Design and Analysis of Fractal Antennas
Jaymin J. Modi1, Trushit K. Upadhyaya2, Vaidehi M. Patel3
1
M.Tech. Student of CSPIT, Changa, Gujarat - India. Email: jayminmodi91@gmail.com
Assistant Professor. of CSPIT, Changa, Gujarat - India. Email: trushitupadhyaya.ec@ecchanga.ac.in
3
M.E. Student of SVIT, Vasad, Gujarat - India. Email: patelvaidehi07@gmail.com
2
Abstract: In this work, the design of Sierpinski Model, Koch dipole model and Minkowski fractal geometry, which
are different well-known types of fractal antenna to be used in wireless communications, is presented. For
optimization of antenna's shape and dimensions, we can use properties of fractal shapes at the radiating slots. Here,
analysis and design of these all model’s antenna used for wireless communication systems between 1 to 8 GHz are
presented.
Keyword: Fractal antenna; Sierpinski Model; Koch dipole model; Minkowski fractal geometry.
1.
INTRODUCTION
In recent communication system need small
sized and high performance antennas which can be
operated on different frequency bands. After study of
Fractal geometry it is observed that it can be able to
fulfill these requirements. Mathematically, fractal is a
highly complex and irregular geometry. As its name
implies, it consists of similar-shaped pieces of
different scales, which is very difficult to represent
with a continuous function. Fractal-shaped antenna is
highly dependent upon the geometry. It allows
smaller, multiband and broadband antenna design.
Multiband and miniaturization of a fractal antenna
are possible because of its geometrical characteristics
of self-similarity and space-filling, respectively. Selfsimilarity means that the pattern becomes similar if a
portion is expanded infinitely, while space filling
means that, the effective length increases while
keeping the same area as the number of iteration
increases. The type of fractal antenna is Koch model,
Sierpinski model, Minkowski model and other types.
It has many advantages such as reduced antenna size;
multiband and wideband functionally, improved
antenna performance and better input impedance
matching.
2.
Figure1: Scheme of three stages Sierpinski gasket
antenna
The design of Sierpinski starts with an equilateral
triangle with operating frequency in between 2 GHz
to 5 GHz at various iteration, Four different iterations
of triangular patch are compared in terms of their
radiation pattern, return loss and gain bandwidths.
Usually, it is inverse triangle configuration when
its zero Proceedings stage and dug out a 1/2 side
triangle of 2√3 cm of its length of side of triangle,
when it’s one stage. Figure1 shows a three stages
Sierpinski gasket antenna which is decrease by
degrees. In theory fractal antenna can be infinite
stages, but the stages must be finite in practice. The
material used for this synthesis is FR-epoxy with
permittivity (4.4) and dielectric loss tangent (0.02).
SIERPINSKI MODEL
Fractals have self-similarity in their geometry,
which is a feature where a section of the fractal
appears the same regardless of how many times the
section is zoomed in upon. Self-similarity in the
geometry creates effective antennas of different
scales. This can lead to multiband characteristic
antennas, which is displayed when an antenna
operates with a similar performance at various
frequencies. The generation of the fractal is shown in
Figure 1.
Figure2: Sierpinski triangle
The initial estimation of the fractal geometry
is obtained by,
fn  k
c
cos 2  n
h
 x
W1   
 y
1
2

0

1
 x 2
W2    
 y 0

1
 x  3
W3    
 y  0

1 
4 3  x 
2  y 
3 
1 
1
4 3  x    2 
2  y   0 
3 

0  x   1 
    6
1  y   2 

 3
3
For the iteration 3 the length of side of each
triangle is √3/4 cm, €r = 4.4, the result BW for
VSWR<2 (i.e. 1.0485) is 140 MHz (5.4%) at the
center frequency of 2.6GHz and return loss is 32.5dB.
3.
KOCH DIPOLE MODEL
A Koch curve is generated by replacing the
middle third of each straight section with a bent
section of wire that spans the original third. Each
iterations adds length to the total curve which results
in a total length that is 4/3 the original geometry
In this example, we take h2= 2/3 h1.
We begin by calculating the similarity
factor, δ, from the ratio of the resonant frequencies
desired: f2 = 2.4 GHz and f3 = 5.0 GHz. We obtain δ
= 2.08 ≈ 2, which will allow for a very simple and
symmetric fractal pattern. Each triangular structure of
the Sierpinski is twice as large as its sub-structure.
Since the height of the triangular structure resonating
at f2 is h2 = λ2/2 = 3.05 cm, the height of the
monopole is calculated to be h = 2h2 = 6.1 cm.
The number of iterations needed to generate
the required fractal is η max = 4. It should be noted
that although the complete fractal structure will
resonate at two other frequencies, f1 and f4, they are
simply included in the pattern to provide continuity.
Because of this, it’s f1 ≈ f2/3.5.
Table 1: Design parameter for Sierpinski monopole
Design Parameter
value
Similarity factor δ
2
Height h
6.1cm
Flare angle α
60˚
Max. Iterations η max
4
Figure 4: Geometry of Koch monopole
4
Lkoch  h   
3
n
1

 x'   s 0  x 
W1    
 
 y '   0 1  y 
s

1
1

 x'   s cos   s sin   x   1 
W2    
    s 
 y '   1 sin  1 cos   y   0 
s
s

1
 x'   s cos 
W3    
 y'   1 sin 
 s
1
 x'   s
W4    
 y'   0

1

sin   x   1 2 
s
    1

1
cos   y   s sin  
s


0  x   s  1 
s
 
1  y   0 



s
S = 2(1+ Cos θ)
The similarity dimension is obtained as
D
log 4
log[ 2(1  cos  )]
Figure 3: Reflection coefficient for a 4-iteration
Sierpinski monopole (Z0 =50Ω)
Figure 5: Design of Koch monopole
Where;
(a)
(b)
h, is the monopole antenna’s height,
l, the effective length,
n, its iteration number,
s, scaling factor.
Considering the stated equations for 2.45
GHz, the calculated h and l is 30.6 mm. After
optimization, l is equal to 44 mm, which is slightly
larger compared to proposed conventional design
equation.
Figure 7: The initial generator model for creating
fractal patch antenna. (a) 0 iteration, (b) 1st iteration
1 c 

f  
4  L1  L2 
c = free space velocity of light, 3 × 108 m/s
f = frequency of operation
(L1 + L2) = x; where x = total length
x
c
3  108

