 7.5 Height of absorber (mm) (c) Fig. 4.11 4 (d) Convergence of the Pareto front as a function of the number of generations for absorbing performance versus absorber height. (a) Initial population. (b) After 5 generations. (c) After 20 generations. (d) After 200 generations. 65 8 (a) (b) 0.5 (c) 0.45 Reflection cost 0.4 (d) 0.35 0.3 0.25 0.2 0.15 0.1 Fig. 4.12 4 4.5 5 5.5 6 6.5 Height of profile (mm) 7 7.5 8 Final converged Pareto front of absorbing performance versus absorber height. The insets show four sample designs on the Pareto front. 66 (a) (b) 0.7 (c) 0.6 Reflection cost 0.5 0.4 (d) 0.3 0.2 0.1 0.15 0.2 0.25 0.3 Normalized Weight 0.35 0.4 Weight of the absorber Fig. 4.13 Final converged Pareto front of absorbing performance versus absorber weight. The insets show four sample designs on the Pareto front. 67 It should be noticed that inset (d) is very similar in shape to inset (d) in Fig. 4.12. However, instead of trimming the top off in order to reduce the height, the weight consideration results in designs that become progressively skinnier, as shown by insets (c), (b) and (a). Nevertheless, the shapes still preserve the sharp sidewalls as those presented in Fig. 4.12. 4.5 Summary Optimized shapes for a corrugated absorber under near-grazing incidence have been investigated using GA. First, GA was applied to design corrugated coating depending on incident polarizations. The designed absorber shape for the vertical polarization resembled a triangular profile, while that for the horizontal polarization resembled a rectangular profile. The optimized shapes were compared to canonical planar and triangular shaped designs, and were shown to have better absorbing performance. The sensitivity of the designs to variations in shape and incident angles were also tested, and the results showed reasonable tolerance. A physical interpretation for the optimized shape was presented. It was shown that the sharp sidewalls of the resulting shape effectively changed the incident polarization from horizontal to the vertical case, thus facilitating wave absorption. The Pareto GA has also been applied to efficiently map out absorbing performance versus absorber height. The non-dominated sorting method was used 68 to combine the two costs for each solution by means of the Pareto ranking. A sharing scheme was implemented to avoid the solutions on the Pareto front from converging to a single point in the cost space. The converged Pareto front showed that better absorbing performance must be traded off against absorber height. Similar conclusions were also found for the absorbing performance versus absorber weight. 69 Chapter 5 Design of Electrically Small Wire Antennas Using a Pareto Genetic Algorithm 5.1 Introduction In Chapter 2 and 3, the basic GA for microstrip antenna design was introduced. In Chapter 4, the more advanced Pareto GA was investigated to solve the more challenging multi-objective problem. In this chapter, the developed methodology is applied to the design of electrically small wire antennas. As the size of wireless devices shrinks, the design of electrically small antennas is an area of growing interest [42,43]. By the classical definition, an electrically small antenna is one that can be enclosed in a volume of radius r much less than a quarter of a wavelength. It is well known that the bandwidth of an electrically small antenna decreases as the third power of the radius [44-46]. Much research has been carried out to increase the bandwidth of small antennas using structures such as folded design, disk-loaded monopole, inverted-L or inverted-F designs, multi-armed spiral and conical helix [47-50]. Fig. 5.1 shows the bandwidth performance of these antennas in terms of the normalized antenna size kr, where k=2?/? is the wave number. Recently, Altshuler reported on the use of a GA in designing electrically small wire antennas [9]. Instead of using a 70 regular shape, he used GA to search for an arbitrary wire configuration in 3-D space that results in maximum bandwidth for a given antenna size. While much of the small antenna research has been focused on antenna bandwidth, antenna miniaturization also impacts antenna efficiency. The objective of this chapter is to apply GA in the design optimization of electrically small wire antennas, taking into account of bandwidth, efficiency and antenna size. To efficiently map out this multi-objective problem, the Pareto GA [2,4] is utilized. 3dB Bandwidth (%) Fundamental Limit (L. J. Chu, 1948) 100 Goubau [50] 50 Dobbins & Rogers [49] 30 Foltz & McLean [48] 20 0.6 Fig. 5.1 0.8 1.2 Antenna size (kr) 1/4? ? monopole 1.6 Achievable bandwidth in terms of antenna size and some examples of small antennas. 71 The concept of divided range multi-objective [51] is employed to accelerate convergence in the GA process. In this GA approach, the multisegment wire structure similar to the one used in [9] is employed. The Numerical Electromagnetics Code (NEC) [52] is used to predict the performance of each wire structure. Then an optimal set of designs is generated by considering bandwidth, efficiency and antenna size. To verify the GA results, several GA designs are built, measured and compared to the simulation. Physical interpretations of the GA-optimized structures, showing the different operating principles depending on the antenna size are also provided. The performance curve achieved by the GA approach is compared against the well-known fundamental limit for small antennas [44-46]. To more easily assess the performance of the antennas, the efficiency-bandwidth product is normalized by the antenna size in order to represent the antenna performance as a single figureof-merit [54]. Finally, further improvement of the GA results is attempted by exploring additional geometrical design freedoms to better approach the fundamental limit. This chapter is organized as follows. In Section 5.2, the details of the GA implementation are described. Section 5.3 describes the GA designs and the measurement verification of the results. In Section 5.4, the GA results are compared to the fundamental limits. In Section 5.5, other design freedoms are 72 explored to further improve performance. Section 5.6 provides conclusions gathered from this research. 