The Pennsylvania State University

The Graduate School

College of Engineering

FRACTAL SHAPED ANTENNA ELEMENTS FOR

WIDE- AND MULTI- BAND WIRELESS APPLICATIONS

A Thesis in

Engineering Science and Mechanics

by

K.J. Vinoy

 2002 K.J. Vinoy

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

August 2002

We approve the thesis of K.J. Vinoy.

Date of Signature

Vijay K. Varadan Distinguished Alumni Professor of Engineering Science and Mechanics and Electrical Engineering Thesis Advisor, Chair of Committee

Vasundara V. Varadan Distinguished Professor of Engineering Science and Mechanics and Electrical Engineering

Jose A. Kollakompil Senior Research Associate

Douglas H. Werner Associate Professor of Electrical Engineering

James K. Breakall Professor of Electrical Engineering

Judith A. Todd Professor of Engineering Science and Mechanics P.B. Breneman Department Head Chair

ABSTRACT

The use of fractal geometries has significantly impacted many areas of science and engineering; one of which is antennas. Antennas using some of these geometries for various telecommunications applications are already available commercially. The use of fractal geometries has been shown to improve several antenna features to varying extents. Yet a direct corroboration between antenna characteristics and geometrical properties of underlying fractals has been missing. This research work is intended as a first step to fill this gap.

In terms of antenna performance, fractal shaped geometries are believed to result in multi-band characteristics and reduction of antenna size. Although the utility of different fractal geometries varies in these aspects, nevertheless they are primary motives for fractal antenna design. For example, monopole and dipole antennas using fractal Sierpinski gaskets have been widely reported and their multiband characteristics have been associated with the self-similarity of the geometry. However this qualitative explanation may not always be realized, especially with other fractal geometries. A quantitative link between multiband characteristics of the antenna and a mathematically expressible feature of the fractal geometry is needed for design optimization. To explore this, a Koch curve is chosen as a candidate geometry, primarily because its similarity dimension can be varied from 1 to 2 by changing a geometrical parameter (indentation angle). Extensive numerical simulations presented here indicate that this variation has a direct impact on the primary resonant frequency of the antenna, its input resistance at this frequency, and the ratio of the first two resonant frequencies. In other words, these antenna features can now be quantitatively linked to the fractal dimension of the geometry. This finding can lead to increased flexibility in designing antennas using these geometries. These results have been experimentally validated.

The relationship between the fractal dimension and multiband characteristics of fractal shaped antennas has been verified using other fractal geometries as well. The physical appearance of fractal a binary tree can be varied by either changing the branching angle, or using different scale factors between the lengths of the stem and branches of the tree.

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While the change in angle does not affect its fractal dimension, the scale factor does. A similar trend is observed in the multiband characteristics of monopole antennas using these geometries. This confirms similar findings based on Koch curves. It may however be mentioned that this correlation between multiband nature of the antenna and the fractal dimension of the geometry could not yet be linked across different geometries having the same dimension.

Apart from these theoretical findings, this research is also directed towards designing antennas with unconventional features. The multiband characteristic of the Sierpinski gasket antenna has been experimentally modified, for octave bandwidth using a ferroelectric substrate material along with an absorber layer. It has also been shown that these antennas can be converted to a conformal configuration with relative ease, without loosing much of its bandwidth. However the presence of absorbers cause some loss in energy, necessitating a compromise between bandwidth and efficiency of the antenna.

The space filling nature of Hilbert curves lead to significant reduction in antenna size. This has been explored numerically and validated experimentally. One of the advantages of using fractal geometries in small antennas is the order associated with these geometries in contrast to an arbitrary meandering of random line segments (which may also result in small antennas). However this fact has not been used in antenna design thus far. In this work, approximate expressions for designing antennas with these geometries have been derived incorporating their fractal nature. Numerical simulations presented here also indicate that antenna size can be further reduced by superimposing one fractal geometry (Koch curve) along line segments of another (Hilbert curve).

Due to the presence of a large number of closely placed line segments, antennas using Hilbert curves can be designed for reconfigurable radiation characteristics with the inclusion of few additional line segments and RF switches. Although these switch positions are not optimized for specific performance, they offer immense potential for designing antennas with novel characteristics.

To conclude, the research work reported here is a numerical and experimental study in identifying features of fractal shaped antennas that could impart increased flexibility in the design of newer generation wireless systems.

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TABLE OF CONTENTS

List of Figures...... viii List of Tables ...... xiii Acknowledgements...... xiv Chapter 1 Introduction ...... 1 1.1 Background...... 1 1.1.1 Fractal Geometries...... 1 1.1.2 Engineering Applications of Fractals...... 5 1.1.3 Fractals in Antenna Engineering...... 6 1.1.4 Fractal Shaped Antenna Elements ...... 6 1.2 Motivation for Present Research...... 8 1.3 Thesis Organization ...... 10 Chapter 2 Techniques for Analysis and Characterization of Antennas ...... 12 2.1 Introduction...... 12 2.2 Theory and Design of Small Antennas ...... 12 2.2.1 EMF Method for Antenna Impedance ...... 13 2.2.2 Issues in Design of Small Antennas...... 14 2.3 Fundamentals of Modeling Techniques Used...... 16 2.3.1 Method of Moments (MoM)...... 16 2.3.2 Finite Difference Time Domain (FDTD) method...... 22 2.4 Experimental Set up for Antenna Measurements ...... 28 Chapter 3 Sierpinski Gasket Geometry for Multiband and Wideband Antennas...... 30 3.1 Introduction...... 30 3.2 Sierpinski Gasket Fractal Geometry ...... 31 3.3 Basic Antenna Configurations using Sierpinski Gaskets...... 33 3.4 Sierpinski Gasket Monopole Antenna ...... 35 3.4.1 Effects of Fractal Iteration Levels...... 35 3.4.2 Effects of Apex Angles...... 41 3.4.3 Use of Geometries not Strictly Self-similar...... 43 3.4.4 Effect of Dielectric Support on Antenna Performance...... 49

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3.5 Wideband Antenna...... 53 3.6 Conformal Antenna Configuration ...... 58 3.7 Summary...... 61 Chapter 4 Influence of Fractal Properties on the Characteristics of Dipole Multi-band Antennas Using Koch Curves...... 62 4.1 Introduction...... 62 4.2 Fractal Properties and Generalization of Koch Curves...... 64 4.2.1 IFS for the Standard Koch Curve...... 65 4.2.2 IFS for Generalizations...... 66 4.3 Antenna Modeling Studies...... 69 4.3.1 Dipole Antenna Model...... 69 4.3.2 Results of Numerical Simulations ...... 69 4.3.3 Characteristics of Multiband Antenna ...... 76 4.3.4 Effect of Changing Feed Location...... 76 4.4 Fractal Features in Antenna Properties...... 80 4.4.1 Lowest Resonant Frequency...... 80 4.4.2 Multi-band Characteristics...... 82 4.5 Experimental Validation ...... 82 4.6 Designing Antennas with Optimal Performance ...... 87 4.7 Summary...... 89 Chapter 5 Multi-band Properties of Antennas with Fractal Canopies ...... 95 5.1 Introduction...... 95 5.2 Fractal Nature of Tree...... 95 5.3 Antenna Characteristics using Fractal Tree ...... 100 5.3.1 Parametric Studies by Increasing Fractal Iteration ...... 101 5.4 Parametric Study by Changing Branching Angle of the Fractal Tree ...... 104 2.1.1 Effect of Variation of Branch Length Ratio ...... 106 5.5 Summary...... 107 Chapter 6 Hilbert Curves for Small Resonant Antennas ...... 108 6.1 Introduction...... 108 6.2 Fractal Properties of Hilbert Curves ...... 109 6.3 Antenna Configurations Using Hilbert Curves...... 112

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6.4 Simulation Studies of Hilbert Curve Antenna ...... 114 6.5 Implementation Issues for Dipole Antenna Configuration...... 122 6.6 Design Formulation ...... 122 6.6.1 Modeling of Antennas Neglecting Dielectric Loading...... 124 6.6.2 Model with Dielectric Loading Included...... 127 6.7 Reconfigurable Antennas...... 130 6.8 Patch antenna with Hilbert Curve Geometry...... 134 6.9 Doubly Fractal Hilbert-Koch Antenna Geometry...... 137 6.10 Summary...... 141 Chapter 7 Conclusions and Future Work...... 142 7.1 Conclusions...... 142 7.2 Future Directions ...... 144 References...... 146 Appendix...... 152 Vita

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LIST OF FIGURES

Fig. 1.1 Some common examples of fractals...... 2 Fig. 2.1 Geometry of a cylindrical dipole antenna...... 19 Fig. 2.2 Yee cell geometry used in FDTD algorithm ...... 25 Fig. 2.3 Set up for antenna measurements in an anechoic chamber ...... 29 Fig. 3.1 Two approaches for the generation of Sierpinski gasket geometry...... 32 Fig. 3.2 IFS for the generation of a strictly self-similar Sierpinski gasket geometry...... 32 Fig. 3.3 Antenna configurations with fractal Sierpinski gasket...... 34 Fig. 3.4 Monopole antenna configuration with printed Sierpinski gasket geometry...... 35 Fig. 3.5 Different fractal interations studied for the antenna performance. The overall height in each case is kept at 8 cm. The apex angle is 60°. The antennas are printed on a RO 3003 substrate for validation...... 36 Fig. 3.6 Simulated return loss characteristics of the antennas shown in Fig. 3.5...... 37 Fig. 3.7 Measured return loss characteristics of the antennas printed on Duroid RO 3003 substrate...... 37 Fig. 3.8 Radiation patterns of Sierpinski gasket monopole antenna for four different iterations of the geometry. The patterns are measured at 615 MHz and 1.75 GHz and 3.6 GHz in an anechoic chamber. Patterns at two orthogonal planes are plotted, in planes parallel and normal to the geometry...... 40 Fig. 3.9 Measured return loss for fractal Sierpinski monopole antenna with different apex angles (where the feed is connected). All geometries are of first iteration (simple triangles)...... 41 Fig. 3.10 Radiation patterns of triangular monopole antenna for different flare angles. The patterns are measured at and 1.75 GHz and 3.6 GHz in an anechoic chamber. Patterns in the plane parallel to that of the geometry are plotted...... 42 Fig. 3.11 Generalized Sierpinski gasket geometry that is not strictly self-similar. This geometry is still self-affine, can be generated with IFS and hence fractal...... 44 Fig. 3.12 Photographs of different iterations of geometry generated with IFS. The scale factor (height ratio of bottom triangle to the upper ones) in each iteration is 2:1. ... 44 Fig. 3.13 Photographs of fabricated geometries obtained using different IFS. Height ratios: top right-1:1, bottom left- 1:2, bottom right- 2:1. All these can be generated from the same simple triangle shown here for comparison...... 45 Fig. 3.14 Return loss of antennas shown in Fig. 3.12...... 45 Fig. 3.15 Return loss of antennas shown in Fig. 3.13...... 46 Fig. 3.16 Radiation patterns for three different iterations (shown in Fig. 3.12)of the

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modified geometry. The height ratio in each iteration remain 2:1. Patterns are plotted for 2nd and 3rd resonances, in two orthogonal planes...... 47 Fig. 3.17 Radiation patterns of monopole antennas shown in Fig. 3.13. Patterns on the left side are for second resonance and on the right are for 3rd resonance. Each plot has patterns in two orthogonal planes for the same antenna...... 48 Fig. 3.18 Simulated return loss for 3rd iteration Sierpinski monopole antenna on different dielectric substrates...... 50 Fig. 3.19 Return loss characteristics of the Sierpinski gasket monopole antenna with a BST substrate. The widening of bandwidth obtained using an absorber is not taken up in this simulation study...... 51 Fig. 3.20 Simulated radiation patterns of 3rd iteration fractal Sierpinski monopole antenna printed on Alumina (εr = 9.8) Duroid (εr = 2.2) and FR-4 (εr = 4.4) substrates. These are θ-polarized and are plotted for the respective resonant frequencies...... 52 Fig. 3.21 Simulated total gain of the antennas at a constant direction (φ= 0, θ = 40) for all substrates, plotted against frequency...... 53 Fig. 3.22 Sierpinski monopole antenna configuration with improved impedance bandwidth...... 54 Fig. 3.23 Improvement in input bandwidth obtained when the monopole antenna is backed with an absorber...... 55 Fig. 3.24 radiation patterns in planes parallel (left) and normal (right) to the geometry for a wideband Sierpinski gasket monopole antenna. The effect of having an absorber backing is compared. Each plot is normalized with the maximum...... 57 Fig. 3.25 Conformal wideband antenna configuration. The ground plane entends beyond the area of the printed antenna geometry...... 58 Fig. 3.26 Input characteristics of the conformal antenna configuration...... 59 Fig. 3.27 Radiation patterns at few indicative frequencies for the conformal wideband antenna...... 60 Fig. 4.1 Geometrical construction of standard Koch curve...... 64 Fig. 4.2 Generalized Koch curves of first four iterations with two different indentation angles. The length of subsections for a given iteration is a function of angle of indentation...... 67 Fig. 4.3 Change in unfolded (stretched-out) curve length obtained by the variation of indentation angles of the generalized Koch curve geometry. The parametric curves are for different fractal iterations. The end-to-end distance in all these cases is of unit length...... 68 Fig. 4.4 Variation of similarity dimension of the generalized geometry for various indentation angles...... 68 Fig. 4.5 Configuration of symmetrically fed Koch dipole antenna. 4th iteration Koch

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curves with indentation angle θ = 60° form each arm of the antenna...... 69 Fig. 4.6 Input resistance for dipole antennas with Koch curves of different indentation angles...... 72 Fig. 4.7 Input reactance of dipole antennas with Koch curves of different indentation angles...... 74 Fig. 4.8 Variation of input resistance of the dipole antennas with generalized Koch curves of various fractal iterations...... 75 Fig. 4.9 Variation of resonant frequencies of dipole antennas with generalized Koch curves of various fractal iterations. The resonant frequencies for each resonance of all cases converge to that on linear dipole when the indentation angle approaches zero...... 77 Fig. 4.10 Current distribution on a dipole antenna with standard Koch curve geometry of 3rd iteration, at its various resonant frequencies...... 78 Fig. 4.11 Radiation patterns of the Koch dipole antenna at its resonant frequencies...... 78 Fig. 4.12 New feed location for matching the input impedance to standard value...... 79 Fig. 4.13 The normalized resonant frequency of generalized Koch dipole antennas of different fractal iterations...... 81 Fig. 4.14 Input resistance of the generalized Koch dipole antennas of different fractal iterations plotted against reciprocal of the fractal dimension...... 82 Fig. 4.15 The ratio of first two resonant frequencies of multi-band Koch dipole antennas as a function of fractal dimension of the geometry...... 83 Fig. 4.16 Photographs of Koch curve dipole antennas: Set 1 for comparison on the effect of the fractal iteration...... 85 Fig. 4.17 Return loss of dipole antennas shown in Fig. 4.16...... 85 Fig. 4.18 Photographs of Koch curve dipole antennas: Set 2 for comparison on the effect of indentation angles...... 86 Fig. 4.19 Return loss of dipole antennas shown in Fig. 4.18...... 86 Fig. 4.20 Primary resonant frequencies for dipole antennas with Koch curves of various indentations...... 87 Fig. 4.21 Ratio of resonant frequencies for dipole antennas with Koch curves of various indentations...... 87 Fig. 4.22 A generalized Koch curve generated with different indentation angle at each iteration stage. In this case, θ=20°, 30°, 40° (starting from the innermost) are used...... 88 Fig. 4.23 Input reactance of dipole antennas based on two sets of generalized Koch curves. All antennas have arms spreading 10 cm, and have 2nd generation Koch curve. The indentation angles for different iterations are different as marked in the plots...... 90

x

Fig. 4.24 Input resistance at the first resonance of dipole antennas based on generalized Koch curves. All antennas have an arm length of 10 cm, and have 2nd generation Koch curve. The angle at the inner IFS is the parametric variation while the x-axis is for the angle of the outer IFS...... 92 Fig. 4.25 Input reactance of dipole antennas based on two sets of generalized Koch curves. All antennas have arms spreading 10 cm, and have 3rd generation Koch curve. The angle at each iteration is different as marked in the plots. The indentation angle for the innermost stage is the same (60° in all these cases)...... 94 Fig. 5.1 Various iterations of fractal binary tree...... 96 Fig. 5.2 3rd iterated fractal binary tree with different branching angles ...... 97 Fig. 5.3 3rd iterated binary trees with different length ratios for arm to stem...... 98 Fig. 5.4 Variation of the fractal dimension with length ratio of branch to stem of a binary tree ...... 100 Fig. 5.5 Monopole antenna using a binary tree geometry...... 101 Fig. 5.6 Input impedance characteristics of binary tree monopole antenna with fractal geometries of various iterations...... 102 Fig. 5.7 Radiation patterns of a typical tree monopole antenna at its various resonances...... 103 Fig. 5.8 Variation of first resonant frequency of monopole antennas with binary trees with branching angles ...... 104 Fig. 6.1 First four iterations of Hilbert curve geometry. The segments used to connect copies of the previous iteration are shown in dashed lines...... 109 Fig. 6.2 Generation of Hilbert curve using L-systems...... 111 Fig. 6.3 A conceptual evolution of a fractal Hilbert curve dipole antenna...... 113 Fig. 6.4 Different antenna configurations using Hilbert curves...... 114 Fig. 6.5 Computed input impedance for dipole antennas with the first four iterations of Hilbert curves. All antennas have outer dimensions of 7 cm, and are modeled with wire of 1.3 mm diameter...... 115 Fig. 6.6 Simulated radiation patterns of the 4th Hilbert curve dipole antenna at various resonant frequencies at different φ planes. Due to symmetry, only one half is shown here. The antenna occupies outer dimension of 7 cm x 7 cm, and modeled with wire of 1.3 mm diameter. The resonant frequencies are: 352, 922, 1233, and 1715 MHz...... 117 Fig. 6.7 Current distribution along the 4th iterated Hilbert curve geometry at its various resonances...... 118 Fig. 6.8 Current along the length of the Hilbert curve when the feed is connected at the center of the geometry. These plots are for the resonant frequencies of the antenna...... 119

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Fig. 6.9 Designing antennas for a specified frequency with various iteration curves. .. 120 Fig. 6.10 Designing antennas for a specified frequency with various resonant frequency options. It may be observed that the radiation characteristics vary only slightly in these cases...... 121 Fig. 6.11 Measured performance of a dipole antenna with Hilbert curve geometry. .... 123 Fig. 6.12 Composition of a HCA with iteration order 3. The short-circuited parallel wire sections and connection wire sections are shown separately...... 124 Fig. 6.13 Comparison of resonant frequencies calculated using the formulation with NEC simulations for different iterations of the geometry. The side of the circumscribed square is varied...... 126 Fig. 6.14 Comparison of resonant frequencies calculated using the formulation with NEC simulations for different iterations of the geometry. The wire diameter is varied. 127 Fig. 6.15 Coplanar strip transmission line – physical parameters ...... 128 Fig. 6.16 Goemetry for reconfigurable antenna...... 131 Fig. 6.17 VSWR of the reconfigurable antenna. Antenna for Case 1 is shown in Fig. 6.16. Case 2 and Case 3 are for antenna performance when switches 1 and 2 are broken respectively...... 132 Fig. 6.18 The radiation pattern (θ-plane) of the base Hilbert curve fractal antenna used in this study. The plots shown are at φ=0° and φ=90°. Only a half is shown because of apparent symmetry...... 132 Fig. 6.19 Radiation patterns as of antenna on its plane, as the additional segments are attached. The additional segments in the geometry are shown with thicker lines for clarity. These can be realized in practice by turning ON or OFF switches connected in the additional arms...... 133 Fig. 6.20 Patch antenna with Hilbert curve geopmetry ...... 134 Fig. 6.21 Measured S11 of the antenna...... 135 Fig. 6.22 Measured S21 of the antenna after calibrating with standard gain antenna. The gain of the standard antenna varies from 8 to 11 dBi within this band. Peak gain of this antenna at 2.4 GHz is 6.5 dBi, 2.8 GHz: 4 dBi, and 3.6 GHz: 4.5 dBi...... 135 Fig. 6.23Radiation patterns of the patch antenna with Hilbert curve geometry in two orthogonal planes...... 136 Fig. 6.24 Examples of hybrid fractal geometry with Koch curves superimposed on Hilbert curves. For dipole antennas using these, the feed is located at the point of symmetry on the curve...... 137 Fig. 6.25 Input impedance simulated using NEC for different iterations of Koch curves superimposed on 3rd iteration Hilbert curve...... 139

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LIST OF TABLES

Table 1-1 Fractal dimension of some geometries shown in Fig. 1.1...... 4 Table 4-1 Primary (first) resonant frequencies for dipole antennas with Koch curves for various iterations obtained by numerical simulations...... 75 Table 4-2 Geometric interval between resonant frequencies of generalized Koch dipole antenna. Although the ratios are different for each interval, they remain the same for different fractal iterations...... 84 Table 4-3 Resonant frequencies of dipole antennas with 2nd generation generalized Koch curves. The indentation angle in each generation stage is different. The dipole arms are of length 10 cm in all cases...... 91 Table 5-1 Resonant Frequency and Input resistance (at resonance) for binary tree monopole antennas with different fractal iterations. The angle between branches is 120° and the scale factor is 0.5...... 103 Table 5-2 Periodicity of resonant frequencies of monopole antennas with fractal binary tree geometry. The band placement remains nearly the same for all branching angles ...... 105 Table 5-3 Multiband characteristics of monopole antennas with 4th generation fractal tree geometry...... 106 Table 6-1 Comparison of results from design formulation with NEC results ...... 126 Table 6-2 Comparison of formulation with experimental results for Hilbert curve dipole antennas printed on dielectric substrate ...... 129 Table 6-3. Characteristics of the radiated beam of the reconfigurable antenna...... 132 Table 6-4 Physical length in meters for the hybrid fractal geometry with Koch curve superimposed on line segments of fractal Hilbert curve. The first row of data corresponds to the basic Hilbert curve itself...... 138 Table 6-5 The primary resonant frequency of the antenna determined by numerical simulations with hybrid fractal geometry consisting of different iterations of Hilbert curve and Koch curve...... 140 Table 6-6 Multi-band characteristics of the hybrid fractal antenna consisting of 3rd iterated Hilbert curve with different Koch curves superimposed. The ratios of successive frequencies are shown in brackets. The first row of data corresponds simple Hilbert curve geometry...... 140

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ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to Professor Vijay Varadan for his valuable guidance and financial support throughout this research work. I also appreciate insightful comments and suggestions from Professor Vasundara Varadan. Special thanks are also due to Dr. K.A. Jose for continued encouragements and support. Professors J.K. Breakall and D.H. Werner have been meticulous in their review of this research work. Their suggestions are deeply appreciated.

I was lucky enough to make new friends at State College, both from within the Center and outside. Their patience and continuous encouragements has indeed made my stay here very fruitful. Anil Tellakula of HVS Technologies has been especially helpful in doing the antenna measurements. My thanks to all of them.

Several people have impacted my career positively, but for which I could not have reached where I am right now. Many of my long time friends have inspired my quest for higher learning. It is a great pleasure to place on record my gratitude towards all of them.

I am also thankful to members of my family, especially my parents for their constant appreciation and continued support.

