1. 66th
International Astronautical Congress, Jerusalem, Israel. Copyright ©2015 by the International Astronautical Federation. All rights reserved.
IAC-15-B2.5.1 Page 1 of 5
IAC-15-B2.5.1
Pythagorean Tree Fractal for Multiband Patch Antenna.
Advait Kulkarni
Beihang University, China, contactadvait@gmail.com
Miniaturization of satellite systems demands development of optimized miniature electronic components. Use of
simplistic multiband antennas is becoming a pressing issue. Microstrip patch antennas are quote commonly used for
their low structural profile, robust features and ease of production. However, very often patch antennas face narrow
bandwidth and low power handling. This paper, aims at developing a multiband patch antenna using fractal
mathematics for structural modifications. The design presented in this paper is based on a Pythagorean equations.
The model uses, teflon (dielectric constant = 2.1) for substrate and metallize using perfect electric conductor for
simulation purposes. In the base antenna resonance is achieved at 8GHz, while in the first iteration of fractal
progression, two resonances at 8.2 GHz and 9.2 GHz are achieved. Broadside radiation pattern is not the same as
base antenna, however this problem can be solved employing an array configuration.
I. INTRODUCTION
ULTIBAND nature of small profile antennas is
becoming a very important feature for
antenna in any configuration. While patch antennas
remain a very widely used option, it has limitations
of narrow band and low power handling [1]. Many
techniques for improving bandwidth of patch
antennas have been employed in the past. In recent
times use of fractal geometry for antenna
development has been pursued very actively. A
fractal antenna uses the self- similar design to
maximize the length, or increase the perimeter, of
the material that is being used as a conductor. This
is done within a given total surface area or volume
[2]. Thus an antenna can be designed to be under
original size or volume constrains while showing
broadband or multiband behavior. Improved
standing wave ratio performance on a reduced
physical area can also be obtained using a fractal
progression [3].
Conventionally sierpenski equation, or Koch
progression are used for fractalizing a patch
antenna. Work with sierpenski patch and
monopole antenna has been presented in [4] by
Ja’afar. In this work multiple resonances have
been proven using different feeding techniques. In
another work [5], by Nagpal and Marwaha, a novel
E shape to progress a fractal shape and achieve
multiband behavior for regular patch antenna. In
their final iteration, they achieve 3 distinct
resonating frequencies, with a comparatively
higher bandwidth.
The presented research is based on [6], which
uses a Pythagorean equation for progression of the
fractal structure. Use of Pythagorean equation is
able to retain the original feed point regardless of
iteration level. This makes radiation impedance
analysis comparatively easy. Simplicity of the
structure makes this approach a suitable candidate.
In [6], operating frequency is 2.45GHz. However
the presented paper concentrates on X band
frequencies. A novel approach, different from [6]
has been used for achieving the required behavior.
Different feeding techniques have been tried with
and two major feeds have been compared in the
paper. Antenna modelling and full wave
simulations have been done in Computer
Simulation Technology – Microwave Studio.
II. THEORY OF PATCH ANTENNA DESIGN
This section will cover a short introduction to
design of conventional patch antennas. Figure 1
shows basic design of a patch antenna. Following
set of equations were used for design of the patch.
These formulae are quite widely used, however
there is need of model optimization for frequencies
beyond 5GHz, which was also the case in this
research.
M

2. 66th
International Astronautical Congress, Jerusalem, Israel. Copyright ©2015 by the International Astronautical Federation. All rights reserved.
IAC-15-B2.5.1 Page 2 of 5
Figure1: Microstrip patch antenna iteration 0
Following are the equations used for patch
antenna design, based on work in [7]. The patch
dimensions control operating frequency and other
features of the antenna. Width of the patch is
calculated as
(1)
Where is the operating frequency, is
permeability of vacuum, is permittivity of
vacuum, is dielectric constant of the substrate.
Length of patch is given by
(2)
Where,
(3)
And,
(4)
Base patch in this research has been derived
from the above equations. Microstrip feed line
and co-axial pin feed were tested for their
performance. Matching in this case has to be
done with input impedance of the patch.
Design of coaxial pin is based on [8-10].
II.I Feeding methods
Coaxial feed and micro-strip feed were tested
for best performance. Initially a 50ohm microstrip
was used to feed the patch. However, mismatch
between patch impedance and line impedance,
resonance was not achieved. Patch impedance was
calculated and was matched to 50 ohm using a
quarter wavelength transformer following the work
in [7].