5.2 Pareto GA Approach The basic antenna configuration considered in this chapter is shown in Fig. 5.2. The antenna consists of M connected wire segments. Each segment of the antenna is confined in a hemispheric design space with a radius r and an infinite ground plane. r Fig. 5.2 Configuration of the multi-segment wire antenna used in the GA design. The three design goals are: broad bandwidth, high efficiency and small antenna size. The Pareto GA is employed to efficiently map out this multi- 73 objective problem. The advantage of using the Pareto GA over the conventional GA is that a wide range of solutions corresponding to more than one objective can be mapped by running the optimization only once. Binary Chromosome Antenna Shape 1 0 0 0 1 0 1 1 0 0 1 1 Fig. 5.3 0 1 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 Encoding of the wire configuration into a binary chromosome. In this GA implementation, the hemispheric design space is evenly n discretized into 2 grid points, and the location of each joint of the antenna is encoded into an n-bit binary string, as shown in Fig. 5.3. Thus, the total number of bits in the chromosome is nM when M connected wire segments are used. The three costs associated with these design goals are: 74 Cost 1 = 1 ? Antenna Bandwidth Theoretical Bandwidth Limit (5.1) Cost 2 = 1 ? Efficiency Cost 3 = Antenna Size ( kr ) In the above definition, the theoretical bandwidth limit of 1/((1/kr)+(1/kr)3) derived in [46] is used. NEC [52] is used to predict the antenna performance in order to compute the cost functions. Multiple Linux machines are used in parallel to carry out this computation. After evaluating the three cost functions of each sample structure using NEC, all the samples of the population are ranked using the non-dominated sorting method [40]. Here, the higher the rank (1 denotes the highest rank), the better the solution. This method assigns rank1 to the non-dominated solutions of the population. The term non-dominated solution means that there are no other solutions that are superior to this solution with respect to all design objectives. Then the next non-dominated solutions among the remaining solutions are assigned rank2. The process is iterated until all the solutions in the population are ranked. Based on the rank, a reproduction process is performed to refine the population into the next generation. The set of rank1 solutions is termed the Pareto front. By favoring the higher-ranked solutions in the reproduction process, the Pareto GA tries to push the Pareto front as close to the optimum solution in the cost space as possible. 75 For the crossover operation, a two-point crossover scheme involving three chromosomes is used. For numerical stability, a geometrical check is applied to prevent the wires from intersecting one another. In order to avoid the solutions from converging to a single point, a sharing scheme as described in [41] is performed to generate a well-dispersed population. In the sharing process, the rank is modified by penalizing those members on the front that are too close to each other in the cost space. This is accomplished by multiplying a niche count (mi) to the assigned rank. The niche count is calculated according to: 1 mi = Np Np j =1 Sh (dij ) (5.2) where the Np is the number of rank-1 members and the sharing function, Sh(dij), is a function of the cost distance between solutions expressed as: Sh (dij ) = 2? 1 dij = dij dshare if dij < dshare (5.3) if dij > dshare ( Cost 1(i) ? Cost 1( j) )2 + ( Cost 2 (i) ? Cost 2 ( j) )2 + ( Cost 3 (i) ? Cost 3 ( j) )2 The sharing function increases linearly if the other members on the front are closer than dshare from a chosen member i in the cost space. Consequently, 76 those members that have close-by neighbors in the cost space are assigned lower ranks in the reproduction process. The standard Pareto GA did not always give satisfactory results in this problem, since it is much harder for small-sized antennas to converge than for large-sized antennas. Thus, large-sized antennas usually dominate the whole population after several generations of the GA process. Cost1 (Size) Range 1 Range 2 Cost3 (Efficiency) Range 3 Cost2 (Bandwidth) Fig. 5.4 Divided range multi-objective GA approach. 77 As a result, the final Pareto front contains only antenna designs with sizes kr?0.45. To avoid this bias, the concept of the divided range multi-objective GA [51] is employed in this implementation. As shown in Fig. 5.4, the size is partitioned into multiple ranges and carry out the Pareto GA on each range individually. After each range converges to an optimal solution, the populations from all ranges are merged and the combined population is optimized in the last step of this process. This algorithm shows much improved performance for this more difficult multi-objective problem. Using the scheme good results for smallsized antennas (kr<0.45) as well as large-sized ones (kr>0.45) can be achieved. 5.3 GA-Optimized Results 5.3.1 GA Optimized Designs This section investigates the optimal antenna shapes that produce the best efficiency-bandwidth (EB) product for a given antenna size. Seven wire segments are used in the antenna configuration. The 3-D hemisphere with radius r is discretized into 215 points and the locations of the 7 wire segments are encoded as a binary chromosome of 7� bits. In addition, an extra three bits are added for choosing 8 different wire conductivities varying from 1�6 (graphite) to 5.7�7 (copper). The population size is chosen to be 2000 and the population is divided into four sub ranges (0.29?kr<0.38, 78 0.38?kr<0.46, 0.46?kr<0.54 and 0.54?kr<0.63). Each of four ranges has a population size of 500. A crossover probability of 0.8, a mutation probability of 0.1 and a dshare distance of 1 are used. The target design frequency is chosen at 400 MHz and an infinite ground plane is assumed in the simulation. All antennas are designed to match to a 50 ? impedance. The total computational time is about 20 hours using four Pentium IV 1.7GHz machines running in parallel. Figs. 5.5 (a), (b) and (c) show the designs in the population with a rank of 1 at respectively, the initial, 200 and 1000 generations of the Pareto GA process. The three axes are bandwidth, efficiency and antenna size. Each dot represents a particular rank-1 design. In the initial generation, only a few rank-1 solutions exist. After 200 generations, many more rank1 solutions appear. After 1000 generations, a large portion of solutions (770 over 2000) is on the Pareto front. The solutions are relatively well spread out over the Pareto front due to the sharing operation. 