Journal of Software Engineering and Applications, 2011, 4, 129-136
doi:10.4236/jsea.2011.42014 Published Online February 2011 (http://www.SciRP.org/journal/jsea)
Copyright © 20 11 SciRes. JSEA
Application of Genetic Algorithm for Optimization
of Important Parameters of Magnetically Biased
Microstrip Circular Patch Antenna
Naveen K. Saxena1, Mohd A. Khan2, Nitendar Kuma r3, Pr a deep K. S. Pourush1
1Department of Physics, Agra Col lege, Agra, India; 2Department of Electronics & Communication, Anand Engineering College,
Agra, India; 3Solid State Physics Laboratory , Delhi, India.
Received January 19th, 2011; revised February 9th, 2011; accepted February 15th, 2011.
ABSTRACT
The application of Genetic Algorithm (GA) to the optimization of important parameters (Directivity, Radiated Power,
Impedance etc.) of magnetically biased microstrip antenna, fabricated on ferrite substrate, is reported. The fitness func-
tions for the GA program have been developed using cavity method for the analysis of microstrip antenna. The effect of
external magnetic biasing has also been incorporated in the fitness function formulation as effective propagation con-
stant. Using stochastic based search method of GA the common characteristics of electro-magnetic were entertained
which cannot be handled by other optimization techniques. The genetic algorithm was run for 500 generations. The
computed results are in good agreement with the results obta ined experime ntally.
Keywords: Cavity Method, Circular Ferrite Microstrip Antenna, Genetic Algorithm, Magnetically Biased
1. Introduction
In this work, a precise and effective approach is applied
to calculate important parameters of circular patch an-
tenna. Microstrip patch antennas of all shapes are widely
used in communication systems where their small size,
conformal geometry and low cost can be used to advan-
tage. Due to the recent availability of low loss, commer-
cial microwave ferrites there is an increasing interest in
the performance of the patch antennas printed on ferrite
substrates. Although some work [1-6] have been per-
formed for microstrip antenna with GA approach for the
patch antennas without magnetic biasing but analysis of
almost all importa nt par a meters for ferr ite substrate u nder
magnetic biasing for circular patch antenna is new one.
Present analysis also incorporate the dispersion effects
due to magnetic field biasing in the form of effective
propagation constant (k) which is not discussed in the
referenced articles. Some similar referenced works [7-11]
also have done mathematically or by conventional me-
thods for opti mizatio n b ut this te chnique i s rather pr ecise,
accurate and sensitive to optimize parameters of patch
antenna as well as other type of antenna also.
There are many optimization techniques frequently
using for the same work. The four methods used to opti-
mization are: 1) Broyden–Fletcher–Goldfarb–Shanno
(BFGS) 2) Davidon–Fletcher–Powell (DFP) 3) Nelder–
Mead downhill simplex (NMDS) 4) Steepest descent.
The above four algorithms are quite sensitive to the start-
ing values of the amplitude weights. These algorithms
quickly fall into a local minimum, because their theo-
retical development i s b ase d on fi nd in g the mi nimum of a
b owl-shaped objective function. Each algorithm was given
25 different starting values and the results were averaged.
GAs were introduced by Holland [12] and were applied
to many practical problems by Goldberg [13,14]. It is well
known that search technique, the genetic algorithm is a
parallel, robust and probabilistic search technique that is
simply and easily implemented without gradient calcula-
tion, compare with the conve ntional gradient base search
procedure. Most important of all, the GA proposed also
provides a mechanism for global search that is not easily
trapped in local optima. The GA proposed here an adap-
tive mutation rate str a tegy.
2. Genetic Algorithm (GA)
Genetic Algorithm (GA) is a stochastic based search me-
thod that can handle the common characteristics of elec-
tro-magnetic which cannot be handled by other optimiza-

