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Central Force Optimization: Nelder-

Mead Hybrid Algorithm for RectangularMicrostrip Antenna DesignK. R. Mahmoud

a

a

Electronics and Communications Department, Faculty ofEngineering , Helwan University , Helwan, Egypt

Published online: 31 Oct 2011.

To cite this article:K. R. Mahmoud (2011) Central Force Optimization: Nelder-Mead HybridAlgorithm for Rectangular Microstrip Antenna Design, Electromagnetics, 31:8, 578-592, DOI:

10.1080/02726343.2011.621110

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Electromagnetics, 31:578592, 2011

Copyright Taylor & Francis Group, LLC

ISSN: 0272-6343 print/1532-527X online

DOI: 10.1080/02726343.2011.621110

Central Force Optimization: Nelder-Mead HybridAlgorithm for Rectangular MicrostripAntenna Design

K. R. MAHMOUD1

1Electronics and Communications Department, Faculty of Engineering,

Helwan University, Helwan, Egypt

Abstract In this article, an efficient global hybrid optimization method is proposedcombining central force optimization as a global optimizer and the Nelder-Meadalgorithm as a local optimizer. After the final global iteration, a local optimization can

be followed to further improve the solution obtained from central force optimization.The convergence capability of the hybrid central force optimizationNelder-Mead

approach is compared with other recent evolutionary-based algorithms using 13benchmark functions grouped into unimodal and multimodal functions. In addition,

the proposed algorithm is used to calculate accurately the resonant frequency andfeed-point position of rectangular microstrip patch antenna elements with various

dimensions and various substrate thicknesses. It is found that, in addition to decreasingthe required evaluation number and the required processing time, excellent results are

obtained.

Keywords central force optimization, Nelder-Mead algorithm, rectangular microstripantenna, resonant frequency, feed position

1. Introduction

In the past few decades, nature-inspired computation has attracted more and more at-

tention to tackle complex computational problems. Among them, the most successful

are evolutionary algorithms (EAs), which draw inspiration from evolution by natural

selection. There are several different types of EAs, including genetic algorithms (GAs),

genetic programming (GP), evolutionary programming (EP), and evolutionary strategies

(ES) (Abraham et al., 2006). In recent years, a new kind of computational intelligence

known as swarm intelligence (SI), which was inspired by collective animal behavior, has

been developed. The SI includes different algorithms. The first one is particle swarm

optimizer (PSO), which gleaned ideas from swarm behavior of bird flocking or fish

schooling (Kennedy & Eberhart, 1995). Another SI algorithm is ant colony optimization

(ACO), which was developed based on ants foraging behavior (Dorigo et al., 1996).

Recently, bacterial foraging behavior, known as bacterial chemotaxis (BC), has served

as the inspiration of two different stochastic optimization algorithms. The first is the BC

algorithm, which was based on a BC model (Muller et al., 2002). The way in which

Received 18 January 2011; accepted 3 July 2011.

Address correspondence to Korany R. Mahmoud, Helwan University, Electronics and Com-munications Department, 1 Sherif Street, Faculty of Engineering, Helwan 11792, Egypt. E-mail:[email protected]

578


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580 K. R. Mahmoud

algorithm, proposed in Mahmoud (2010), to optimize a bow-tie antenna for a 2.45-GHz

RFID reader. The BSO-NM algorithm has produced results better than those generated

by other algorithms.

The hybrid CFO-NM algorithm and its implementation details are introduced in

Section 2. The problem formulation for both benchmark functions and the MSA are

illustrated in Section 3. The simulation results are given in Section 4, and, finally,

Section 5 presents the conclusions.

2. Hybrid CFO-NM Algorithm

The CFO is a gradient-like deterministic algorithm that explores DS by flying a group

of probes whose trajectories are governed by equations analogous to the equations

of gravitational motion in the physical universe (Formato, 2007, 2008, 2009a, 2009b,

2010). In the physical universe, objects traveling through three-dimensional space become

trapped in close orbits around highly gravitating masses, which is analogous to locating

the maximum value of an objective function. In the CFO metaphor, mass is a user-

defined function of the value of the objective function to be maximized. In general,

CFO starts with user-specified initial probes positions and accelerations distributions.

The initial probes are considered here to be uniformly distributed on each coordinate,

and the initial acceleration vectors are usually set to zero.