79 (a) Size (kr) 0.6 0.55 0.5 0.45 0.4 0.35 0.3 2 1.8 1.5 1 1.6 1.4 Efficiency log10(%) 0.5 Bandwidth log10(%) 0 (c) 0.6 C Size (kr) 0.55 0.5 B 0.45 0.4 A 0.35 0.3 (b) 2 1.8 1.5 1 1.6 Size (kr) 0.6 1.4 Efficiency log10(%) 0.55 0.5 0.5 0 Bandwidth log10(%) 0.45 0.4 0.35 0.3 2 1.8 1.5 1 1.6 1.4 Efficiency log10(%) 0.5 0 Bandwidth log10(%) Fig. 5.5 Convergence of the Pareto front as a function of the number of generations in terms of bandwidth, efficiency and antenna size. (a) Initial generation. (b) After 200 generations. (c) After 1000 generations. 80 5.3.2 Verification of the GA-Optimized Results To verify the GA results, three GA-optimized designs are selected from the Pareto front (at points A (kr=0.32), B (kr=0.42) and C (kr=0.5)) and are shown in Fig. 5.6. The smallest sample, at point A, somewhat resembles a helix, while the largest sample C resembles a complicated loop where the end of the wire is connected to the ground plane. These three designs were built and their performances were measured. Copper wire of radius 0.5mm is used, and a 1.6m � 1.6m conducting plate as the ground plane is used. Fig. 5.7(a) is a photo of design B and Fig. 5.7(b) is the resulting return loss (dB) as a function of frequency by simulation and measurement. Except for a slight (3%) shift in the resonant frequency, the simulation and measurement results show nearly the same bandwidth (about 5.3% based on |S11| ? -3dB). Fig. 5.7(c) is the resulting efficiency of the antenna. The standard Wheeler cap method [55,56] is used to measure the efficiency. The measured efficiency matches the simulation well at the resonant frequency of 400 MHz, as indicated by the arrow in Fig. 5.7(c). At other frequencies, the agreement is also good except in the neighborhood of 385MHz. The presence of the large efficiency dip based on the measured data is due to an anti-resonance in the antenna, as the Wheeler cap method fails near this anti-resonance. 81 (a) kr=0.34 (c) kr=0.50 (b) kr=0.42 Fig. 5.6 Three samples from the Pareto front (a) kr=0.34, (b) kr=0.42 and (c) kr=0.50. 82 (a) (c) 0 100 -1 90 -2 80 -3 70 Efficiency (%) S11 (dB) (b) -4 -5 -6 -7 60 50 30 -8 20 -9 10 -10 350 360 370 380 390 400 410 420 430 440 450 0 350 Simulation 360 370 380 390 400 410 420 Frequency (MHz) Frequency (MHz) Fig. 5.7 Measurement 40 (a) Photo for antenna B, which has an antenna size of kr=0.42. (b) Return loss and (c) efficiency versus frequency of antenna B. The efficiency measurement was done using the Wheeler cap method. 83 430 440 450 Similar good agreements were also found for antennas A (kr=0.34) and C (kr=0.50). The results are summarized in Table 5.1. It is noted that both the achievable bandwidth and efficiency drop as the antenna size is reduced. Measured BW (3dB) Simulated BW (3dB) Measured Eff (%) Simulated Eff (%) Antenna A 2.1 % 2.5 % 84 % 88 % Antenna B 5.3 % 5.5 % 92 % 94 % Antenna C 8.5 % 9.8 % 94 % 97 % Table 5.1 Bandwidth and efficiency for the sample antennas A, B and C by measurement and simulation. 5.3.3 Physical Interpretation of GA-Optimized Design By examining the GA-optimized antenna structure in Fig. 5.6(c), we see that the end of the antenna is connected to the ground plane. Most of the antennas with 0.45<kr<0.65 have this characteristic, which is similar to a folded monopole antenna. Since a folded monopole has four times the input impedance of a standard monopole [57], the GA-designed antennas use this basic structure to boost up the impedance of the antenna to approach 50 ?. As we examine at the GA-optimized structures for even smaller-sized antennas (kr<0.45) such as those in Figs. 5.6(a) and (b), most of the antennas are shorted to the ground plane at the joint between the first and second segments 84 from the feed. This turns the first segment into an inductive feed. Segments 2 through 7 become the radiating part of the antenna, carrying most of the current. The strength of the inductive coupling depends on the distance between the first and second segments. Since inductive coupling can greatly increase the input impedance, the GA finds this as an optimized structure for very small-sized antennas (kr<0.45), which need a large impedance step-up to get to 50 ?. This concept is currently investigated in more detail in order to design very small antennas [58]. 5.4 Comparison to Fundamental Limit In this section, the GA results are compared to the fundamental limits for small antennas. Fig. 5.8 depicts all of the GA-optimized designs on the Pareto front plotted in the 3-D bandwidth, efficiency and antenna size space. A least squares fit is used a to create a surface that best fits the GA results. To more easily interpret the results, the 3-D plot is projected onto three planes, and the results are shown in Figs. 5.9(a), 5.9(b) and 5.9(c). Then these GA results are compared to the well-known fundamental limit using a combination of equations in [45] and [46]: BW = 1 Eff 譗 Where Q = (5.4) 1 1 + 3 kr (kr) 85 Antenna size (kr) C B A Efficiency log10(%) Fig. 5.8 Bandwidth log10(%) Pareto front of the GA designs after convergence. The surface is generated using a least squares fitting to best fit the GA results shown as dots. The curves based on (5.4) are shown in Figs. 5.9(d)-5.9(f). Fig. 5.9(a) shows the maximum bandwidth curve achievable by the GA designs as a function of antenna size for different efficiencies. As expected, for a given efficiency, the achievable bandwidth decreases as the antenna size is reduced. Also, the higher the efficiency, the lower the achievable bandwidth. It is similar to the trend of the 86 fundamental limit in Fig. 5.9(d). However, the GA performance is lower than the fundamental limit. Fig. 5.9(b) is the projection of GA designs on the antenna size versus efficiency plane. For a given bandwidth, the achievable efficiency decreases as the antenna size is reduced. This trend can also be observed in Fig. 5.9(e). Fig. 5.9(c) is the projection of the GA results on the bandwidth versus efficiency plane. For a given antenna size, the tradeoff between antenna bandwidth and antenna efficiency can be clearly seen. Again, the trend on this graph is similar to the fundamental limit in Fig. 5.9(f). To make the antenna performance easier to assess, the figure-of-merit ? suggested in [54] is used: ?