# The Effect of Indentation Angle of Koch Fractal Boundary  on the Performance of Microstrip Antenna

Authors: P. Nageswara Rao; N. V. S. N. Sarma
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

The effect of indentation angle of Koch fractal boundary applied to a triangular patch on the resonant behavior and bandwidth is presented. It is shown that the resonant frequency can be controlled by changing the indentation angle of the boundary. With the experimental verification, it is established that for an indentation angle of30∘
more bandwidth is obtained compared to conventional  60∘.


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## Body

## 1. Introduction

It is required to design and fabricate antennas which are compact in size, low profile,
conformal, and having multiband characteristics especially in military and commercial applications
[1]. There are various techniques available in the literature to design compact
antennas like providing shorting pins at regular intervals, or slots on the
patch, and so forth. However, the use of fractals to design compact antennas is
a recent technique. The fractal antennas are not only of compact size but also operate
at multiple bands. Hence, fractal antennas more or less satisfy many of the
aforesaid properties. The term fractal means broken or irregular fragments
originally introduced by Mandelbrot to describe complex shapes that possess an
inherent self-similarity or self-affinity in their geometrical structures. The main idea of using fractal geometries in the design of microstrip antenna is to
increase the effective electrical length through which current travels. This is
generally referred to as space filling property of fractal and upon whichthe
size reduction of  the patch depends.Conventional approach to the
analysis and design of microstrip antennas is to use Euclidean geometry. Since fractal
geometry is an extension of classical geometry, its introduction provides
antenna designers an opportunity to explore a new and innovative antenna design.
There are an  infinite number of fractal geometries available, but a few can be
applied to microstrip antennas to observe their behavior particularly on the
size reduction and multiband operation. Sierpinski carpet and Sierpinski gasket
antennas are familiar fractal geometries for their multiband operation
[2, 3].
The Koch curve is one of the most familiar fractal curves which can be used as
wire antennas to reduce the size and for multiband operation. The behavior of
dipoles using Koch curves of different fractal
dimension is presented by
Vinoy et al. [4].
Borja and Romeu [5] have presented the behavior of Koch curve
fractal boundary microstrip antenna with different iterations in the localized
and nonlocalized modes, and they described the behavior of the same as high-directivity
antenna. The effect of varying the
indentation depth of Minkowski loop wire antenna on its resonant frequency is presented
in [6]. The use of fractal boundary antenna in multistandard terminals
like LAN, GSM1800, and UMTS applications
is presented in [7]. The Indentation angle used for the boundary by the authors
[5, 7] is only 60°. To the best of authors’ knowledge, the other
possible indentation angles are not considered till now to study the resonant
behavior and bandwidth especially as boundary for the patch. It is shown that
the less complex shapesfilling less space is more effective at lowering
resonant frequency [8], and optimizing the Koch fractal boundary using genetic
algorithm can reduce the size of the antenna up to 55% [9], but they are
confined only to wire antennas. When the
indentation angle of the fractal boundary applied to triangular patch is changed,
very interesting points are observed about the resonant behavior and the
bandwidth of the patch. In the present work, the variation of resonant
frequency and bandwidth with the indentation angle of Koch fractal boundary is
investigated. The experimental results are compared with MoM-based IE3D simulation
results.


## 2. Antenna Geometry

The
proposed fractal boundary microstrip antennas are obtained by replacing each
side of triangular patch antenna by Koch curve of varying indentation angles.
It is to study the effect of indentation angle of fractal boundary on the resonant
frequency and bandwidth of the antenna.The
fractal dimension is one of the important properties of fractals. It is a
measure of the space filling ability of  a fractal curve. There are different
versions of fractaldimensions available in the literature, but the most easily
understood definition for the fractal dimension is self- similarity
dimension. It is defined as(1)D=−lognlog(1/s),where n = number of copies, s = scaling factor.For the standard
Koch curve, indentation angle(θ)=60°, n=4, s=3, and D=1.261.
If the indentation angle is made variable, the scaling factor becomes function of θ and is given by(2)1s=12(1+cosθ). Thus,
geometries with varying fractal dimension can be obtained using this generalization.
In the present work, the generalized Koch curve is used as boundary for the
triangular patch antenna and the behavior is presented in detail. The geometry
of the antennas is shown
in Figures 1–3.First iterated Koch fractal
boundary microstrip antennas with indentation angles (a) 10°, (b) 20°,
(c) 30°, (d) 40°, (e) 50°, and (f) 60°.
(a)(b)(c)(d)(e)(f)Second iterated Koch fractal boundary microstrip antennas with
indentation angles (a) 10°, (b) 20°, (c) 30°,
(d) 40°, (e) 50°, and (f) 60°.
(a)(b)(c)(d)(e)(f)Third iterated Koch fractal boundary microstrip antennas
with indentation angles (a) 10°, (b) 20°, (c) 30°,
(d) 40° , (e) 50°, and (f) 60°.
(a)(b)(c)(d)(e)(f)


