Int. J. of Information Systems and Communications Vol. 1, No. 1, June 2011

Copyright Mind Reader Publications

www.ijisc.yolasite.com

Design of Microstrip Antennas Using

Differential Evolution Algorithm

Arindam Deb1, Jibendu Sekhar Roy

1, and Bhaskar Gupta

2

1School of Electronics Engineering

KIIT University, Bhubaneswar, Orissa, India

e-mail: adebfet@kiit.ac.in, drjsroy@rediffmail.com

2Electronics and Telecommunication Engineering Department

Jadavpur University, Kolkata, WB, India

e-mail: gupta_bh@yahoo.com

Abstract

Differential evolution (DE) algorithm is applied to the design of

single-layer and multi-layer microstrip antennas. For the design of

single-layer co-axial probe-fed microstrip antenna, fitness function

is obtained from cavity model theory of microstrip antenna, whereas

for the design of multi-layer aperture-coupled microstrip antenna,

transmission line model is used to obtain fitness function for DE. In

both the cases, results obtained using DE, are verified with the

results, obtained using genetic algorithm (GA) and particle swarm

optimization (PSO).

Keywords: Differential evolution algorithm, microstrip antenna, probe-fed,

aperture-coupled, genetic algorithm, particle swarm optimization.

1 Introduction

Microstrip antennas are the most attractive for microwave communication

systems including wireless communication systems due to their low cost, light

weight, low profile, conformability to planar and non-planar surfaces and ease of

A. Deb et al. 2

fabrication [1-4]. Widely used configuration is a single layer microstrip antenna,

excited by a co-axial probe-fed microstrip antenna (PFMA). The basic form of a

microstrip antenna is a radiating metallic patch, fabricated on a dielectric substrate

and backed by a ground plane. Main drawback of this single layer microstrip

antenna is its narrow bandwidth. In order to enhance the bandwidth of microstrip

antennas, many feeding techniques have been developed, out of which aperture-

coupled microstrip antennas (ACMSA) are more attractive, because large number

of dimensional parameters can be optimized to achieve a particular requirement [2,

5-7]. The basic form of an ACMSA is a radiating patch on a dielectric substrate,

fed by a microstrip line on another substrate, through a small aperture or slot on a

ground plane. Design of PFMA and ACMSA, using DE, is presented in this

paper. Differential algorithm (DE) is a variant of genetic algorithm (GA), which

attempts to replace the crossover operator in GA by a special type of differential

operator for reproducing offspring in the next generation. It is a combination of

original GA, evolutionary programming and evolutionary strategies [8-10]. DE

has better performance compared to GA and recently, attention has been given to

apply DE in antenna problems [11-23], though in most of the cases DE is applied

to antenna array optimization problems. In case of probe-fed microstrip antenna,

evaluation of the fitness function is done by analyzing the antenna structure by

cavity model analysis [1-2]. In this paper, for the design of multilayered ACMSA,

the fitness function for DE is obtained using transmission line model of microstrip

antenna [1-2]. Parameters like patch length(a), patch width (b), slot length (La)

and slot width (Wa), width of microstrip line (W) and open stub length (Ls) are

optimized for the desired resonant frequency. To implement DE for ACMSA, an

initial population of target vectors is defined. Each target vector contains different

design parameters. For each parameter, a lower and upper limit is defined. Initial

parameter values are randomly selected, uniformly in the interval. The weighted

difference of any two of the parameters to the third vector is added to form a

donor vector whose components enter into the trial vector with certain probability.

Now the trial vector is compared with the target vector and with a better fitness

and it is admitted to the next generation. For both probe-fed microstrip antenna

and aperture coupled microstrip antenna, results obtained using DE are verified by

the results, obtained using binary coded GA and particle swarm optimization

(PSO), which are very popular optimization techniques [24-27]. All the

optimization techniques are implemented using MATLAB.

2 Design Steps for PFMA and ACMSA Using DE

The steps for the design of PFMA and ACMSA, using DE, can be summarized as

follows:

3 Design of Microstrip Antennas Using Differential Evolution Algorithm

Step 1: Define an initial population of target vectors. Each target vector contains

different parameters of the design. For each parameter, a lower limit and an upper

limit are defined ( XiL< Xi, j(0) < Xi

U ).

Step 2: Randomly select the initial parameter values uniformly on the intervals

( XiL, Xi

U ).

Step 3: For each target vector, select other parameter vectors randomly.

Step 4: Add the weighted difference of any two of the parameter to the third

vector to form a donor vector.

Vk,n(t+1) = Xm,n(t) +F * (Xi,n(t) – Xj,n(t)) (1)

The scaling factor F scales the difference of two vectors and adds it to the third

one, which can range from 0 to 2.

Step 5: Components of donor vector enter into trial vector with probability CR.

Tk,n(t+1) =Vk,n(t+1) if rand (0,1) < CR

= Xk,n(t) otherwise (2)

Step 6: The trial vector is compared with the target vector and with a better

fitness, is admitted to the next generation.

