An overview of fractal antenna geometries

1. AN OVERVIEW OF FRACTAL ANTENNA
GEOMETRIES
Rupleen Kaur
Dept. of Electronics and
Communication Engg.
Guru Nanak Dev University,
Regional Campus Gurdaspur.
Punjab, India
rupleeenkaur@gmail.com
Sahil Saini
Dept of Electronics and
Communication Engg.
Guru Nanak Dev University,
Regional Campus Gurdaspur.
Punjab, India
sahilbadwals@gmail.com
Satbir Singh
Dept. of Electronics and
Communication Engg
Guru Nanak Dev University,
Regional Campus Gurdaspur.
Punjab, India
satbir1_78@yahoo.co.in
Abstract: Currently there has been a keen interest in
designing of an antenna in the fields of wireless
communication because of overgrowing demand of
telecommunicationservices. A low profile antenna having
capability of operating at multiple frequencies is the need
of today. Fractal antennas have small size and supports
multiband and wideband frequencies because of their self
similarity and space filling properties. Fractal structures
are virtual combinations of capacitors and inductors.
These combinations make the antennas so that they have
many different resonances which can be chosen and
adjusted by choosing the proper fractal design. In this
paper an overview is provided about various parameters
that affect the performance of fractal antennas. We have
also explained geometries of fractal antennas for various
communication applications.
Keywords: Fractal antenna, Fractal geometries, wireless
communication.
I. INTRODUCTION
The word fractal was first devised by Benoit
Mandelbort in the year 1975. It has been derived froma
Latin word “fractus” meaning fractured or broken. The
fractal geometries are generated from the complex
structures occurring in nature. In the year 1988, Nathan
Cohen built the first fractal antenna. These antennas are
designed using simple fractal geometries which have
self similar and space filling properties. Self similar
property associated with fractal geometry enables to
design different parts of antenna that look similar to
each other when viewed at different scale. Space filling
property reduces the size of antenna when compared to
other traditional antennas. Therefore by using fractal
geometries a compact antenna that can be operated at
different frequencies can be obtained.
Antennas with fractal geometries are quiet attractive
due to their low weight, compact size, multiband nature
and easy manufacturing. The geometries of fractal
antenna are difficult to define using Euclidean
geometries. The fractal antennas undergo number of
iteration forming a copy of the parent. Hence these
antennas are also known as “Natural Antennas” because
their geometry resembles natural occurring phenomena
such as branches of trees, rivers, galaxies etc.
The antenna that undergoes number of iterations is
called deterministic fractal antenna. These antennas are
designed using broken lines known as generators. The
segments forming broken line are substituted by the
generator forming first iteration according to the
algorithm. The step is repeated infinitely resulting in
fractal geometries. The iteration function system
decides the number of iterations [1].
II. DIMENSIONS OF FRACTAL
GEOMETRY
Dimensions can be defined as the number of
parameters or co-ordinates of an object. There are
different parameters for dimensions of fractal
geometries such as self similarity, topological
dimension, and box counting dimension. Here self
similarity dimension is considered to define fractal
geometries. The self similarity dimension is defined as:
𝐷𝑠 = log 𝑁/ log(
1
𝑠
) ... (1)
N is the number of self similar copies.
s is the scale factor.
Fractal Antennas have repeating patterns, therefore
despite of using the word “repeat” we use “iterate” and
the process of repeating pattern is called iteration. The
iterative function is given by:
𝑊( 𝑥) = 𝐴𝑥 + 𝑡 = [ 𝑎 𝑏
𝑐 𝑑
] [ 𝑥1
𝑥2
] + [
𝑒
𝑓] ... (2)
Where 𝐴 = [
(
1
𝑠
) 𝑐𝑜𝑠𝜃 − (
1
𝑠
) 𝑠𝑖𝑛𝜃
(
1
𝑠
) 𝑠𝑖𝑛𝜃 (
1
𝑠
) 𝑐𝑜𝑠𝜃
] ... (3)
Here a, b, c, d are defined by rotation and scaling of
initial geometry and e and f denote the translation.

2. Fractal geometries have two main components:
1. Initiator: The basic geometry of fractal antenna.
2. Generator: Shape that we get after different
iterations.
Fractal antennas provide better input impedance and
can be used in devices where space is a major issue
because of space filling property. These antennas can
show multiband properties due to self similar design.
By using self similar property a number of copies can
be found within the whole geometry. Hence fractal
geometries do not have any characteristic size and
shape. Despite of having so many advantages fractal
antennas are difficult to fabricate and sometimes
provide lower gain [2].
III. PARAMETERS DETERMINING
THE PERFORMANCE OF
FRACTAL ANTENNA
A fractal microstrip patch antenna consists of a
ground plane, substrate and a patch with fractal defects.