= Eff � BW 2 Q (5.5) where Q is given in (5.4) and an extra factor of 2 is used to account for the loaded Q. Using this expression, the fundamental limit on ? is always 1 for antennas of arbitrary sizes. All of the GA-optimized designs are re-plotted using this figureof-merit in Fig. 5.10(a). As a reference, the disk-loaded monopole from [48] is plotted on the same figure. As can seen in Fig. 5.10(a), most of the GA designs are ?<0.5. The smaller the antennas, the worse the performance of the GA designs. Next the way to further improvement of the GA designs is investigated. 87 (a) 50 40 50 40 30 30 Bandwidth (%) Bandwidth (%) (d) 20 10 Eff Eff Eff Eff 1 0.3 0.35 0.4 0.45 0.5 = = = = 0.55 40% 55% 70% 85% 20 10 1 0.3 0.6 Eff Eff Eff Eff 0.35 Antenna size (kr) 0.4 0.45 0.5 = = = = 0.55 40% 55% 70% 85% 0.6 Antenna size (kr) (b) (e) Efficiency (%) 80 BW BW BW BW = = = = 100 5% 10% 20% 30% 80 Efficiency (%) 100 60 40 20 0.3 0.35 0.4 0.45 0.5 0.55 60 40 20 0.3 0.6 BW BW BW BW 0.35 Antenna size (kr) 0.4 0.45 0.5 = = = = 0.55 5% 10% 20% 30% 0.6 Antenna size (kr) (f) 100 100 80 80 Efficiency (%) Efficiency (%) (c) 60 40 20 kr kr kr kr 1 = = = = 0.34 0.42 0.50 0.59 10 20 30 40 20 40 50 Bandwidth (%) Fig. 5.9 60 kr = kr = kr = kr = 1 0.34 0.42 0.50 0.59 10 20 30 40 50 Bandwidth (%) (a) Projection of the Pareto front onto the size and efficiency plane. (b) Projection of the Pareto front onto the size and bandwidth plane. (c) Projection of the Pareto front to the bandwidth and efficiency plane. (d) through (f) show the corresponding fundamental limit based on (5.4). 88 5.5 Further Improvement on GA Designs To bring the figure-of-merit of the designs even closer to the fundamental limit, additional design freedoms are explored to the original 7-wire configuration. First, the number of segments is increased up to 16 wires. However, these results show almost no improvement compared to the original 7wire ones. Next, the selection of characteristic impedance is permitted to vary from 1 ? to 300 ?, instead of requiring a fixed 50 ? for the input port. This is the assumption used in the work of Altshuler [9], who assumes that a perfect impedance transformer is available. When allowed this freedom, the GA produced the results plotted in Fig. 5.10(b). As the graph shows, using variable characteristic impedance only gives a slight improvement in performance over the original design plotted in Fig. 5.10(a). Then a multi-arm configuration is examined to improve antenna performance. Two arms are used for the antennas structure and the resulting figure-of-merits are shown in Fig. 5.10(c). It shows good improvement for largesized antennas (kr>0.45). Using the multi-arm configuration, the antenna efficiency is increased while the antenna bandwidth is preserved. This is achieved by spreading the current on multiple branches and lowering the power dissipation. However, the multi-arm design does not show much improvement for small-sized antennas (kr<0.45). The number of arms is further increased to four, but adding more number of arms does not show more improvement over the two89 arm design. It is due to the difficulty in packing a multi-arm structure in a limited design space. Finally, two different wire radii are allowed in each design as an additional degree of design freedom. The resulting antenna has one wire radius for the lower portion of the antenna and another radius for the upper portion. Stepping the wire radius has the effect of increasing the input impedance, similar to the way a radius step up is used in a folded monopole design. Since it is known that the NEC version 2 used in the simulation does not accurately model the wire radius change, the radius change is limited to less than two. Fig. 5.10(d) shows the result of the design. It shows improved performance for both small-sized antennas and large-sized antennas. Based on these preliminary results, it appears that the use of targeted design freedoms can further improve the performance of the GA-optimized designs toward the fundamental limit. 5.6 Summary The Pareto GA has been applied to design electrically small wire antennas by considering antenna bandwidth, efficiency and size. Wire structures comprising of multiple segments were considered. The key advantage of using the Pareto GA method is that a whole series of optimal designs of varying size, bandwidth and efficiency can be generated efficiently in a single GA run. 90 �- 7-segment wire (a) 1.2 Fundamental Limit (2/Q) 1 Figure-of-merit (?) [48] 0.8 0.6 C B A 0.4 0.2 0 0.3 0.35 0.4 0.45 0.5 Antenna size (kr) 0.55 0.6 0.65 �- Variable characteristic impedance (b) 1.2 Fundamental Limit (2/Q) Figure-of-merit (?) 1 [48] 0.8 0.6 0.4 0.2 0 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Antenna size (kr) Fig. 5.10 Small antenna performance using the definition of ? = (Eff � BW) / Theoretical BW Limit. (a) Original 7-wire configuration. (b) Variable input impedance. 91 �- Multi-arm (c) 1.2 Fundamental Limit (2/Q) 1 [48] Figure-of-merit (?) 0.8 0.6 0.4 0.2 0 0.3 0.35 0.4 0.45 0.5 Antenna size (kr) 0.55 0.6 0.65 �- Radius step up (d) 1.2 Fundamental Limit (2/Q) 1 Figure-of-merit (?) [48] 0.8 0.6 0.4 0.2 0 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Antenna size (kr) Fig. 5.10 (Cont' d) Small antenna performance using the definition of ? = (Eff � BW) / Theoretical BW Limit. (c) Multi-arm configuration. (d) Multiple wire radii. 92 By incorporating the concept of the divided range GA, a well-formed Pareto front in terms of the three objectives was achieved. To verify the GA results, several antennas based on the GA designs were built and their bandwidth and efficiency were measured. Both the bandwidth and efficiency measurements agreed well with the simulation for all the sample antennas. Based on the Pareto front, it was observed that for a given antenna size, broader antenna bandwidth must be traded off against lower antenna efficiency. The results also showed that the maximum achievable bandwidth and efficiency both decrease as the antenna size is reduced. The performance achieved by the GA designs was also compared against the well-known fundamental limit for small antennas. The resulting GA designs followed the trend of the fundamental limit, but were about a factor of two below the limit. To further improve the performance of the GA-designed antennas, other design freedoms were explored such as variable characteristic impedance, multiarm wires and multiple wire radii. Results showed that the use of targeted design freedoms could further improve the optimization performance toward the fundamental limit. 