The fitness function used for computation is

F= (1/N) ∑| Reflection coefficient at frequency i|

(3)

Where the summation is done over the frequency range of interest and N is the

number of frequency points, used in computation.

3 Design of PFMA Using DE

In cavity model analysis, a rectangular microstrip antenna is considered as a

rectangular electromagnetic cavity, bounded at its top and bottom by electric walls

and its sides by a magnetic wall. Cavity is a leaky cavity, where, leakage of

electromagnetic fields produces radiation by the structure. The geometry of a

rectangular PFMA and its equivalent circuit are shown in Fig. 1(a) and Fig. 1(b)

respectively. In Fig. 1(b), the microstrip antenna is represented by a parallel L-C-

R resonator circuit with inductance L1 and capacitance C1 and resistance R1. Lf is

the inductance of the co-axial probe.

A. Deb et al. 4

Figure 1(a): Probe-fed microstrip antenna. Figure 1(b): Equivalent circuit.

If the length of the antenna is L, the resonant frequency of the antenna is [1]:

fr = c/(2L √Єf) (4)

Where, “c” is the speed of light in free space, Єf is the effective permittivity of the

substrate [1], “h” is the height of the substrate and (x0,y0) represents the co-

ordinates of the probe feed. In this design, PTFE substrate with dielectric constant

2.4, height 1.6 mm and loss tangent of 0.001 is considered. The electric field

within the antenna is assumed to have only a vertical component and can be

expressed as a summation of ortho-normal functions [1-2]:

Ez(x,y)= jwμ0I0 ∑∑Ψmn(x,y) Ψmn(x0,y0) sinc(mπDx/2L) sinc(nπDy /2W) (5)

k2-kmn

2

where

Ψmn(x,y)= √(єmєn/(LW) cos(mπx/L) cos(nπy/W) (6)

and єp =1 for p=0 and єp =2 otherwise

Summation for both m and n ranges from zero to infinity. For practical realization

the maximum value of both m and n are taken to be 5. Contribution of other

higher order modes is assumed to be negligible. I0 is the feed point current,

(x0,y0) represents position of probe feed, k is the wave number and kmn

=(mπ/L)2+(nπ/W)

2. Dx and Dy represents the dimensions of probe when the cross

sectional area of the inner conductor is taken to be a rectangle. The input

impedance at the feed point can be calculated as:

Zin=-hEz(x0,y0)/I0 (7)

This input impedance computed would be purely reactive because the above

calculation does not take into account the losses associated within the antenna.

Basically, two losses, the power lost through radiation (Pr) and the loss within the

substrate is considered for further calculations by a modified loss tangent (δeff) as,

5 Design of Microstrip Antennas Using Differential Evolution Algorithm

δeff= Pr/(ωWT ) + 0.001 (8)

Where, ω is the operating frequency in radians and WT represents energy stored

within the antenna.

The modified value of “k” is written as

k2=k0

2 єr(1-j δeff) (9)

Using the modified value of “k”, the input impedance Zin is re-evaluated and once

Zin in calculated, the return loss (in dB) is calculated as:

S11 =20 log10 | (Zin-Z0)/(Zin+Z0)| (10)

Where, Z0 is the characteristic impedance of the feed line.

Aim is to design a PFMA at resonance frequency of 2.4 GHz. Optimization of the

PFMA is carried out separately using Differential Evolution, Particle Swarm

Optimization and Genetic algorithm and the optimized values of antenna

dimensions are length of the antenna L = 37.8 mm, width of the antenna W = 27.2

mm, co-ordinates of probe feed (x0,y0 ) are (16.55, 9.8). The optimization for

impedance matching is done for a frequency range of 2.38-2.42 GHz. Each of the

optimization schemes is run for 100 times and the fitness values at each

generation were averaged over 100 runs. 40 frequency points in the range of 2.38-

2.42 GHz were chosen to evaluate the fitness function as given in (3). In Fig. 2,

best fitness obtained using DE is compared with that obtained using GA and PSO.

Figure 2: Variation of best fitness with number of generations.

A. Deb et al. 6

Similarly in Fig. 3, the mean fitness obtained using DE, PSO and GA was

compared.

Figure 3: Variation of mean fitness with number of generations.

The return loss plot of the PFMA as obtained by each of the three optimization

schemes is shown in Fig. 4. The resonance frequency of the antenna is 2.4 GHz.

Bandwidth, obtained using DE is 30 MHz.

Figure 4: Variation of return loss with frequency obtained using DE, PSO and GA.

The radiation pattern of the PFMA at 2.4 GHz, obtained using DE, is evaluated and is

shown in Fig. 5.

7 Design of Microstrip Antennas Using Differential Evolution Algorithm

Figure 5: Radiation pattern of the PFMA.