CFO finds the maxima of an objective function f .x1; : : : ; xNd/ by flying a set of

probes through the DS along trajectories computed using the gravitational analogy. In

an Nd-dimensional real-valued DS, each probe p with position vector R.p; i;j / 2 RNd

experiences acceleration A.p;i;j/at discrete time step j , given by

A.p;i;j/ DGNpXkD1kp

UM.k;j/ M.p; j /M.k; j / M.p;j/

R.k;i;j/ R.p;i;j/

vuut NdXmD1

R.k;m;j/ R.p;m;j/2

; (1)

where Np is the total number of probes; p D 1; : : : ; N p is the probe number; j D

0 ; : : : ; N t is the time step (iteration); G is the gravitational constant; R.p; i;j / is theposition vector of probe p at step j ; M.p;j/ D f .R.p;i; j // is the fitness value at

probe p at time step j ; and are the CFO exponents ( D D 2) (Formato, 2007,

2008, 2009a, 2009b, 2010); and U. / is the unit step function. The unit step U. / is

essential because it creates positive mass, thus insuring that CFOs gravity is attractive.

Each probes position vector at step j will be updated according to the following

equation:

R.p;i;j/ D R.p;i;j 1/ C1

2A.p;i;j 1/t2; j 1; (2)

wheret is the unit time step increment. After the probes position updating, the probe

may fly outside the DS. Therefore, it should be enforced to stay inside the desired


5/16

Hybrid CFO-NM Algorithm for MSAs Design 581

domain of interest. There are many possible probe retrieval methods. A useful one is

the reposition factor, Frep, which plays an important role in a voiding local trapping

(Formato, 2009a). If the probe R.p; i;j / falls below R mini , then it is assigned to be

R.p;i;j/ DRmin

i C Frep:

R.p;i;j 1/ Rmin

i

: (3)

But, if the probe R.p; i;j / is greater than Rmaxi , then

R.p;i;j/ D Rmaxi Frep:

Rmaxi R.p;i;j 1/

; (4)

whereRmini andRmaxi are the minimum and maximum values of thei th spatial dimension

corresponding to the optimization problem constraints. In general, the range ofFrep is

set from 0 to 1, or it may be variable (Qubati et al., 2010). In this article, the reposition

factor is set to be 0.05. After every kth step, the DS size is adaptively reduced around

the probes location with the best fitness Rbest, where the DSs boundary coordinates willbe reduced by one-half coordinate-by-coordinate basis (Formato, 2010). Thus,

KRmini D Rmini C

1

2

Rbest R

mini

; (5)

KRmaxi D Rmaxi

1

2

Rmaxi Rbest

: (6)

Just the global deterministic algorithm is completed; the NM local optimization technique

is followed to finely optimize the results. NM-based local optimization approaches are

among the most popular approaches for solving unimodal problems. While these methods

are sometimes appropriate, they are not effective for problems that contain several local

minima and for problems where the DS is highly discontinuous or convoluted. For these

types of problems, heuristic global search approaches, such as the CFO algorithm, are

more effective. But such methods as CFO are inefficient for fine-tuning solutions once a

near-global minimum is found. For problems that contain several local minima, a hybrid

approach starting with a global method and then fine-tuning with a local method may

be more attractive. Therefore, with the CFO-NM hybrid method, the advantages of both

methods will be combined.

The NM simplex algorithm attempts to minimize a scalar-valued nonlinear function

of Nd real variables using only function values without any derivative information(Nelder & Mead, 1965). The NM method thus falls in the general class of direct

search methods. It is based on the comparison of the function values at the ( NdC 1)

vertices for Nd-dimensional decision variables. The selection of these points can be

prescribed, but random selection allows the potential to fully investigate the merit space.

The simplex method essentially has four possible steps during each iteration: reflection,

contraction in one dimension, contraction around the low vertex, and expansion. Four

scalar parameters must be specified to define a complete NM method: coeffcients of

reflection (), expansion (), contraction (), and shrinkage (). The nearly univer-

sal choices used in the standard NM algorithm are D 1, D 2, D 0:5, and

D 0:5. More details about the NM algorithm is illustrated in Nelder and Mead

(1965). Figure 1 shows a flowchart diagram of the main steps of the hybrid CFO-NM

algorithm.


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582 K. R. Mahmoud

Figure 1. Flowchart showing the main steps of the hybrid CFO-NM algorithm.