93 Chapter 6 Conclusions In this dissertation, GA has been applied to three classes of EM design problems: microstrip antennas, low-profile microwave absorbers and electrically small wire antennas. In Chapter 2, GA-optimized patch shapes for broadband microstrip antennas on thin FR-4 substrate was investigated. The optimized shape showed a four-fold improvement in bandwidth compared to the standard square microstrip. This result was verified by laboratory measurement. The basic operating principle of the optimized shape can be explained in terms of a combination of two-mode operation and ragged edge shape. Chapter 3 examined GA-optimized patch shapes for multi-band microstrip antennas. It was shown that for dual-band operation, a frequency ratio ranging from 1:1.1 to 1:2 between the two bands can be achieved using the GA methodology. Tri-band and quad-band microstrip shapes were also generated, and the resulting antennas performed well at the design frequencies. All results were verified by laboratory measurements. In Chapter 4, GA-optimized shapes for corrugated microwave absorbers under near-grazing incidence were investigated. Corrugated coatings were studied 94 for different incident polarizations, and a physical interpretation for the optimized shapes was presented. The Pareto GA was applied to efficiently map out the absorbing performance versus absorber height. The converged Pareto front showed that better absorbing performance must be traded off against absorber height. Chapter 5 described the use of the Pareto GA to design electrically small wire antennas by considering the bandwidth, efficiency and antenna size. Using the concept of divided-range GA, a well-formed Pareto front in terms of all three objectives was achieved. The performance achieved by the GA designs was compared against the well-known fundamental limit for small antennas. The resulting GA designs followed the trend of the fundamental limit but were below the limit by approximately a factor of two. Further improvement of the GAdesigned antennas was tried by incorporating additional design freedoms. In this dissertation, it has been demonstrated that the GA methodology is very effective in solving real-world EM design problems. It has also been shown that GA is highly adaptable to a variety of design problems. However, a detailed understanding of the actual physics of a particular problem is essential for the success of GA. The physics of the problem should be incorporated into such GA process as chromosome encoding and geometrical filtering to achieve fast convergence and satisfactory solutions. Another consideration is that the total computation time needed for the GA is usually much longer than deterministic 95 algorithms. To solve the time issue, parallel computation was implemented in this dissertation. Other ways to boost the convergence rate should also be further explored. Advanced GA techniques such as the Pareto optimization and the divided range optimization have been employed in this dissertation to solve multiobjective EM problems. These advanced techniques showed highly efficient capability to generate an entire set of optimal solutions with a single optimization run. Consequently, it became possible to utilize such methodology to map out the performance bounds associated with a given problem. The resulting performance bounds provided valuable information on the design of very complex EM systems. In future research, both microstrip and wire type antenna structures should be further investigated for new applications. In particular, the study of electrically small wire antennas is currently being extended to HF communications, where the physical size of antennas is usually very large. Some other candidate EM problems are listed below. Design of antennas for ultra-wideband (UWB) Applications: The recent emergence of UWB communication systems requires new types of antennas that can operate over a frequency bandwidth of more than 100%. Furthermore, UWB systems require antennas to have not only flat amplitude but also linear phase response over the broad bandwidth. Additional design concerns include low 96 cross-polarization and high efficiency. These criteria for UWB systems are very challenging and have not been fully addressed. Design of multiple-input multiple-output (MIMO) communications systems: Multiple antennas have recently been investigated to increase channel capacity and service quality in mobile communication systems. Some preliminary research has been done to design polarization and pattern diversity antennas for mobile handset terminals in MIMO systems. One challenging problem yet to be addressed is the design of diversity antennas with a small antenna form factor. Design of on-glass antennas for vehicular applications: On-glass antenna design for automobiles is a costly and time-consuming engineering process. The design methodology developed in this dissertation can be applied to generate optimized printed antenna designs for vehicle rear windows (on sedans) or vehicle side windows (on SUVs and minivans). Design of microwave absorbers: Chapter 4 in this dissertation examined microwave absorber shapes to reduce the RCS of military targets. A possible commercial application could be the design of absorbers to minimize EM interference in RF systems. 