## 3. Numerical Results and Discussion

The antennas are analyzed using Zeland IE3D
electromagnetic simulator and are printed on RT Duroid substrate having
thickness of 3.2 mm and relative permittivity of 2.33. All the antennas are fed
using probe feed at aposition, where input impedance is 50 ohms. The return
loss characteristics are measured with HP 8719A vector network analyzer.Because of manufacturing constraints, only first three iterations are
considered. Hence, these are only prefractals not actually fractals. In this paper, two cases have been considered:
one with all the antennas having same initial side length of 49.3 mm and
changing indentation angle and iteration of the boundary. The 2nd case is to consider the antennas operating at the same
frequency changing the indentation angle and iteration.Case 1.
When the indentation angle of
the Koch fractal curve applied to the triangular patch as boundary is varied,
it is observed that the resonant frequency of the antenna can be systematically
controlled. The shift in the resonant frequency is observed not only with
indentation angle but also with iteration of the fractal boundary as can be
observed from Figures4–6.
From Table1, it
can be inferred that when the indentation angle is changed from 10° to 60°, the resonant frequency is changed from 2492 MHz to 2311 MHz for
the first iterated fractal antenna, where as for the 2nd iterated antenna it is
changed from 2475 to 2116 MHz, and for 3rd iterated antenna it is changed from
2468 to 2048 MHz. All these antennas are circumscribed in a circle of radius
28.4 mm that is a/3,
where “a” is the initial side length of the triangle. The percentage reduction
of resonant frequency for different iterations is listed in Table 2.
It is interesting to note an important observation that the
bandwidth of these antennas also changes with indentation angle. For all the
three iterations, it is demonstrated that more bandwidth is obtained for fractal boundary antenna with an indentation angle of 30° compared to the
conventional angle of 60°. The higher bandwidth is
because of higher value of area to perimeter ratio. The above-simulated values are
experimentally verified. The photographs of the antennas are shown in
Figure 7.
Figure8 shows
the measured S11 characteristics of 1st, 2nd, and 3rd iterated fractal boundary
microstrip antennas with indentation angle 30° and 60°.
The comparison between measured and simulated results is shown in
Table 3.
The area and perimeter of generalized
fractal boundary microstrip antenna can be calculated using the relation given
by (3) for the 1st iterated fractal antenna. As said earlier, the higher bandwidth is
because of higher value of area to perimeter ratio. This is verified for a
particular case of 1st-iteration fractal boundary antenna by changing
indentation angle with initial side length of 49.3 mm: (3)Perimeter=12a2(1+cosθ),Area=34a2+3a2sinθcosθ[2(1+cosθ)]2,a = initial side length of triangular patch, θ = indentation angle.Table 1
Resonant frequency and bandwidth of proposed antennas.
Indentation angle1st iteration2nd iteration3rd iterationfr
(MHz)BW(MHz)fr (MHz)BW(MHz)fr (MHz)BW(MHz)10249257.2247555.6246855.220248859.2245057.6243455.630247470.2240058.223806340242461.6232754226451.650236859222849.6216147.560231156211643.2204839.2Table 2
Percentage reduction of resonant frequency with iteration and
indentation angles.
Iterationfr for indentation
angle 10° (MHz)fr for indentation angle 60° (MHz)Shift (MHz)% reduction of fr1st249223111817.26%2nd2475211635914.5%3rd2468204842017%Table 3
Comparison of simulated and measured results.
AntennaResonant frequency (MHz)10 dB Imp bandwidth (MHz)Indentation angleIterationSimulatedMeasuredSimulatedMeasured30°1st2474247670.28230°2nd2400240258.26630°3rd23802386636360°1st23122298566060°2nd21162118434760°3rd204820493942Table 4
Area and perimeters of
1st iterated fractal boundary antenna with different
indentation angles (a=49.3mm).
Indentation angleArea (mm2)Perimeter (mm)Ratio of area to
perimeterBW (MHz)1011351497.657.22012161537.959.23012831588.1270.2401344167861.65013891807.7596014051977.1356Figure 4
S11 characteristics of 1st iteration
Koch fractal boundary microstrip antennas with
different indentation angles.Figure 5
S11 characteristics of 2nd iteration Koch fractal boundary microstrip
antennas with different indentation angles.Figure 6
S11 Characteristics of 3rd iteration Koch fractal boundary microstrip
antennas with different indentation angles.Fabricated fractal boundary patch antennas
with indentation angles of 30° and 60°: 1st iteration, 2nd  iteration, and 3rd iteration.
(a)(b)(c)(d)(e)(f)(a) MeasuredS11 characteristics fractal boundary antenna with
indentation angle 30°.
(b) Measured S11
characteristics fractal boundary
antenna with indentation angle 60°.
(a)(b)Case 2.
The above-mentioned antennas are made
to operate at a fixed frequency of 2.5 GHz, while varying the indentation angle
and iteration. The statement of getting less bandwidth for an indentation angle
of 60° is still valid even
for the antennas operating at the same frequency as can be observed from Table 5.
For example, the bandwidth of 3rd iterated fractal antenna with indentation
angle of 60° is only 48 MHz, whereas it is 72 MHz for 30°,
both are operating at the same frequency of 2.5 GHz. Hence,
the indentation angle of the boundary has significant effect on the performance
of antenna in terms of both resonant frequency and bandwidth. As the fractal
dimension and indentation angle are related, the resonant frequency and
bandwidth of fractal boundary can be controlled by varying the fractal
dimension of the boundary.Table 5
Bandwidth comparison of antennas operating at the same
frequency of 2.5 GHz.
Indentation angleBandwidth (MHz)1st iteration2nd iteration3rd iteration105952602060676230647072406267665058646260545648


## 4. Conclusion

The variation of resonant frequency and
bandwidth with indentation angle for first 3 iterations of Koch fractal
boundary microstrip antenna is presented. It is observed that the resonant
frequency decreases with change in indentation angle apart from change in the
iteration. Further, the bandwidth also changes with indentation angle and hence
with fractal dimension of the fractal boundary. It is concluded that better
bandwidth can be obtained with different indentation angle than conventional 60°,
keeping all the antennas circumscribed within the radius of a/3 (“a” is the initial side length of triangular patch).
Hence, the indentation angle can be taken as one of the key parameters for the
design of Koch fractal boundary microstrip antennas since all the parameters
are being changed with the indentation angle. A very good agreement between
simulated and experimental results is obtained.



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*Source: 387686-2008-05-18.xml*