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CHAPTER 1 INTRODUCTION

Modern telecommunication systems require antennas with wider bandwidths and smaller dimensions than conventionally possible. This has initiated antenna research in various directions, one of which is by using fractal shaped antenna elements. In recent years several fractal geometries have been introduced for antenna applications with varying degrees of success in improving antenna characteristics. Some of these geometries have been particularly useful in reducing the size of the antenna, while other designs aim at incorporating multi-band characteristics. Yet no significant progress has been made in corroborating fractal properties of these geometries with characteristics of antennas. The research work presented here is primarily intended to analyze geometrical features of fractals that influence the performance of antennas using them.

Several antenna configurations based on fractal geometries have been reported in recent years [1] – [4]. These are low profile antennas with moderate gain and can be made operative at multiple frequency bands and hence are multi-functional. In this work the multi-band (multifunctional) aspect of antenna designs are explored further with special emphasis on identifying fractal properties that impact antenna multi-band characteristics. Antennas with reduced size have been obtained using Hilbert curve fractal geometry. Further more, design equations for these antennas are obtained in terms of its geometrical parameters such as fractal dimension. Antenna properties have also been linked to fractal dimension of the geometry. To lay foundations for the understanding of the behavior of such antennas, the nature of fractal geometries is explained first, before presenting the status of literature on antennas using such geometries.

1.1 BACKGROUND

1.1.1 Fractal Geometries

The term fractal was coined by the French mathematician B.B. Mandelbrot during 1970’s after his pioneering research on several naturally occurring irregular and fragmented

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geometries not contained within the realms of conventional Euclidian geometry [5]. The term has its roots in the latin word fractus which is related to the verb fangere (meaning: to break) [6]. These geometries were generally discarded as formless, but Mandelbrot discovered that certain special features can be associated with them. Many of these curves were recognized well before him, and were often associated with mathematicians of yesteryears. But Mandelbrot’s research was path-breaking: he discovered a common element in many of these seemingly irregular geometries and formulated theories based on his findings.

Two examples of naturally occurring fractal geometries are snow-flakes and boundary of geographic continents. Several naturally occurring phenomena such as lightning are better analyzed with the aid of fractals. One significant property of all these fractals is indeed their irregular nature. Some examples of fractals are given in Fig. 1.1. Most of these geometries are infinitely sub-divisible, with each division a copy of the parent. This special nature of these geometries has led to several interesting features uncommon with Euclidean geometry.

Cantor set Koch curve

Sierpinski gasket Sierpinski carpet

Fig. 1.1 Some common examples of fractals.

Mandelbrot defines the term fractal in several ways. These rely primarily on the definition of their dimension. A fractal is a set for which the Hausdorff Besicovich

2

dimension strictly exceeds its topological dimension. Every set having non-integer dimension is a fractal [5]. But fractals can have integer dimension. Alternately, fractal is defined as set F such that [7]:

F has a fine structure with details on arbitrarily small scales,

F is too irregular to be described by traditional geometry

F having some form of self-similarity (not necessarily geometric, can be statistical)

F can be described in a simple way, recursively, and

Fractal dimension of F greater than its topological dimension

Dimension of a geometry can be defined in several ways, most of these often lead to the same number, albeit not necessarily. Some examples are topological dimension, Euclidean dimension, self-similarity dimension and Hausdorff dimension. Some of these are special forms of Mandelbrot’s definition of the fractal dimension. However the most easily understood definition is for self-similarity dimension. To obtain this value, the geometry is divided into scaled down, but identical copies of itself. If there are n such copies of the original geometry scaled down by a fraction f, the similarity dimension D is defined as:

log n D = ( 1.1)  1  log   f 

For example, a square can be divided into 4 copies of ½ scale, 9 copies of ⅓ scale, 16 2 1 copies of ¼ scale, or n copies of /n scale. Substituting in the above formula, the dimension of the geometry is ascertained to be 2.

The same approach can be followed for determining the dimension of several fractal geometries. The dimension of geometries shown in Fig. 1.1 are listed in Table 1-1.

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Table 1-1 Fractal dimension of some geometries shown in Fig. 1.1.

Geometry Dimension

Cantor set 0.6309

Koch curve 1.2619

Sierpinski gasket 1.5850

Sierpinski carpet 1.893

Although this approach is very convenient for many such geometries, all fractals are not amenable for this approach. Such is the case with most plane-, or space-filling fractals. In these cases more mathematically intensive definitions such as Hausdorff dimension are required.

Non-integer dimension is not the only peculiar property of fractals. A glossary of terms used in describing properties of fractals is introduced next. A self-similar set is one that consists of scaled down copies of itself. Many fractal geometries are self-similar, a property which makes easier to accurately compute their Hausdorff dimension. In order to define self-similarity mathematically, first the concept of contraction is introduced. A map ψ:Rn→ Rn is a contraction if there exists some constant number c∈ (0,1) so that the inequality

ψ (x) −ψ (y) ≤ c x − y ( 1.2)

n holds for any x,y ∈ R . For a natural number m ≥ 2, and a set of m contractions {ψ1, ψ2, n n …, ψm} defined on R , a non-empty compact set V in R is self-similar if

m

V = ψ i (V ) ( 1.3) Ui=1

Each such division of the geometry is termed an iteration. Fractal geometries are generally infinitely sub-divisible. Self similar sets defined by linear contractions are

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called Self-affine sets. Other properties associated with fractal geometries include scale- invariance, plane-filling or space-filling nature, and lacunarity. Lacunarity is a term coined to express the nature of area fractal having hollow spaces (“gappiness”) [8]. Plane filling fractals are those that tend to fill an area (or space, in more general terms) as the order of iteration is increased.

Some of these properties are qualitatively linked to the features of antenna geometries using them. It is envisaged that the above description of these properties would shed light into a better understanding of such connection. In the following sub-sections a brief introduction is provided on the use of fractal in science and engineering, and antenna engineering in particular.

1.1.2 Engineering Applications of Fractals

Ever since they were mathematically re-invented by Mandelbrot, fractals have found widespread applications in several branches of science and engineering. Disciplines such as geology, atmospheric sciences, forest sciences, physiology have benefited significantly by fractal modeling of naturally occurring phenomena. Several books and monographs are available on the use of fractals in physical sciences. Fracture mechanics is one of the areas of engineering that has benefited significantly from the application of fractals [9].

The space filling nature of fractal geometries has invited several innovative applications. Fractal mesh generation has been shown to reduce memory requirements and CPU time for finite element analysis of vibration problems [10].

One area of application that has impacted modern technology most is image compression using fractal image coding [11] - [13]. Fractal image rendering and image compression schemes have led to significant reduction in memory requirements and processing time.

In electromagnetics, scattering and diffraction from fractal screens have been studied extensively. Diffracted fields from self-similar Sierpinski gasket in the Franhauffer zone have been shown to be self-similar [14]. The study of wave interactions with such self- similar structures has later been termed fractal electrodynamics [15] - [17]. Some of these studies indicate that scattering patterns from fractal geometries have features of those geometries imprinted on these. More recently fractal geometries have also been

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used in frequency selective screens [18] - [20]. The self-similarity of the fractal geometry has been attributed to the dual band nature of their frequency response. Surface impedance of metallic patterns of fractal geometries on a dielectric slab has also been characterized [21].

Fractal antenna arrays and fractal shaped antenna elements have evolved in 1990’s. An overview of the literature on the use of fractals in antenna engineering is given next.

1.1.3 Fractals in Antenna Engineering

The primary motivation of fractal antenna engineering is to extend antenna design and synthesis concepts beyond Euclidean geometry [22] - [23]. In this context, the use of fractals in antenna array synthesis and fractal shaped antenna elements have been studied.

Obtaining special antenna characteristics by using a fractal distribution of elements is the main objective of the study on fractal antenna arrays. It is widely known that properties of antenna arrays are determined by their distribution rather than the properties of individual elements. Since the array spacing (distance between elements) depends on the frequency of operation, most of the conventional antenna array designs are band-limited. Self-similar arrays have frequency independent multi-band characteristics [24]. Fractal and random fractal arrays have been found to have several novel features [25].

Variation in fractal dimension of the array distribution has been found to have effects on radiation characteristics of such antenna arrays. The use of random fractals reduce the fractal dimension, which leads to a better control of sidelobes [23]. Synthesizing fractal radiation patterns has also been explored [26]. It has been found that the current distribution on the array affects the fractal dimension of the radiation pattern. It may be concluded that fractal properties such as self-similarity and dimension play a key role in the design of such arrays.

1.1.4 Fractal Shaped Antenna Elements

As with several other fields, the nature of fractal geometries has caught the attention of antenna designers, primarily as a past-time. However with the deepening of understanding of antennas using them several geometrical and antenna features have been

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inter-linked. This has led to the evolution of a new class of antennas, called fractal shaped antennas.

Cohen, who later established the company Fractal Antennas Inc. is among the first to get into the bandwagon. He has tried the usefulness several fractal geometries experimentally. Koch curves, Minkowski curves, Sierpinski carpets are among them [3]. These geometries have a large number of tips and corners, a fact that would help improve antenna efficiency. Fractal trees were explored for the same reason, and were found to have multiband characteristics [27]. Self-similarity of the geometry is qualitatively associated with the multiband characteristics of these antennas.

Fractal shaped dipole antennas with Koch curves are generally fed at the center of the geometry. By increasing the fractal iteration, the length of the curve increases, reducing the resonant frequency of the antenna. The resonance of monopole antennas using these geometries below the small antenna limit has been reported by Puente et al [4]. They have also studied the shift in resonant frequency by increasing the fractal iteration order. However detailed studies indicated that this reduction in resonant frequency does not follow the same pace as the increase in length with each subsequent iteration [28]. As the fractal iteration is increased the feature length gets smaller. There seems to be a limit in the minimum feature length that influences antenna properties.

Several other self-similar geometries have also been explored for multiband antenna characteristics [29] - [31], [1]. Sierpinski gaskets have been studied extensively for monopole and dipole antenna configurations [32]. The self-similar current distribution on these antennas is expected to cause its multi-band characteristics [33]. It has been found that by perturbing the geometry (thereby removing its self-similarity) the multiband nature of these antennas can be controlled [34]. Variation in the flare angle of these geometries have also been explored to change the band characteristics of the antenna [32].

Efforts have also been made to improve bandwidths of these antennas. A stacked antenna configuration with multiple layers of fractal geometries has been found to have some effect in this regard [35]. This configuration has also been made conformal to

7

improve the utilization of the antenna.

Similar to Sierpinski gaskets, Sierpinksi carpets have also been used in antenna elements [36]. This geometry is also used as microstrip patch antenna with multiband characteristics [37]. The antenna characteristics such as the peak gain is reported to improve by replacing a rectangular geometry with this fractal.

To summarize the survey on fractal antenna engineering, key aspects of using fractals in antennas are presented here. For fractal arrays, several novel synthesizing algorithms have been developed to tailor radiation patterns. It has been established that random fractals can be used for better control of sidelobe levels. Multi-band operation and a certain extent of frequency independence are possible with such array designs.

The advantages of using fractal shaped antenna elements are manifold. These geometries can be used to design smaller sized resonant antennas. The antenna radiation efficiency is thought to have improved by large number of bends and corners in many of such fractals. These geometries can lead to antennas with multi-band characteristics, often with similar radiation characteristics in these bands.

1.2 MOTIVATION FOR PRESENT RESEARCH

This thesis work is a study on the suitability of some fractal geometries for multifunctional and small antenna applications. Although many fractal geometries are identified, and mathematically studied for long, their applications into electromagnetics is fairly recent. Within the past decade few fractal geometries have been proposed as antenna elements and special antenna characteristics (?) introduced by the use of these geometries have been widely acclaimed. It has been claimed that the non-euclidean nature of these geometries would lead to equally non-conventional antenna performance, and the self-similarity of the geometry is the cause of multi-band characteristics of the resulting antenna. This thesis aims at furthering the understanding the effectiveness of these geometries in antennas, and to bring about true advantages of their use in antenna engineering.

In the process several questions lingering among the community of antenna engineers are addressed, some of which partially. These questions arise when one compares the so

8

called “fractal antennas” using Sierpinski gaskets with other conventional antenna structures such as bow-tie antennas. We have studied monopole, dipole, and planar configurations of these geometries, and it has been found that the effect of fractal iteration is not reflected in the antenna properties. Apart from slight shift in frequencies (due to the removal of metallization), operational bands of the resulting multi-band antennas are quite similar to that obtained with a simple triangular geometry. This poses questions regarding the existence of the newly-found area of fractal antennas. To address this issue, several other fractal geometries are studied to see if true mathematical properties of these geometries can somehow be related to characteristics of the resulting antenna.

This led to a second misconception prevailing regarding design of “small antennas” using fractal geometries such as Koch curves. The low resonant frequency obtained using these geometries has been associated with their fractal nature! Even this notion is questionable in the context of other small antennas, such as meander line antennas. The use of the meander line geometry can effectively reduce the antenna size, as much as with fractal geometry such as Koch curve. This poses the second question: why use fractal geometry in antennas at all? The answer to this question may be stated this way: compared to other irregular geometries, the use of fractals brings in a kind of mathematical order into the antenna shape; the effectiveness of the use of such an order is there for one to explore. As a first step in this direction, the design equations for dipole antenna using Hilbert curves are obtained in terms of their geometrical parameters.

Just as many of the fractal geometries are visually identifiable with vividly differing mathematical features, their impact on the characteristics of the resulting antennas are also distinctively different. It required a great mathematician like Mandelbrot to codify the mathematical properties of these seemingly isolated shapes. Hence to look for common features for all fractal geometries in electromagnetics in general, or, antenna engineering in particular, would be futile until a large number of them are analyzed exhaustively. To choose few from an ever-increasing list of fractal geometries and their variants may not be sufficient in this context. However constraints on time and other resources limit such an exhaustive search, and what is presented here must be considered

9

as one of the first steps in understanding the true implications of using fractal geometries in electromagnetics.

In this study, fractal geometries such as Sierpinski gasket, Hilbert curves, Koch curves, and trees and few generalizations of some of these are explored. Some distinguishing features of their use demonstrated by corresponding antenna characteristics are presented separately. A common characteristic feature is hard to come by, yet some of these could be identified and related to properties of fractals. This required a certain extend of generalization of geometries such as Koch curves with different indentation angles, or binary trees with different branching angles and scaling ratios. These generalizations result in varying mathematical properties for the generic geometry, which could then be related to antenna characteristics. It has been shown in this work that apart from bringing in the necessary mathematical order, the use of these fractal geometries can be exploited in antenna design by incorporating their fractal definition into conventional antenna design formulae.

1.3 THESIS ORGANIZATION

The rest of this thesis is organized into six more chapters. Chapter 2 presents fundamental ideas used in this research work. Several fractal antennas have the potential of being “small antennas”. Hence a brief account of the definition and features such small antennas is discussed. A basic framework of the numerical techniques used in simulating antenna properties, such as the moment method and the finite difference time domain technique is briefly introduced. In later chapters commercial simulation packages based on these theories, such as NEC and XFDTD are used. A description of antenna measurement techniques used in the experimental study is also presented there.

Ideally one would like to have formulations for the dissertation presented first, followed by implementation issues, results obtained, and a discussion based on these. However due to the varied nature of fractal geometries studied here, antenna designs using them differ in their appearance and performance. Even the advantages obtained using various fractal geometries differed. Hence in this work, the chapters are arranged geometry-wise. However towards the end, common features from all these approaches would be

10

summarized.

Various forms of antennas using fractal Sierpinski gasket geometry are discussed in Chapter 3. Detailed numerical and experimental investigations reveal flaws in the current general understanding of antennas using them. Design of VHF/UHF antennas and wideband antennas using these geometries along with certain special materials are also described.

Different variants of Koch curves are recently used in antenna design. Some of these are said to result in “small antennas” with improved characteristics. A relationship between fractal properties of generalized Koch curves and characteristics of resulting antennas is obtained in Chapter 4 by curve fitting approach. This opens up newer vistas to incorporating fractal features in antenna design.

Another fractal geometry studied by way of generalization is tree configurations. Binary trees with various branching angles and scaling ratios are studied in Chapter 5. A direct relationship between geometrical properties and antenna resonance characteristics is obtained in this case also. This generalizes to a certain extent the findings based on generalized Koch curves presented in the Chapter 4.

Yet another fractal geometry useful in the antenna design is Hilbert curves. Their use can reduce the antenna size beyond achievable though the use of Koch curves and other geometries. However there are difficulties associated with such features ignored in previous investigations. Remedial approaches to overcome this are presented in Chapter 6, along with a novel application of this geometry in realizing antennas with reconfigurable radiation characteristics.

A summary of the conclusions drawn from the research towards this thesis work is presented in Chapter 7. Some directions for future research in this area to further understand antenna features achievable using fractal geometries are also mentioned in Chapter 7. It is expected that results and discussions presented in this dissertation would enable a true understanding of “fractal antennas” by shedding some misconceptions prevailing in recent literature.

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CHAPTER 2 TECHNIQUES FOR ANALYSIS AND CHARACTERIZATION OF ANTENNAS

2.1 INTRODUCTION

This Chapter presents a basic framework for the theoretical understanding of the effectiveness of fractal shaped antenna elements. Many such elements operate as small antennas. Hence a brief discussion is included here on the merits of such antennas. Apart from this, numerical techniques used in simulating the antenna properties, such as the moment method (MoM) and the finite difference time domain (FDTD) technique are also briefly introduced. This is done for comprehensiveness as these techniques are the core of commercial simulation packages such as NEC and XFDTD used in later Chapters for the analysis of the performance of fractal shaped antennas. A description of the procedure of characterizing antennas also provided in this Chapter.

2.2 THEORY AND DESIGN OF SMALL ANTENNAS

The quest for smaller sized resonant antennas has been on for decades. Several of antenna design principles may have to be modified while dealing with small-sized antennas. By convention, a small antenna is defined as one occupying a fraction (typically <1/6) of the wavelength [38]. The primary concern in their design is in the impedance matching. This is better explained in the context of dipole antennas. As the length of a dipole antenna is reduced, the real part of its input impedance approaches zero, while the imaginary part tends to be an extremely large negative number. This causes major reflections at the input terminal as the transmission line connected to it generally has a standard characteristic impedance (50Ω).

The reactive part of the impedance is contributed by the induction fields in the near-zone of the antenna [39]. The resistive part on the other hand may be attributed to various loss mechanisms present in and around the antenna, including the radiation “loss”. Losses due to the finite conductivity of the antenna structure, and that due to currents induced on

12

nearby structures including ground contribute to the antenna input resistance.

2.2.1 EMF Method for Antenna Impedance

In order to get a physical understanding, an approximate derivation for the input impedance of a linear antenna is presented here [39]. The Poynting theorem on a closed surface S surrounding an antenna is given by

1 E× H* ⋅dS = 2 jω()W −W + P (2.1) 2 ∫S m e where Wm-We is the time averaged net reactive energy stored within the volume bounded by S, and P is the total power flow through S. If one can find equivalent voltages and currents that can replace the LHS in the above equation, the antenna impedance can be obtained as their ratio. For a thin cylindrical antenna of length 2l and radius a the current on the antenna is assumed to be

I = Im sinβ ()l − z (2.2)

It also assumed that this current is distributed symmetrically around the axis of the cylinder, leading to a current distribution of

I J = (2.3) z 2πa

The vector potential is given by

− jβR µ I l e A = 0 0 sin β ()z′ − l dz′ (2.4) z 4π ∫−l R

For the purpose of evaluating the vector potential the current is assumed to be concentrated at the center. The electric field is then given by [39]

− jβr − jβR1 − jβR2 ηI0  e e e  Ez = − j 2 cos βl − −  (2.5) 4π  r R1 R2 

2 2 2 2 2 2 2 2 2 where r = x + y + z R1 = x + y + ()z − l R2 = x + y + ()z + l .

* * At a surface immediately surrounding the antenna, Jz can replace Hφ and (2.1) can be

13

rewritten as

−1 l −1 l E (a, z)J *adφdz = E (a, z)I *dφdz = 2 jω()W −W + P (2.6) 2 ∫−l z z 2 ∫−l z z e m r

At the input terminals, this leads to

1 I I *Z sin2 βl = 2 jω()W −W + P (2.7) 2 0 0 in m e r

The input impedance is then

1 l Z = − E I * sin β z − l dz (2.8) in * 2 ∫−l z 0 () I0I0 sin βl

Substituting for Ez, and simplifying for wire radius zero, one can approximate the real part of the input impedance to be

η l  sin βz sin β (l − z) sin β (l + z) R0m = ∫ 2cos βl − − sin β ()z − l dz (2.9) 4π −l  z l − z l + z 

The reactive component can be obtained as

η l  cos βr cos βR  X = cos βl − 1 sin β z − l dz (2.10) 0m ∫−l   () 2π  r R1 

This shows that by properly choosing the antenna length, its reactance can be made zero. It may be noted that when the radius of the antenna approaches zero, the resonant length of the antenna may be obtained as 2l= λ/2.

2.2.2 Issues in Design of Small Antennas

It is often preferred to operate an antenna around its resonance to obtain proper impedance match with the transmission line. There are several artificial ways of obtaining similar input characteristics without actually making the dipole λ/4 long. One is to use reactive elements at the input that cancel out the reactance of the antenna. However this approach often limits the bandwidth of operation of the antenna.

Another method extends the length of the dipole, without actually making it longer. Meandered and zigzag lines are examples of this approach. Bends and corners

14

introduced by this modification adds to the inductance of the line, but does not significantly impact the real part of impedance. Hence the study on small resonant antennas is still ongoing and antenna applications of several fractal geometries have been approached with this objective.

An important issue that remains is the low bandwidth, which in other words is a high quality factor. At resonance, the quality factor is defined as [40], [41]

2ωW Q = (2.11) P

where W is larger of We or Wm are the time averaged electric or magnetic energy stored in the near field of the antenna. P is the radiated power from the antenna. The factor of 2 is used to incorporate the effects of adding additional network components required to make the antenna resonant. This value is reduced by the losses in the tuning elements. It is also stated that for large Q, the antenna bandwidth is defined as the reciprocal of Q [40]. The minimum Q of a small antenna is obtained as [42]:

1 1 Qmin = + (2.12) 2()βa 3 ()βa

where a is the smallest possible sphere that encloses the antenna. In general Q is evaluated by treating the antenna as a one port device and obtaining its RLC equivalent circuit [43].

Another issue that concerns the design of small antennas is radiation efficiency. This is defined as [38]

Radiation PF Radiation efficiency = (2.13) Radiation PF + Loss PF

Several techniques are available for evaluating the radiation efficiency of small antennas. One of the improved techniques is to enclose the antenna is a special waveguide fixture, by a transmission measurement using a reference antenna [44]. The efficiency is calculated from measured S-parameters.

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2.3 FUNDAMENTALS OF MODELING TECHNIQUES USED

This research work relies to a great extent on the modeling studies using commercially available EM simulation softwares such as G-NEC and XFDTD. An understanding of the fundamental theory behind these numerical techniques is essential in creating the antenna models and to evaluate the results obtained with them. Hence a brief exposition into the theory behind these techniques is presented in this section.