97 Appendix A Wheeler Cap Method for Measuring the Efficiency of Microstrip Antennas In this Appendix, some work into the Wheeler cap method for measuring the efficiency of microstrip antennas is described. Since antenna efficiency characterization is an important issue in several of the chapters in this dissertation, this material is included here for completeness. A.1 Introduction The Wheeler cap method is a simple and well-known technique for measuring antenna efficiency [55,56]. The method involves making only two input impedance measurements of the antenna under test: one with a conducting cap enclosing the antenna and one without. The antenna efficiency is then estimated based on either a parallel or a series RLC circuit model for the antenna. Pozar and Kaufman reported on the use of this method for measuring the efficiency of microstrips [59]. Even though it is generally believed that a microstrip antenna is more appropriately modeled as a parallel circuit [14,15,60], their measurement results did not support the parallel RLC model, and they concluded that the loss mechanism in the microstrip is similar to that of a series 98 RLC circuit. Even recently microstrip antenna efficiency has been measured using a series RLC circuit model [61]. In this Appendix, the Wheeler cap method for measuring the efficiency of microstrip antennas is revisited. The main interest stems from the need to characterize the efficiency of a class of miniaturized microstrip antennas that was designed using a GA [62]. Here three main findings are presented. First, it is shown that the parallel RLC model is indeed a more appropriate model to use than the series model for microstrips. The results are corroborated by numerical simulations using the commercial software ENSEMBLE. Second, the role of interior cap modes is investigated and an optimal shape of the Wheeler cap for microstrips is proposed. Finally, the Wheeler cap method is applied to investigate the efficiency of the GA-designed miniaturized microstrip antennas. A.2 Antenna Circuit Model In the Wheeler cap method, two input resistance measurements are needed to obtain the efficiency of the antenna. One is the input resistance before using the cap, Rbefore, and the other is the input resistance R after putting the cap on, Rafter. Since this method is valid only for antennas with a simple dominant loss mechanism, we should know whether a given antenna follows the series circuit model or the parallel circuit model. If the test antenna works more like the series circuit model, then R decreases after applying the cap, and the efficiency is calculated by the following expression: 99 Eff = Rbefore ? Rafter PR RR = = PR + PL RR + RL Rbefore ( A.1) where PR is the total radiated power, PL is the power loss, RR is the radiation resistance and RL is the loss resistance. If the test antenna works more like the parallel circuit model, R should increase after applying the cap, and the efficiency can be calculated using this expression: Eff = Rafter ? Rbefore PR RL = = PR + PL RR + RL Rafter ( A.2) As a test, this Wheeler cap method is applied to measure a standard square-shaped microstrip antenna built on lossy FR-4 substrate with thickness of 1.6mm. The test antenna?s dimensions are 36mm � 36mm, and it has a resonant frequency of 2 GHz. For the Wheeler cap, a conducting rectangular cap (10cm � 5cm � 10cm) is used to completely enclose the test microstrip. Then an HP 8753C network analyzer is used to gather the input impedance of the test antenna. As is stated in [56,59], a perfect contact between the cap and the ground plane is critical for an accurate measurement. Therefore, aluminum tape is used to shield the slight gap between the cap and the ground plane. Fig. A.1(a) shows the resulting input resistance before and after applying cap, marked as a dashed line and a solid line, respectively. At the resonant frequency, it shows Rafter of 113 ohms and Rbefore of 65 ohms. Since the test antenna is represented by a parallel RLC circuit 100 model the cap should increases the input resistance. Based on the parallel RLC model, the Wheeler cap measurement shows an efficiency of 34%. (b) (a) Measurement (FR-4) Input Resistance (?) ENSEMBLE Simulation (FR-4) cap 140 120 140 ------ Before ?? After Rafter ------ Before ?? After 120 100 100 Rbefore 80 80 60 60 40 40 20 20 0 0 -20 1.8 Fig. A.1 1.85 1.9 1.95 2 2.05 2.1 2.15 Frequency (GHz) 2.2 2.25 2.3 -20 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 Frequency (GHz) 2.2 2.25 2.3 Input resistance of a standard square-shaped (36mm�mm) microstrip built on FR-4 before (------) and after (? ??) using the Wheeler cap size of 10cmm�cmm�m. (a) Measured (b) ENEMBLE simulated. Next, the measurement result is verified with two different numerical simulations using the full wave EM simulator ENSEMBLE [63]. In the first simulation, the Wheeler cap is modeled to rigorously predict Rafter and Rbefore. To model the conducting rectangular cap, we generate a cavity on top of the microstrip. The side of the cavity that faces the microstrip is opened. The simulated input resistance is shown in Fig. A.1(b). Similar to Fig. A.1(a), the result shows that the input resistance is actually increased after the cap is placed. This again shows that the loss mechanism in the microstrip more closely 101 resembles a parallel circuit model. Based on the parallel circuit model, the efficiency computed by this Wheeler cap simulation method is 41%. In the second simulation, the gain of the test antenna is computed with and without dielectric and metal loss. Note that metal loss occurs not only on the patch but also on the ground plane, and this should be taken into account when modeling the microstrips. The efficiency is then calculated using eq. (A.3), which gives us a value of 32%. Eff = Gwith loss ( A.3) Gwithout loss The comparisons of two simulation and measurements in terms of frequency are shown in Fig. A.2(a). These two simulation results are reasonably close to the measurement results over the frequency range of interest. The efficiency using the series RLC model is plotted as thin solid line. This shows that the parallel RLC circuit model is more appropriate to measure the efficiency of microstrip. This Wheeler cap method is also applied to a microstrip which is built on a low loss substrate, RT Duroid 5880 (loss tangent of around 0.001). This test microstrip has dimensions of 17.5mm�mm and operates at near 5.3GHz. This measurement shows a 92% efficiency, and this compares favorably to corresponding simulation results (95% in the gain simulation, and 98% in the Wheeler cap simulation) as shown in Fig. A.2(b). The small deep at the frequency of 5.12GHz is caused by an excitation of a