 32.6mm
4 f 4  2.3  109
So, (L1 + L2) = 32.6 mm L1≈ L2≈ 16.3mm
Figure 6: S11 and bandwidth with the variation of the
Koch monopole’s substrate length (Ls)
The miniaturization of the fractal antenna is
exhibited by scaling each iterations to be resonant at
the same frequency. The miniaturization of the
antennas shows a greater degree of effectiveness for
the first several iterations. The amount of scaling that
is required for each iterations is reduced as the
number of iterations increase. The total length of the
fractals at resonance is increasing, while the height
reduction is reaching an asymptote. Therefore, it can
be concluded that increased complexity of the higher
iterations are not advantageous. The miniaturization
benefits are achieved in the first several iterations.
4.
After optimization:
L1 = L2 = 24.6 mm
This equation is being used to determine all
necessary dimension of the microstrip patch antenna.
The most important parameters required for the
design of this antenna are the width and length (L1
and L2) and the size to reduce each of the side of the
patch antenna. Figure 8 shows schematic diagram of
the patch monopole antenna on the top and bottom
layer, with modified ground plane to improve the
impedance bandwidth and radiation performances at
high frequency.
MINKOWSKI FRACTAL GEOMETRY
The Minkowski fractal antenna was
designed using Roger RO4003c substrate (εr = 3.38)
with a thickness (h) of 0.813 mm. This substrate was
been chosen as it provides better performance in high
frequency operation. The shape of patch which is
generated the initial generator model at each side of
patch antenna are shown for without iteration (zero),
first, and second iteration.
The width and the length, L1 and L2 of the
patch can then be calculated as:
Figure 8: Schematic diagram of the patch monopole
antenna with a modified Minkowski fractal geometry
and a modified ground plane.
A. First Iteration
The first iteration structure was designed by
reducing each side of the antenna. Figure 8 shows the
generator model for creating fractal start from zero to
first iteration. As mentioned above, the values of both
L1 and L2 are 24.6 mm. hence, the value of LP2 is 8.2
which is one-third from the value of L1 and L2. After
optimization, value of LP2 was found to be 8 mm.
Figure 10: Simulated return losses for various LP3 =
3.45 mm and 3.1 mm.
C. Radiation pattern for different frequency
Figure 9: Simulated return losses for modified ground
plane and full ground plane of Minkowski fractal
antenna.
Figure 9 shows that the antenna resonated at
two different frequencies. For the first resonant
frequency, it is cover frequency from 1.780 GHz –
2.735 GHz, which covers for 3G, WLAN and
WiMAX frequency. The second resonant frequency
covers for UWB application, which resonated from
5.66 GHz to 8.127 GHz. In Figure 9 shown the result
for S11 when the antenna is designed with full
ground plane and modified ground. It shows that,
wide bandwidth can be achieved with modified
ground plane.
B. Second Iteration
The second iteration structure was designed
by reducing each side of LP1. Figure 8 (top layer)
shows the initial shape of Minkowski fractal
geometry antenna, transforming up to the second
iteration. The size of LP3 was calculated to observe
the variation of the operating frequency for each
band. First, the length of LP3 is reduced to 3.45 mm.
Then in second attempt, the length of LP3 is
decreased by 0.35 mm and made LP3 3.1 mm. The
result for both parameters is shown in Figure 10.
Gain in
horizontal
plane
(simulated)
Radiation
efficiency
(simulated)
-
5.0 dBi
4.394 dB
-
98.8 %
90%
Fractals have space-filling geometries that
can be used as antennas to effectively fit long
electrical lengths into small areas. This concept has
been applied to patch antennas. Through
characterizing the fractal geometries and the
performance of the antennas, it can be observed that
increasing the fractal dimension of the antenna leads
to a higher degree of miniaturization.
6.
REFERENCES
[1] Douglas H. Werner and Suman Gangguly, “An
[2]
Table 2: Comparison Performance for Different
Parameters 1st Iteration and 2nd Iteration.
Parameter
Operating
Frequency
1st
Full Ground
6.921-6.642 GHz
Iteration Plane
Modified
1.780-2.735 GHz
Ground Plane
5.66-8.127 GHz
2nd
LP3 =3.45mm
2.006-2.582 GHz
Iteration
4.971-5.488 GHz
LP3 =3.1mm
1.662-2.615 GHZ
5.77-5.669 GHZ
5.
Parameter
First
resonant
frequency
Second
resonant
frequency
[3]
[4]
[5]
[6]
RESULT AND CONCLUSION
Sierpinski
monopole
Koch
monopole
Minkowski
Fractal
2.4 GHz
915 MHz
2.25GHz
[7]
[8]
5.0 GHz
-
6.1GHz
[9]
Impedance
Bandwidth
8.3%
(2.4GHz)
and 4.4%
(5.0 GHz)
14.4%
-
Input
Impedance
50Ω
22.5+
j12.6Ω
-
[10]
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[12] Introduction
to Fractal Geometry, Martin
Churchill, 2004.
[13] Ramavath Ashok Kumar, Y ogesh Kumar
Choukiker, S K Behera, “Design of Hybrid
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