Using a 50 ohm coaxial feed with a probe
offset was also tested. Variation in the probe
position varied the input impedance. Different
probe offset values were tried and most suitable
probe position was selected.
TABLE 1
Iteration 0 design parameters
Variable Description Value
Fr Operation Frequency 8 GHz
W Patch Width 15.06 mm
L Patch Length 11.68 mm
Wg Width of substrate 30 mm
Lg Length of substrate 35 mm
Zt Width of lamda/4 transformer 0.9357 mm
Lamda Wavelength 25.86 mm
Pin off In case of coax feed, we use this
pin offset for the best impedance
match in iteration 0 (depicted in
figure 2)
2.5 mm
This table shows the design parameters for iteration 0
patch antenna
In table 1, parameters for figure 1 have been
detailed. This design using the quarter wavelength
transformer to match the input impedance is based
on [7].
A similar design has been made with a pin feed
and varying the position, match of input impedance
is done. Pin location has been depicted in figure 2.
Both the feed techniques have been used for
iteration 0 and 1, results are presented in the
following sections.
Parameters value may vary from the theoretical
derivation. These variations are based on

3. 66th
International Astronautical Congress, Jerusalem, Israel. Copyright ©2015 by the International Astronautical Federation. All rights reserved.
IAC-15-B2.5.1 Page 3 of 5
simulation results and optimization.
III. PYTHAGOREAN FRACTAL
This section details the procedure of fractal
patch antenna. Pythagorean equation has been used
in this case of progressing the fractal. At every
iteration we check the validity of Pythagorean
condition. In this research, we concentrate on first
iteration; since it fulfils the operational frequency
requirements.
The presented antenna design was implemented
in a Teflon substrate; Er = 2.1. Simulations were
done using PEC for metallization. A lower
substrate dielectric constant leads to narrow
resonance bandwidth [1], which is an advantage
since it leads to a good separation between the
resonating frequencies.
To begin fractalizing the structure we consider
width of the patch (W); as hypotenuse of the
Pythagorean triangle. Using figure 2 as reference,
we assume values of angle a and angle b, such that,
(5)
From this combination of angles and one side,
we derive width of patch 2 and patch 3;
(6)
(7)
Figure2: Rectangular patch antenna with iteration 1 of
using triangle equations. The variables used for annotations
match the variables as described in table 2.
For ease of analysis, angles ‘a’ and ‘b’, were
defined as variables in the simulation model.
TABLE 2
Iteration 1 design parameters
Variable Description Value (mm)
W Patch Width 15.06
L Patch Length 11.68
W2 Width of patch 2 13.04
L2 Length of patch 2 9.83
W3 Width of patch 3 7.53
L3 Length of patch 3 5.68
Pin_off Pin offset used for
iteration 1
0.5
This table shows design parameters for iteration 1 patch
antenna with pin feed.
IV. SIMULATION RESULTS
IV.I Iteration 0
The design was modelled and simulated in
CST-Microwave studio, 2014. In case of pin feed;
different pin offset positions were tested for best
impedance match. Both feeding techniques have
been detailed in table 3. For strip line feed,
resonant frequency is 8.014GHz and for coax feed,
it is 8.002 GHz; with a return loss of -36.09dB in
the first case and -31.37dB in the second case.
TABLE 3
Simulation Results Iteration 0 comparison
Variable Description Value
Fr Operation Frequency 8.014 GHz
Return
loss
Return loss at the resonant
frequency
-36.09dB
VSWR Standing wave ratio at
resonant frequency
1.031
Input
impedance
Input impedance at
resonance
56.51ohm
Fr (Pin) Operation Frequency for
pin feed
8.002 GHz
Return
loss (Pin)
Return loss at the resonant
frequency for Pin feed
-31.37dB
This table shows the results for iteration 0 patch antenna
with strip line feed.
IV.II Iteration 1
For first iteration design, both feeding
techniques are compared. Parameter values are the

4. 66th
International Astronautical Congress, Jerusalem, Israel. Copyright ©2015 by the International Astronautical Federation. All rights reserved.
IAC-15-B2.5.1 Page 4 of 5
same in both cases. For the first iteration, pin offset
is set at 0.5mm. This variation is to get the
appropriate match of input impedance. For first
iteration 1 and strip line feed, resonance are
obtained at 8.002 GHz and 9.064 GHz; with return
loss of -11.18 dB and -20.92 dB respectively.