2.3.1 Method of Moments (MoM)

The electromotive force (emf) method for calculation of antenna impedance mentioned in Section 2.2.1 is applicable to a very limited number of antenna configurations. Numerical techniques are essential in more general antenna structures, especially when complicated geometry such as fractals is involved. One very common method in electromagnetics, suitable for antenna analysis as well as for scattering and diffraction problems is the method of moments (MoM) [45]. MoM is a very powerful and versatile technique which can be applied to linear, planar, as well as three-dimensional structures [46]. The method involves segmentation of the antenna structure and choosing suitable basis functions to represent currents on these segments. A set of equations is generated by enforcing the boundary conditions with a suitable set of testing functions. This results in a matrix whose order is proportional to the number of segments on which the current distribution is represented. The solution to the problem is found by inverting this matrix. A formal mathematical structure of MoM is presented below.

2.3.1.1 MoM Basics

It is assumed that when a source of excitation q is impressed upon a system, it manifests in a response p. These parameters q and p could for example represent voltage and current of an electrical circuit. These physical parameters are related by a system dependent linear operator L,

L( p) = q (2.14)

In many situations, such as in antennas where the voltage function is known, while currents on the segments are unknowns. A salient feature of MoM is the hypothesis that

16

it is possible to represent p by a set of linear basis functions pi:

p = ∑ ai pi (2.15) i

where ai are coefficients associated with basis functions. The substitution of (2.15) in (2.14) yields

  L∑ ai pi  = q (2.16)  i 

These coefficients are not known a priori and must be determined. MoM overcomes this

problem by assuming a set of testing or weighting functions tm. For the method to be fruitful, both the weighting functions and basis functions mentioned above must be linearly independent. Weighting functions are employed to define the inner product with respect to (2.16):

  tm ,L∑ ai pi  = tm ,q (2.17)  i 

Commutative property of linear operators permits the above equation to be re-written as:

∑ ai tm ,L( pi ) = tm ,q (2.18) i

This equation is in fact a set of linear equations which can be symbolically expressed in the matrix form:

Bm×nAn×l = Cm×l (2.19) where:

 t1,L( p1) t1,L( p2 ) L t1,L( pn )    t ,L( p ) t ,L( p ) t ,L( p ) B =  2 1 2 2 L 2 n  (2.20)  M M M M     tm ,L( p1) tm ,L( p2 ) L tm ,L( pn ) 

T A = []a1 a2 L an (2.21)

17

T B = []t1,q t2 ,q L tm ,q (2.22)

Assuming that B-1, the inverse of B exists, the set of coefficients associated with basis functions can be evaluated from

−1 An×1 = Bn×lCm×1 (2.23)

For square matrices, the inverse exists only if the matrix is non-singular. Coefficients of the column vector A is used to express the response of the system (2.15) as

n p = ∑ ai pi (2.24) i=1

It may be pointed out that in the context of antenna analysis, the matrix equation in (2.19) readily becomes

ZI = V (2.25) so that the MoM solution essentially involves inverting the impedance matrix Z. For very simple problems, MoM yields solutions identical to exact ones. However for several practical situations, the solution is found through an iterative procedure. This computational intensity is a constraint in the application of MoM to some problems, especially at higher frequencies. Yet for several antenna structures such as fractal shaped antennas discussed in later Chapters, MoM is probably the optimum analysis approach involving integral equations.

The integral-equation approach can lead to the current distribution on antennas fed by a source. Another objective of this approach is to obtain the input impedance of the antenna [47]. Hence a brief discussion of the integral equation approach is given next.

2.3.1.2 MoM for Solving Antenna Problems

Several antenna problems can be analyzed using integral equation techniques. The method of moments introduced previously is an useful tool for implementing this approach. Its usefulness is described here with the help of an example where the method is applied to analyze a center-fed cylindrical antenna of radius a and length 2l as shown

18

in Fig. 2.1. Since the electric field is continuous at the surface of the conductor,

Ez,ρ=a−da = Ez,ρ=a+da (2.26)

These are the tangential components of fields just inside and just outside the conductor.

z

. ()x,y,z

2l y

x

2a

Fig. 2.1 Geometry of a cylindrical dipole antenna.

Similarly at the end faces of the wire:

Eρ,z=l−dl = Eρ,z=l+dl (2.27)

It is assumed that the length l of the antenna is much larger than a and the radius is very small compared to wavelength. In such case, the effect of end faces is neglected, and the currents at z=±l are taken equal to zero. Then the electric field on the surface of the conductor is

Ez = ZI z (2.28)

where, Z is the impedance of the conductor due to the skin effect expressed in Ω/m and Iz is the total current on it.

The vector electric potential A is used to obtain the electric field outside the conductor:

c2 E = − j ∇(∇ ⋅ A) − jωA (2.29) ω

Outside the conductor, E has only a z-directed component. The current on the conductor is in the z-direction, and the vector potential also has only a z component. Hence (2.29) reduces to:

19

ω  ∂2 A  E = − j  z + β 2 A  (2.30) z 2  2 z  β  ∂z 

Substituting (2.30) in (2.28), and making use of (2.26) the wave equation may be obtained:

∂2 A jβ 2 z + β 2 A = ZI (2.31) ∂z2 z ω z

This wave equation may be solved as the sum of a complementary function and a particular integral:

Az = Ac + Ap j jZ z (2.32) = − ()C1 cos βz + C2 sin βz + ∫ I(z′)sin β (z − z′)dz′ c c 0

As the antenna is symmetrically excited,

I z (z) = I z (−z) (2.33)

Az (z) = Az (−z) (2.34)

If Vin is the voltage applied at the terminals of the antenna, C2 in (2.32) is evaluated to be

Vin/2.

It is also possible to calculate the vector potential due to current on the antenna:

jωt l I e− jβr µe z1 Az = ∫ dz1 (2.35) 4π −l r

2 2 where z1 is a point on the conductor of the antenna, and r = ρ + ()z − z1 . Using these, the solution of the wave equation may be re-written as

jωt l I e− jβr z jcµe z1 Vin ∫ dz1 = C1 cos βz + sin β z − Z ∫ I(z′)sin β (z − z′)dz′ (2.36) 4π −l r 2 0

This is Hallen’s integral equation. Its solution leads to the current on the conductor which can obtained in terms of the antenna dimensions and the impedance (due to skin

20

effect) of the conductor. For very good conductors, the last term in (2.36) vanishes. This equation can be solved by the method of moments.

A simplified first order solution to (2.36) when the antenna is made of a good conductor is given by [47]:

 2l  2ln sin β l − z + b jV  ()1  I = in a (2.37) z 2l  2l  120ln  2ln cosβl + d  a  a 1  where

b1 = F1(z)sin βl − F1(l)sin β z + G1(l)cosβz − G1(z)cosβl (2.38)

l − jβr1 ()cosβz1 − cosβl e d1 = F1(l) = − ∫ dz1 (2.39) −l r1

2   2  2    z   1  a   a   F1(z) = −F0z ln1−    + F0z ln  1+   +1 1+   +1   l   4   l − z    l + z         (2.40) l − jβr F0z1e − F0z − ∫ dz1 −l r

G0z = sinβ z − sinβl (2.41)

F0z = cosβz − cosβl (2.42)

The terms involving G in (2.38) are evaluated from (2.40) with the F terms replaced with corresponding terms in G.

Once the current distribution on the antenna is determined, its input impedance is obtained as:

Vin Zin = (2.43) Iin where Iin =Iz(0). A first order approximation for the input impedance of the antenna is

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 2l  2ln cosβl + d 2l  1  Z = − j120ln a (2.44) in  2l  a  2ln sinβl + b   a 1 

It may however reiterated that the above equations for current and impedance are for a very simplified antenna structure and feed configuration. Yet this gives an idea of significance of various design parameters on the performance of the antenna.

2.3.1.3 Using G-NEC for Antenna Analysis

Several antenna configurations studied in this work are simulated using the moment method based software G-NEC available from Nittany Scientific Inc. This has NEC-4 as the antenna analysis core and a convenient user interface. The antenna structure is modeled with wires. The segmentation of these wires is determined by the maximum frequency of interest, and is often restricted to a maximum length of λ/20 at this frequency. The radius of segments should not exceed half the segmentation length. The software can generate data for currents in all segments of the antenna, input impedance, and radiation characteristics.

2.3.2 Finite Difference Time Domain (FDTD) method

Although MoM based software is convenient for several cases, it does not incorporate the effects of the dielectric support, if present. Some of the antenna configurations studied during this work used dielectric substrates to improve the antenna bandwidth. Since the properties of these novel materials could be varied by a large order of magnitude, the simulation package to be used should be able to handle these accurately. For example, the moment method based simulation packages such as G-NEC do not handle dielectric materials. In contrast finite difference time domain (FDTD) approaches rely on Maxwell’s equations in the difference equation form, and hence are not expected to cause major problems as the dielectric constant of material systems is increased. With this view an FDTD based commercial EM simulation package XFDTD from Remcom Inc. in State College, PA was used in some of the antenna simulations presented in later chapters. In the following FDTD theory which forms the core of this simulation package

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is briefly introduced [48].

2.3.2.1 FDTD Formulation

The FDTD formulation is based entirely on Maxwell’s equations in the time domain in the differential equation form [48]. Starting with the curl equations, which are rewritten in a form convenient for use with FDTD:

∂H 1 σ * = − ∇×E − H (2.45) ∂t µ µ

∂E −σ 1 = E + ∇× H (2.46) ∂t ε ε

where σ* is the magnetic conductivity. The inclusion of this along with the more common material properties such as ε, µ, and σ would facilitate restricting the field quantities to electric and magnetic field intensities (E and H) only.

It is possible to replace the time and space derivatives used in the above equations with differences as:

∂f lim f (x,t ) − f (x,t ) f (x,t ) − f (x,t ) ≡ 2 1 ≈ 2 1 (2.47) ∂t ∆t → 0 ∆t ∆t

∂f lim f (x ,t) − f (x ,t) f (x ,t) − f (x ,t) ≡ 2 1 ≈ 2 1 (2.48) ∂x ∆x → 0 ∆x ∆x

One key issue of using this approach is the Courant stability condition which limits the time step in terms of the space steps as:

1 1 ∆t ≤ (2.49) v 1 1 1 + + (∆x) 2 (∆y) 2 (∆z) 2

where v is the velocity of EM waves in the medium within the model space with the

lowest velocity. The spatial cell size is often limited to λm/20 where λm is the wavelength in this material corresponding to the highest frequency of interest.

When these center difference equations are applied to Maxwell’s equations, E and H are

23

interleaved spatially and temporally. This leads to a leap frog computation of the field quantities.

Another important step in the formulation is to rewrite these curl equations into their respective spatial components. For example, time derivatives of the x component of E and H in a linear non-conducting medium are:

∂E 1  ∂H ∂H y  x  z  =  −  (2.50) ∂t ε 0  ∂y ∂z 

∂H 1  ∂E y ∂E  x  z  =  −  (2.51) ∂t µ 0  ∂z ∂y 

These expressions can be easily extended to y and z components. Replacing derivatives with differences, for propagation in the z direction (1-D transmission line analysis):

n− 1 n− 1 H 2 (i, j, k) − H 2 (i, j −1, k)   z z − E n (i, j, k) − E n−1 (i, j, k) 1  ∆y  x x = (2.52)  n− 1 n− 1  ∆t ε 2 2 0 H (i, j, k) − H (i, j, k −1)  y y −  ∆z 

n n Ez (i +1, j,k) − Ez (i, j,k)  n+ 1 n− 1 2 2  − H y (i, j,k) − H y (i, j,k) 1 ∆x =   (2.53) ∆t µ En (i, j,k +1) − En (i, j,k)  0 x x −  ∆z 

These simplified formulae can be extended easily to lossy materials with additional terms. The convention of using ijk indices in the xyz co-ordinate directions are shown in Fig. 2.2. This approach of nomenclature of cells in the Cartesian system is attributed to Yee [49]. With this background on the FDTD theory we proceed to explaining the issues related with modeling antennas using this approach.

2.3.2.2 FDTD for Antenna Analysis

FDTD theory can be used to simulate several antenna properties including its input impedance, gain, efficiency, and radiation patterns [48]. In FDTD calculations one uses a Gaussian pulse of maximum amplitude 1 V to facilitate computations for a wide

24

frequency range. The field at the feed location is given as

V (n∆t) E n (i, j, k) = − (2.54) z ∆z

where the Gaussian voltage source V(t) is given by:

2 V (t) =1.0e −α (t −β∆t) (2.55)

with β = 32 and α = (4β∆t)2 and the pulse is truncated for t<0 and t>2β∆t. The value of βis modified if the model space has dielectric materials. In such cases, its value should be 32 ε r where εr is the highest dielectric constant value within the model space. The current at the feed location can be approximated by taking the line integral in the discretized form:

1 1 1 1 n+ 2 n+ 2 n+ 2 n+ 2 I z (n∆t) = [H x (i, j −1, k) − H x (i, j, k)]∆x + [H y (i, j, k) − H y (i −1, j, k)]∆y (2.56)

Fig. 2.2 Yee cell geometry used in FDTD algorithm

The self-admittance of the antenna can be obtained by dividing the Fourier transform of the driving point current by that of the Gaussian voltage pulse feeding the antenna. The incident power in the frequency domain is given by:

* Pin (ω) = Re[V (ω)I (ω)] (2.57)

25

Similarly, the dissipated power within a region in the model space is given by

2 σ∆x∆y P = σ []E (ω) dv = [][]E (ω)∆z 2 = G V (ω) 2 (2.58) diss ∫∫∫ z ∆z z z

where G is the lumped conductance of a cell. A ratio of equations (2.57) and (2.58) would lead to the efficiency of the antenna:

P −P Efficiency= in diss (2.59) Pin

Far zone fields are computed in an effort to plot the radiation pattern and gain of the antenna. These are computed using discrete Fourier transform of the time harmonic tangential fields on a closed surface surrounding the FDTD geometry, updated at each time step. The excitation is a Gaussian pulse, to enable wide frequency range, thus providing frequency domain currents. From these currents vector potentials for each far zone angle of interest are computed and are stored for the post processing. This approach saves memory requirement compared to storing the currents at all the faces of the model space

Time harmonic vector potentials used in this computation are:

N(ω) = J s (ω) exp( jkr′⋅rˆ)ds′ (2.60) ∫s′

L(ω) = M s (ω) exp( jkr′⋅rˆ)ds′ (2.61) ∫s′

with the electric and magnetic surface currents given by

J s (ω) = nˆ × H(ω) (2.62)

M s (ω) = −nˆ ×E(ω) (2.63)

where rˆ is the unit vector to the far zone field point, r′ is the vector to the source point of integration, and s’ is the closed surface surrounding the antenna. From the vector potentials, the far zone fields are computed using:

− ηN + L E = j exp(− jkR) θ φ (2.64) θ 2λR

26

− ηN + L Eφ = j exp(− jkR) φ θ (2.65) 2λR

where η is the impedance of free space, R is the distance from origin to the far zone field point, and λ is the wavelength at the frequency of interest. From the far zone field thus computed, the antenna gain is calculated as

E (ω,θ ,φ) 2 /η Gain(θ ,φ) = F (2.66) Pin / 4π

2.3.2.3 Using XFDTD for Antenna Analysis

The formulation of FDTD with emphasis on antenna analysis problems have been described in the previous two sub-sections. XFDTD is a commercial package based on the above formulation. Relevant issues for modeling antennas using this software are presented here.

Depending on the maximum frequency of interest, and material properties involved, cell sizes are chosen. For example, if fmax = 15 GHz, and εr = 4 for the substrate material, the work sheet is as follows:

Wavelength at 15 GHz in vacuum = 2 cm

Wavelength in substrate medium = 1 cm

Using 20 cells per wavelength, cell dimension = 0.5 mm

Next the model volume is determined based on these cell dimensions and the physical size. To prevent reflections due to the discontinuity at the boundary, it is preferred to leave some vacuum cells between the antenna geometry and the boundary of the modeling space. Hence for an antenna with 128 mm height, and 69 mm width, on a substrate of 1.5 mm, the following final dimensions are chosen: 161 x 21 x 261 in x,y,z directions respectively

The next step is to draw the antenna object. Some primitive objects are available in the software. For example quadrilaterals can be used to draw triangles. In addition rectangular sheets can be used for metallization such as ground plane. Rectangular box

27

can be used for dielectric sheets. The interface layer of the substrate with air is modeled separately for better results.

The feed (stimulus type) is determined, and a proper feed arrangement is chosen. The most common method is to use wires, a primitive structure available in the software.

The boundary conditions are defined next. If radiation characteristics are not required, the ground plane can be modeled as a PEC boundary. All other walls of the space are modeled as absorbing boundaries, to simulate the free space effect.

Since in most cases we need the S parameters, FFT of the time domain data is required. The FFT base is chosen depending of the resolution required in the frequency domain.

Now the model is ready for simulation. This is done with Run CalcFDTD option. The results can either be plotted using the software itself, or can be ported in ASCII format to other plotting routines.

2.4 EXPERIMENTAL SET UP FOR ANTENNA MEASUREMENTS

Antenna characteristics are measured using the network analyzer HP 8510 C. It is very convenient and easy to obtain antenna input characteristics using a network analyzer. This is because this equipment measures both amplitude and phase simultaneously. The return loss (S11), VSWR, or input impedance are measured using single port calibration.

The radiation characteristics of antennas can be measured in an anechoic chamber using a network analyzer. The experimental set up is shown in Fig. 2.3. Since the distance between transmitting and receiving antennas is approximately 3 meters, an amplifier is often used in the transmitter side. An RF amplifier HP 8347 A (100 KHz – 3 GHz) or a microwave amplifier HP 8349B (2-20 GHz) can be used depending on the frequency of interest. When a standard antenna is used as the transmitter and test antenna as the receiver, S21 is measured using the analyzer.

The advantage of using a network analyzer for this measurement is that a band of frequencies can be swept in a single measurement, thus saving time. Custom-software is used to control the positioner as well as to download data from the network analyzer. This software is also capable of generating radiation pattern plots at individual

28

frequencies, or S21 versus frequency at different look angles of the test antenna.

Wideband Antenna

Antenna under test

Positioner

Amplifier Network Analyzer

Controller Computer

Fig. 2.3 Set up for antenna measurements in an anechoic chamber

A comparison method is adopted for antenna gain measurements. This assumes the return loss of the test antenna is very small and comparable with that of a standard antenna. First the standard antenna is placed on the positioner, aligned with the transmitter. S21 is measured using the network analyzer. The network analyzer may be normalized with this measurement data. Next the standard antenna at the receiver is replaced with the test antenna without disturbing the rest of the set up. The test antenna is aligned at the direction of its peak by controlling the positioner. S21 is measured for the test antenna. The difference in these S21 measurements is added to the known gain of the standard antenna to obtain the gain of the test antenna.

29

CHAPTER 3 SIERPINSKI GASKET GEOMETRY FOR MULTIBAND AND WIDEBAND ANTENNAS

3.1 INTRODUCTION

The Sierpinski gasket is perhaps the most widely studied fractal geometry for antenna applications. Sierpinski gaskets have been investigated extensively for monopole and dipole antenna configurations [2]. The self-similar current distribution on these antennas is expected to cause its multi-band characteristics [33]. It has been found that by perturbing the geometry the multi-band nature of these antennas can be controlled [34], [50]. Variation of the flare angle of these geometries have also been explored to change the band characteristics of the antenna [32].

Efforts have also been made to improve the bandwidths of these antennas. A stacked antenna configuration with multiple layers of the fractal geometry has been found to be useful in this regard [35]. In addition, this configuration has been made conformal to improve the utilization of the antenna.

Antennas using this geometry have their performance closely linked to conventional bow-tie antennas. However some minor differences can be noticed in their performance characteristics. As a prelude to investigate this, the fractal nature of the geometry is explained first in Section 3.2. Basic antenna configurations using this geometry are presented in Section 3.3. The multi-band characteristics of monopole antenna configuration using various geometries are compared in Section 3.4. The performance of these antennas is further affected by the nature and presence of dielectric supports. The effect of dielectric support on the antenna performance is studied using numerical simulations and the results are also included here.

It has been found that the multi-band nature of the antenna can be transformed into wideband characteristics by using a very high dielectric constant substrate and suitable absorbing materials. This antenna configuration is explained in Section 3.5. This antenna configuration can be converted into a conformal antenna relatively easily, and

30

the experimental results in this case are included as Section 3.6. The concluding remarks from this Chapter are summarized in Section 3.7.

3.2 SIERPINSKI GASKET FRACTAL GEOMETRY

Sierpinski gasket is one of the most common fractal geometries. The generation of this geometry is explained using Fig. 3.1. Although the geometry presented here consists of equilateral triangles, the description here holds good for any triangular geometry. One can explain its generation in two ways: the multiple copy approach, or the decomposition approach. In the first, one starts with a small triangle. Two more copies of this triangle (same size) are generated and and attached to the original triangle. This process can be done n number of times, n being the order of the fractal iteration. In the decomposition approach, one starts with a large triangle encompassing the entire geometry. The midpoints of the sides are joined together, and a hollow space in the middle is created. This process divides the original triangle to three scaled down (half sized) versions of the larger triangle. The same division process can be done on each of the copies. After n such divisions, the geometry shown in the figure is obtained.

Affine transformations, of which similarity transformations form a convenient sub-class, are important characteristics of fractal geometries. These involve scaling, rotation and translation. These transformations can be expressed in the mathematical form as:

 x  r cosθ − s cosφ x   x0  W   =    +   (3.1)  y r sin θ s sin φ  y  y0 

In the above equation, r and s are scale factors, θ and φ correspond to rotation angles and

x0 and y0 are translations involved in the transformation. If r and s are both reductions (r,s <1) or both magnifications (r,s >1), the transformation is self-affine. If r = s and θ = φ, the transformation is self-similar.

First the generation of ‘strictly self-similar’ Sierpinski gasket is considered. Starting with an equilateral triangle of unit length side the transformations involved to get the next iterated geometry are:

31

Multiple Copy Approach

Decomposition Approach

Fig. 3.1 Two approaches for the generation of Sierpinski gasket geometry.

 x  0.5 0  x  W1  =    (3.2)  y  0 0.5 y

 x  0.5 0  x  0.5 W2   =    +   (3.3)  y  0 0.5 y  0 

 x  0.5 0  x   0.25  W3  =    +   (3.4)  y  0 0.5 y 0.433

It is assumed that the origin of the co-ordinate system is at the bottom left corner of the triangle, and the x-axis pass through the base (bottom) side of the triangle. The transformations W1, W2, W3 are indicated pictorially in Fig. 3.2. The second geometry is obtained with a union of these three transformations:

W3 W3 …. W1 W1

W2 W2

Fig. 3.2 IFS for the generation of a strictly self-similar Sierpinski gasket geometry.

W (A) = W1(A) ∪W2 (A) ∪W3 (A) (3.5)

32

This process can be recursively used for generation of the higher iterations of the geometry. e.g.,

A1 = W (A) (3.6)

A2 = W (A1) = W ()W (A) (3.7)

This process of generation of the geometry is very convenient in the context of computer platforms, and is often called multiple reduction copy machine (MRCM). In mathematics, these are referred to as iterated function systems (IFS)

The decomposition approach is an easier tool in calculating the fractal dimension of the geometry. There are at least ten different ways the dimension of a fractal geometry s defined. The fractal similarity dimension of this geometry is:

log(N) log 3 D = = = 1.585 (3.8) log( 1 ) log 2 f

This formula makes use of the fact that there are three copies of a triangle, each scaled down to half the size of the original, in each stage of fractal iteration. This formula requires the geometry to be strictly self-similar. However, this is not a necessary condition for a geometry to be fractal.