3. Problem Formulation

3.1. Test Functions

To fully evaluate the performance of the hybridized CFO-NM algorithm, a large set of

standard benchmark functions, which are given in Table 1, was first employed, indicating

the optimum minimum value (fmin) for each function. The set of 13 benchmark functions

can be grouped into unimodal functions (f1 to f5), multimodal functions (f6 tof10), and

low-dimensional multimodal functions (f11 to f13). The obtained optimum minimum


7/16


Table 1

Benchmark functions

Test function Nd S Neval fmin

Unimodal functions

f1.x/ DnX

iD1x2i 30 100; 100n 150,000 0

f2.x/ DnX

iD1jxi j C

nYiD1jxi j 30 10;10n 150,000 0

f3.x/ DnX

iD1

0@ iX

jD1xj

1A

2

30 100; 100n 250,000 0

f4.x/ D maxifjxi j; 1 i ng 30 100; 100n 150,000 0

f5.x/ Dn

1X

iD1

100

xiC1 x2i

2 C .xi 1/2 30 30;30n 150,000 0

Multimodal functions, many local maxima

f6.x/ DnX

iD1

xi sin

pjxi j

30 500; 500n 150,000 12,569.5

f7.x/ DnX

iD1

x2i 10 cos.2xi /C 10

230 5:12; 5:12n 250,000 0

f8.x/ D20 exp0@0:2

vuut 1n

nXiD1

x2i

1A exp

1

n

nXiD1

cos2x i

!C 20C e 30 32; 32n 150,000 0

f9.x/ D 14000

30XiD1

.xi 100/2 nY

iD1cos

xi 100p

i

C 1 30 600; 600n 150,000 0

f10.x/ D n

(10 sin2.y1/C

29XiD1

.yi 1/2 1C 10 sin2.yi C 1/

C.yn 1/2)

C30XiD1

u.xi ; 10; 100;4/

yiD 1C 14

.xi C 1/

u.xi ;a;k;m/ D 8 a0 a xi ak.xi a/m xi < a

30 50;50n 150,000 0

Multimodal functions, few local maxima

f11.x/ D 4x21 2:1x41C 1

3x61 C x1x2 4x22 C 4x42 2 5;5n 1,250 1.0316285

f12.x/ D

x2 5:142

x21 C 5

x1 6

2C 10

1 1

8

cos x1 C 10 2 5;10 0; 15 5,000 0.398

f13.x/ D1C .x1 C x2C 1/2

19 14x1 C 3x21 14x2 C 6x1x2 C 3x22

30C .2x C 1 3x2/218 32x1 C 12x

2

1C48x2

36x1x2

C27x2

2

2 2;2n 10,000 3


8/16

584 K. R. Mahmoud

results are compared with other optimization algorithms, such as GA, PSO, and GSO,

in addition to stand-alone CFO, following the same evaluation number used in He et al.

(2009).

3.2. Rectangular Microstrip Patch Antennas

In rectangular MSA design, it is important to determine the resonant frequency of the

antenna accurately, because it has a narrow bandwidth and can only operate effectively

in the vicinity of the resonant frequency (Bahl & Bhartia, 1980). In addition, the feed-

point calculation in this type of antenna is difficult, especially when the antenna size

is drastically small. The resonant frequency of rectangular MSA is generally calculated

using either an analytical technique or numerical methods. The analytical techniques offer

both simplicity and physical insight but depend on several assumptions and approxima-

tions that are valid only for thin substrates, normally of the order of h=o 0:0815,

where h is the thickness of dielectric substrate and o is the free-space wavelength.

Forh=o > 0:0815, the properties of patch antenna change drastically (Bahl & Bhartia,1980). Therefore, in this case, the designers are forced to obtain the accurate resonant

frequency using a trial-and-error method or by using numerical methods that usually

require considerable computational time and costs. Recently, the resonant frequencies

of both electrically thin and thick rectangular MSAs are obtained by using different

algorithms (Guney & Sarikaya, 2007; Gollapudi et al. 2008, 2011; Lohokare et al. 2009;

Kalinli et al., 2010).