For model with coax feeding, resonance
are obtained at 7.966 GHz and 9.022 GHz; with
return loss of -12.63 dB and -20.18 dB
respectively.
TABLE 4
Simulation Results Iteration 1 comparison
Variable Description Value
Fr1 Operation Frequency
stripline
8.022 GHz
Fr2 Operation Frequency
stripline
9.064 GHz
Return
loss1
Return loss at the FR1
stripline
-11.18 dB
Return
loss2
Return loss at the FR2
stripline
-20.92 dB
Fr1 Operation Frequency
COAX
7.966 GHz
Fr2 Operation Frequency
COAX
9.022 GHz
Return
loss1
Return loss at the FR1
COAX
-12.63 dB
Return
loss2
Return loss at the FR2
COAX
-20.18 dB
This table shows the results for iteration 1 of the
fractalised patch antenna comparing strip line feed and Coax
feed.
IV.III Iteration 1: Radiation Patterns
It is observed that results using both coaxial
feed and strip line feed are very close. When
compared coaxial feed demonstrates better input
impedance match, and hence lower return loss.
Based on this, radiation pattern for coaxial fed
antenna have been detailed below.
Figure 6 Varying theta vs degrees at phi = 0 deg ; Frequency
7.966
Figure 7 Varying theta vs degrees at phi = 90 deg ;
Frequency 7.966
Figure 8 Varying theta vs degrees at phi = 0 deg ;
Frequency 9.022
Figure 9 Varying theta vs degrees at phi = 90 deg ;
Frequency 9.022

5. 66th
International Astronautical Congress, Jerusalem, Israel. Copyright ©2015 by the International Astronautical Federation. All rights reserved.
IAC-15-B2.5.1 Page 5 of 5
It is clearly seen from radiation pattern
depictions that at no frequency the radiation
pattern is completely broadside. However it is seen
that even if the radiation is not perfectly broadside,
antenna is radiating in the upper quadrant. To get a
higher directive radiation pattern different
solutions can be applied.
V. CONCLUSIONS
This paper presents with a novel solution of
designing a multiband patch antenna using fractal
structure. The structure presented in this paper is
simple to conceive, model and manufacture. This
paper presents a solution which works equally well
with coaxial and strip-line feeds. Radiation pattern
exhibited by the structure, are close to broadside,
and there is a scope for improving directivity and
gain by using different solutions. In future, this
structure will be used as a single element for an
array configuration.
REFERENCES
[1] D. M. Pozar., Microwave Engineering, 4th
Edition, John
Wiley & Sons, New York, 2012.
[2] N. Cohen, “Fractal Antenna Applications in Wireless
Telecommunications”, IEEE Electronics Industries
Forum of New England, pp 43-49, 1997
[3] R. Garg, P. Bhartia, I. Bahl and A. Ittipiboon, “Multiband
Antenna Design Handbook”, Artech House.
[4] Ja’afar. A, “Sierpenski Gasket Patch and Monopole
Antenna”, M.S. Thesis, Universiti Teknologi Malaysia,
Malaysia, 2005.
[5] A. Nagpal, S. Dillon, A. Marwaha, “Multiband E-
Shaped Fractal Microstrip Patch Antenna with DGS for
Wireless Applications”, 5th International Conference on
Computational Intelligence and Communication
Networks (CICN), 2013, pp 22-26.
[6] S. Yadav, K. Vyas and S. Kumar, “A Pythagoras tree
shape Fractal Antenna for Multiband Applications”,
International Journal of Emerging Technology and
Advanced Engineering, vol 3, Issue 12, Dec 2013
[7] V. Rajeshkumar, K. Priyadarshani, D. Devakirubai, C.
Ananthi and P. Snekha, “Design and Comparative Study
of Pin feed and Line feed Microstrip Patch antenna for X
band Applications”, International Journal of Applied
Information Systems, Vol 1, Isuue 5, Feb 2012.
[8] T.A.Millikgan, “Modern Antenna Design”, 2nd
ed., IEEE
Press, John Wiley and Sons inc.
[9] R. Garg, P. Bharatia, I. Bahl and A. Ittipobiin,
“Microstrip Antenna Design Handbook”, Artech House,
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[10] Md. Tanvir Ishtaique-ul- Huquel , Md. Kamal Hosain,
Md Shihabul Islam and Md.Al-Amin Chowdhury,
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Antennas with Optimum Parameters for X band
Applications”, IEEE proceedings of IJACSA 2011.