3.3 BASIC ANTENNA CONFIGURATIONS USING SIERPINSKI GASKETS

Several antenna configurations have been discussed in the literature using this geometry. These include monopole, dipole, patch and several varients of these [1], [2], [51]. Some of these configurations are shown in Fig. 3.3. In dipole and monopole configurations, the characteristics of the antenna have been qualitatively related to geometrical features of the underlying fractal patterns. Such a close relationship is hard to come by for the patch configuration, although this being conformal, has several aesthetic advantages from applications point of view.

In the Sierpinski gasket monopole antenna, the fractal geometry is printed on an ungrounded dielectric substrate. This is then placed perpendicular to a ground plane. Typically low dielectric constant substrates such as RT Duroid and FR-4 are used as the

33

substrate material, while an aluminum sheet is preferred for the ground plane.

Monopole

Dipole Patch

Fig. 3.3 Antenna configurations with fractal Sierpinski gasket.

Two Sierpinski gaskets are printed on the ungrounded substrate so as to face each other at their apex, to form the dipole configuration. In this case, the feed is split between the two geometries. There is no ground plane present, making the antenna of low profile. Dielectric materials similar to the previous case may be used. It may be noticed that the antenna configuration is very similar to a printed bow-tie antenna.

In the patch configuration, the Sierpinski gasket geometry is placed parallel to the ground plane, as done in the case of microstrip patch antennas. Either one, or two Sierpinski gasket geometries (similar to monopole or dipole antennas mentioned above) can be used for the purpose. Often a multilayered configuration is used to obtain good input impedance characteristics. This is also accomplished by spacing the substrate with an air gap, above the ground plane. A probe feed is convenient if only one gasket is used. However if two are present (similar to a printed bow-tie) either a probe feed, or a microstrip in-line feed with a balun can be used. The performance of monopole antennas and their variants are discussed in the following sections.

34

3.4 SIERPINSKI GASKET MONOPOLE ANTENNA

The schematic of a typical Sierpinski gasket monopole antenna is shown in Fig. 3.4. The antenna characteristics are studied both experimentally and numerically. A commercial finite difference time domain software (XFDTD) is used for numerical simulations. Fundamentals of the underlying theory of this method have been presented in Chapter 2.

Substrate

Ground Plane

Fig. 3.4 Monopole antenna configuration with printed Sierpinski gasket geometry.

Several modeling and experimental studies are done to understand the role played by these geometries in the design of these antennas. These include the effects produced by changes with the fractal iteration, and apex angle of the triangles. A similar study is also conducted for similar geometries that do not have a strict geometrical self-similarity.

3.4.1 Effects of Fractal Iteration Levels

Both numerical and experimental studies have been pursued to study the effects the fractal iterations on the performance of the monopole antenna configuration. The substrate used in these studies was RT Duroid RO 3003, commercially available from Rogers Corporation, USA. This material has a dielectric constant of 3.0, and a thickness of 1.5 mm. In all cases, the total height of the geometry remained the same at 8 cm and apex angle at 60°. These geometries are shown in Fig. 3.5.

The simulated return loss characteristics of these antennas are shown in Fig. 3.6. It may be observed that the lower resonant frequencies of the antennas remain unperturbed by

35

the increase in the iteration order. This is consistent with the physics of the geometrical resonance of the antenna structure. For example, the lowest resonance corresponds to the largest triangle, which remain the same in all cases. However, the remaining resonances are higher order ones, which once again are controlled by the same dimensions. At higher frequencies however the performance of these antennas differ, especially in their radiation characteristics.

Iteration: 1 2 3 4

Fig. 3.5 Different fractal interations studied for the antenna performance. The overall height in each case is kept at 8 cm. The apex angle is 60°. The antennas are printed on a RO 3003 substrate for validation.

The measured return loss characteristics of the antennas fabrication is shown in Fig. 3.7. As mentioned earlier, no specific trend in change in resonance can be deciphered from these curves. It is believed that the measured and simulated characteristics differ due to the effects of rectangular gridding used in the simulations. Such grids do not accurately represent triangular geometries, especially at intersections of triangles. The radiation patterns are plotted at resonant frequencies of the antenna (Fig. 3.8) also do not show significant differences, except at higher resonances. These patterns are shown in planes parallel and normal to the plane of the geometry.

36

0

-2

-4

-6

-8

-10

-12

-14 Return Loss (S11) in dB

-16

-18

-20 0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09 3.5E+09 4.0E+09 4.5E+09 5.0E+09 Frequency (Hz)

Order:#1 #2 #3 #4 Fig. 3.6 Simulated return loss characteristics of the antennas shown in Fig. 3.5.

0

-5

-10

-15

-20

-25 Return Loss (S11) in dB -30

-35

-40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (GHz)

# 1 # 2 # 3 # 4 Fig. 3.7 Measured return loss characteristics of the antennas printed on Duroid RO 3003 substrate.

37

(a) 615 MHz –in plane

(b) 615 MHz normal plane

38

(c) 1.75 GHz in plane

(d) 1.75 GHz normal plane

39

(e) 3.6 GHz in plane

(f) 3.6 GHz Normal plane

Fig. 3.8 Radiation patterns of Sierpinski gasket monopole antenna for four different iterations of the geometry. The patterns are measured at 615 MHz and 1.75 GHz and 3.6 GHz in an anechoic chamber. Patterns at two orthogonal planes are plotted, in planes parallel and normal to the geometry.

40

3.4.2 Effects of Apex Angles

A similar approach is also followed to study the effect of changing the apex angle on the antenna performance. Changes in flare (apex) angle have been reported to change the multi-band characteristics of the monopole antenna using Sierpinski gaskets [32]. In the present study, all models are of the same height, and only the first iteration geometry (solid triangle) is used in this comparison. The angles studied are 45°, 60°, and 75°. The measured characteristics are compared in Fig. 3.9. These indicate a characteristic shift in resonance towards the lower side, as the angle is increased. Radiation patterns for these antennas in a plane along the geometry is shown in Fig. 3.10.

0

-5

-10

-15

-20

-25 Return Loss (S11) in dB -30

-35

-40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (GHz)

Angle = 45 deg. 60 deg. 75 deg. Fig. 3.9 Measured return loss for fractal Sierpinski monopole antenna with different apex angles (where the feed is connected). All geometries are of first iteration (simple triangles).

The above results indicate that multi-band characteristics of the simple triangular geometry itself can be modified by changing the flare angle. This, in effect throws into contention the statement that multi-band characteristics are introduced by the self- similarity of the fractal geometry [33].

41

Fig. 3.10 Radiation patterns of triangular monopole antenna for different flare angles. The patterns are measured at and 1.75 GHz and 3.6 GHz in an anechoic chamber. Patterns in the plane parallel to that of the geometry are plotted.

42

3.4.3 Use of Geometries not Strictly Self-similar

In the two pervious parametric studies the height of the geometries are kept constant. As shown in Section 3.4.1, changing the Sierpinski gasket geometry by increasing fractal iteration does not alter the antenna characteristics significantly, especially at the lower resonant frequencies. This often leads to a question: what is the true advantage of using a fractal geometry such as Sierpinski gasket, when a simple triangle monopole, or a conventional bow-tie would perform comaprably. Similar studies mentioned in Section 3.4.2, conclude that variations in multi-band characteristics can also be obtained for monopole antennas with simple triangular geometries.

It is also possible to generate geometries such that the height of one triangle (after each division) is more than the other two [34], [50]. This modification results in variants that are not strictly (geometrically) self-similar. To illustrate this point, the iterated function system (IFS) that defines the properties of the geometry is discussed next.

The IFS of the Sierpinski given in Section 3.2 gives identical copies of the original geometry at each iteration. The scale factors in these transformations are identical, leading to geometrically identical copies of the original. Such strict self-similarity is not essential for a geometry to be fractal. One such geometry is shown in Fig. 3.11. Although we start with the same triangle as in Fig. 3.2, compared to the ‘strictly self- similar transformations there, the copies in this case are not identical, and are not 2 ‘geometrically similar’ to the parent. In this example, we take h2= /3 h1. The transformations in Fig. 3.11 are:

 x   1 1  x  W   =  2 4 3   (3.9) 1 y 2  y   0 3  

 x   1 −1  x   1  W   =  2 4 3   +  2  (3.10) 2  y 2  y     0 3   0

 x  1 0 x   1  W   = 3   +  6  (3.11) 3   1    2   y 0 3  y  3 

This IFS can be used to generate the fractal geometry in Fig. 3.11. However for reasons

43

mentioned previously, these transformations are not strictly self-similar, but this fractal geometry is self-affine.

W3 W3 h W W1 1 2h/3

W W2 2

Fig. 3.11 Generalized Sierpinski gasket geometry that is not strictly self-similar. This geometry is still self-affine, can be generated with IFS and hence fractal.

Two sets of parametric studies are conducted in this context. Photographs of fabricated antenna geometries are shown in Fig. 3.12 and Fig. 3.13. In the first, the ratio of height of bottom triangle and top two triangles in each iteration is 2:1. Another geometry with this ratio reversed is included in the set in Fig. 3.13. Their performance are compared with a simple triangle, and a strictly self-similar geometry of the same iteration. The input characteristics of these antennas are compared. Measured Return losses of these antennas are shown in Fig. 3.14 and Fig. 3.15.

Fig. 3.12 Photographs of different iterations of geometry generated with IFS. The scale factor (height ratio of bottom triangle to the upper ones) in each iteration is 2:1.

44

Fig. 3.13 Photographs of fabricated geometries obtained using different IFS. Height ratios: top right-1:1, bottom left- 1:2, bottom right- 2:1. All these can be generated from the same simple triangle shown here for comparison.

0

-5

-10

-15

-20

-25 Return Loss (S11) in dB -30

-35

-40 0123456 Frequency (GHz)

full triangle 2nd iteration, scale ratio 2:1 3rd iteration, scale ratio 2:1

Fig. 3.14 Return loss of antennas shown in Fig. 3.12.

45

0

-5

-10

-15

-20

-25

Return Loss (S11) in dB -30

-35

-40 0123456 Frequency (GHz)

full triangle 3rd iteration, scale ratio 1:2 3rd iteration, scale ratio 2:1 3rd iteration scale ratio 1:1

Fig. 3.15 Return loss of antennas shown in Fig. 3.13.

Shown in Fig. 3.14 is the return loss of antennas of the first set, where different fractal iterations are compared. The first resonances in all cases coincide, while the second is different for both iterated geometries compared to the simple triangle. Additional observations can be made based on these results. Since there is no distinctive geometrical feature for deciding the third resonance of the second geometry, it coincides with that of the simple triangle. However for the 3rd iterated geometry, there is a geometrical feature that can be associated for the 3rd resonance, which makes its performance for this resonance distinctive. Radiation patterns of these antennas shown in Fig. 3.16 do not show significant variation between geometries of different fractal iterations.

These observations are further emphasized in Fig. 3.15. The heights of sections of geometries are different for the second resonant onwards. As shown, frequencies corresponding to the second resonance are distinct for each geometry. Hence it is concluded from this study that in order to truly affect the multi-band characteristics of antennas of this kind, a significant perturbation in geometry is required. Radiation patterns of these antennas for their respective second and third resonances are shown in Fig. 3.16. Each plot has patterns in two orthogonal planes, normal and parallel to the

46

plane of geometry of the antenna. Thus radiation patterns are also modified by this approach. It is also conjectured that since this variation affects the self-similarity of the geometry, it also affects the fractal dimension, a point explored further.

Fig. 3.16 Radiation patterns for three different iterations (shown in Fig. 3.12)of the modified geometry. The height ratio in each iteration remain 2:1. Patterns are plotted for 2nd and 3rd resonances, in two orthogonal planes.

47

Fig. 3.17 Radiation patterns of monopole antennas shown in Fig. 3.13. Patterns on the left side are for second resonance and on the right are for 3rd resonance. Each plot has patterns in two orthogonal planes for the same antenna.

48

3.4.4 Effect of Dielectric Support on Antenna Performance

This FDTD simulation study focuses on the effect of dielectric substrates on the input and radiation characteristics of the fractal monopole antenna. To simplify the computational complexity a reduced size model space is used. Furthermore to facilitate an easy comparison of the true effects of the dielectrics, all cases employed the same substrate thickness.

The antenna geometry studied is similar to the one in Fig. 3.4. The dimensions and material properties used are given below:

Height of triangle: 80 mm

Apex angle of triangle: 60°

Substrate thickness: 2 mm

Configuration: Monopole

Ground plane: 10 cm x 10 cm

Substrates studied: RT Duroid 5880 (εr = 2.2) FR-4 (εr = 4.4)

Alumina (εr = 9.8) BST (εr = 75)

The extent of the ground plane is limited to reduce computational complexity. A comparison of the return loss characteristics is shown in Fig. 3.18. These show a consistent trend of reduction in the resonant frequency as the dielectric constant is increased. However, as shown in Fig. 3.19, when higher dielectric constant materials like BST is used the resonance dips become increasingly clustered. Such a behavior is instrumental in getting wideband input characteristics for antennas using high dielectric constant substrates.

49

0

-5

-10

-15

-20

-25 Return Loss (S11) in dB

-30

-35

-40 024681012 Frequency (GHz)

Duroid (2.2) FR-4 (4.4) Alumina (9.8) Fig. 3.18 Simulated return loss for 3rd iteration Sierpinski monopole antenna on different dielectric substrates.

To explore changes in the radiation characteristics, more specifically, effects on antenna gain, their radiation patterns are simulated . The patterns in the θ-plane (parallel to the plane of the geometry) are plotted between 0-90° at every 5° intervals. In Fig. 3.20 patterns for antennas on Alumina, Duroid and FR-4 substrates are plotted. These patterns are plotted at resonant frequencies for these individual cases. Although there is a change in the shape of the radiation patterns in these cases, the overall performance such as peak gain does not seem to vary much. Fig. 3.21 shows a comparison of the total gain at 40° to the zenith, plotted against frequency.

50

0

-5

-10

-15

-20 Return Loss (S11) in dB

-25

-30 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (GHz)

Fig. 3.19 Return loss characteristics of the Sierpinski gasket monopole antenna with a BST substrate. The widening of bandwidth obtained using an absorber is not taken up in this simulation study.

(a) Alumina

10

5

0

-5

-10

-15

-20

-25

-30 0 102030405060708090

705 MHz 1.64 GHz 2.81 GHz 3.47 GHz 5.21 GHz 6.51 GHz 7.74 GHz

51

(b) RT Duroid 5880

10

5

0

-5

-10

E-theta (dBi) -15

-20

-25

-30 0 153045607590 Angle (deg.)

705 MHz 1.92 GHz 3.69 GHz 6.62 GHz 11.69 GHz

(c) FR-4

10

5

0

-5

-10

E_theta (dBi) -15

-20

-25

-30 0 153045607590 Angel (deg.)

705 MHz 1.81 GHz 3.29 GHz 5.87 GHz 9.76 GHz Fig. 3.20 Simulated radiation patterns of 3rd iteration fractal Sierpinski monopole antenna printed on

Alumina (εr = 9.8) Duroid (εr = 2.2) and FR-4 (εr = 4.4) substrates. These are θ-polarized and are plotted for the respective resonant frequencies.

52

10

5

0

-5 Total gain (dBi) -10

-15

-20 024681012 Frequency (GHz)

Duroid FR-4 Alumina Fig. 3.21 Simulated total gain of the antennas at a constant direction (φ= 0, θ = 40) for all substrates, plotted against frequency.

3.5 WIDEBAND ANTENNA

Ferroelectric materials such as barium strontium titanate (BST) are well suited for the microwave phase shifter applications in phased array antennas [52]. Their use as antenna substrates can potentially reduce the antenna size in addition to result in an electronically tunable microstrip antenna [53]. The change in dielelectric constant of these materials can be more than 50% depending on the composition of BST and its applied bias field. A microstrip antenna fabricated on BST is found to change its operating frequency on the by application of a bias electric field and this could find many applications in phased array antennas and smart antennas [54]. Another important characteristic of this class of material is that a wide range of dielectric values are obtained by suitably choosing the stoichiometric composition [55].

A fractal monopole antenna is designed and developed in high dielectric constant BST substrate and its radiation performances were evaluated. The antenna configuration is similar to the one in Fig. 3.4, with the substrate being BST. A large number of distinctive but smaller bands of frequencies, particularly in the region of 1 GHz to 10 GHz are shown to have good input impedance characteristics as compared to the finite number of bands obtained with previously mentioned materials (e.g., Fig. 3.18). The BST substrate used in this antenna configuration has a dielectric constant of 75.

53

A further investigation of similar materials with different dielectric constants to better understand this behavior is done and it has been found out that a similar return loss characteristic can be obtained for all dielectric ferroelectric materials considered. This leads us to the conclusion that this behavior is due to the waves excited on the dielectric slab itself. Therefore, it should be possible to modify the antenna characteristics if one could modify the field distribution on the dielectric without significantly affecting the radiation characteristics of the antenna.

The closely clustered multiple bands in the return loss characteristics of fractal antennas on BST substrates could be controlled and their input characteristics could be smoothened by placing an absorber behind the substrate (Fig. 3.22). The thickness of the absorber layer is minimized if a chiral absorber [55] is used. Our research center (CEEAMD) has, over the past decade, come up with various configurations for such absorbers and studied various aspects of their design, development and characterization [55] – [58].

Fig. 3.22 Sierpinski monopole antenna configuration with improved impedance bandwidth.

The improvement in bandwidth obtained in such a wideband antenna configuration

54

shown in Fig. 3.22 is presented in Fig. 3.23. Commercially available AN-73 is used as the absorbing layer in this case. The ripple in the characteristics of the antenna is evened out and the measured input impedance of the antenna shows that it is truly wideband. A properly matched absorbing material behind the substrate could bring down these surface waves. The return loss of the antenna is well below -10 dB (VSWR ~ 2.) for frequencies ranging from 1.5 to 13 GHz. It may however be pointed out that a large increase in bandwidth is not observed when low dielectric constant substrate is used along with the absorber. However widening of the bandwidth is obtained when BST substrates with a wide range of dielectric constants.

0

-5

-10

-15

-20

-25 Return Loss (S11) in dB -30

-35

-40 13579111315 Frequency (GHz) No Absorber With AN 73

Fig. 3.23 Improvement in input bandwidth obtained when the monopole antenna is backed with an absorber.

The radiation characteristics of this new configuration of antenna are comparable with other monopole antennas discussed earlier. The radiation pattern of the antenna is measured in an anechoic chamber with automated measurement systems using a network analyzer. These measurements use a sweep frequency source, and the results are reasonably consistent within the band. Hence radiation patterns for six indicative frequencies are shown in Fig. 3.24. In view of the wide band nature of the antenna, only a few indicative frequencies are shown here. Patterns for elevation planes parallel to and

55

normal to the plane of the geometry are shown here.

(a) 2 GHz

(b) 4 GHz

(c) 6 GHz

56

(d) 8 GHz

(e) 10 GHz

(f) 12 GHz

Fig. 3.24 radiation patterns in planes parallel (left) and normal (right) to the geometry for a wideband Sierpinski gasket monopole antenna. The effect of having an absorber backing is compared. Each plot is normalized with the maximum.

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3.6 CONFORMAL ANTENNA CONFIGURATION

In the previous section, a Sierpinski gasket antenna on BST substrate with another layer of absorber below it has been shown to have wideband characteristics. Such performance is nevertheless obtainable with many other existing antenna configurations. What makes this configuration unique is its adaptability to a conformal arrangement as shown in Fig. 3.25. In this configuration the ground plane is behind the absorber, parallel to it. This antenna can now be attached to any metallic surface with minimal interference to its outer profile. As shown in Fig. 3.26, the wideband characteristics of the antenna prevailed even after this modification.

The antenna is fabricated on a substrate with a dielectric constant of 50. In this configuration the return loss remains well below -10 dB for frequencies ranging from 1 GHz to 10 GHz. This corresponds to VSWR better than 2.2. Hence the antenna can be operated anywhere in the L, S, or C bands and partly in the X-band.

Fractal Pattern Absorber (AN-73)

Ferroelectric Substrate (BST)

Ground Plane

Feed

Fig. 3.25 Conformal wideband antenna configuration. The ground plane entends beyond the area of the printed antenna geometry.

58

0

-5

-10

-15

-20 Return Loss (S11) dB

-25

-30 024681012 Frequency (GHz)

Fig. 3.26 Input characteristics of the conformal antenna configuration.

The radiation patterns of this antenna at a few indicative frequencies are given in Fig. 3.27. It may be noted that the antenna configuration is not symmetrical, except in two octants, on either side of the plane perpendicular to the antenna geometry and along feed direction. Therefore, the radiation patterns are given only for these regions. It may be pointed out that the beam direction is neither normal to the antenna nor always exactly fixed, as with the multi-band Sierpinski gasket antenna at its multiple resonant frequencies [1], [2].

With a wideband characteristics, moderate gain and being conformal this antenna has potential in many military and civilian applications. These antennas could be used in many military systems in addition to telecommunication applications. For example, VHF/UHF antennas currently in use pose severe operational disadvantages due to their large sizes. Often the use of such antennas considerably curtails the freedom of movement of the personnel. Even the setting up of the communication system itself takes precious time to unfold and deploy these antennas. An antenna attached conformal to the vehicle or on the back-pack of the personnel therefore has tremendous military potential. The antenna however requires some modifications for applications in for HMMWV,

59

soldier antennas, PCS antennas and advanced electronics warfare (EW) systems. A comparison of this antenna with widely used spiral antennas is in order. This antenna has linear polarization, while spiral antennas are designed for circular polarization. However the gain of this antenna is lower than that of many commercial spiral antennas. However it is envisaged that this configuration is useful in designing antennas for lower frequencies for which spiral antennas are not commercially available.

Azimuth Elevation 2 GHz

5 GHz

10 GHz

Fig. 3.27 Radiation patterns at few indicative frequencies for the conformal wideband antenna.

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3.7 SUMMARY

In this Chapter various antenna configurations using Sierpinski gasket fractal geometry has been studied. It has been observed that the performance of the monopole configuration is hardly affected by fractal iteration. The multiband characteristics of this type of antenna can be influenced by changing the apex angle of the triangular geometry. These two observations together contradict the concept that the self-similarity of the geometry is the primary cause of multi-band characteristics of the antenna. As would be shown in later chapters, the fractal dimension of the geometry has a more significant role in the multiband characteristics of the antenna. The fact that this antenna characteristic can be affected significantly by having Sierpinski geometries that are not geometrically self-similar (e.g., by dividing the triangles at different heights than midway) adds to this conviction.

Wideband monopole and conformal antennas with the geometry printed on BST substrates backed by an absorber have been shown to have nearly an octave bandwidth. The measured radiation characteristics of these antennas are also presented here.

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CHAPTER 4 INFLUENCE OF FRACTAL PROPERTIES ON THE CHARACTERISTICS OF DIPOLE MULTI-BAND ANTENNAS USING KOCH CURVES

4.1 INTRODUCTION

Apart from Sierpinski gaskets, yet another fractal geometry pursued by many antenna researchers is the Koch curve. Several variants of this geometry have been used as dipole, monopole, loop and patch antennas with varying relative advantages[3], [4], [28], [59], [60] [61]. Koch monopoles are among the first antennas based on a fractal geometry designed as small sized antennas. In addition to being small, these geometries can potentially lead to multiband antenna characteristics [28].