In Guney and Sarikaya (2007), a combination of artificial neural networks with an

adaptive-network based fuzzy interference system (ANFIS) is considered for calculating

resonant frequency of a rectangular MSA. The biogeography-based optimization (BBO)

as a new bioinspired optimization technique was proposed by Lohokare et al. (2009) for

accurate determination of resonant frequency and feed-point calculation of a rectangular

microstrip antenna. Kalinli et al. (2010) presented a new approach based on feed-

forward artificial neural networks trained with a Levenberg-Marquardt parallel ant colony

optimization (PACO) algorithm to determine the resonant frequencies of the rectangular,

circular, and triangular MSAs. Gollapudi et al. (2008) applied the bacterial foraging

algorithm (BFA) to calculate the resonant frequency of a rectangular microstrip antenna.

Further, bacterial foraging is applied to the calculation of the feed point of a microstrip

antenna. Moreover, some modification of the BFA is done for faster convergence; the BFA

is oriented by PSO to combine both algorithms advantages, which is called the bacterial

swarm optimization (BSO) algorithm. This combination aims to make use of PSO ability

to exchange social information and BFA ability in finding a new solution by eliminationand dispersal. The hybridized optimization technique called velocity modulated bacterial

foraging optimization (VMBFO) was tested by Gollapudi et al. (2011) to calculate the

resonant frequency of a rectangular microstrip antenna.

To illustrate the capabilities of the proposed CFO-NM algorithm as an optimization

technique in antenna design, the edge extension (W) is optimized to calculate accurately

the resonant frequency of electrically thin and thick rectangular MSAs. In addition,

the input resonant resistance of a probe-fed rectangular MSA is optimized to 50

by searching the appropriate feed-point position (yo). The results are compared with

the theoretical and experimental results reported by other scientists (Guney & Sarikaya,

2007; Gollapudi et al. 2008, 2011; Lohokare et al. 2009; Kalinli et al., 2010).

Figure 2 illustrates a rectangular patch of width W and lengthL over a ground plane

with a substrate of thickness h and a relative dielectric constant. The resonant frequency


9/16


Figure 2. Configuration of a rectangular MSA.

of a rectangular patch antenna driven at its fundamental TM10 mode was given in Kara

(1996) as

frc D co

2.L C 2W /p"e.W /; (7)

whereco is the velocity of electromagnetic waves in free space, and "e.W / is the effectivedielectric constant, which is obtained from Schneider (1969);

"e.W / D "r C 1

2 C

"r 1

2

s1 C 10

h

W

: (8)

Then, the edge extension can be calculated from Hammerstad (1975):

W D 0:412h

"e.W / C 0:300W

h C 0:264

"e.W / 0:258

W

h C 0:813

: (9)

This expression is completely based on the substrate thickness (h). As the substrate

thickness increases, the expressions for the calculation of resonant frequencies begin

to lose accuracy and applicability. Therefore, the proposed algorithm is considered to

optimize widthW to obtain the optimum value to the experimental resonant frequency

(frm) using the following objective function:

Objective_function D min jfr j; (10)

fr D 100%

1 frc

frm

; (11)


10/16

586 K. R. Mahmoud

wherefr is the resonance frequency percentage error. By obtaining the objective values,

it will be simple to move each probe (W ) to its next location. After this process is

carried out for each probe, it will be repeated until the number of time steps is finished.

The NM algorithm will then start with the CFO-optimized edge extension value (W).

Also, the hybrid CFO-NM technique will be used to search for the feed-point position

(yo) to match the input impedance of the MSA to 50 . The feed point is calculated

using the following equation (Richards et al., 1981):

Rinc .y D yo/ D Rin.y D 0/ cos2

Lyo

: (12)

The input resistance percentage error (Rin) between the required resonant input resis-

tance (Rinm D 50 ) and the calculated input resistance (Rinc ) at the feed-point position

yo is calculated by

Rin D 100%

1

Rinc

Rinm

: (13)

The feed position yo is optimized to obtain the required resonant input resistance Rinmusing the following objective function:

Objective_function D min jRinj: (14)

4. Simulation Results

4.1. Test Functions

The CFO algorithm is tested on a set of unimodal functions in comparison with other

algorithms (He et al., 2009). Table 2 shows that the stand-alone CFO generated sig-

nificantly better results than the GA on all the unimodal functions f1f5. From the

comparisons between CFO and the PSO, it can be seen that CFO had significantly

better performance onf4 andf5. However, the CFO algorithm yielded statistically worse

results on the benchmark functions f1 and f2 and a comparable result on f3 compared

to PSO. In addition, CFO performs better than GSO on f3, f4, and f5. Figure 3 shows

the normalized fitness versus the iterations number for the 13 functions using CFO.