cavity mode (TM111) by a Wheeler 102 cap. These results confirm the validity of our Wheeler cap measurements. In [59], Pozar and Kauffman reported Rafter < Rbefore in their measurement of the simple microstrip, and they concluded that the loss mechanism in the microstrip is similar to the a series circuit model. It should be noted that this is probably due to the influence of interior cap modes when using a large-sized Wheeler cap. In the next section, cap modes and their dependence on the size of the cap will be discussed. (b) (a) 100 100 Measured Measured (Series) Wheeler Cap Simulation Gain Simulation 90 90 80 70 Efficiency (%) Efficiency (%) Efficiency (% ) 80 -10dB bandwidth 60 50 40 30 60 50 40 30 20 20 10 10 0 1.96 1.98 2 2.02 2.04 2.06 0 5.1 2.08 Frequency (GHz) Fig. A.2 -10dB bandwidth 70 Measured Measured (Series) Wheeler Cap Simulation Gain Simulation 5.15 5.2 5.25 5.3 5.35 5.4 Frequency (GHz) Wheeler cap measured efficiency based on parallel circuit model (??), Wheeler cap measured efficiency based on series circuit model (??Purple), efficiency by Wheeler cap simulation (------) and efficiency by gain simulation (------). (a) microstrip built on FR-4 substrate (b) microstrip build on Duroid. 103 5.45 A.3 Effect of Cap Dimensions Wheeler recommended that the cap radius be around 1/6 of a wavelength to cause no change in the current distribution on the antenna [55,56]. However, for microstrip antennas, a larger size Wheeler cap may have to be used to enclose an extended substrate or to enclose a microstrip array. Cap Height: 8.5cm ------ Before ?? After 700 600 17�5�cm (2.83�41�83?) 500 400 ------ Before ?? After 800 Input Resistance (?) 800 Input Resistance (?) Cap Height: 2.0cm 900 900 300 200 700 600 17�0�cm (2.83�33�83?) 500 400 300 200 100 100 0 0 4.6 4.8 5 5.2 5.4 5.6 4.6 5.8 Frequency (GHz) Fig. A.3 4.8 5 5.2 5.4 Frequency (GHz) 5.6 5.8 Measured input resistance of a standard square-shaped (36mm x 36mm) microstrip built on Duroid before (------) and after (??) using (a) the Wheeler cap size of 17cmm x 17cmm x 8.5cm. (b) using the Wheeler cap size of 17cm x 17cm x 2cm. Fig. A.3(a) is an example of the effect using a larger size (17cm�5cm�cm) Wheeler cap for the same microstrip (17.5mm�mm) built on low loss substrate. In a small Wheeler cap, interior cap modes exist, but at such high frequencies that they do not significantly interfere with the resonant 104 frequency of the microstrip as shown in Fig. A.1(a) and (b). On the other hand, a larger Wheeler cap, shown in Fig. A.3(a), creates interior cap modes near the resonant frequency of the microstrip, which causes a deviation in the input resistance value. This can cause an inaccurate efficiency measurement when using the Wheeler cap method. Thus, the interior cavity modes should be as sparse as possible. This may be achieved by using a smaller cap size. However, the finite size of the microstrip substrate restricts the minimum size of the cap. If we take a detailed look at the interior cavity modes, only TM modes are dominant in the Wheeler cap since the microstrip works similarly to a horizontal magnetic current parallel to the ground plane. f TM MNP 1 2? � M? a 2 N? + b 2 P? + c 2 ( A.4) Fig. A.4(a) shows the interior cavity mode-spectrums for a Wheeler cap sizes of 17cm x 8.5cm x 17cm. The solid line is the measured input resistance and the dashed line is mode-spectrum calculated using eq. (A.4) [64]. The measured cavity modes compare well with the ones by calculation. The index N is associated with the cap height while the other indexes M and P are associated with the length and the width of the cap, respectively. 105 (a) Cavity Mode Spectrum of 8.5cm Cap Height 2 1 0.5 (b) 3 3.5 4 4.5 5 5.5 5 5.5 TM114 TM313 TM214 2.5 TM212 TM113 2 TM111 0 1.5 TM112 Log10 [Rin (?)] 1.5 Cavity Mode Spectrum of 2.0cm Cap Height Log10 [Rin (?)] 2 1.5 1 0.5 0 1.5 Fig. A.4 2 2.5 3 3.5 4 Frequency (GHz) 4.5 Interior cavity modes spectrums for two Wheeler cap size. The upper spectrum is for the Wheeler cap size of 17cm x 17cm x 8.5cm.The lower spectrum is for the Wheeler cap size of 17cm x 17cm x 2.0cm. 106 Then the cavity mode-spectrum is observed by reducing only the height of the Wheeler cap since the planar profile of the microstrip prevents us from decreasing the other two dimensions. As shown in Fig. A.4(b), the mode spectrum of the 2cm height Wheeler cap (17cm x 2cm x 17cm) is sparser than the mode spectrum of the 8.5cm one. This is due to the fact that a cap height reduction makes the mode spectrum sparser by pushing the interior modes with index N=1 to much higher frequencies. Sparser mode spectrums provide more space in which to make Wheeler cap measurements. This theory is applied to the test by measuring the input resistance of the same microstrip shown in Fig. A.3(a) but using a reduced cap height of only 2.0cm (0.33?). The results shown in Fig. A.3(b), indeed show the sparser cavity mode-spectrum as expected. However, the resonant frequency with cap is about 91MHz lower than the resonant frequency without cap. One solution is to shift the capped resonant frequency up to 91MHz to compensate the frequency shift by cap effect [65,66]. After the frequency compensation, an efficiency value of 94% with this reduced height (17cm x 2.0cm x 17cm) Wheeler cap is obtained, which is closer to the simulation value than the efficiency of 72.5% by using the size of Wheeler cap. This shows that reduced height wheeler caps are helpful in obtaining more accurate efficiency values without sacrificing the length and width of the cap. 107 A.4 Results for Efficiency of Miniaturized Microstrips In this section, the Wheeler cap method described above is applied to measure the efficiency of our miniaturized microstrips. A GA is previously applied to minimize the size of a microstrip patch while keeping its bandwidth as broad as possible [62]. (a) (b) 100 50 Efficiency(%) Efficiency (%) 40 35 30 25 20 15 90 Standard Square Microstrips 80 Efficiency (%) Efficiency(%) -Measurement -Simulation 45 GA Microstrips 70 60 50 40 30 10 20 5 10 0 40 50 60 70 80 90 0 40 100 % from regular size Fig. A.5 GA Microstrips -Measurement -Simulation 50 60 70 80 90 100 % from regular size (a) built on FR-4 substrate in terms of % from the regular size and (b) the microstrips built on Duroid. The