Application of Genetic Algorithm for Optimization of Important Parameters of Magnetically Biased
Microst r ip Circular Patch Antenna
130
tion techniques. A GA has several advantages over the
traditional numerical optimization, including the facts
that it optimizes with continuous or discrete parameters,
doesn’t require derivative information, works with a large
number of variables, well suited for parallel computers,
provides a list of optimum parameters not just a single
solution and works with numerically generated data, ex-
perimental data, or analytical functions.
A chromosome in a computer algorithm is an array of
genes. Each chromosome has an associated cost function
assigned to the relative merit. The algorithm begins with
a large list of randomly generated chromosomes. Cost
function is evaluated for each chromosome. Gene s are the
basic building blocks of a genetic algorithm. A gene is a
binary encoding of a parameter. The populations which
are able to reproduce best fitness are known as parents.
Then the GA goes into the production phase where the
parents are chosen by means of a selection process. The
most fitted members of the population are assigned the
highest pro bability of being selected for mating. T he two
most common ways of choosing mates are roulette wheel
and tournament selection. The selected parents reproduce
using the genetic algorithm operator called crossover. In
crossover random points are selected. When the new
generation is co mplete, the pro ces s of cro ssover is stopped.
Mutation has a secondary role in the simple GA operation.
Mutation is needed because, even though reproduction
and crossover effectively search and recombine extant
notions, occasionally they may become overzealous and
lose some potentially useful genetic material. In simple
GA, mutation is the occasional random alteration of the
value of a string position. When used sparingly with re-
production and crossover, it is an insurance policy against
premature loss of important notions. Mutation rates are of
the order of one mutation per thousand bit transfers. Ac-
cording to the probability of mutation, the chromosome
are chosen at random and any one bit chosen at random is
flipped from “0” to “1” or vice versa. After muta tion has
taken place, the fitness is evaluated. Then the old genera-
tion is replaced completely or partially. This process is
repeated. After a while all the chromosome and asso-
ciated fitness become same except for those that are
mutated. At this point the genetic algorithm has to be
stopped [15-16].
3. Structure & Theory of Antenna
Structure of microstrip circ ular patch antenna is depicted
in Figure 1. Here “a” and “ae” are the radius and effective
radius of microstrip patch respectively. Patch has been
modeled on LiTi ferrite substrate of thickness “h”. The
dielectric constant and saturation magnetization (
4
s
M π
)
of substrate is 17.5 and 2200 Gauss respectively.
ferrite substrate
radiant source
ground pl a n e
extern al
shea th
central
conductor
h
a
a
e
Figure 1. Schematic diagram of microstrip circular patch
antenna.
It has been established that, for a biased ferrite slab, a
normal incident plane wave may excite two types of
waves (ordinary and extraordinary wave). In the case of
normal incident magnetic field biasing ordinary wave is
same as the plane wave in the dielectric slab. On the oth-
er hand, the extraordinary wave is a TE mode polarized
parallel to the biasing direction with its phase propaga-
tion constant Ke [17-20].
eeff eff
w
Kc εµ = ×
(1)
d eff
w
Kc ε = (2)
22
eff
k µ
µµ
−
=
(3)
22
1om
o
ww
ww
µ = +− (4)
22
m
o
ww
k
ww
=− (5)
whe re
and 4
oo ms
wH wM γ γπ = =
where o H is the bias field, 4s M π is the saturation
magnetization,
γ
is the gyromagnetic ratio as
γ
=
2.8 MHZ./Oe. To obtain good performance, there are
many feeding methods, such as CPW in the ground feed-
ing microstrip antenna, and CPW with stub patch feeding
slot antenna. Considering impedance matching of patches
coaxial feeding is generally preferred. Thus the far zone
expressions for circular patch microstrip antenna are ob-
tained as follo ws:
( )
( )( ) { }
11
sin cos
cos
2 cos
sinsin
jkr
n
t
nn
kh
kaVe
Ej n
r kh
J kaJ ka
θ
θ
ϕθ
θθ
−
+−
=
×−
(6)

Application of Genetic Algorithm for Optimization of Important Parameters of Magnetically Biased
Microst r ip Circular Patch Antenna
131
( )
( )( ) { }
11
sin cos
cos
2 cos
sinsin
jkr
n
t
nn
kh
kaVe
Ej n
r kh
J kaJ ka
ϕ
θ
ϕθ
θθ
−
+−
=
×+
(7)
whe re
12
om
dom
www
KK ww
±

+
== 



k
4. Application of Genetic Algorithm to the
Microstrip Antenna and Computed Result
All the vital parameters like thickness of the substrate,
bias magnetic field, radius, dielectric constant etc. were
coded into 5 bit scaled binary coding as the requirement
of fitness function. The Roulette wheel selection was
used fo r GA p op ulat ion. The genet ic algorithm was run
for 500 generations. The probability of crossover was
varied fr om 0.7 to 0.85 and the probability of mutation
was varied from 0.001 to 0.002. The fitness functions
expressions of antenna used for optimization are:
 Fitness Function: 1—Effective Radius
1
2
2
1log 1.7726
2
eff
ha
aa af h
π
π

 

=++

 





(8)
 Fitness Function: 2—Radiation Power
22
22
00
sin
2
radt t
in
A
PEE rdd
Z
ππ
θϕ θθφ

= +


∫∫
(9)
 Fitness Function: 3—Dir ec tivity
( )
21
120 4
g effrad
D kaP

=××

×

(10)
 Fitness Function: 4—Input Impedance
( )
1
4
in rad
ZP =×
(11)
 Fitness Function: 5—Quality Factor
111
trdc
QQQQ
=++
(12)
whe re
( )
12
22
1
eff
rrad
d
c
ak
Qcf hP
QLT
Qh f
π
π µσ
×

= ×

×

=
= ×
 Fitness Function: 6—Band wi d t h
( )
( )
( )
12
1
t
s
BQs
−
=×
(13)
The antenna parameters have been characterized by a
particular o f combinatio n of input variable s like d ielectric
constant, patch radius and substrate thickness of the fer-
rite which is determined using cavity model. The GA
consists of five components. These are the random num-
ber generator, a fitness evaluation unit and genetic oper-
ators for reproduction, crossover and mutation operations.
The flow chart with proposed initial values of GA com-
ponents, for optimization of parameters of microstrip
antenna, is shown in Fig ure 2.
5. Results and Calculations
Obtained Graphs (Figures 3-8) show the va r iatio n of b e st,
mean and expected values of radiation power of antenna.
During calculation GA program at every generation cal-
culate expected value, mean and best value, then plot
them for the corresponding parameter fitness functions.
Every graph has a certain generation points above which
convergence become very slowly and variation among
mean and best values become negligible.
All graphs (Figures 3-8) show the appreciate variation
in mean values but in best value, carry a very little varia-
tion due to big generatio n attempt which precise or accu-
rate the desired result. This big generation amount (500)
has been applied to removing the inaccuracy in the best
Fig ure 2. Flow chart of genetic algorithm applied for the
optimization of parameters.