4 Design of ACMSA Using DE

The geometry of ACMSA is shown in Fig. 6. For the design of ACMSA, PTFE

substrate of height 1.57 mm, dielectric constant of 2.32 and loss tangent of 0.001

is used. In order to obtain fitness function for DE the design of the ACMSA,

transmission line model is used [1-2]. The equivalent transmission line model of

the ACMSA of Fig. 5 is shown in Fig. 7. Various parameters like patch length (a),

patch width (b), slot length (La) and slot width (Wa), width of microstrip line (W)

and open stub length (Ls) are to be optimized for the desired resonant frequency.

Figure 6: Geometry of aperture-coupled microstrip antenna.

A. Deb et al. 8

Figure 7: Equivalent transmission line model of ACMSA.

A population size of 30 is used in DE with 80 iterations per generation and ran for

50 times. The coupling of the patch to the aperture is described by an impedance

transformer of turn ratio n1 roughly equal to the fraction of patch current

intercepted by the slot to the total patch current. The patch impedance is

determined at the centre of the slot and its value can be obtained from the simple

transmission line model. The antenna is designed to operate in frequency range of

5.470GHz – 5.725GHz, which is the frequency range of high performance local

area network-type 2, generally known as HIPERLAN/2 and used mainly for

indoor wireless communication with high data rate.

The input impedance of the transmission line can be expressed as [2]

Zin=n22/(n1

2Ypatch+Yap) - jZom Cot(âmLs) (11)

where n2 is the turns ratio of the transformer used to describe the coupling of the

patch to the microstrip line and its expression in terms of various dimensions can

be found in [2] and n1=La/b .

Where J0 is Bessel function of order zero and

k1 = k0 √ ( rf - res - rem) (12)

k2 = k0 √( res + rem - 1) (13)

βs = k0 √res βm = k0 √rem (14)

Where W, єrem, ßm, Zom are the width, effective permittivity, phase constant and

characteristic impedance of the microstrip line and Wa, єres, ßs and Zos are the

same parameters of the slot line respectively.

The aperture susceptance Yap can be obtained as follow as:

Yap = -j2Yos Cot(βs La /2) (15)

9 Design of Microstrip Antennas Using Differential Evolution Algorithm

S11=20log10{(Zin-Zom)/(Zin+Zom)} (16)

Where Zom is the characteristic impedance of the microstrip feed line.

Directivity D can be obtained as:

D=2b2/(15 Gr λo

2) (17)

λo is the free-space wave length

Gain of the antenna can be obtained as

G = ηD (18)

Where η is the antenna efficiency

A population size of 30 is used in DE with 80 iterations per generation. The best

individual obtained with the parameter values:

a=15mm, b=12.87mm, La=5.68mm, Wa=3.05mm, W=2.54 mm and Ls=6.84 mm.

Where „a‟ is the length of the patch, „b‟ is the width, „La‟ is the length of the slot,

„Wa‟ is the width of the slot, „W‟ is the width of the feed line and „Ls‟ is the stub

length.

In binary coded GA, a population size of 30 is chosen. The crossover probability

was taken to be 0.6 and the mutation probability as 0.05. The tournament selection

is used for reproduction, which is a very important step in GA. In Particle swarm

optimization, the inertia factor, local acceleration and global acceleration constant

was taken to be 0.5 and the swarm size was taken to be 30.

The natures of best fitness and mean fitness, obtained using DE are compared

with those obtained using MATLAB GA and PSO in Fig. 8 and Fig. 9

respectively. 300 frequency points in the range of 5.470-5.725 GHz were chosen

for evaluation of fitness function given in (3). In Fig. 8, it can be seen that the GA

has achieved a higher fitness function value compared to DE and PSO owing to its

drawback of local minima trapping. The convergence time is also higher in the

case of GA compared to DE and PSO. In case of DE and PSO the convergence

time is almost same and the best fitness values are much better compared to GA.

A. Deb et al. 10

Figure 8: Comparison of best fitness

Figure 9: Comparison of mean fitness

In Fig. 10, the return loss of the antenna, obtained using DE, is compared with the

results obtained using GA and PSO. The resonance frequency of the antenna is

5.59GHz. The -10dB return loss bandwidth of the ACMSA, obtained using DE is

360 MHz which is about 6.5% of the centre frequency.

11 Design of Microstrip Antennas Using Differential Evolution Algorithm

Figure 10: Return losses of ACMSA obtained using DE, PSO and GA.

The radiation pattern of the ACMSA is plotted in Fig.11.

Figure 11: Radiation pattern of the ACMSA.

In Fig. 11, in both the planes, radiations in the lower hemisphere are due to the

aperture on the ground plane.

A. Deb et al. 12

5 Conclusion

The method of designing single-layer microstrip antenna and complex structure

like multi-layer aperture coupled microstrip antenna, using differential evolution

algorithm, is presented in this paper. DE scheme, used for antenna design, is

DE/rand/1. Results are verified by measurement. In case of DE and PSO the

convergence time is almost same and the best fitness values are much better

compared to GA. Fitness value is better for DE(0.0023) compared to PSO(0.01).

DE requires lesser control variables compared to GA and PSO and is easier to

tune it to a particular optimization problem. It also requires lesser population size

compared to other optimization techniques.

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