In one of the first approaches in the antenna design using fractal shaped geometries, Cohen has reported a dipole antenna configuration using variants of Koch curves [3]. These are fed at the center of the geometry. By increasing the fractal iteration, the length of the curve increases, reducing the resonant frequency of the antenna. The scaling property associated with Koch geometry has been used in [62] for fast computation of input impedance of these antennas with higher fractal iteration order. This is based on MoM and uses the fact that the length of the curve increases by a factor 4/3 at each additional order of iteration. Although this approach works well with the first few iterations of the geometry, its effectiveness for antennas with higher iterations of the geometry is doubtful.

The resonance of monopole antennas using these geometries below the small antenna limit has been reported by Puente et al [4]. These authors have studied the shift in resonant frequency by increasing the fractal iteration order. However detailed studies indicated that this reduction in the resonant frequency does not follow the same pace as the increase in the unfolded curve length obtained by each subsequent iteration [28]. As the fractal iteration is increased the feature length gets smaller. There seems to be a minimum feature length that influences antenna properties.

62

Research using these geometries, discussed above, have concentrated on introducing these geometries into the realm of antenna design, without getting into design ideas and have concentrated on few standard geometries. The only exception is the works by Werner et al [60], [63] where optimization of antenna properties by modifying the geometry using genetic algorithm. It may however be pointed out that this approach is computationally intensive as it requires extensive evaluation of antenna properties for each candidate geometry.

In the present work however, we attempt to extract fractal features that contribute to the variation of antenna performance. An important fractal property that is considered is the fractal dimension. As will be explained later, by changing the indentation angle of the Koch curve geometry (in other words IFS) the fractal dimension changes. An extensive study presented here shows a direct relationship between the antenna characteristics and this variation [64]. It is expected that the use of these ideas would significantly reduce the computational intensity of optimization approaches such as the ones mentioned above. The geometries studied here may be considered as a special case of those presented in [60] and [63] by making the length of the middle segment set to zero.

In order to explain these research findings in perspective, the variations in fractal geometries considered are presented in the next section. The results of numerical simulations using NEC with these geometries are described in Section 4.3. After summarizing these results, the relationship between the fractal dimension and antenna properties are extracted in Section 4.4. Some of these antenna models are experimentally validated in Section 4.5. Based on these studies, it has been found that the indentation angle of Koch iteration may be varied to design multi-resonant antennas with variable frequency intervals. Hence, as a generalization, a set of non-recursive geometries is studied in Section 4.6. A brief summary of the new findings presented in this Chapter are made in Section 4.7.

63

4.2 FRACTAL PROPERTIES AND GENERALIZATION OF KOCH CURVES

Fractal geometries used in this Chapter were originally introduced by the Swedish mathematician Helge von Koch in 1904 [6]1. This geometry is significant as this could lead to several other generalizations. The geometric construction of the basic curve is shown in Fig. 4.1. To distinguish this from generalizations introduced later, this geometry will be referred to as the standard Koch curve for the rest of the discussions.

The geometric construction of the standard Koch curve is fairly simple. One starts with a straight line, called the initiator. This is partitioned into three equal parts, and the segment at the middle is replaced with two others of the same length. This is the first iterated version of the geometry and is called the generator. The process is reused in the generation of higher iterations.

Iteration: 1 2 3 4

Fig. 4.1 Geometrical construction of standard Koch curve.

1 H. von Koch, “sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire,” Arkiv for Matematik 1, pp. 681-704, 1904.

64

It may be recalled that each segment in the first iterated curve is ⅓ the length of the initiator. There are four such segments. Thus for nth iterated curve the unfolded (or 4 n stretched out) length of the curve is ( /3) . This is one important property that would be useful in the design of antennas of this geometry.

4.2.1 IFS for the Standard Koch Curve

An iterative function system (IFS) can be effectively used to generate the standard Koch curve. A set of affine transformations forms the IFS for its generation.

Let us suppose that the initiator (unit length) is placed along the x-axis, with its left end at the origin. The transformations to obtain the segments of the generator are:

1  x′  3 0 x  W1  =    (4.1)  ′ 1    y  0 3  y

′ 1 1  1  x  3 cos60° − 3 sin 60°  x   3  W2   =    +   (4.2)  ′ 1 1      y   3 sin 60° 3 cos60°  y 0

1 1 1  x′  3 cos60° 3 sin 60° x   2  W3   =    +   (4.3)  ′ 1 1    1   y  − 3 sin 60° 3 cos60° y  2 sin 60°

′ 1 2  x   3 0 x   3  W4   =    +   (4.4)  ′ 1      y  0 3  y 0

The generator is then obtained as

A1 = W (A) = W1(A) ∪W2 (A) ∪W3(A) ∪W4 (A) (4.5)

This process can be repeated for all higher iterations of the geometry. It may be observed that the (straight line) distance between the start and end points of curves of all iterations is the same.

It may be observed that scale factors and rotation angles in these transformations are such that they lead to a self-similar geometry, a fact that is visually apparent. The similarity dimension of the geometry can thus be calculated as

65

log 4 D = =1.26186 (4.6) log3

This follows from the observation that at each iteration there are four identical copies of the original geometry that are scaled down by a factor of 3.

4.2.2 IFS for Generalizations

In the proposed generalizations studied as part of this work, the rotation (indentation) angle is made a variable. This leads to generalization of IFS in the following form:

1  x′  s 0 x  W1  =    (4.7)  ′ 1    y  0 s  y

′ 1 1 1  x   s cos θ − s sin θ x   s  W2   =    +   (4.8)  ′ 1 1      y   s sin θ s cos θ  y 0

1 1 1  x′  s cos θ s sin θ x   2  W3  =    +   (4.9)  ′ 1 1    1   y  − s sin θ s cos θ y  s sin θ

′ 1 0 s−1  x   s  x   s  W4   =    +   (4.10)  ′ 1      y  0 s  y  0  where the scale factor s is angle dependent and is given by

1 1 = (4.11) s 2()1+ cosθ

This ensures the distance between the start and end points for all iterations is the same. It may be easily verified that this formulation degenerates to the standard Koch curve for θ=60°.

The generator can be obtained similar to eq. (4.5). These affine transformations in the generalized case also lead to a self-similar fractal geometry. The similarity dimension is obtained as

log 4 D = (4.12) log[]2()1+ cosθ

66

Thus, geometries with varying fractal similarity dimension can be obtained using this generalization. Examples of the geometries for various iterations with two different angles of indentation are shown in Fig. 4.2. These show the difference in the plane filling nature associated with the change in angle, which may be associated with the change in dimension. The change in the unfolded curve length for each iteration as a function of the indentation angle is shown in Fig. 4.3. Similarly, the dimension as a function of the indentation angle is shown in Fig. 4.4. It may be observed that for angle θ = 0, the curve is linear and its dimension is 1, and for θ=90° a geometry of sufficiently large iteration tend to fill a triangle thereby approaching a dimension of 2.

θ = 30° θ = 70°

θ θ

Fig. 4.2 Generalized Koch curves of first four iterations with two different indentation angles. The length of subsections for a given iteration is a function of angle of indentation.

67

25

20

15

10 Unfolded Curve Length 5

0 0 102030405060708090 Indentation Angle (deg.)

Fractal Iteration: n=1 n=2 n=3 n=4 n=5 n=6

Fig. 4.3 Change in unfolded (stretched-out) curve length obtained by the variation of indentation angles of the generalized Koch curve geometry. The parametric curves are for different fractal iterations. The end- to-end distance in all these cases is of unit length.

2

1.9

1.8

1.7

1.6

1.5

Dimension 1.4

1.3

1.2

1.1

1 0 102030405060708090 Indentation Angle

Fig. 4.4 Variation of similarity dimension of the generalized geometry for various indentation angles.

68

4.3 ANTENNA MODELING STUDIES

4.3.1 Dipole Antenna Model

Dipole antennas with arms consisting of Koch curves of different indentation angles and fractal iterations are simulated using a moment method based software G-NEC. A typical dipole antenna using 4th order iteration curves with an indentation angle of 60° and with the feed located at the center of the geometry is shown in Fig. 4.5. Similar geometries with various fractal iterations and indentation angles are used for the simulation studies presented in subsequent sections. The radius of wire segments constituting the antenna model is 0.1 mm. The segmentation length used in the NEC model is taken as approximately 0.5 mm uniform in all cases. Each dipole arm has an end-to-end length of 10 cm. Resonant frequencies of dipole antennas using several iterations for geometries with different angles of indentation are determined by extensive numerical simulations with these models.

Feed point

Fig. 4.5 Configuration of symmetrically fed Koch dipole antenna. 4th iteration Koch curves with indentation angle θ = 60° form each arm of the antenna.

4.3.2 Results of Numerical Simulations

The input and radiation characteristics of this type of antenna have been widely studied by experiments as well as numerical simulations. In the present attempt the primary objective is to obtain a connection, if any, between the fractal dimension of the geometry and antenna characteristics. Since these antennas are small in terms of operational wavelength, their radiation performance is not expected to change significantly. Hence only the input characteristics of the antennas are examined in the following discussions.

In Fig. 4.6, the input resistances of the antennas with generalized Koch curves with different iterations are plotted. Geometries with only few selected indentation angles are plotted to make the plots readable. Similarly the input reactances for the corresponding

69

cases are given in Fig. 4.7. These indicate a reduction in resonant frequencies as the indentation angle is increased as well as for increasing fractal iteration. This is further apparent in Table 4-1 where the first resonant frequencies of the antennas are tabulated.

The input resistance at the resonant frequency also changes by these modifications. In Fig. 4.8 these variations are plotted for various iterations of the fractal. For angle θ=0, these antennas all degenerate to identical linear dipoles with a resonant input resistance of about 72Ω. As the angle is increased and the fractal iteration order is increased, this value is reduced significantly. It may be observed that it is always preferable to match the antenna impedance to a standard value (50Ω). Hence this approach of generalization may be used to design antennas with the required input characteristics at a specified frequency. In other words, the indentation angle may be used as a design parameter.

70

(a) Iteration: 1

4000

3500

3000

2500

2000

1500 Input Resistance 1000

500

0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

0 20 40 60 80

(b) Iteration: 2

4000

3500

3000

2500

2000

1500 Input Resistance 1000

500

0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

20 40 60 80

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(c) Iteration: 3

4000

3500

3000

2500

2000

1500 Input Resistance 1000

500

0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

20 40 60 80

(d) Iteration: 4

4000

3500

3000

2500

2000

1500 Input Resistance 1000

500

0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

20 40 60 80

Fig. 4.6 Input resistance for dipole antennas with Koch curves of different indentation angles.

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(a) Iteration: 1

2000

1500

1000

500

0

-500 Input Reactance -1000

-1500

-2000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

0 20 40 60 80

(b) Iteration: 2

2000

1500

1000

500

0

-500 Input Reactance -1000

-1500

-2000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

20 40 60 80

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(c) Iteration: 3

2000

1500

1000

500

0

-500 Input Reactance -1000

-1500

-2000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

20 40 60 80

(d) Iteration: 4

2000

1500

1000

500

0

-500 Input Reactance -1000

-1500

-2000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

20 40 60 80

Fig. 4.7 Input reactance of dipole antennas with Koch curves of different indentation angles.

74

Table 4-1 Primary (first) resonant frequencies for dipole antennas with Koch curves for various iterations obtained by numerical simulations.

Indentation Resonant frequencies for vairous iterations of Koch curve Angle Deg. 1 2 3 4 10 716.4 713.5 711.7 710.1 20 704.9 693.9 687.8 685.8 30 686.1 662.4 649.6 643.2 40 660.2 618.8 595.1 589.1 50 627.9 565.6 528.6 512.5 60 590.4 505.1 453.8 427.9 70 549.5 441.2 376 337.2 80 510.6 381.3 304.6 256.6

80

70

60

50

40

30

20 Input Resistance (ohms)

10

0 0 1020304050607080 Indentation Angle (deg.) Fractal Iteration: 1 2 3 4

Fig. 4.8 Variation of input resistance of the dipole antennas with generalized Koch curves of various fractal iterations.

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4.3.3 Characteristics of Multiband Antenna

The antenna input characteristics at higher resonant frequencies are also altered by the change in angles. These are plotted in Fig. 4.9 for the first four iterations of the geometry. It may be noticed that for very small indentation angles, these antennas behave similar to linear dipoles. However as the angle is increased, the periodicity of these multiple resonances changes. It may be argued that the indentation angle of the antenna can be changed for appropriate positioning of the resonant frequencies of the antenna. This would be further explored in the next Section.

Other antenna characteristics at these multiple resonances are worth exploring. Current distributions on the dipole antenna based on the standard Koch curve, for its resonant frequencies are plotted in Fig. 4.11. The corresponding radiation patterns are shown in Fig. 4.12. It may be observed that the radiation patterns of the antenna at these resonances behave like a regular linear dipole. Since the antenna size reduction factors for these geometries are not significant, their second (and higher) resonant frequencies are substantially higher than the primary resonance. Therefore the maxima of current distribution on the antenna are sufficiently apart to cause multiple lobes in the radiated beam. Hence these antennas are studied here primarily for the placement of their resonant frequencies.

4.3.4 Effect of Changing Feed Location

Previous sub-sections indicated that the indentation angle of the geometry can be used as a control parameter for either the input resistance at the primary resonance, or the multi- band characteristics of the antenna. To enhance the design flexibility, one may choose to associate the indentation angle for multi-band characteristics, and resort to manipulating the location of the feed for impedance matching. This asymmetric feed positioning is often used for linear dipole antennas [65]. In such a case, the current at the new input terminals is given by

76

6000

Fourth resonance

5000

4000 Third resonance

3000

Second resonance

2000 Resonant Frequency(MHz)

1000 First resonance

0 0 20406080 Indentatation angle (deg.)

Iteration: 1 Iteration: 2 Iteration: 3 Iteration: 4

Fig. 4.9 Variation of resonant frequencies of dipole antennas with generalized Koch curves of various fractal iterations. The resonant frequencies for each resonance of all cases converge to that on linear dipole when the indentation angle approaches zero.

77

Fig. 4.10 Current distribution on a dipole antenna with standard Koch curve geometry of 3rd iteration, at its various resonant frequencies.

Fig. 4.11 Radiation patterns of the Koch dipole antenna at its resonant frequencies.

78

  L  Iin = Im sinβ − d  (4.13)   2 

where d is the distance of the new feed location from the center and Im is the maximum current on the antenna. For half wave dipole, this reduces to

Iin = Im cos βd (4.14)

The input resistance for a dipole antenna can now be obtained in terms of radiation

resistance Rr as (without including the ohmic losses):

2 Im Rin = 2 Rr (4.15) Iin

For half wave dipoles with asymmetric feed, this reduces to

R R = r (4.16) in cos2 βd

The input resistance increases as the feed is moved away from the center. Near the ends of the dipole, this assumes a very large value.

This approach can now be extended to the Koch curve dipole antennas where the input resistance is below standard values. As an example, the 3rd iterated standard Koch dipole antenna is considered. The input resistance of this antenna is 28Ω when the feed is located at the center. By moving it to a new location, the standard value of 50 Ω can be obtained. The new feed location for one such geometry is shown in Fig. 4.12.

Feed Location

Fig. 4.12 New feed location for matching the input impedance to standard value.

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4.4 FRACTAL FEATURES IN ANTENNA PROPERTIES

In the last two sections it has been established that the fractal dimension of the generalized Koch curve is dependent on the indentation angle, and that the antenna properties can be linked to the indentation angle of the Koch curve geometry. It is now apparent that these two, when put together lead to significant results in fractal antenna engineering. The results presented here establish a direct correspondence between the fractal dimensions of the geometry and the performance of the antennas constructed using them. These aspects are explored further here.

4.4.1 Lowest Resonant Frequency

As shown in Section 4.2, the generalization of Koch curves offers a unique opportunity to study a set of similar fractal curves with continuously varying fractal dimension. The fractal dimension of the geometry changes from 1 to the maximum value of 2, as the angle is increased from 0 to 90°. In the study on the performance of dipole antennas using these geometries, it has been found that input characteristics of the antenna change significantly by these modifications.

First, resonant frequencies of antennas with these generalized Koch curves are considered. The data in Table 4-1 is used to obtain a relationship between change in resonant frequency and variation in fractal dimension. These results are plotted in Fig. 4.13. The horizontal axis is chosen to be the reciprocal of the fractal dimension to demonstrate a nearly linear relationship. The normalization is done with respect to a linear dipole with the same end-to-end length. In this case, dipoles with an arm length of 10 cm having a resonant frequency of 720.2 MHz is used as the reference. It may be recalled that the resonant frequency of a wire antenna is dependent on the radius of the wire. Hence this normalization is valid if a corresponding translation in the wire diameter used in the design.

It is now possible to obtain a design equation for this type of antenna using the order of fractal iteration n and the dimension D of the geometry as inputs. Using a curve fitting approach, to fit the parametric curves in Fig. 4.13, we get

80

  n −1 ln D fK = fD 1− exp   (4.17)   n  D 

where fD is the resonant frequency of the linear dipole antenna with the same (end-to-end) length as the Koch curve.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2 Normalized Resonant Frequency Normalized 0.1

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1/Fractal dimension Fractal Iteration: 1 2 3 4

Fig. 4.13 The normalized resonant frequency of generalized Koch dipole antennas of different fractal iterations.

Similar approximations can also be obtained for the input resistance at the resonant frequency of the antenna. This is plotted in Fig. 4.14. It may be recalled that all iterations reduce to linear dipole (dimension=1) for the indentation angle equals 0. Curve fitting approach is used to obtain an approximate formula for the input resistance as:

2  ln D R = R 1− ()1+ 0.9ln n (4.18) in in0    D 

where Rin0 is the input resistance of a linear dipole. In short, if the primary objective is to design a small resonant antenna, the above equations may be used to obtain the geometrical parameters for the desired resonant frequency and input resistance of the antenna. Similar expressions may be obtained separately if there is change in dimensions of the components of the geometry.

81

80

70

60

50

40

30 Input Resistance

20

10

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1/Fractal dimension

Fractal Iteration: 1 2 3 4

Fig. 4.14 Input resistance of the generalized Koch dipole antennas of different fractal iterations plotted against reciprocal of the fractal dimension.

4.4.2 Multi-band Characteristics

From Fig. 4.9 it is quite clear that the period between multiple resonant frequencies of the antenna differs as the indentation angle of the geometry is increased. These results are summarized in Table 4-2. The ratios of successive resonant frequencies are listed in the table for comparison. It may be recalled that due to the closeness of the geometry to a linear dipole, these ratios are not converging to a unique value as with some other fractal antennas. Although these ratios are different for each interval, they remain nearly constant for different fractal iterations of the geometry of the same dimension. The ratios

of first two resonant frequencies (fr2/fr1) are plotted against the dimension of the geometry in Fig. 4.15 to show its almost linear relationship to the fractal dimension.

4.5 EXPERIMENTAL VALIDATION

Several dipole antennas with Koch curve geometry have been fabricated to verify the conclusions derived from the above discussions. All these antennas have been fabricated on Duroid RO 3003 dielectric substrate (εr = 3, tanδ = 0.0013) with 1.5 mm thickness.

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All these antennas have two arms each with an end-to-end length of 10 cm. The input characteristics of these antennas have been measured using a network analyzer (HP 8510 C). To overcome the issue of current balancing at the input terminals, ferrite beads are used on the coaxial cable feeding the antenna.

Photographs of these antennas are shown in Fig. 4.16 and Fig. 4.18. The first set is used to make a comparison between various iterations of the geometry. The measured return loss of antennas shown in Fig. 4.16 are plotted in Fig. 4.17. The first three iterations of Koch curves and a linear dipole of the same length are compared here. These indicate a downward shift in the resonant frequency of these antennas as the fractal iteration is increased. Similarly, return loss characteristics for antennas with different indentation angles are shown in Fig. 4.19. All geometries in this case are of 2nd iteration. The primary resonance frequencies of these antennas are plotted against the reciprocal of the fractal dimension in Fig. 4.20. However it may be noticed that the slope of this line is understandably different from that in Fig. 4.13 due to the fact that this antenna is printed on a dielectric substrate. Similarly, the ratio of first two resonant frequencies is plotted against the fractal dimension in Fig. 4.21.

4

3.5

3

2.5

Frequeny Ratio Frequeny 2

1.5

1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Fractal dimension

Fig. 4.15 The ratio of first two resonant frequencies of multi-band Koch dipole antennas as a function of fractal dimension of the geometry.

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Table 4-2 Geometric interval between resonant frequencies of generalized Koch dipole antenna. Although the ratios are different for each interval, they remain the same for different fractal iterations

Indentation Fractal Fractal Angle Dimension Iteration fr2/fr1 fr3/fr2 fr4/fr3 1 3.07 1.68 1.40 2 3.07 1.68 1.40 10 1.006 3 3.07 1.68 1.40 4 3.06 1.68 1.40 1 3.06 1.68 1.40 2 3.06 1.67 1.40 20 1.023 3 3.06 1.67 1.40 4 3.05 1.67 1.40 1 3.05 1.68 1.41 2 3.03 1.67 1.40 30 1.053 3 3.02 1.67 1.40 4 3.04 1.67 1.40 1 3.03 1.67 1.41 2 3.00 1.66 1.40 40 1.099 3 2.99 1.66 1.40 4 2.97 1.65 1.40 1 3.00 1.67 1.41 2 2.96 1.65 1.40 50 1.165 3 2.94 1.64 1.39 4 2.91 1.63 1.39 1 2.96 1.66 1.42 2 2.91 1.63 1.39 60 1.262 3 2.86 1.60 1.38 4 2.80 1.60 1.37 1 2.90 1.65 1.43 2 2.82 1.59 1.38 70 1.404 3 2.76 1.57 1.37 4 2.68 1.55 1.35 1 2.80 1.61 1.44 2 2.70 1.55 1.37 80 1.625 3 2.62 1.52 1.35 4 2.50 1.50 1.34

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Fig. 4.16 Photographs of Koch curve dipole antennas: Set 1 for comparison on the effect of the fractal iteration.

0

-5

-10

-15

-20

-25

Return Loss (s11) in dB -30

-35

-40 00.511.522.53 Frequency (GHz)

Linear Koch-1 Koch-2 Koch-3

Fig. 4.17 Return loss of dipole antennas shown in Fig. 4.16.

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Fig. 4.18 Photographs of Koch curve dipole antennas: Set 2 for comparison on the effect of indentation angles.

0

-5

-10

-15

-20

-25

Return Loss (S11) in dB -30

-35

-40 00.511.522.53 Frequency (GHz)

Koch-2-20 Koch-2-40 Koch-2-60 Koch-2-80

Fig. 4.19 Return loss of dipole antennas shown in Fig. 4.18.

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1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2 Normalized Resonant Frequency Normalized

0.1

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1/Dimension

Fig. 4.20 Primary resonant frequencies for dipole antennas with Koch curves of various indentations.

4

3.8

3.6

3.4

3.2

3

2.8

2.6

Ratio of Resonant Frequencies of Resonant Ratio 2.4

2.2

2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 Dimension

Fig. 4.21 Ratio of resonant frequencies for dipole antennas with Koch curves of various indentations.

4.6 DESIGNING ANTENNAS WITH OPTIMAL PERFORMANCE

As mentioned previously, in recent works by Werner et al, an optimization approach by modifying the antenna geometry using genetic algorithm has been suggested [60], [63].

