FRACTAL SHAPED ANTENNA ELEMENTS FOR WIDE- AND MULTI- BAND

169 Pages·2002·2.55 MB·English

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 (ε = 9.8) Duroid (ε = 2.2) and FR-4 (ε = 4.4)
r r r
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
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