It is found that all functions are approximately converged before the first 50 iterations.

Therefore, in the hybrid CFO-NM algorithm, the first 50 iterations will be executed usingCFO, and the remainder evaluation number will be performed by the NM algorithm. The

obtained results in Table 2 indicated that the CFO-NM algorithm converges better than

other algorithms except functions f1 and f2 optimized by PSO.

Multimodal functions with many local minima are regarded as the most difficult

functions to be optimized, since the number of local minima increases exponentially as

the function dimension increases. From Table 2, it is clearly seen that for the multimodal

tested benchmark functionsf6f10, the CFO-NM markedly outperformed GA, PSO, and

GSO. The only exception is the Rastrigin function (f6/ in which the GSO algorithm

slightly outperformed to the CFO-NM algorithm by a very small percentage of 1.255 E

07. The other set of benchmark functions f11f13 are also multimodal but in low

dimensions (Nd D 2), and they have only a few local minima. From the comparison

shown in Table 2, it can be found that the CFO-NM algorithm outperformed GA and


11/16

Table2

Comp

arisonofCFOandCFO-N

Mw

ithotheralgorithmsonbenchmarkfunctions

Presentresults

Previouslypublishedresults

CFO

CFO-N

M

Test

functio

n

fmin

GAa

PSOa

GSOa

CFOb

Bestfitness

Time(sec)

Bestfitness

Time(sec)

Unimodalfunctions(otheralgorithms:averageof1,0

00runs)

f1

0

3.7

11

3.

6927E37

1.9

481E8

2.6

592E02

3.4

006198317872E0

2

84.5

3

8.8

659725907481E25

13.2

3

f2

0

0.5

771

2.

9168E24

3.7

039E05

4.0

0E08

4.0

4774459401E0

4

90.4

8

9.4

690892316E05

9.2

2

f3

0

9,7

49.9

145

1.1

979E03

5.7

829

6.0

0E08

7.2

00253384852E0

3

150

5.

890192E09

37.1

9

f4

0

7.9

610

0.4

123

10.7

E02

4.2

0E07

8.7

56509522136E0

3

74.9

18

6.

406404160714E03

10.4

f5

0

338.5

616

37.3

582

49.8

359

2.1

7187E02

11.7

22674316984

73.6

2

1.

12593331E07

11.3

8

Multimodalfunctions,manylocalm

axima(otheralgorithms:averageof1,00

0runs)

f6

12,5

69.5

12,5

66.0

977

9659.6

993

12,

569.

4882

12,5

69.4

852

12,5

69.4

457090996

74.4

5

12,5

69.4

866181729

11.3

5

f7

0

6.5

09E01

20.7

863

1.0

179

3.5

2E06

9.8

9943525291968E0

1

127.5

6

6.

89346922442722E23

7.8

5

f8

0

0.8

678

1.3

404E03

2.6

548E05

1.5

0E07

2.7

380954126433E0

2

77.8

4

1.

49080747746666E11

7.3

6

f9

0

1.0

038

0.2

323

3.0

792E02

2.0

0124

5.3

3728986617E0

4

76.5

9

4.

9960036108132E15

6.9

1

f10

0

4.3

572E02

3.9

503E02

2.7

648E11

1.0

585E01

7.2

09336298027E0

3

88.4

2

1.

93295817450142E21

22.8

5

Multimodalfunctions,fewlocalm

axima(otheralgorithms:averageof50runs)

f11

1.0

316285

1.0

298

1.0

160

1.0

31628

1.0

31607

1.0

3082061159916

0.3

52

1.

03162845348

988

0.2

31

f12

0.3

98

0.4

040

0.4

040

0.3

979

0.3

98

0.3

97892565359964

1.2

74

0.

39799999999

9999

0.2

21

f13

3

7.5

027

3.0

050

3

3

30.8

612320701999

2.4

8

3.

00000000000

0003

0.2

2

a

He

etal.(2009).

b

For

mato(2010).

Bold

faceindicatesthebestobtainedfitnessvalue.