Various parameters determine the performance of the
antenna, some of which are explained below.
a) Effects of Substrate
It is seen that the bandwidth of the antenna depends
upon various parameters of the substrate. The quality
factor Q varies inversely with the impedance bandwidth
of the patch antenna.
Therefore various parameters of substrate such as
thickness and dielectric constant can be changed to
obtain different values of Q. Here Q is
Q = Energy stored/Power lost ... (4)
The bandwidth also depends upon the thickness of
the substrate.As the thickness of the substrate increases
the bandwidth also increases. On the contrary when εr
decreases, bandwidth increases.
However a thick substrate results in poor radiation
efficiency. Also thick substrates give rise to fictions
radiation and radiation from the probe feed also
increases. Therefore this can be counted as a limitation
in obtaining an octave bandwidth.
b) Effect of Parasitic Patches
A parasitic patch can be defined as a patch that is
placed close to the feed patch. The presence of parasitic
patch excites the feed patch through coupling between
the two patches. If the resonance frequencies of the two
patches are in close proximity to each other, then a
wide bandwidth can be obtained because the VSWR is
of responses that results in broad bandwidth.
c) Effect of Multilayer
Configurations
When two or more patches of the dielectric
substrate are placed together on different layers, they
form multilayer configurations. A multilayer
configuration shows a decline in radiation pattern when
compared to single layer. One major drawback is the
increase in height which is not advantageous in the
applications where space is a major issue. This
configuration also increases back radiation.
Fig.1. Multilayer Configuration[1]
Multilayer configuration yields broad bandwidth but
shows an increment in the height of the antenna. The
layers are separated either by foam or air gap in
between [3, 4].
IV. FRACTAL GEOMETRIES
In the year 1975, Mandelbort presented a new
geometry which was different from Euclidean designs,
these geometries are known as Fractal geometries. The
word ‘fractal’ coined from the Latin word ‘frangere’
meaning broken or fractured. Various fractal
geometries are discussed below [5].
1) Sierpinski Carpet
In Sierpinski Carpet a simple square patch at zeroth
order iteration is iterated number of times. In first order
iteration a square having dimensions one third of the
main square is subtracted from the center of the square.
This process is repeated infinite times in order to get
next order iterations. The pattern is repeated in a
symmetrical manner such that each etched square is one
third in dimension of the sharing square. The fractal
geometry of Sierpinski Carpet is shown in fig.2 [6]
Fig.2 Sierpinski Carpet upto3rd
iteration [6]

3. Dong –You Choi (2013) designed a miniaturized
microstrip Sierpinski Carpet fractal antenna upto 2nd
iteration. Experimental results of iterations shows that
the size reduces to 31% and 32% corresponding to 1st
and 2nd iteration. These antennas can be used in
meterological satellite communication systems (18-
12.5 GHz), Radar and navigation services, Bluetooth,
WiMax applications, UMTS etc [7].
2) Koch Curve
In the year 1998, von Koch monopole antenna
improved various features such as radiation resistance,
bandwidth and resonance frequency when compared to
conventional antennas. A Koch Curve is designed by
substituting the middle third straight section with a bent
section of wire. A length is added to the total curve
after each iteration. The fractal geometry of Koch
Curve is shown in fig.3 [8]
Fig.3. Koch Curve upto 3rd
iteration [8]
Mustafa Khalid Tahir (2007) designed a cross
dipole antenna merged with Koch Curve geometry. It is
seen that the new designed antenna is low profile and
shows multiband performance as compared to
traditional antennas.These antenna designs can be used
in mobile communication systems, LAN, GSM etc [9].
3) Minkowski Curve
In 1907, a German mathematician, Hermann
Minkowski devised a new fractal shape called
Minkowski Sausage and later known as Minkowski
Curve. This fractal geometry reduces the size of
antenna and also increases the efficiency by occupying
the volume with electrical length. The fractal geometry
of the fractal design is shown in fig.4 [10].
Fig.4. Minkowski Curve upto 3rd
iteration[10]
Piyush Dalsania, Brijesh Shah, Trushit Upadhya and
Ved Vyas Dwivedi (2012) designed a square patch
fractal antenna using Minkowski geometry. The
experimental results show reduction in size and
multiband nature of antenna. The antenna can be used
in aeronautical radio navigation (2.7-2.9 GHz) and
maritime radio navigation (9-9.3 GHz) [11].
4) Hilbert Curve
This geometry is also known as Space Filling Curve
since it fills the area it occupies. In this geometry each
consecutive iteration consists of previous four copies of
iteration. The geometry is simple as the curves can be
drawn easily and the lines of the geometry do not
intersect with each other. The fractal geometry of
Hilbert Curve is shown in fig.5 [12].
Fig.5. Hilbert Curve upto4th
iteration [12]
Huang (2010) designed an Inverted F Antenna
(IFA) using Hilbert geometry. It was observed that the
size reduces 77% when compared to traditional
antenna. These antennas can be used in wireless sensor
network applications [13].