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They use a slightly different IFS, with a each subsequent iteration having five copies of the previous. Since this does not seem to use any a priori knowledge of the behavior for the geometry, it is bound to be computationally intensive.

As discussed in previous sections, we have noticed that the fractal dimension plays a key role in deciding antenna operational frequency (or frequencies). The geometries used thus far were all generated recursively by using an IFS. If one were to break this rule, antenna properties may be tailored with better flexibility. However the resulting antenna structure may not be called truly fractal.

The geometries used in this section are not generated by a strictly recursive algorithm. We use the generalization presented in section 4.2.2 in a hybrid fashion. For example, we may choose a different angle θ at each level of iteration. For example, one may choose indentation angle 20° for the first stage, 30° for the second, 40° for the next, etc. However all segments at the same stage of iteration are of the same size and shape. Hence the geometry appears to be self-similar. This geometry is shown in Fig. 4.22.

Fig. 4.22 A generalized Koch curve generated with different indentation angle at each iteration stage. In this case, θ=20°, 30°, 40° (starting from the innermost) are used.

Extensive numerical simulation studies have been performed to explore the usefulness of this approach. A few representative cases are presented here. The results prove that the resonant frequencies of the antenna can be shifted using this approach. In Fig. 4.23, two plots, one for indentation angle of 40° for the inner iteration, and the other for 80° are given. The outer iteration angle is varied from 0 to 80° in steps of 20° for these plots. The change in resonant characteristics may be observed. The resonant frequencies are listed in Table 4-3. Another interesting observation is that although the curves having angle combinations 80°-40° and 40°-80° have the same unfolded length, they do not share the same characteristics. These are further evident from the input resistance plot of Fig. 4.24. In this plot, the indentation angle for the inner generator is the parametric

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variation while the x-axis is for the angle of the outer generator.

Similar variations in input characteristics are also obtained for antennas with 3rd iteration geometries are shown in Fig. 4.25. Only the case with indentation angle of 60° for the innermost generator is considered. Each of the plots here corresponds to a different angle for the outermost generator.

Following this approach one can have two curves with the same length, but with different set of resonant frequencies. These differences are found to be more pronounced as the order of iteration is increased. Hence this approach offers a scheme of designing antennas based on Koch curves suiting the requirements in terms of input resistance and resonant frequency. It may be concluded from this study one can design multi-band antennas with considerable flexibility in choosing individual resonant frequencies.

4.7 SUMMARY

In this Chapter, multiple resonant frequencies of a fractal element antenna using Koch curves is related for the first time to the fractal dimension of the geometry. The similarity dimension of the Koch curve geometry can be varied by changing the indentation angle used in the recursive IFS. This is found to have a direct bearing on the input characteristics of these antennas. The primary resonant frequency, the input resistance at this resonance, and the ratio of first two resonant frequencies, have all been directly related to the fractal dimension of the geometry. Curve-fit expressions are also obtained for the performance of the antenna at its primary resonance, in terms of fractal iteration and fractal dimension. The antenna characteristics are studied using extensive numerical simulations. A few selected cases are experimentally verified.

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2000

1500

1000

500

0

-500 Input Reactance

-1000

-1500

-2000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (MHz)

0-40 20-40 40-40 60-40 80-40

2000

1500

1000

500

0

-500 Input Reactance

-1000

-1500

-2000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (MHz)

0-80 20-80 40-80 60-80 80-80

Fig. 4.23 Input reactance of dipole antennas based on two sets of generalized Koch curves. All antennas have arms spreading 10 cm, and have 2nd generation Koch curve. The indentation angles for different iterations are different as marked in the plots.

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Table 4-3 Resonant frequencies of dipole antennas with 2nd generation generalized Koch curves. The indentation angle in each generation stage is different. The dipole arms are of length 10 cm in all cases.

Resonant Frequencies Indentation Angle Ratios of Resonant Frequencies

Inner Outer fr1 fr2 fr3 fr4 fr2/fr1 fr3/fr2 fr4/fr3 0 20 705 2157 3615 5078 3.060 1.676 1.405 40 660 1999 3348 4716 3.029 1.675 1.409 60 590 1748 2910 4130 2.963 1.665 1.419 80 511 1432 2300 3311 2.802 1.606 1.440 20 20 694 2121 3549 4981 3.056 1.673 1.403 40 650 1964 3284 4621 3.022 1.672 1.407 60 580 1714 2847 4035 2.955 1.661 1.417 80 497 1395 2247 3236 2.807 1.611 1.440 40 20 662 2012 3354 4690 3.039 1.667 1.398 40 619 1861 3093 4332 3.006 1.662 1.401 60 550 1619 2666 3748 2.944 1.647 1.406 80 469 1308 2093 2991 2.789 1.600 1.429 60 20 610 1839 3043 4221 3.015 1.655 1.387 40 570 1696 2792 3870 2.975 1.646 1.386 60 505 1468 2386 3317 2.907 1.625 1.390 80 427 1177 1858 2619 2.756 1.579 1.410 80 20 554 1648 2689 3663 2.975 1.632 1.362 40 515 1513 2451 3336 2.938 1.620 1.361 60 455 1299 2075 2828 2.855 1.597 1.363 80 382 1031 1599 2194 2.699 1.551 1.372

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80

70

60

50

40

30 Input Resistance

20

10

0 0 102030405060708090 Indentation Angle (outer) Indentation Angle (Inner): 0 20 40 60 80

Fig. 4.24 Input resistance at the first resonance of dipole antennas based on generalized Koch curves. All antennas have an arm length of 10 cm, and have 2nd generation Koch curve. The angle at the inner IFS is the parametric variation while the x-axis is for the angle of the outer IFS.

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-2000 0 500 1000 1500 2000 2500 3000 Frequency (MHz) 60-20-0 60-20-20 60-20-40 60-20-60 60-20-60

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-2000 0 500 1000 1500 2000 2500 3000 Frequency (MHz)

60-40-20 60-40-40 60-40-60 60-40-80

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-1000

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-2000 0 500 1000 1500 2000 2500 3000 Frequency (MHz)

60-60-20 60-60-40 60-60-60 60-60-80

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0

-500 Input Reactance

-1000

-1500

-2000 0 500 1000 1500 2000 2500 3000 Frequency (MHz)

60-80-20 60-80-40 60-80-60 60-80-80

Fig. 4.25 Input reactance of dipole antennas based on two sets of generalized Koch curves. All antennas have arms spreading 10 cm, and have 3rd generation Koch curve. The angle at each iteration is different as marked in the plots. The indentation angle for the innermost stage is the same (60° in all these cases).

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CHAPTER 5 MULTI-BAND PROPERTIES OF ANTENNAS WITH FRACTAL CANOPIES

5.1 INTRODUCTION

In previous Chapters several well-known fractal geometries have been considered for antenna design. The Sierpinski gasket is perhaps the most widely studied fractal geometry for antenna elements. A generalization of Koch curves in Chapter 4 has led to the conclusion that the fractal dimension is a key parameter influencing input characteristics of the antenna. A viable means of verifying this hypothesis could not be obtained for either Sierpinski Gasket or Hilbert curves antennas. Hence different fractal geometries with called fractal trees are considered here. These geometries are also known as fractal canopies and Pythagoras trees.

Fractal binary trees presented in this Chapter have several features common with other fractals such as Koch curves. But their branching nature offers a significant variation, and is expected to cause some difference in antenna performance. In this case, tree branching in two directions is considered. Hence the geometry is referred to as a binary tree in this work. The fractal nature of the geometry is explained first in Section 5.2. Antenna modeling studies with this geometry are presented in Section 5.3. Several parametric variations of the geometry are considered here. The conclusions from this Chapter are summarized in the last Section.

5.2 FRACTAL NATURE OF TREE

The geometries discussed thus far in this work have generally been self similar. In all those, one starts with a simple geometry, makes certain number of (connected) scaled copies of the starting object at each consecutive iteration. However the approach taken for the generation of trees here is somewhat different. One starts with a “stem” and allows one of its ends to branch off in two directions. In the next stage of iteration, each of these branches are allowed to branch off again, and the process is continued infinitely as shown in Fig. 5.1 [66]. In contrast to a ternary tree studied in [29], this is termed a

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“binary tree”.

n=1 n=5

n=2 n=4

n=3 Fig. 5.1 Various iterations of fractal binary tree

The primary objective of studying this geometry here is to verify the hypothesis developed about the relation between antenna resonance characteristics and the fractal dimension of the geometry. Hence two generalizations are attempted to the basic geometry in the present study. The first is to vary the angular separation of the branches at every iteration stage. The second approach is to vary the relative lengths of the branch in one stage relative to the next. These are portrayed in Fig. 5.2 and Fig. 5.3. It may be recalled that in both cases, the geometry can be generated by a recursive algorithm, after the first branching is made. To make the computation of fractal properties convenient, we propose to use only line segments for all branches of the geometry. Compared to Mandelbrot’s tree structure, the one used here does not involve the width of the constituting segments.

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Branch angle 2θ = 60° 120° 180°

Fig. 5.2 3rd iterated fractal binary tree with different branching angles

The generation algorithm of this geometry is conveniently expressed in terms of L- systems. This uses a pointer on the graphics screen often referred to as a “turtle”. In this approach., a string of symbols with the following notations are used for pointing and moving the turtle.

F: for moving forward a step (drawing a line),

+: to turn left by a fixed angle (θ),

−: to turn right by a fixed angle (θ).

[: save the turtle state for branching

]: restore the previously saved turtle state

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Branch:stem = 0.9

Branch:stem = 0.6 Branch:stem = 0.3

Fig. 5.3 3rd iterated binary trees with different length ratios for arm to stem.

The initial string of symbols is called an axiom. We start with the stem of the tree:

Axiom: F

Production rule: FÆ F[+F][-F] (5.1)

It is possible to vary the scale factor between the length of the stem and branches. The transformations required to obtain branches of the geometry in such case may be expressed as follows:

1 1  y′  s cosθ − s sin θ y 0 W1  =    +   (5.2)  ′ 1 1      z   s sin θ s cosθ  z  1

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 y′ 1 cos θ 1 sin θ y 0 W   = s s   +   (5.3) 2  ′  1 1      z   s sin θ s cos θ z  1

In Fig. 5.1, the scaling is by a factor of 2, and the branching half-angle θ is 45°. This angle is varied between 15° to 60° in Fig. 5.2. The length ratio is varied between 0.2 to 0.6 in Fig. 5.3 for binary trees of the same iteration stage.

The fractal dimension D for the geometry shown in Fig. 5.1 is obtained using [7]:

D D  2   1    + 2  =1 (5.4)  3   3

This assumes the branched structure replaces the original stem such that the total length of the new stem and one branch equals the length of the previous branch. Thus the new 2 1 stem is reduced by a factor of /3 and the branch is /3, so that they have a length ratio of ½. The dimension of this geometry is 1.395.

Since the branching angle has no direct role in determining the lengths of the segments, the dimension of all geometries in Fig. 5.2 remain the same. However, as the scale factors are changed, the fractal dimension is also changed. For a length ratio x:1 between branches and the stem, the following expression may be satisfied for the fractal dimension:

D D  1   x    + 2  = 1 (5.5) 1+ x  1+ x 

The variation obtained for the fractal dimension is plotted in Fig. 5.4. It may be mentioned that as the ratio x:1 increases, the two halves of the tree tend to overlap each other. This may lead to higher number for the dimension of the geometry meeting the above condition.

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2

1.9

1.8

1.7

1.6

1.5

1.4 Fractal Dimension 1.3

1.2

1.1

1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Length Ratio

Fig. 5.4 Variation of the fractal dimension with length ratio of branch to stem of a binary tree

5.3 ANTENNA CHARACTERISTICS USING FRACTAL TREE

Monopole antennas with binary tree structures have been modeled using NEC. All wire segments have the same diameter (0.1 mm) and segmentation is approximately 0.5 mm long. The feed for this monopole is placed at the intersection of the geometry with the ground plane. An infinitely extending perfectly conducting ground plane is used in the numerical simulations. For easy comparison, the length of the curve (defined here as the shortest distance along the curve, for a unique path starting from the base of the stem, ending at the tip of the last branch) is kept at 10 cm in all these cases.

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Feed Location

Ground plane

Fig. 5.5 Monopole antenna using a binary tree geometry

5.3.1 Parametric Studies by Increasing Fractal Iteration

Since the length of the curve (as defined above) is kept the same (=10 cm), the overall spreading of the antenna changes by fractal iteration. It is worth studying its impact on the input characteristics of the antenna. A typical antenna geometry is shown in Fig. 5.5. The angle between branches is kept at 120°. In this case, the length ratio of the branch to stem after an iteration is kept at 0.5. The input characteristics of the monopole antenna are plotted in Fig. 5.6. It may be observed that the 2nd and 3rd resonances for most of these curves are very close to each other. However the first resonant frequency varies significantly with the number of iterations. It is conjectured that this is due to the nature of the reactance contribution of branching junctions. The variation of first resonant frequency and the input resistance at these frequencies are tabulated in Table 5-1. It may be recalled that the input resistance would be higher if dipole antenna configuration is used.

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Input Resistance 600

400

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0 0 1000 2000 3000 4000 5000 6000 Frequency (MHz) Fractal Iteration: 1 2 3 4 5 6 Fig. 5.6 Input impedance characteristics of binary tree monopole antenna with fractal geometries of various iterations.

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Table 5-1 Resonant Frequency and Input resistance (at resonance) for binary tree monopole antennas with different fractal iterations. The angle between branches is 120° and the scale factor is 0.5.

Fractal Resonant Input Iteration Frequency Resistance

1 606 27.2

2 541 20.3

3 496 16.7

4 465 14.6

5 443 13.3

6 431 12.5

It may however be noted that the higher order resonances of this antenna do not lead to radiation patterns that are not similar to the first resonance. Hence the usefulness of this antenna as a true multband antenna is questionable. The radiation characteristics of a typical tree antenna at various resonant frequencies are plotted in Fig. 5.7.

Fig. 5.7 Radiation patterns of a typical tree monopole antenna at its various resonances.

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5.4 PARAMETRIC STUDY BY CHANGING BRANCHING ANGLE OF THE FRACTAL TREE

A similar parametric study is conducted by changing the branching angle of the tree structure. The angle is varied from 30° to 180° in steps of 30°. The first resonant frequencies for antennas with these geometries are plotted in Fig. 5.8. This shows an optimum angle is possible in the design for antenna with lowest resonant frequency. However such an optimum value does not exist for other bands (higher resonances), as the frequencies vary monotonously with branching angle. Thus the multi-band characteristics of these antennas remain nearly the same (except for the above discrepancy in the first resonant frequency) in all cases. An example of this is shown for 2nd to 4th generation geometries in Table 5-2.

580

560

540

520

500

480

460

Resonant Frequency (MHz) Frequency Resonant 440

420

400 0 20406080100120140160180 Branching Angle (deg.) Fractal Iteration: 2 3 4 5 6 Fig. 5.8 Variation of first resonant frequency of monopole antennas with binary trees with branching angles

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Table 5-2 Periodicity of resonant frequencies of monopole antennas with fractal binary tree geometry. The band placement remains nearly the same for all branching angles .

Ratios of Resonant Frequencies Fractal Branching

Iteration Angle f2/f1 f3/f2 f4/f3 f5/f4 f6/f5

2 30 4.02 1.65 1.49 1.28 1.21

60 4.20 1.64 1.43 1.34 1.19

90 4.26 1.62 1.44 1.35 1.18

120 4.27 1.60 1.45 1.35 1.17

150 4.24 1.59 1.44 1.35 1.16

180 4.18 1.58 1.42 1.35 1.16

3 30 4.31 1.64 1.49 1.28 1.21

60 4.58 1.63 1.49 1.27 1.21

90 4.66 1.61 1.50 1.27 1.20

120 4.66 1.59 1.51 1.26 1.20

150 4.60 1.57 1.50 1.26 1.20

180 4.49 1.56 1.50 1.29 1.18

4 30 4.51 1.65 1.51 1.24 1.24

60 4.87 1.63 1.50 1.24 1.24

90 4.99 1.61 1.51 1.23 1.24

120 4.96 1.59 1.52 1.20 1.27

150 4.85 1.58 1.52 1.22 1.25

180 4.70 1.57 1.52 1.22 1.25

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2.1.1 Effect of Variation of Branch Length Ratio

The antenna characteristics are also studied for variation in the length ratio between branches and the stem in each stage of iteration. It may be recalled that this variation results in change in fractal dimension. The resonant characteristics of the antennas based on the 4th generation geometry with a branching angle of 120° are presented in Table 5-3.

There is a significant variation in the multiband characteristics. However, the nature of this variation is not always monotonous.

Table 5-3 Multiband characteristics of monopole antennas with 4th generation fractal tree geometry.

Scale Fractal Resonant Frequencies Frequency Ratios

Ratio Dimension fr1 fr2 fr3 fr4 fr5 fr2/fr1 fr3/fr2 fr4/fr3 fr5/fr4

0.7 1.47 416 2430 4025 5266 7413 5.84 1.66 1.31 1.41

0.6 1.43 436 2371 3824 5443 6915 5.44 1.61 1.42 1.27

0.5 1.40 465 2305 3669 5575 6669 4.96 1.59 1.52 1.20

0.4 1.36 504 2213 3691 5190 7108 4.39 1.67 1.41 1.37

0.3 1.31 555 2123 3802 5111 6591 3.82 1.79 1.34 1.29

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5.5 SUMMARY

The primary objective of this chapter is to verify the conclusions derived in the previous chapter where the characteristics of dipole antennas based on Koch curve are related to fractal properties of the geometry. In this chapter, another fractal geometry is considered where similar variation in fractal properties is achievable. It is assumed that the fractal dimension of binary trees presented here does not change with branching angle, but does change with scale factor in successive iterations. Accordingly, the multi-band properties of the monopole antenna designed using these structures does not change significantly with the first variation, but changes with the second. This rule is applicable for all resonances above the first. Hence it is concluded that the antenna multi-band properties may be related to fractal dimension.

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CHAPTER 6 HILBERT CURVES FOR SMALL RESONANT ANTENNAS

6.1 INTRODUCTION

One of the fundamental advantages of using a fractal geometry in antennas is reducing the size of a resonant antenna. This is very evident in dipole and monopole antennas using fractal Koch curves [28], and some of their modifications in the form of closed loops, and Minkowski curves [59]. The ability of these geometries to pack longer curves within relatively smaller area is the salient aspect in their use in antennas. Being a plane- filling geometry, Hilbert curves can enclose longer curves for a given area than Koch curves [67]. Hence dipole and monopole antennas using Hilbert curves are expected to be smaller. The fractal nature Hilbert curves and antenna geometries using them are presented in Section 6.2 and 6.3 respectively.

The self-similarity of the fractal geometry is often loosely linked to the multi-band nature of the antenna. It is found that the radiation characteristics of the antenna is self-similar for the first few resonant frequencies. Although it may not be possible to generalize this statement, the antenna may be designed for use in multi-functional applications. These have been covered in Section 6.4.

There are several implementation aspects to be considered while designing antennas using these geometries. The reduction in size does not come without a prize. In this case it is the reduced radiation resistance and hence a lower radiation efficiency. A simple approach to overcome these problems is presented in Section 6.5.

An advantage of using fractal geometries like this over other small antenna configurations such as meander line antennas is the ordered nature of the fractal geometry. Although this may not have a direct impact on the antenna properties, this does have the potential of simplifying the numerical complexity in antenna analyses. A design formulation of the antenna using its fractal nature is developed in Section 6.6.

Yet another advantage of the Hilbert curve is the possibility of obtaining antennas with reconfigurable radiation characteristics. A few additional line segments are used along

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with the original fractal geometry to change the characteristics of the radiation beam of the antenna (Section 6.7). In order to explore the suitability of this geometry in a patch configuration, an antenna is designed and characterized. The performance of this antenna is presented in Section 6.8.

Further size reduction of the antenna can be achieved by superimposing other fractal geometries such as Koch curves on to Hilbert curves. The characteristics of these novel doubly-fractal geometry antennas are introduced in Section 6.9. A brief summary of the Chapter is presented in Section 6.10.

6.2 FRACTAL PROPERTIES OF HILBERT CURVES

The first few iterations of Hilbert curves are shown in Fig. 6.1. It may be noticed that each successive stage consists of four copies of the previous, connected with additional line segments. This geometry is a space-Filling curve, since with a larger iteration, one may think of it as trying to fill the area it occupies. Additionally the geometry also has the following properties: self-Avoidance (as the line segments do not intersect each other), Simplicity (since the curve can be drawn with a single stroke of a pen) and self- Similarity (which will be explored later). Because of these properties, these curves are often called an FASS curves [68].

Iteration: 1 2 3 4

Fig. 6.1 First four iterations of Hilbert curve geometry. The segments used to connect copies of the previous iteration are shown in dashed lines

The generation algorithm of this geometry is commonly expressed in terms of L-systems. In this representation, a string of symbols with the following notations are used.

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F: for moving forward a step,

+: to turn left by a fixed angle (90° in this case),

−: to turn right by a fixed angle (90° in this case).

The initial string of symbols is called an axiom. We start with two of such axioms L and R as shown in Fig. 6.2.

L0 = +F − F − F + (6.1)

R0 = −F + F + F − (6.2)

To get the iteration of next order, one proceeds with the following production rules:

L1 = +RF − LFL − FR + (6.3)

R1 = −LF + RFR + FL − (6.4)

It may be observed that both these rules lead to the same geometry with different orientations, such that one is a mirror image of the other. A recursive approach may be used to generate higher iterations of the geometry from these:

Lm+1 = +Rm F − LmFLm − FRm + (6.5)

Rm+1 = −Lm F + Rm FRm + FLm − (6.6)

Fractal properties such as self similarity, fractional dimension can be ascertained for the Hilbert curves by going through the steps described here. It may be noticed from Fig. 6.1 that this geometry is not exactly self-similar, since additional connection segments are required when an extra iteration order is added to an existing one. But the contribution of this additional length is generally small compared to the overall length of the geometry, especially when the order of iteration is large. Since ideally one can have a high number of iterations, the effect of these small extension-lines are disregarded, the geometry is considered self-similar.

A similar ambiguity prevails in determining the dimension of the geometry also. The topological dimension of the curve is one, since it consists only of line segments. However, the dimension of a fractal curve can be defined in terms of a multiple copy

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algorithm [68]. The similarity dimension D is defined as

L0 L1

R0 R1

Fig. 6.2 Generation of Hilbert curve using L-systems.

log N D = (6.7) log f where N is the number of copies and f the scale factor of consecutive iterations. The dimension of Hilbert curve is:

 4n −1   4n  log  log   4n−1 −1  4n−1  log 4 D =   ≈ (for large n)   = = 2 (6.8)  2n −1   2n  log 2 log  log   n−1   n−1   2 −1  2 

The similarity dimension of this curve approaches an integer value (two) because of the approximation involved, when a large fractal order is considered. But if we consider the length and number of line segments 1st and 2nd iterations, the dimension is

15 log D = 3 =1.465 (6.9) 2 7 log 3

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The corresponding numbers in the new two iterations are 1.694 and 1.834. These numbers point to the fact that the dimension of the geometry is still a fractional number, albeit approaching 2.

