A NOVEL APPROACH FOR DESIGNING FRACTAL ANTENNAS
David C. Ni
Direxion Technology, 9F, 177-1 Ho-Ping East Road Section 1, Taipei, Taiwan, R.O.C.
Chou Hsin Chin
Department of Electro-Physics, National Chiao Tung University, Hsin Chu, Taiwan, R.O.C.
Keywords: Antenna, Fractal, Herman Ring.
Abstract: Designing fractal antennas for broadband communications, various modifications and optimisations to the
fractal patterns, such as Koch or Sierpinski, are adopted for better frequency responses. In this article, we
explore a new approach by defining a set of complex functions, z
q
ΠC
i
, where C
i
=exp(i*g
i
(z))[(z-v
i
)/(1-v
i
z)],
and then using the shapes of the solution domains of the functions directly as patterns of fractal antennas.
1 INTRODUCTION
Design of fractal antennas is currently targeted for
highly desirable characteristics such as compact size,
low profile, conformal, multi-band and broadband,
as described in (Cohen,1995), (
Gianvittorio, 2002), and
(Werner). Most of the designers adopt operations
such as translation, rotation, iterations, etc. on the
fractal generator motifs, such as Koch, Minkoski,
Cantor, Torn Square, Mandelbrot, Caley Tree,
Monkey’s Swing, Sierpinski Gasket, Julia etc. for
the creation of the self-similar shapes. To further
improve the frequency responses, they applied
modifications on the created shapes, such as in
(Puente, 2000). Recently, new approaches, such as
Generic Algorithm, are studied for handling antenna
optimisation on multi-dimensional parameters
(
Altshuler, 2002). However, these approaches do not
provide the initial conditions, namely, the original
shapes for optimisation.
In this article, we explore a novel approach based on
the methodologies used in the area of dynamic
systems in conjunction with fractal geometry as
described in (Mandelbrot, 1977), (Milnor, 2000),
and
(Falconer, 1990). We define a set of complex
functions, z
q
ΠC
i
, based on relationship of moving
and observing entities. By solving the functions for a
given domain on the complex plane, we obtain a
solution domain based on the criteria of function
convergence (Ni, 2006). Then we extract the internal
and external contours of the shapes of the solution
domains and directly use them as topologies of
antenna or antenna arrays.
Of our particular interest, we adopt fractal shapes
known as Herman Rings for the fractal antennas.
Herman Rings are characterized by fractal internal
and external contours. We observe the broadband
characteristics from these antennas.
2 FUNCTIONS
We define the function set, f = z
q
ΠC
i,
which may
have the following forms:
where z is a complex variable, q is an integer, and C
i
has following form:
here v
i
is the complex conjugate of v
i
. We propose
this form based on the following form known in the
theory of special relativity by A. Einstein:
The z
q
term in Equation (1) has implication of time
and is used to ensure that the function may converge.
The term
f = z
q
C
1
C
2
f = z
q
C
1
C
2
C
3
and so on (1)
C
i
=ex
p(
i*
g
i
(
z
))[(
z-v
i
)
/
(
1-v
i
z
)]
(
2
)
1/(1 – v
2
/c
2
)
1/2
(3)
exp(i*g
i
(z)) (4)
157
C. Ni D. and Hsin Chin C. (2007).
A NOVEL APPROACH FOR DESIGNING FRACTAL ANTENNAS.
In Proceedings of the Second International Conference on Wireless Information Networks and Systems, pages 157-160
DOI: 10.5220/0002148701570160
Copyright
c
SciTePress
In Equation (2) represents the phase, where g
i
(z) is a
complex function. A given domain can be a domain
of complex numbers, x+yi, with (x
2
+y
2
)
1/2
1
The function f will go through iteration as:
Here, n is a positive integer indicating the order of
iteration.
3 DOMAINS
The domains shown in the following figures are the
normalized solution domains. Figure 1a and 1b show
the impact of iteration number and phase on the
solution domains.
Figure 1 a: Bow-tie like Fractal Domain.
Figure 1 b: Bow-tie like Fractal Domain.
These two domains show asymmetrical features.
Figure 2 shows a nebular-like domain, which may be
seen as an antenna array. Figure 3 shows a Spiral-
like fractal domain, which has potentially broadband
radiation characteristics.
Figure 2: Nebular-like Fractal Domain.
Figure 3: Spiral-like Fractal Domain.
Other solution domains look like some natural
objects, such as shells (Figure 4). This domain
shows external and internal structures or contours
varying with iteration orders and locations. In short,
the proposed functions have richness of solution
domains, which may look like the objects in the
nature.
Figure 4:Shell-like Fractal Domain.
f
n
(z) = f f
n-1
(5)
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158
4 ANTENNAS
We adopted commercially available tools and
platforms, such as Matlab and Zeland (IE3D), for
domain creation, contour capture, and antenna
performance analysis. Figure 5 shows a set of 772
data points in the Nebular-like fractal domain (Fig.2)
imported to a Matlab GUI, which treats each data
point as antenna, and plots the radiation pattern. The
fractal domain is shown at center of the plotted
pattern.
Figure 5: Radiation Pattern of an Antenna array.
To demonstrate the use of the solution domains for
broadband fractal antennas, we select the domain as
in Figure 6 for building a patch antenna on 1.6 mm
thick FR4 substrate. The ground plane size of FR4
sample is about 4cm x 4cm.
Figure 6: Spiral-like Fractal Domain.
We fed the signals at the centre of this Spiral-like
antenna and observed the activated modes as shown
in Figure 7. Matched modes were observed down to
1 GHz range close to the simulated results. It was
noticed that high harmonic modes were suppressed
from these samples. Potentially, we can construct a
broadband antenna with bandpass characteristics
based on the observations.
Figure 7: Measurement on S
11
of Spiral-like Antenna.
By optimising the ground plane, we are able to
obtain –10 dB S
11
floor from 3 GHz to 10 GHz for
these fractal antennas (Fig. 8).
Figure 8: Simulated S
11
of Spiral-like Antenna.
We also observed multi-mode radiation patterns
through the frequency bands although the patterns
are rather stable through the bands. Figure 9 shows
the E-field patterns at 3.6 GHz, 5.4 GHz, and 8.5
GHz. The simulated scalar and vector current
distributions show that the areas of radiation on
fractal antenna are not changed too much over the
observed frequencies.
A NOVEL APPROACH FOR DESIGNING FRACTAL ANTENNAS
159
Figure 9: Simulated Radiation Patterns.
The asymmetrical nature of the fractal domains
shows radiation distribution toward to one direction.
Figure 10 shows this observation when the Spiral-
like antenna is on a square ground plane. The gain is
about two fold of uniform radiation patterns.
Figure 10: Simulated Radiation Patterns.
5 CONCLUSIONS
We have explored a new approach to develop fractal
antennas by computing a set of complex functions,
z
q
ΠC
i
, based on the methodologies in the area of
dynamic systems. The topological solution domains
are directly used as antenna elements or arrays. By
selecting the desirable domains, we can potentially
build broadband antennas with characteristics of
frequency filtering. This approach provides detailed
patterns initially for fast optimisation.
ACKNOWLEDGEMENTS
We appreciate Professor Hsue of National Taiwan
University of Science and Technology for his
advices and helps on the measurements of the
sample antennas.
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