587


12/16

588 K. R. Mahmoud

Figure 3. Normalized fitness values versus the iterations number for the benchmark functionsusing CFO algorithm. (color figure available online)

PSO algorithms and was comparable to GSO for these sets of functions. In summary, the

search performance of the CFO-NM algorithm is better than other algorithms, especially

for multimodal functions. It should be noted that the good results obtained in Formato

(2010) are due to determining where along the diagonal of the orthogonal the probe

array would be placed by selecting the best value of the parameter , which start from

0 to 1. That means a total of 11 runs were made with 0 1 in incrementsof 0.1 to determine the best value that corresponds to the best fitness, which means

excessive runs should be performed first for each optimized problem to determine its

correspondingbest.

Comparing CFO with CFO-NM, it is found that the hybrid CFO-NM algorithm

performs better than CFO in addition to decreasing the average required processing time

to 13.72% of the CFO required time. Table 2 indicates the required processing time on a

Dell Latitude D530 (Core 2 Due Intel Processor 2 GHz, 2-GHz RAM, Malaysia) to get

the results for each function.

4.2. Rectangular Microstrip Patch Antennas

To calculate accurately the resonant frequency of rectangular MSAs, the edge extensionhas first been optimized by the CFO algorithm with an evaluation number of 1,280,

which is the same evaluation number used in Gollapudi et al. (2008). It is found that for

the 16 MSAs described in Table 3, better results are obtained using the stand-alone CFO

algorithm compared to ANFIS, PACO, BBO, and BFA algorithms; where the absolute

resonance frequency percentage error fr ranges from 6.79208E06 to 4.57334E08,

as depicted in Table 4. It should be noted that the VMBFO algorithm performs better

than CFO due to increasing the number of evaluations from 1,280 to 30,000. From the

comparisons between CFO and BSO, it can be seen that CFO is slightly better than

BSO. Then, the hybrid CFO-NM algorithm is used with a total evaluation number of

760 (640 evaluations using the CFO algorithm and the remaining 120 using the NM

algorithm), which required 0.259329 sec to get the results compared to 0.368698 sec

to get the result using the CFO algorithm. It is found that in addition to decreasing


13/16


Table 3

Measured resonant frequency results and dimensions for

electrically thin and thick rectangular MSAs

Patch no.

Measured

fr (GHz) L (mm) W (mm) yo (mm) "r h (mm) h=d

1 7.74 12.90 8.50 4.15 2.22 0.17 0.0065

2 8.45 11.85 7.90 4.10 2.22 0.17 0.0071

3 3.97 25.00 20.00 6.83 2.22 0.79 0.0155

4 7.73 11.83 10.63 3.90 2.22 0.79 0.0326

5 4.6 10.00 9.10 3.75 10.2 1.27 0.0622

6 5.06 18.60 17.20 5.94 2.33 1.57 0.0404

7 4.805 19.60 18.10 6.27 2.33 1.57 0.0384

8 6.56 13.50 12.70 4.25 2.55 1.63 0.0569

9 5.6 16.21 15.00 5.28 2.55 1.63 0.0486

10 6.2 14.12 13.37 4.75 2.55 2.00 0.0660

11 7.05 12.00 11.20 4.25 2.55 2.42 0.0908

12 5.8 14.85 14.03 4.60 2.55 2.52 0.0778

13 5.27 16.30 15.30 4.70 2.50 3.00 0.0833

14 7.99 10.18 9.05 3.70 2.50 3.00 0.1039

15 6.57 12.80 11.70 3.40 2.50 3.00 0.1263

16 5.1 15.80 13.75 5.82 2.55 4.76 0.1292

the number of evaluations to 40% and the required time to 30%, excellent results are

obtained where the absolute percentage errorfr ranges from 2.22E14 to absolute zero,

as illustrated in Table 4. It is very clear that the results of CFO-NM show better agreement

with the experimental results as compared to the previously published results. From the

comparison shown in Table 4, it can be found that the proposed CFO-NM algorithm

is comparable to BSO-NM algorithm.

For the feed-point position calculation, the 50 ohms of input resistance can be

obtained by varying the distance from the radiating edge of the MSA element to the

feed location (yo). In this section, a TM10 mode rectangular microstrip patch antenna

having length L D 0:906 cm, width W D 1:186 cm, substrate relative permittivity

"r D 2:2, and operating at 10 GHz frequency is considered. Table 5 shows the re-sults of a feed-point position and input resistance percentage error Rin comparison

between previously published results and those obtained using CFO and hybrid CFO-

NM algorithms. It is found that the position of the feed point is 0.59343105939 cm by

using CFO technique, giving a resistance percentage error of 4.46E06. From Table 5,

it is clearly seen that CFO markedly outperformed BFA, BBO, and BSO. However,

using the CFO-NM technique with a lower number of evaluations, the obtained feeding

position is found to be 0.5934310628 cm, giving an input resistance percentage error of

8.88E14.