It may however be noted that the topological dimension of the curve is one. This anomaly is not surprising, since the curve consists only of line segments, yet tending to fill a planar area. It may however be mentioned that the self similarity dimension of this curve is of integer value (two) because of the approximations involved in arriving at the value, and may therefore be taken as an approximate number. This allows one to consider the curve as fractal.

6.3 ANTENNA CONFIGURATIONS USING HILBERT CURVES

It would be interesting to study the properties of a new antenna with reference to various existing, and more familiar antennas. In this context, a schematic of the thought process leading to the Hilbert curve dipole antenna is given in Fig. 6.3. The half-wave dipole has a figure of 8 radiation pattern in the elevation plane and a circular pattern in the azimuth plane. The antenna gain is 2.15 dBi (over isotropic radiator), and is resonant when the arms are approximately quarter wavelength long. The biconical antenna is a broadband variant for the common dipole [69]. Shown in Fig. 6.3 is a practical geometrical approximation of this with triangular sheets, called the bow-tie antenna. This antenna can even be simulated with wires along its periphery. Puente et al [2] have used a bow- tie as the base model for explaining the properties of the Sierpinski gasket fractal antenna with multi-band radiation characteristics.

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Bow-tie Sierpinski Gasket

Half-wave Dipole

Meander Hilbert line Curve Fig. 6.3 A conceptual evolution of a fractal Hilbert curve dipole antenna.

It should be possible to reach the Hilbert curve antenna (HCA) starting with the half- wave dipole, with meander line antenna as the intermediary. This approach fits in well with the primary goal of development of such an antenna, i.e., a smaller resonant antenna. Meander line antennas were considered a significant step in that direction, where length of the antenna can be made much smaller than the wavelength [70]. Accordingly, the resonant frequency of HCA can be obtained by the same approach as for meander line antennas. This approximate design formulation is presented later in Section 6.6.

Other antenna configurations such as monopole and patch (Fig. 6.4) are also possible with this geometry. However for monopoles, the geometry has to be placed entirely above the ground plane. This may require additional line segments. Only dipole configuration is pursued extensively in this Chapter.

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Substrate Ground Plane Ground Plane

Monopole Dipole Printed Antenna

Fig. 6.4 Different antenna configurations using Hilbert curves.

6.4 SIMULATION STUDIES OF HILBERT CURVE ANTENNA

Antennas using these geometries can be easily modeled using NEC. The geometry is assumed to be consisting of wire segments. The input cards for the NEC modeling are generated using a recursive fractal generation algorithm. This results in a fast turn- around time for the modeling. Theoretical fundamentals of NEC used in the numerical simulations were presented in Chapter 2.

In a simple dipole antenna configuration using the Hilbert curves, the feed is located at the center of the geometry. Typical input impedance characteristics of dipole antennas with first four iterated Hilbert curve geometries are shown in Fig. 6.5. This shows multiple resonances for the antenna. Also, notice that both these antennas occupy an area of 7 cm x 7 cm. A visual comparison of the two input characteristics indicates significant reduction in resonant frequencies. This is because of the increase in length of the curve as the fractal iteration is increased. However it may be also be pointed out that such reduction is not linear. As one keep increasing the iteration, the resonant frequencies tend to asymptotically decrease. It has been reported for antennas with Koch curves that this may be due to the increased insignificance of the smallest feature size of these line segments [28]. Similar phenomena is expected for Hilbert curves also, since the area in which the curves are inscribed is kept unchanged.

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1000

900

800

700

600

500

400 Input Resistance 300

200

100

0 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (MHz)

Iteration: 1 2 3 4 2000

1500

1000

500

0 0 500 1000 1500 2000 2500 3000 3500 4000 -500 Input Reactance

-1000

-1500

-2000 Frequency (MHz)

Iteration: 1 2 3 4 Fig. 6.5 Computed input impedance for dipole antennas with the first four iterations of Hilbert curves. All antennas have outer dimensions of 7 cm, and are modeled with wire of 1.3 mm diameter.

The radiation characteristics of the antenna with the 4th iterated geometry at its resonant frequencies (352 MHz, 922 MHz, 1.233 GHz, and 1.715 GHz) are presented in Fig. 6.6. This shows that the radiation patterns for the first two resonances nearly overlaps and are therefore “similar”. For antennas with higher iterated geometry, more such frequencies with similar radiation patterns are feasible.

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The current distribution in these wire segments constituting the antenna is also studied for various iterations and their respective resonant frequencies. Color coded current distributions as obtained by NEC simulations are shown for various cases in Fig. 6.7. For a simple comparison of the current distributions at various resonances of these geometries are shown separately in Fig. 6.8.

In these studies, the antenna size is restricted to 7 cm x 7 cm area. The small antenna limit for this size is around 500 MHz. It may be observed that all these geometries have their first resonance below this limit. Hence it may be concluded that the use of a Hilbert curve geometry results in a small dipole antenna.

It may be observed that if one were to design an antenna of this type, (for a set of predefined resonance frequencies) there are several options available. For example if it is required that the antenna has a resonance at 620 MHz. Geometries with different fractal iterations can have the same resonant frequency, but with a different inscribing area. In this case, 2nd, 3rd, and 4th iterated geometries with areas 16 x 16 cm2, 10.5 x 10.5 cm2, and 7.8 x 7.8 cm2 have their second resonance at this frequency. However, as the size is reduced, the operational bandwidth is severely curtailed. The VSWR in these cases are shown in Fig. 6.9.

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(a) φ = 0

(b) φ = 90°

Fig. 6.6 Simulated radiation patterns of the 4th Hilbert curve dipole antenna at various resonant frequencies at different φ planes. Due to symmetry, only one half is shown here. The antenna occupies outer dimension of 7 cm x 7 cm, and modeled with wire of 1.3 mm diameter. The resonant frequencies are: 352, 922, 1233, and 1715 MHz.

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1st Resonance 3rd resonance

2nd Resonance 4th resonance

Fig. 6.7 Current distribution along the 4th iterated Hilbert curve geometry at its various resonances.

Alternatively one can choose any of the resonances for geometries with a fixed iteration. As an example, if one wants to use a 3rd iterated curve, the geometry inscribed in 4.2 x 4.2 cm2 area has its first resonance at this frequency, while that with 10.5 x 10.5 cm2 area has its second resonance, and another with 16 x 16 cm2 area has its third resonance here. However the difference between these cases is a significant change in input characteristics. The VSWR in these cases are plotted in Fig. 6.10. Recall that the radiation characteristics are nearly the same for these options.

From the input characteristics of the antenna in Fig. 6.5, it may be observed that at the first resonance, the input resistance of the antenna is very low. This can potentially lead

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to low radiation efficiency, consistent with other such small antennas. One approach to match input resistance of the antenna with the feed cable impedance is by using a feed location away from the point of symmetry. In such cases, a better VSWR is possible for the antenna.

1.000

0.100

Current Magnitude 0.010

0.001 0 20 40 60 80 100 120 140 160 180 200 Segment # on Curve

1st 2nd 3rd 4th

Fig. 6.8 Current along the length of the Hilbert curve when the feed is connected at the center of the geometry. These plots are for the resonant frequencies of the antenna.

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(a) 2nd iteration geometry: 16 cm x 16 cm

(b) 3rd iteration: 10.5 cm x 10.5 cm

(c) 4th iteration: 7.8 cm x 7.8 cm

Fig. 6.9 Designing antennas for a specified frequency with various iteration curves.

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(a) 3rd iterated: 4.2 cm x 4.2 cm, 1st resonance

(b) 3rd iterated: 10.5 cm x 10.5 cm, 2nd resonance

(c) 3rd iterated: 16 cm x16 cm, 3rd resonance

Fig. 6.10 Designing antennas for a specified frequency with various resonant frequency options. It may be observed that the radiation characteristics vary only slightly in these cases.

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6.5 IMPLEMENTATION ISSUES FOR DIPOLE ANTENNA CONFIGURATION

Although modeled with wire elements, fabrication of the antenna in this configuration is nearly impossible, especially for smaller dimension prototypes. Hence the antenna geometry is printed on a substrate. Low dielectric constant substrates such as FR-4 or RT Duroid 3003 are used as the support structure. As these substrates are electrically thin, they are expected to have only a small dielectric loading effect on the antenna performance. As a result, the resonant frequency of the antenna is shifted towards lower frequencies. Although NEC cannot model this effect directly, other software are capable of modeling dielectric loading effects on antenna performance. However, a uniform loading term can be introduced in the NEC input to approximately incorporate this effect.

The antenna is fed using a coaxial cable. This poses problems of balancing the currents at the input terminals. Hence a set of ferrite beads are used outside the coaxial cable to increase the inductance of the outer conductor, to reduce current flow though it. The measured input characteristics of the antenna are shown in Fig. 6.11.

6.6 DESIGN FORMULATION

It is now possible to obtain approximate design equations for this type of antenna. The approach for the design formulation is based on that followed for resonant meander line antennas [70]. In this approach, the inductances of the turns of the meander line are calculated, considering them as short circuited parallel-two-wire lines. The self inductance of an imaginary straight line connecting all these turns is then added to this to get the total inductance. This is then compared with the inductance of a regular half wavelength dipole. Since dipole antennas with approximately half wavelength are resonant (their capacitive and inductive reactances cancel each other), and assuming that the input capacitive reactance for a dipole antenna remains unchanged by reducing its apparent length by introducing turns, the resonant condition for this antenna is derived.

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0

-5

-10

-15

-20 Return Loss (S11) in dB

-25

-30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (GHz)

Fig. 6.11 Measured performance of a dipole antenna with Hilbert curve geometry.

The approach reported in [70] for the meander line dipole antenna can readily be extended for the Hilbert curve antenna (HCA) [71]. The definition of self inductance of a straight line for meander line antenna is replaced here with the total inductance of the line segments otherwise unaccounted (not forming short-circuited parallel wire sections). Another important assumption is that the capacitances of the dipole configurations remain the same in all cases. The geometry of a 3rd iterated Hilbert curve is split into these components in Fig. 6.12.

For an HCA with outer dimension of l and order of fractal iteration n, the length of each line segment d is given by:

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Parallel wire section with length = d and diameter = b Short circuit terminations for the Parallel wire sections Connection segments

Fig. 6.12 Composition of a HCA with iteration order 3. The short-circuited parallel wire sections and connection wire sections are shown separately.

l d = (6.10) 2n −1

The geometries for the first four iteration orders are shown in Fig. 6.1. Further, in an HCA geometry of order n, there are

n−1 m = 4 (6.11)

short circuited parallel wire sections, each of length d. Similarly, as shown in Fig. 6. 12, the segments not forming the parallel wire sections amount to a total length of

2n−1 s = (2 −1)d (6.12)

6.6.1 Modeling of Antennas Neglecting Dielectric Loading

The characteristic impedance of a parallel wire transmission line consisting of wires with diameter b, spacing d is

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η 2d Z = log (6.13) 0 π b where η is the intrinsic impedance of free space. This can be used to calculate the input impedance at the ends of the line, which is purely inductive.

Z L = 0 tan βd (6.14) in ω

There are m such sections. The self inductance due to a straight line of length s as defined in (6.12) is:

µ0  8s  Ls = slog −1 (6.15) π  b 

Substituting (6.13) in (6.14) and using (6.15), the total inductance is therefore

µ0  8s  η 2d LT = slog −1 + m log tan βd (6.16) π  b  πω b

To find the resonant frequency of the antenna, this total inductance is equated with that of a resonant half-wave dipole antenna (with approximate length l=λ/2). This leads to the condition of primary resonance of the Hilbert curve antenna as:

η 2d µ  8s  µ λ  2λ  m log tan βd + 0 slog −1 = 0 log −1 (6.17) πω b π  b  π 4  b 

It should however be noted that regular dipole antennas resonate when the arm length is a multiple of quarter wavelength. Thus, by changing the resonant length related terms on the RHS of equation (6.17), one can obtain all the resonant frequencies of the multi-band HCA. Therefore the first few resonant frequencies of the HCA can be obtained from:

η 2d µ  8s  µ kλ  8 kλ  m log tan βd + 0 slog −1 = 0 log −1 (6.18) πω b π  b  π 4  b 4 

where k is an odd integer. It may be noted that d, m, and s, in (6.18) are expressed in terms of fractal iteration order in equations (6.10 - 6.12) above. Also, this expression does not account for higher order effects and hence may not be accurate at very high resonant modes.

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The validation of this expression is examined by NEC simulations. A few cases with varying geometrical parameters are modeled and the resonant frequencies calculated by the above formulation are compared with the results obtained with NEC simulations in Table 6-1. A comparison for different iterations of the fractal geometry, where different wire diameters are considered, is shown in Fig. 6.13. A similar comparison, when the area of the geometry is changed, is shown in Fig. 6.14. These cases are studied to explore the versatility of the design formulation.

Table 6-1 Comparison of results from design formulation with NEC results

Outer Wire Fractal Resonant NEC % dimension diameter iteration Frequency simulation Difference (mm) (mm) (MHz) (MHz) 50 1 2 789 704 12.07 50 1 3 486 510 4.7 80 1 2 483 433 11.55 80 1 3 295 304 2.96 80 2 2 499 445 12.13 80 2 3 308 328 6.1 100 2 1 544 500 8.8 100 2 2 395 351 12.53 100 2 3 243 252 3.57

10000

1000

Frequency (MHz) 100

10 0 2 4 6 8 10 12 14 16 18 20 Side of Outer Square (cm) #2 NEC4 #3 NEC4 #4 NEC4 #2 Formulation #3 Formulation #4Formulation

Fig. 6.13 Comparison of resonant frequencies calculated using the formulation with NEC simulations for

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different iterations of the geometry. The side of the circumscribed square is varied.

400

350

300

250

200 Resonant Requency Resonant

150

100 0 0.2 0.4 0.6 0.8 1 wire radius (mm)

# 2 # 3 # 4 #2 NEC2 # 3 NEC2 #4 NEC2

Fig. 6.14 Comparison of resonant frequencies calculated using the formulation with NEC simulations for different iterations of the geometry. The wire diameter is varied.

6.6.2 Model with Dielectric Loading Included

The previous approach is extended here for modeling antennas including the dielectric loading effect. It may be recalled that the expression for the resonant frequency of the antenna derived in Section 6.6.1 is valid only for free-standing antennas. The approach we introduce to derive the condition for the resonant properties of Hilbert curve antennas printed on a dielectric substrate, is to consider sections of the strip as terminated parallel strip transmission lines. The characteristic impedance of two parallel strips of negligible thickness t printed on a dielectric of height h, and dielectric constant εr, as shown in Fig. 6.15 in terms of complete elliptic integral of the first kind (K) is given by [72]:

n K(k) Z = 0 (6.19) D ′ ε eff K(k )

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b

a t

h

Fig. 6.15 Coplanar strip transmission line – physical parameters

and the effective dielectric constant

ε r −1 K(k′) K(k1 ) ε eff = 1+ (6.20) 2 K(k) K(k1′)

where

a k = (6.21) b

π a  sinh  4 h k =   (6.22) 1 π b  sinh   4 h 

k′ = 1− k 2 (6.23)

2 k1′ = 1− k1 (6.24)

The ratio of elliptic integrals is given by an approximate formula [72]:

K(k) 1  1+ k + 4 4k  1 K(k) = ln2  for ≤ k ≤ 1 and1 ≤ ≤ ∞ (6.25)  4  K(k′) 2π  1+ k − 4k  2 K(k′)

K(k) 2π 1 K(k) = for 0 ≤ k ≤ and 0 ≤ ≤ 1 (6.26) K(k′)  1+ k + 4 4k  2 K(k′) ln2   4   1+ k − 4k 

The εeff computed above represents the dielectric constant of an equivalent (infinite) medium in which the conductor is embedded, to result in the same transmission line properties as the line with dielectric of height h and dielectric constant εr. It should

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however be noted that there is no ground plane in the configuration currently studied.

This quantity (εeff) is therefore also useful in determining the propagation characteristics of the transmission line. i.e.,

β β m = (6.27) ε eff

Applying these modifications into eq. (6.17), we get the condition for the resonance of the Hilbert curve antenna printed on a dielectric substrate as:

Z D µ0  8s  µ0 kλ  8 kλ  m tan β m d + slog −1 = log −1 (6.28) ω π  b  π 4  b 4 

Changes in the capacitance of the equivalent dipole antenna, due to the presence of the dielectric medium is not considered in the present study. To incorporate this effect the approach followed for modeling parasitics in spiral inductors may be used [73]. But in general these effects are studied when a ground plane is present below the substrate. In the current antenna configuration there is no such ground plane present.

The resonant frequencies obtained through the above formulation are compared with experiments. A table of comparison is given below (Table 6-2). These antennas are fabricated on a Duroid RO 3003 board, with 1.5 mm line segments. These results show a reasonable match between the two, and hence it is concluded that the above formulation may be used as an empirical design equation for antennas of this type.

Table 6-2 Comparison of formulation with experimental results for Hilbert curve dipole antennas printed on dielectric substrate

Resonant Frequencies Iteration 3 Iteration 4

fr1 fr2 fr1 fr2 fr3 Computed 319.6 953.5 193.4 580 966 Measured 316 864 230 635 815 % Difference 1.131 9.39 -18.92 -9.48 15.63

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6.7 RECONFIGURABLE ANTENNAS

Apart from being smaller, another important advantage of the Hilbert curve antenna is the large number of self-similar segments. This paves the way for incorporating RF switches in series with the meandering length of the antenna. This approach results in frequency tuning characteristics. As the antenna is exposed to significant changes in environmental factors its frequency characteristics could alter considerably, but this can be brought back to the original values by controlling these switches [74], [75]. In the current study it is however assumed that the switches perform ideally. When turned ON, the switch is assumed to produce zero insertion loss, and when turned OFF, it produces infinite isolation. We have chosen an antenna with outer dimensions of 10.5 cm having its second resonant frequency at 620 MHz. The feed location for best impedance match with a 50Ω transmission line is shown in Fig. 6.16. The VSWR for this antenna is shown in Fig. 6.17 (Case1 ). Case 2 correspond to the situation when switch 1 is turned off. Similarly, case 3 is for switch 2 to be turned off. RF switches used to realize these antennas in practice can be either based on either PIN diodes, or micro electromechanical systems (MEMS). The latter would be particularly useful at higher microwave frequencies. This approach is significant in imparting frequency agility to antenna characteristics.

The Hilbert curve fractal antenna offers yet another adaptive capability, this time in its radiation characteristics. In this second approach, shown in Fig. 6.19, switches are connected along with additional segments between existing arms of the antenna. In this case, the frequency characteristics of the antenna are not significantly altered by the action of the switches. However, as shown in Fig. 6.19, a wide range of radiation characteristics can be obtained by just 2 additional components. The additional short circuit segments are added to this selectively for the present simulation studies of the reconfigurable antenna. The antenna is placed on the xy plane. The change in the xy (φ) plane is dominant, hence patterns in this plane are plotted in each case. The patterns in other planes are shown in Fig. 6.18 for the unperturbed case for the sake of comparison. The beam directions and beam widths in the φ plane are compared in Table 6-3. Peak directions 1 and 2 are opposite to each other, caused due to the symmetry of the

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geometry. However, gain in these directions vary slightly, because of the asymmetry in the location of the feed. The beam width shown in the last column is for the highest peak in the radiation pattern of the antenna. These indicate that by the addition or removal of just two line segments, the antenna radiation characteristics can be changed significantly. This approach is significant when the antenna is exposed to an interfering environment, and the system would want to adaptively change its radiation characteristics, without resorting to extensive signal processing.

In the simulation results presented here for both reconfigurable antenna types, the effects of the switch non-idealities on the antenna performance are not considered. Hence any available RF switches, depending on the specific frequencies of interest can be used in the final design. At high frequencies however, MEMS based switches are expected to give better performance.

Feed position

Switch 2 position

Switch 1 position

Fig. 6.16 Goemetry for reconfigurable antenna

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9

7

VSWR 5

3

1 580 600 620 640 660 680 Frequency in MHz

Case 1 Case 2 Case 3 Fig. 6.17 VSWR of the reconfigurable antenna. Antenna for Case 1 is shown in Fig. 6.16. Case 2 and Case 3 are for antenna performance when switches 1 and 2 are broken respectively.

Fig. 6.18 The radiation pattern (θ-plane) of the base Hilbert curve fractal antenna used in this study. The plots shown are at φ=0° and φ=90°. Only a half is shown because of apparent symmetry.

Table 6-3. Characteristics of the radiated beam of the reconfigurable antenna.

Case Study Peak Dir. 1 Gain Peak Dir. 2 Gain ± 3dB BW a 0 1.56 177 1.81 83 b 18 1.28 193 0.95 107 c 19 1.55 195 1.33 100 d 63 1.74 254 2.35 92

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(a)

(b)

(c)

(d)

Fig. 6.19 Radiation patterns as of antenna on its plane, as the additional segments are attached. The additional segments in the geometry are shown with thicker lines for clarity. These can be realized in practice by turning ON or OFF switches connected in the additional arms.

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6.8 PATCH ANTENNA WITH HILBERT CURVE GEOMETRY

The use Hilbert curve geometry as a meandered patch antenna is also explored. Significant advantage of this configuration is that this leads to a dual band conformal antenna. It may be contended that the bends and corners of the geometry would add to the radiation efficiency of the antenna thereby improving its gain. The antenna configuration studied for the purpose is shown in Fig. 6.20. The feed connection is similar to microstrip probe feed and is located somewhat away from the point of symmetry of the geometry. The measured S11 (Fig. 6.21) and S21 (Fig. 6.22) of the antenna indicate its multiband characteristics. The radiation patterns of the antenna is plotted in two orthogonal planes at 2.45 GHz in Fig. 6.23. It may be mentioned that the operational bands of this antenna configuration is dependent largely on the outer dimensions, rather than the fractal properties of the inscribed geometry.

Fig. 6.20 Patch antenna with Hilbert curve geopmetry

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0

-5

-10

-15

-20

-25 Return Loss (S11) in dB -30

-35

-40 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz)

Fig. 6.21 Measured S11 of the antenna

0

-5

-10

-15

-20

-25 Measured S21 (dB)

-30

-35

-40 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Frequency (GHz)

Fig. 6.22 Measured S21 of the antenna after calibrating with standard gain antenna. The gain of the standard antenna varies from 8 to 11 dBi within this band. Peak gain of this antenna at 2.4 GHz is 6.5 dBi, 2.8 GHz: 4 dBi, and 3.6 GHz: 4.5 dBi.

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10

0

-10

-20

-30

-40

-50 0 30 60 90 120 150 180 210 240 270 300 330 360 2.4 Ghz-Cross Pol 2.45 GHz-Cross Pol 2.5 GHz-Cross Pol 2.4 GHz-Co Pol 2.45 GHz-Co Pol 2.5 GHz-Co Pol

0

-5

-10

-15

-20

-25

-30

-35

-40

-45

-50 0 30 60 90 120 150 180 210 240 270 300 330 360

2.4 GHz-Cross Pol 2.45 GHz-Cross Pol 2.5 GHz-Cross Pol 2.4 GHz-Co Pol 2.45 GHz-Co Pol 2.5 GHz-Co Pol

Fig. 6.23Radiation patterns of the patch antenna with Hilbert curve geometry in two orthogonal planes.

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6.9 DOUBLY FRACTAL HILBERT-KOCH ANTENNA GEOMETRY

The primary advantage of using Hilbert curves in antennas is the reduction in the overall size of a resonant antenna. This can be extended further by superimposing Koch curves on to each line segment of the Hilbert curve geometry. It is obvious that this approach would increase the overall curve-length. Few example geometries are shown in Fig. 6.24.