Application of Genetic Algorithm for Optimization of Important Parameters of Magnetically Biased
Microst r ip Circular Patch Antenna
132
Figure 3. Variation of best, mean and expected value of Eff. radius of circular patch antenna.
Figure 4. Variation of best, mean and expected value of radiation power of circular patch antenna.
Figure 5. Variation of best, mean and expected value of directive gain of circular patch antenna.

Application of Genetic Algorithm for Optimization of Important Parameters of Magnetically Biased
Microst r ip Circular Patch Antenna
133
Figure 6. Variation of best, mean and expected value of impedance of circular patch antenna.
Figure 7. Variation of best, mean and expected value of quality facto r of circular patch antenna.
Figure 8. Variation of best, mean and expected value of band width of circular patch antenna.

Application of Genetic Algorithm for Optimization of Important Parameters of Magnetically Biased
Microst r ip Circular Patch Antenna
134
Table 1. Comparison of parameters calculated by GA prog ram and experimentally obtained.
Parameters Opt. Values Exp. Values
Eff. Radius 0.2082 cm 0.2100 cm
Rad . Power 0.2110 mW 0.2000 mW
Directive Gain –10.05 dB –9.55 dB
Impedance 138.88 ohms 100.55 ohms
Quality Factor 30.43% 28.40%
Bandwidth –34.70 dB –38.60 dB
res ult wh ic h c a n b e j ud ge b y expe c ted va l ue gra p h s shown
in every figure. The performance graphs (Figures 6 and 7)
of Input impedance and quality factor show a little bit
variation in best value, which shows the requirement of
more generation attempt but could not be performed
du e to inefficiency of computer program.
Calculated values of parameters of microstrip circular
patch antenna with GA program ha ve been compared with
some theoretical and experi mental results obtained b y oth-
er methods and referenced in the research articles [21-31],
which are in good agreement and giv en in Table 1.
6. Conclusions
Designing the antenna with the optimum values of pa-
rameters over a given frequency range is an example of
an optimization problem. The GA is very precise and
fast compare to other techniques because it encodes the
parameters, and the optimization is done with the en-
coded parameters. To design an antenna with best per-
formance First, the problem should formulated for the
size, shape, and material properties associated with the
antenna. Next, an appropriate mathematical description
that exactly or approximately models the antenna and
electromagnetic waves is applied. Finally, numerical
methods are used for the solution. One problem has one
solution. Finding such a solution has proved quite diffi-
cult, even with powerful computers.
Rather than finding a single solution, optimization im-
plies finding many solutions then selecting the best one.
Optimization is an inherently slow, difficult procedure,
but it is extremely useful when well done. The difficult
problem of optimi zing an ele ctromagn et ics design has only
recently received extensive attention.
In the present communication the application of Ge-
netic Algorithm for the optimization of important pa-
rameters of microstrip circular antenna printed on ferrite
substrate is reported. The fitness functions for the GA
program is developed using cavity method for the analy-
sis of microstrip antenna. The computed graphs and re-
sults show a good agreement with the results obtained
experimentally.
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Application of Genetic Algorithm for Optimization of Important Parameters of Magnetically Biased
Microst r ip Circular Patch Antenna
136
Appendix: List of Symbols
fr = resonant freq uency
λ = wavelength
h = height of substrate
a = radius of patch
aeff = effective radius of patch
εr = dielectric constant
εeff = effective dielectric constant
µr = initial permeabili ty
µeff = effective permeability
µ, κ = permeability tensor components of µeff
Kd = ordinary propagation constant
Ke,± = extraordinary propagation constant
ω = angular fre quency o f incident e-m-waves
ωo = external magnetic field angular frequency
ωm = internal magnetic field angular frequency
Jn+1 = (n+1)th order Bessel’s function of fir st ki nd
Jn-1 = (n – 1)th o rder Bessel’s funct i on of fir st ki nd
Ho = applied bias field
4πMs = saturation magnetization
γ = gyromagnetic ratio (2.8 MHz/Oe.)
Qt = total quality factor
Qr = total quality factor
Qd = total quality factor
Qc = total quality factor
LT = loss tangent of metal of patch (for copper = 0.0005)
σ = conductivity of metal of patch (for copper = 107)
s = voltage standing wave ratio (VSWR)
Prad = radiation power
Zin = input impedance
B = bandwidth
Dg = directive gain
Eθt = E-plane field
Eφt = H-plane field