Generally, the proposed hybrid CFO-NM optimization algorithm shows the ability

to predict the resonant frequency for a new set of antenna and substrate parameters in

addition to determining the appropriate feed position to match the input impedance of

the MSA to the required input resistance.


14/16

Table4

Comparisonofmea

suredandcalculatedresonantfrequenciesofelectricallythin

andthickrectangularMSAs

Previouslypublishedresults

Presentresults

Patchno.

Measured

fr

(GHz)

fr

(%)

ANFISa

fr

(%)

P

ACOb

fr

(%)

BBOc

1,020

evaluations

fr(%

)

BFAd

1,280

evaluations

fr

(%)

VMBFOe

30,000

evaluations

fr

(%)

BSO

1,280

evaluations

fr

(%)

BSONM

760

evaluations

fr

(%)

CFO

1,280

evaluations

fr

(%)

CFO-NM

760

evaluations

1

7.74

4.91E02

2.58E03

6.46E02

0.4

9

3.96E09

3.12E06

0.0

0EC00

6.84264E08

0.0

0EC00

2

8.45

6.51E02

1.07E02

1.70E01

0.1

6

3.52E09

2.08E06

1.

11E14

2.41583E07

1.1

1E14

3

3.97

2.77E02

3.02E02

2.02E02

0.0

8

7.26E09

5.41E07

1.

11E14

4.57334E08

1.1

1E14

4

7.73

4.40E02

1.94E02

9.31E02

0.1

1

3.65E09

1.66E06

1.

11E14

7.13986E07

1.1

1E14

5

4.6

2.61E02

8.70E03

3.91E02

0.0

2

6.00E09

2.77E06

0.0

0EC00

2.98819E06

0.0

0EC00

6

5.06

4.94E02

7.71E02

5.14E02

0.1

5.48E09

2.80E06

0.0

0EC00

1.90196E06

0.0

0EC00

7

4.805

7.83E01

1.44E01

2.29E02

0.0

8

5.63E09

3.06E06

0.0

0EC00

3.00887E06

0.0

0EC00

8

6.56

3.05E03

3.35E02

1.68E02

0.2

1

4.00E09

2.57E06

1.11E14

2.9746E06

0.0

0EC00

9

5.6

9.46E02

1.07E01

6.61E02

0.3

4.35E09

1.97E05

0.0

0EC00

2.44956E06

0.0

0EC00

10

6.2

3.02E01

1.16E01

6.77E02

0.3

3.63E09

1.40E06

0.0

0EC00

1.23598E06

1.11E14

11

7.05

1.56E02

1.13E02

1.21E01

0.2

3.04E09

4.21E06

2.22E14

2.25777E06

0.0

0EC00

12

5.8

1.21E02

9.66E02

1.55E02

0.2

5

3.39E09

2.64E06

0.0

0EC00

5.65174E07

0.0

0EC00

13

5.27

1.44E01

8.54E02

5.88E02

0.0

7

3.48E09

5.27E06

2.22E14

2.46799E06

0.0

0EC00

14

7.99

2.00E02

6.63E02

8.14E02

0.1

7

2.17E09

7.08E06

2.

22E14

6.79208E06

2.2

2E14

15

6.57

1.52E03

2.18E01

1.07E02

0.5

5

2.52E09

3.66E06

2.22E14

1.83461E06

0.0

0EC00

16

5.1

4.31E02

4.12E02

1.96E02

0.2

3

3.24E09

8.29E06

2.

22E14

3.67783E06

2.2

2E14

Totalabsolute

percentageerror

1.68037

1.06798

0.9189

3.3

2

6.53E08

7.09E05

1.55E13

3.32E05

8.8

8E14

aGuneyandSarikaya(2007).

bKalinlietal.(2010).

cLo

hokareetal.(2009).

dGollapudietal.(2008).

eGollapudietal.(2011).

Boldfaceindicatestheminimum

percentageerrorobtainedforeachpatchnumber.

590


15/16


16/16

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