insets in Fig. A.5(a) are the samples of GA-miniaturized microstrips. The achievable bandwidth of these miniaturized antenna drops as the size of the antennas is reduced from 8% to 1.3%. It also shows that even when the size of the patch is reduced to 40% of the regular size, it still maintains a bandwidth of around 1.3%, which is good compared to the microstrip?s small size. However, we thought that investigating the efficiency of these microstrips is crucial due to 108 the high loss in FR-4 substrate. Thus, the efficiencies of these microstrip are observed and the results are shown in Fig. A.5 as the solid line with the efficiencies using Wheeler cap measurements and the dashed line is using gain simulations. The simulated and measured efficiencies are close, showing that our measurements are accurate, even if the results show that our microstrips have very low efficiencies. This is somewhat to be expected due to the high loss on the FR-4 substrate, which has a loss tangent of around 0.025. For this reason, our study is extended to look at how low-loss substrate materials such as Duroid and air increase the efficiency of these miniaturized microstrips. Fig. A.5(b) shows the measured and simulated efficiencies of microstrips using the Duroid substrate with loss tangent of about 0.001. An improved efficiency of more than 65% for all samples can be observed. This gain in efficiency is not without a trade-off. The achievable bandwidth using the Duroid substrate is reduced compared to the bandwidth using the FR-4 substrate. This forces us to evaluate each antenna in terms of both its efficiency and its bandwidth using the antenna?s EB product [66]. Fig. A.6 plots the EB product against the physical antenna size for antennas built on three different substrates: FR-4, Duroid and air substrate, marked as solid, dashed and solid-dotted lines, respectively. Using Fig. A.6, It could be determined which substrate material provides the highest EB product for a given antenna size. 109 7 Air (3.25mm) 5 Bandwidth * Efficiency Efficiency x Bandwidth (EB) 6 Duroid (3.2mm) 4 3 2 FR-4 (1.6mm) 1 0 10 20 30 40 50 60 70 Physical Antenna Size (mm) Fig. A.6 EB Product against physical microstrip patch size for microstrip built on three different substrates. (a) FR-4 (? ??), Duroid (------)and Air (-?-?-?-?). A.5 Summary The Wheeler cap method for measuring microstrip efficiency was revisited, and it was shown that the parallel circuit model is appropriate for the microstrip loss mechanism. The measured efficiency values were verified using a numerical simulation code. Then interior cap modes were investigated, and a way to diminish them using a reduced height Wheeler cap was found. Finally, the reduced-height Wheeler cap method was applied to investigate the efficiency of miniaturized microstrip antennas on various substrate materials. However, this 110 Wheeler cap method is only valid for a limited frequency near a resonance. Also, the method is valid for an antenna that has only one dominant loss mechanism. 111 Bibliography [1] D. G. Luenberger, Introduction to Linear and Nonlinear Programming. Reading, MA: Addison-Wesley, 1973. [2] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. Reading, MA: Addison-Wesley, 1989. [3] J. H. Holland, Adaptation in Natural and Artificial Systems. Ann Arbor: University of Michigan Press, 1975. [4] Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms. New York: John Wiley & Sons, 1999. [5] L. Alatan, M. I. Aksun, K. Leblebicioglu and M. T. Birand, ?Use of computationally efficient method of moments in the optimization of printed antennas,? IEEE Trans. Antennas Propagat., vol. 47, pp.725-732, Apr. 1999. [6] J. M. Johnson, and Y. Rahmat-Samii, ?Genetic algorithms and method of moments (GA/MOM) for the design of integrated antennas,? IEEE Trans. Antennas Propagat., vol. 47, pp. 1606-1614, Oct. 1999. [7] R. L. Haupt, ?Optimum quantised low sidelobe phase tapers for arrays,? Elec. Lett., vol. 31, pp. 1117-1118, July 1995. [8] E. E. Altshuler and D. S. Linden, ?Wire-antenna designs using genetic algorithm,? IEEE Antennas and Propagat. Mag., vol. 39, pp. 33-43, Apr. 1997. [9] E. E. Altshuler, ?Electrically small self-resonant wire antennas optimized using a genetic algorithm,? IEEE Trans. Antennas Propagat., vol. 50, pp. 297-300, Mar. 2002. [10] E. Michielssen, J. M. Sajer, S. Ranjithan and R. Mittra, ?Design of lightweight, broad-band microwave absorbers using genetic algorithm,? IEEE Trans. Microwave Theory Tech., vol. 41, pp. 1024-1031, June-July 1993. 112 [11] M. Pastorino, A. Massa and S. Caorsi, ?A microwave inverse scattering technique for image reconstruction based on a genetic algorithm,? IEEE Trans. Instrumentation Measurement, vol. 49, pp. 573-578, June 2000. [12] Y. Zhou and H. Ling, ?Electromagnetic inversion of Ipswich objects with the use of the genetic algorithm,? Microwave Optical Tech. Lett., vol. 33, pp. 457-459, June 2002. [13] L. J. Chu, ?Physical limitations of omnidirectional antennas,? J. Appl. Phys., vol. 19, pp. 1163-1175, Dec. 1948. [14] J. R. James and P. S. Hall, Handbook of Microstrip Antennas. London: Peter Peregrinus, 1989. [15] D. Pozar and D. H. Schaubert, Microstrip Antennas. New York: The Institute of Electrical and Electronics Engineers, 1995. [16] H. Choo, A. Hutani, L. C. Trintinalia and H. Ling, ?Shape optimisation of broadband microstrip antennas using genetic algorithm,? Elec. Lett., vol. 36, pp. 2057-2058, Dec. 2000. [17] L. C. Trintinalia, ?Electromagnetic scattering from frequency selective surfaces,? M.S. Thesis, Escola Polit閏nica da Univ. de S鉶 Paulo, Brazil, 1992 [18] T. Cwik and R. Mittra, ?Scattering from a periodic array of free-standing arbitrarily shaped perfectly conducting or resistive patches?, IEEE Trans. Antennas Propagat., vol. 35, pp.1226-1234, Nov. 1987. [19] M. Villegas and O. Picon, ?Creation of new shapes for resonant microstrip structures by means of genetic algorithms?, Elec. Lett., vol. 33, pp. 15091510, Aug. 1997. [20] A. Rosenfeld and A. C. Kak, Digital Picture Processing. Academic Press, 1982, 2nd ed. [21] Y. X. Guo, K. M. Luk and K. F. Lee, ?Dual-band slot-loaded shortcircuited patch antenna,? Elec. Lett., vol. 36, pp. 289-291, Feb. 2000. [22] H. Choo, and H. Ling, ?Design of d

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