(a) Basic Hilbert curve Geometry (b) 1st Iterated Koch Superimposed

(c) 2nd Iterated Koch Superimposed (d) 3rd Iterated Koch Superimposed

Fig. 6.24 Examples of hybrid fractal geometry with Koch curves superimposed on Hilbert curves. For dipole antennas using these, the feed is located at the point of symmetry on the curve.

The curve-length for different iterations of Hilbert and Koch curves are shown in Table 6-4. As shown in Fig. 6.25, as the curve length increases, the primary resonant frequency of the corresponding antenna is decreased. The primary resonant frequencies of these antennas are listed in Table 6-5. The higher order resonances of the antenna with

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reference to fractal iterations of both geometries are also explored. These are tabulated in Table 6-6. Although this approach is useful in reducing the resonant frequency of the antenna, secondary effects such as coupling between adjacent elements are expected to cause a reduction in the magnitude of this reduction in resonant frequency. Such reduction is of limited usefulness, as the feature size of individual segments (in wavelengths) gets smaller as the iteration is increased. It may however be mentioned that this approach of reducing the resonant frequency of the antenna affects primarily the imaginary part of its input impedance. The input resistance remains low for such antennas. However the approach discussed previously, i.e., locating the feed away from the point of symmetry of the geometry, is useful to a certain extent in overcoming this difficulty.

Table 6-4 Physical length in meters for the hybrid fractal geometry with Koch curve superimposed on line segments of fractal Hilbert curve. The first row of data corresponds to the basic Hilbert curve itself.

Iterations of Length (m) for Various Iterations of the Hilbert Curve the Koch Curve 1 2 3 4 5 0 0.3 0.5 0.9 1.7 3.3 1 0.4 0.667 1.2 2.267 4.4 2 0.533 0.889 1.6 3.022 5.867 3 0.711 1.185 2.133 4.029 7.822 4 0.948 1.58 2.844 5.373 10.429 5 1.264 2.107 3.793 7.164 13.906

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(a)

2000

1500

1000

500

0

-500 Input Reactance

-1000

-1500

-2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (MHz)

H-K-3-0 H-K-3-1 H-K-3-2 (b)

2000

1800

1600

1400

1200

1000

800

Input Resistance 600

400

200

0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (MHz)

H-K-3-0-Re H-K-3-1 H-K-3-2

Fig. 6.25 Input impedance simulated using NEC for different iterations of Koch curves superimposed on 3rd iteration Hilbert curve.

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Table 6-5 The primary resonant frequency of the antenna determined by numerical simulations with hybrid fractal geometry consisting of different iterations of Hilbert curve and Koch curve.

Iterations Resonant Frequency (MHz) for Various Iteration of the Hilbert Curve of the Koch Curve 1 2 3 4 0 501.7 331.6 216.8 146.1 1 415.7 285.0 191.9 131.5 2 357.4 255.1 176.1 - 3 320.4 237.2 - -

Table 6-6 Multi-band characteristics of the hybrid fractal antenna consisting of 3rd iterated Hilbert curve with different Koch curves superimposed. The ratios of successive frequencies are shown in brackets. The first row of data corresponds simple Hilbert curve geometry.

Iterations Resonant Frequencies (MHz) of the Koch Curve f1 f2 (f2/f1) f3 (f3/f2) f4 (f4/f3) f5(f5/f4) 0 216.9 606.9 (2.798) 932.8 (1.537) 1309.9 (1.404) 1646.1 (1.256) 1 191.9 537.3 (2.799) 819.7 (1.526) 1139.4 (1.390) 1443.0 (1.266) 2 176.1 494.4 (2.807) 751.1 (1.519) 1037.5 (1.381) 1322.2 (1.274)

Another point worth mentioning in this context is that these triangular Koch curves are taken as an indicative example only. Any other geometry may be superimposed in a similar approach. It may be noted that if one were to use other variants of Koch curves such as the one used in [63] (with rectangular segments), the analytical expressions derived in Section 6.6 may be easily extended for obtaining the resonance condition.

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6.10 SUMMARY

In this Chapter the development of antennas using a previously unused fractal geometry is presented. The use of this geometry reduces the size of a resonant antenna beyond achievable with other fractal geometries such as Koch curves. Two novel improvements in antenna characteristics, such as reconfigurable radiation characteristics and a means of further reducing antenna size using a doubly fractal geometry are also developed. Some of the numerical results are validated through experiments. The numerical results presented here indicate that further reduction in resonant frequency is possible with superimposing Koch curve geometries on to segments of Hilbert curves. A patch configuration is also explored. However, the antenna characteristics in this configuration are found to be dictated more by the outer dimensions, than fractal properties of the geometry.

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CHAPTER 7 CONCLUSIONS AND FUTURE WORK

Although relatively new to mathematicians, fractals found widespread use in many branches of science and engineering in a relatively short time. Electromagnetics, and in particular antenna design has also benefited from these novel concepts. Fractal shaped antenna elements are now commercially available for use in several modern telecommunications equipment. However antenna properties are not suitably linked to fractal characteristics hitherto. This research work has addressed this issue, numerically as well as by experimental studies, by identifying features of geometries that influence characteristics of fractals shaped antennas, thereby imparting increased flexibility in the design of newer generation wireless systems. The conclusions derived from this research work are listed below. These are listed in the order they appear in this dissertation, not necessarily in any order of significance.

7.1 CONCLUSIONS

¾ Monopole antennas with Sierpinksi gaskets do not show significant variation in their multiband characteristics when fractal iteration is increased. Although the flare (apex) angle is often changed to vary multiband characteristics of this antenna, this effect is not absent even in simple triangular geometry. This contradicts the notion that self-similarity of the geometry causes multiband characteristics of the antenna. It is however observed that multiband characteristics are rather changed by making the geometry not strictly self-similar.

¾ Wideband and conformal antennas are obtained by starting with Sierpinski monopole antennas. This modification requires a ferroelectric (dielectric constant >50) substrate and an absorber layer. Antennas with octave bandwidth are obtained with this approach.

¾ The fractal dimension of Koch curves can be varied by changing its indentation angle. The resultant geometry retains all fractal properties, including self-similarity. Since the unfolded length of the curve changes by this modification, antenna characteristics

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are affected by it. This has offered a unique opportunity to explore the correlation between antenna characteristics and properties of fractals such as fractal dimension. It has been found that there exists a direct relationship between fractal dimension and antenna properties such as primary resonant frequency, input resistance at resonance, and multiband characteristics. Fractal iteration is independent of dimension, and hence does not affect multiband characteristics of the antenna. Some of these results have since been validated experimentally. This is one of the significant outcomes of this research work.

¾ The use of an arbitrary indentation angle in each iteration stage of fractal Koch curve can impart flexibility in design of multiband antennas. However this approach results in non-recursive geometries. One interesting observation in this study is that although several generic geometries with the same unfolded length and end-to-end length are possible, the multiband characteristics of the antennas using them are different.

¾ Since the inter-dependence of fractal dimension on antenna characteristics can be obtained only for a specific geometry, this relation does not hold good for other fractal geometries, since this could not be validated against other known fractal shaped antennas. However, a similar trend is observed in monopole antennas using fractal binary tree structures. The appearance of these trees can be varied either by changing their branching angles, or by using different scale factors between the stem and branches of tree in each iteration. The fractal iteration does not change by the first modification, but is affected by the change in scale factor. Numerical studies presented here also show that the multiband characteristics of this antenna are affected only when the fractal dimension is changed. This confirms previous findings based on other geometry.

¾ One of the advantages of fractal geometries is in reducing antenna size. Hilbert curves being space filling geometries and having dimension approaching 2, have been shown to reduce antenna size significantly compared to other geometries including previously studied fractals.

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¾ An arbitrary geometry can potentially reduce the antenna size, possibly better than achievable with fractal geometries such as Hilbert curves. However the advantage of fractals over such random approaches is the mathematical order associated with fractals. Fractal properties such as iteration order have been used for the first time in obtaining design equations of dipole antennas using them.

¾ Patch antennas with multiband characteristics is also possible with Hilbert curve geometry. However it has been observed that this antenna configuration is affected more by the outer dimensions of the geometry rather than by its unfolded length of the geometry. Hence for the patch configuration, the antenna size reduction feature of fractals is questionable.

¾ Although larger iterations of Hilbert curves result in significant increase in unfolded curve length, a corresponding improvement is not observed in reducing the resonant frequency of the antenna. The same asymptotic nature is obtained when one fractal geometry (Koch curve) is superimposed on to each line segment of a Hilbert curve geometry. It is believed that this is because the feature size of constituent line segments get very small (in wavelengths).

¾ Fractal geometries such as Hilbert curves have several self-similar groups of closely spaced line segments. By including additional line segments operated with RF switches novel features such as beam reconfigurability has been obtained. Change in beam width as well as peak direction can be achieved by few additional interconnects.

7.2 FUTURE DIRECTIONS

Wideband antenna using Sierpinski gasket has been shown to have octave bandwidth. However currently this is limited to microwave frequencies only. If this can be reduced to VHF/UHF bands, a large number of potential applications arise. However this requires more flexibility in choosing material systems. Often the improvement in bandwidth is obtained at the expense of antenna efficiency, calling for a compromise between the two.

The reduction in antenna size obtained with highly iterated fractal geometries are often associated with low input resistance. Although this can be improved by relocating the

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antenna feed suitably, the radiation efficiency of the antenna is not improved. Hence for this approach to be practical, this may require the use of significantly different fractal geometries.

It would be of interest to have an integrated design approach whereby, given the specifications of antenna performance, a suitable geometry and its dimensions are determined. However this synthesis would require obtaining design algorithms for several fractal geometries.

In this work, the fractal dimension is identified as a key feature influencing antenna characteristics. However this may not be the only one. The quest for other fractal features should give new impetus to the understanding of the behavior of these antennas.

One important advantage of fractal geometries is their ordered nature. However this has not been used in antenna analysis. Several extensive calculations can possibly be simplified by making use of fractal nature of the geometry into the formulation.

The reconfigurable antenna presented here is designed by an intuitive approach. Extensive design optimization is required if one were to synthesize an antenna configuration for a specified radiation characteristics. Development of such optimization algorithms would help popularize this approach.

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APPENDIX

Examples of G-NEC input cards for analyzing various fractal geometries (a) Hilbert curve CM Hilbert curve dipole CM wire radius : 1.3mm CE GW 1 6 0.000000 0.000000 0.000000 0.000000 0.017500 0.000000 0.001300 GW 2 6 0.000000 0.017500 0.000000 0.017500 0.017500 0.000000 0.001300 GW 3 6 0.017500 0.017500 0.000000 0.017500 0.000000 0.000000 0.001300 GW 4 6 0.017500 0.000000 0.000000 0.035000 0.000000 0.000000 0.001300 GW 5 6 0.035000 0.000000 0.000000 0.052500 0.000000 0.000000 0.001300 GW 6 6 0.052500 0.000000 0.000000 0.052500 0.017500 0.000000 0.001300 GW 7 6 0.052500 0.017500 0.000000 0.035000 0.017500 0.000000 0.001300 GW 8 6 0.035000 0.017500 0.000000 0.035000 0.035000 0.000000 0.001300 GW 9 6 0.035000 0.035000 0.000000 0.052500 0.035000 0.000000 0.001300 GW 10 6 0.052500 0.035000 0.000000 0.052500 0.052500 0.000000 0.001300 GW 11 6 0.052500 0.052500 0.000000 0.035000 0.052500 0.000000 0.001300 GW 12 6 0.035000 0.052500 0.000000 0.017500 0.052500 0.000000 0.001300 GW 13 6 0.017500 0.052500 0.000000 0.017500 0.035000 0.000000 0.001300 GW 14 6 0.017500 0.035000 0.000000 0.000000 0.035000 0.000000 0.001300 GW 15 6 0.000000 0.035000 0.000000 0.000000 0.052500 0.000000 0.001300 GS 0 0 1 GE 0, FR 0,1,0,0,710,20 EX 0,8,3,0,1.0,0.0, RP 0 181 1 1000 -90 0 1 1 RP 0 181 1 1000 -90 90 1 1 EN (b) Koch curve CM Koch dipole CM iteration: 3 CE GW 1 8 -0.1 0 0 -0.0962963 0 0 0.0001 GW 2 8 -0.0962963 0 0 -0.0944444 -0.0032075 0 0.0001 GW 3 8 -0.0944444 -0.0032075 0 -0.0925926 0 0 0.0001 GW 4 8 -0.0925926 0 0 -0.0888889 0 0 0.0001 GW 5 8 -0.0888889 0 0 -0.087037 -0.0032075 0 0.0001 GW 6 8 -0.087037 -0.0032075 0 -0.0888889 -0.006415 0 0.0001 GW 7 8 -0.0888889 -0.006415 0 -0.0851852 -0.006415 0 0.0001 GW 8 8 -0.0851852 -0.006415 0 -0.0833333 -0.0096225 0 0.0001 GW 9 8 -0.0833333 -0.0096225 0 -0.0814815 -0.006415 0 0.0001 GW 10 8 -0.0814815 -0.006415 0 -0.0777778 -0.006415 0 0.0001 GW 11 8 -0.0777778 -0.006415 0 -0.0796296 -0.0032075 0 0.0001 GW 12 8 -0.0796296 -0.0032075 0 -0.0777778 -4.33681e-019 0 0.0001 GW 13 8 -0.0777778 -4.33681e-019 0 -0.0740741 -4.33681e-019 0 0.0001 GW 14 8 -0.0740741 -4.33681e-019 0 -0.0722222 -0.0032075 0 0.0001 GW 15 8 -0.0722222 -0.0032075 0 -0.0703704 -4.33681e-019 0 0.0001 GW 16 8 -0.0703704 -4.33681e-019 0 -0.0666667 -4.33681e-019 0 0.0001 GW 17 8 -0.0666667 -4.33681e-019 0 -0.0648148 -0.0032075 0 0.0001

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GW 18 8 -0.0648148 -0.0032075 0 -0.0666667 -0.006415 0 0.0001 GW 19 8 -0.0666667 -0.006415 0 -0.062963 -0.006415 0 0.0001 GW 20 8 -0.062963 -0.006415 0 -0.0611111 -0.0096225 0 0.0001 GW 21 8 -0.0611111 -0.0096225 0 -0.062963 -0.01283 0 0.0001 GW 22 8 -0.062963 -0.01283 0 -0.0666667 -0.01283 0 0.0001 GW 23 8 -0.0666667 -0.01283 0 -0.0648148 -0.0160375 0 0.0001 GW 24 8 -0.0648148 -0.0160375 0 -0.0666667 -0.019245 0 0.0001 GW 25 8 -0.0666667 -0.019245 0 -0.062963 -0.019245 0 0.0001 GW 26 8 -0.062963 -0.019245 0 -0.0611111 -0.0224525 0 0.0001 GW 27 8 -0.0611111 -0.0224525 0 -0.0592593 -0.019245 0 0.0001 GW 28 8 -0.0592593 -0.019245 0 -0.0555556 -0.019245 0 0.0001 GW 29 8 -0.0555556 -0.019245 0 -0.0537037 -0.0224525 0 0.0001 GW 30 8 -0.0537037 -0.0224525 0 -0.0555556 -0.02566 0 0.0001 GW 31 8 -0.0555556 -0.02566 0 -0.0518519 -0.02566 0 0.0001 GW 32 8 -0.0518519 -0.02566 0 -0.05 -0.0288675 0 0.0001 GW 33 8 -0.05 -0.0288675 0 -0.0481481 -0.02566 0 0.0001 GW 34 8 -0.0481481 -0.02566 0 -0.0444444 -0.02566 0 0.0001 GW 35 8 -0.0444444 -0.02566 0 -0.0462963 -0.0224525 0 0.0001 GW 36 8 -0.0462963 -0.0224525 0 -0.0444444 -0.019245 0 0.0001 GW 37 8 -0.0444444 -0.019245 0 -0.0407407 -0.019245 0 0.0001 GW 38 8 -0.0407407 -0.019245 0 -0.0388889 -0.0224525 0 0.0001 GW 39 8 -0.0388889 -0.0224525 0 -0.037037 -0.019245 0 0.0001 GW 40 8 -0.037037 -0.019245 0 -0.0333333 -0.019245 0 0.0001 GW 41 8 -0.0333333 -0.019245 0 -0.0351852 -0.0160375 0 0.0001 GW 42 8 -0.0351852 -0.0160375 0 -0.0333333 -0.01283 0 0.0001 GW 43 8 -0.0333333 -0.01283 0 -0.037037 -0.01283 0 0.0001 GW 44 8 -0.037037 -0.01283 0 -0.0388889 -0.0096225 0 0.0001 GW 45 8 -0.0388889 -0.0096225 0 -0.037037 -0.006415 0 0.0001 GW 46 8 -0.037037 -0.006415 0 -0.0333333 -0.006415 0 0.0001 GW 47 8 -0.0333333 -0.006415 0 -0.0351852 -0.0032075 0 0.0001 GW 48 8 -0.0351852 -0.0032075 0 -0.0333333 2.1684e-018 0 0.0001 GW 49 8 -0.0333333 2.1684e-018 0 -0.0296296 5.2363e-019 0 0.0001 GW 50 8 -0.0296296 5.2363e-019 0 -0.0277778 -0.0032075 0 0.0001 GW 51 8 -0.0277778 -0.0032075 0 -0.0259259 -1.30104e-018 0 0.0001 GW 52 8 -0.0259259 -1.30104e-018 0 -0.0222222 -2.94582e-018 0 0.0001 GW 53 8 -0.0222222 -2.94582e-018 0 -0.0203704 -0.0032075 0 0.0001 GW 54 8 -0.0203704 -0.0032075 0 -0.0222222 -0.006415 0 0.0001 GW 55 8 -0.0222222 -0.006415 0 -0.0185185 -0.006415 0 0.0001 GW 56 8 -0.0185185 -0.006415 0 -0.0166667 -0.0096225 0 0.0001 GW 57 8 -0.0166667 -0.0096225 0 -0.0148148 -0.006415 0 0.0001 GW 58 8 -0.0148148 -0.006415 0 -0.0111111 -0.006415 0 0.0001 GW 59 8 -0.0111111 -0.006415 0 -0.012963 -0.0032075 0 0.0001 GW 60 8 -0.012963 -0.0032075 0 -0.0111111 -8.23994e-018 0 0.0001 GW 61 8 -0.0111111 -8.23994e-018 0 -0.00740741 -9.88471e-018 0 0.0001 GW 62 8 -0.00740741 -9.88471e-018 0 -0.00555556 -0.0032075 0 0.0001 GW 63 8 -0.00555556 -0.0032075 0 -0.00370371 -1.17094e-017 0 0.0001 GW 64 8 -0.00370371 -1.17094e-017 0 -1.76951e-009 -1.33542e-017 0 0.0001 GX 64 100 GS 0 0 1 GE 0 EK PT -1 EX 0 64 8 0 1 0 FR 0 11 0 0 400 10 XQ 0 EN 153

(c) Fractal tree CM binary tree monopole: cm length, level, branch angles, scale ratio CM 0.1 4 60 0.5 CE GW 1 63 0 0.0 0 0 0.0 0.0516129 1e-04 GW 2 51 0 0.0 0.0516129 0.022349 0.0 0.0645161 1e-04 GW 3 25 0.022349 0.0 0.0645161 0.0335236 0.0 0.0580645 1e-04 GW 4 12 0.0335236 0.0 0.0580645 0.0335236 0.0 0.0516129 1e-04 GW 5 6 0.0335236 0.0 0.0516129 0.0307299 0.0 0.05 1e-04 GW 6 6 0.0335236 0.0 0.0516129 0.0363172 0.0 0.05 1e-04 GW 7 12 0.0335236 0.0 0.0580645 0.0391108 0.0 0.0612903 1e-04 GW 8 6 0.0391108 0.0 0.0612903 0.0419045 0.0 0.0596774 1e-04 GW 9 6 0.0391108 0.0 0.0612903 0.0391108 0.0 0.0645161 1e-04 GW 10 25 0.022349 0.0 0.0645161 0.022349 0.0 0.0774194 1e-04 GW 11 12 0.022349 0.0 0.0774194 0.0279363 0.0 0.0806452 1e-04 GW 12 6 0.0279363 0.0 0.0806452 0.0307299 0.0 0.0790323 1e-04 GW 13 6 0.0279363 0.0 0.0806452 0.0279363 0.0 0.083871 1e-04 GW 14 12 0.022349 0.0 0.0774194 0.0167618 0.0 0.0806452 1e-04 GW 15 6 0.0167618 0.0 0.0806452 0.0167618 0.0 0.083871 1e-04 GW 16 6 0.0167618 0.0 0.0806452 0.0139682 0.0 0.0790323 1e-04 GW 17 51 0 0.0 0.0516129 -0.022349 0.0 0.0645161 1e-04 GW 18 25 -0.022349 0.0 0.0645161 -0.022349 0.0 0.0774194 1e-04 GW 19 12 -0.022349 0.0 0.0774194 -0.0167618 0.0 0.0806452 1e-04 GW 20 6 -0.0167618 0.0 0.0806452 -0.0139682 0.0 0.0790323 1e-04 GW 21 6 -0.0167618 0.0 0.0806452 -0.0167618 0.0 0.083871 1e-04 GW 22 12 -0.022349 0.0 0.0774194 -0.0279363 0.0 0.0806452 1e-04 GW 23 6 -0.0279363 0.0 0.0806452 -0.0279363 0.0 0.083871 1e-04 GW 24 6 -0.0279363 0.0 0.0806452 -0.0307299 0.0 0.0790323 1e-04 GW 25 25 -0.022349 0.0 0.0645161 -0.0335236 0.0 0.0580645 1e-04 GW 26 12 -0.0335236 0.0 0.0580645 -0.0391108 0.0 0.0612903 1e-04 GW 27 6 -0.0391108 0.0 0.0612903 -0.0391108 0.0 0.0645161 1e-04 GW 28 6 -0.0391108 0.0 0.0612903 -0.0419045 0.0 0.0596774 1e-04 GW 29 12 -0.0335236 0.0 0.0580645 -0.0335236 0.0 0.0516129 1e-04 GW 30 6 -0.0335236 0.0 0.0516129 -0.0363172 0.0 0.05 1e-04 GW 31 6 -0.0335236 0.0 0.0516129 -0.0307299 0.0 0.05 1e-04 GS 0 0 1 GE 1 -1 0 GN 1 EK EX 0 1 1 0 1 0 FR 0 31 0 0 300 100 RP 0 181 1 1000 -90 0 1 1 RP 0 181 1 1000 -90 90 1 1 EN

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VITA

K.J. Vinoy was born in Ernakulam, India on the 16th of January 1969. He received his B.Tech. degree in Applied Electronics and Instrumentation from University of Kerala, India and M.Tech. degree in Electronics from Cochin University of Science and Technology, India in 1990 and 1993 respectively. In 1994 he joined National Aerospace Laboratories, Bangalore, India and has worked on Computational Electromagnetics till 1998. He joined the Department of Engineering Science and Mechanics, the Pennsylvania State University in January 1999 for doctoral studies. He has been a research assistant at the Center for the Engineering of Electronic and Acoustic Materials and Devices ever since. His research interests include fractal shaped antennas, wave propagation, wave-material interaction, and RF-MEMS for microwave applications.