5) Pythagorean Tree Fractal
In Pythagorean Tree Fractal the geometry starts
with square, called zeroth iteration. When two other
squares are placed upon the first square such that the
corners coincides with the main square then this is
known as second order iteration. The process is
followed by infinite iterations accordingly. The fractal
geometry of Pythagorean Tree Fractal is shown in fig.6
[14].
Fig.6. Pythagorean Tree Fractal upto 3rd
iteration [14]

4. Pourahmadazar (2011) designed an antenna based
upon Pythagorean Tree Fractal geometry. The resultant
antenna was small in size and was more efficient when
compared to other conventional antennas. The antenna
can be used in UWB application [14].
V. CONCLUSION
In this paper various parameters that affect the
performance of fractal patch antenna are discussed.
Also various fractal geometries, their properties and
applications are reviewed. Fractal geometries not only
reduce the size of the antenna but also provide
multiband properties. It is observed that wideband
characteristics can be improved by increasing the
number of iterations. Fractal antennas can have a
promising future in wireless technologies.
REFERENCES
[1] M. Pilevari Salmasi, F.H. Kashani,M.N.Azarmanesh, “A Novel
BroadbandFractal Sierpinski Shaped, Microstrip Antenna”, Progress
in Electromagnetics Research C, Vol.4, pp 179-190, 2008.
[2] Albooyeh, M.,N. Kamjani,andM. Shobeyri,“A novel crossslot
geometrytoimprove impedancebandwidth of microstrip antennas,”
Progress In Electromagnetics Research Letters, Vol. 4, pp 63-72,
2008.
[3] Ray, I., M. Khan, D. Mondal,andA. K. Bhattacharjee, “Effect on
resonant frequency for E-plane mutually coupled microstrip
antennas,” Progress In Electromagnetics ResearchLetters, Vol. 3, pp
133-140, 2008.
[4]C. Puente, J. Romeu, R. Pous, andA. Cardama, “On the behavior
of the Sierpinski multiband fractal antenna,” IEEE Transactions
Antenna Propagation, vol. 46, pp. 517-524, 1998.
[5] Ankita Tiwari, Dr. Mukesh Rattan, Isha Gupta, “Review On:
Fractal Antenna Design Geometries and its Applications”
International Journal ofEngineering and Computer Science, Vol. 3,
Issue 9, pp 8270-8275, 2014.
[6] Sagne D.S.,Batra R.S. and Zade P.L. , “Design of modified
geometry sierpinski carpet fractal antenna array for wireless
communication,” In Proceedings of IEEE International Advance
ComputingConference (IACC), Ghaziabad, India, pp435- 439, 2013.
[7] Dong- YouChoi, “ Design of Modified Sierpinski Fractal Based
MiniaturizedPatch Antenna,” IEEE International Conference on
Information Networking, pp 274-279, 2013.
[8] J. J. Modi, T. K. Upadhyaya and V. M. Patel, “ Review &
Comparisonof Fractal Antennas,”International Journal for Scientific
Research & Development, Vol. 2, Issue 03, pp 1363-1366 ,2014.
[9] Mustafa KhalidTahir,“ FourArm Koch Curve Multiband Cross
Antenna”, IEEEInternational Conferenceon Next GenerationMobile
Application, Services and Technologies, pp 223-228, 2007.
[10] A. Nagpal, S.S. Dhillon andA. Marwaha, “Multiband E-Shaped
fractal microstrip patch antennawith DGSfor wireless applications,”
In Proceedings of International Conference on Computational
IntelligenceandCommunicationNetworks (ICCICN-2013), Mathura,
India, pp. 22-26, 2013.
[11]Piyush Dalsania, Brijesh Shah, Ved Vyas Dwivedi, Trushit
Upadhya“Analysis of MultibandBehaviouron Square Patch Fractal
Antenna”, IEEE International Conference on Communication
Systems and Network Technologies, pp 76-78, 2012.
[12] H.O. Peitgen,J.M. Henriques, L.F. Penedo (Eds.), Fractals in
the Fundamental and Applied Sciences, In proceedings of IFIP
conference of fractals, Amsterdam, North Holland, 1991.
[13] J.T.Huang, J.H. Shiao, and J.M. Wu, “A Miniaturized Hilbert
Inverted-F Antenna for Wireless Sensor Network Applications,”
IEEE Transactions on Antennas AndPropagation, Vol. 58, No.9, pp.
3100-3103, Sep. 2010.
[14] J. Pourahmadazar, C. Ghobadi, and J. Nourinia, “Novel
ModifiedPythagorean Tree Fractal Monopole Antennas for UWB
Applications,” IEEE Antennas And Wireless Propagation Letters,
Vol. 10, pp. 484-487, 2011.