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A new effective side length expression obtained using a modified tabu search algorithm for the resonant frequency of a triangular microstrip antenna

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A New Effective Side Length Expression ObtainedUsing a Modiﬁed Tabu Search Algorithm forthe Resonant Frequency of a TriangularMicrostrip Antenna
Dervis
¸
Karaboga, Kerim Guney, Ahmet Kaplan, Ali Akdagli
ˇ ¨ ˇ
Department of Electronic Engineering, Erciyes University, 38039 Kayseri, Turkey;e-mail: karaboga@zirve.erciyes.edu.tr
Recei
ed 2 August 1996; re
ised 2 December 1996
ABSTRACT: A new, very simple curve-ﬁtting expression for the effective side length ispresented for the resonant frequency of triangular microstrip antennas. It is obtained using( )a modiﬁed tabu search algorithm, and is useful for the computer-aided design CAD of microstrip antennas. The theoretical resonant frequency results obtained using this new effective side length expression are in very good agreement with the experimental resultsavailable in the literature.
1998 John Wiley & Sons, Inc.
Int J RF and Microwa
e CAE
8: 4–10,1998.
Keywords:
microstrip antenna; triangular; resonant frequency; optimization; effective sidelength; tabu search algorithm
INTRODUCTION
Microstrip antennas are among the most popularantenna types, because they are lightweight, havesimple geometries, are inexpensive to fabricate,and can easily be made conformal to the host
body 1
5 . The majority of the studies proposedin this area have concentrated on rectangular andcircular microstrip antennas. However, it is knownthat the triangular patch antenna has radiationproperties similar to those of the rectangularantenna, with the advantage of being physicallysmaller. Triangular microstrip antennas present aparticular interest for the design of periodic ar-rays because triangular radiating elements can bearranged in a manner that allows the designer toreduce signiﬁcantly the coupling between adja-cent elements of the array. This signiﬁcantly sim-pliﬁes array design. In triangular microstrip an-tenna designs, it is important to determine theresonant frequencies of the antenna accurately
Correspondence to: K. Guney
¨
because microstrip antennas have narrow band- widths and can only operate effectively in the vicinity of the resonant frequency. As such, atheory to help ascertain the resonant frequency ishelpful in antenna designs.The resonant frequency of such antennas is afunction of the side length of the patch, thepermittivity of the substrate, and its thickness. A
number of methods 1, 6
14 are available todetermine the resonant frequency of an equilat-eral triangular microstrip patch antenna, as this isone of the most popular and convenient shapes.The experimental resonant frequency results of
this antenna have been reported elsewhere 7,
11 . The theoretical resonant frequency values
presented in the literature 1, 6
14 are not in very good agreement with the experimental re-sults. For this reason, a new, very simple effectiveside length expression is presented in this articlefor an equilateral triangular patch antenna. Theresonant frequencies of this antenna are thenobtained by using this new effective side length
1998 John Wiley & Sons, Inc. CCC 1096-4290
98
010004-07
4
Effecti
e Side Length Expression
5
expression and the relative dielectric constant of the substrate.In this work, ﬁrst, a model for the effectiveside length expression is chosen, then the un-known coefﬁcient values of the expression areobtained by a modiﬁed tabu search algorithm.The tabu search algorithm is a very efﬁcient andﬂexible optimization technique developed espe-
cially for combinatorial optimization problems 15,
16 . But, it has also produced very good solutions
for numerical optimization problems 17
19 . Thetabu search algorithm used here employs anadaptive neighbor production mechanism. There-fore, this algorithm is different from the tabu
search algorithms in the literature 15, 16 .The theoretical resonant frequency results ob-tained using the new, simple effective side lengthexpression presented here are in very good agree-
ment with the experimental results 7, 11 . Guney
¨
20
22 also proposed very simple expressions foraccurately calculating the resonant frequencies of rectangular and circular microstrip antennas.Most of the previous theoretical resonant fre-quency results for triangular microstrip antennas were compared only with the experimental results
reported by Dahele and Lee 7 . In this work, thetheoretical results obtained using the formulaeavailable in the literature are compared with theexperimental results reported by Dahele and Lee
7 , and also Chen et al. 11 .
FORMULATION
For a triangular microstrip antenna, the resonantfrequencies obtained from the cavity model withperfect magnetic walls are given by the formula
6 :2
c
1
22 2
Ž .
f
m
mn
n
1
mn
1
2
Ž .
3
a
r
where
c
is the velocity of electromagnetic wavesin free space,
is the relative dielectric constant
r
of the substrate, subscript
mn
refers to TM
mn
modes, and
a
is the length of a side of thetriangle, as shown in Figure 1.
Ž .
Eq. 1 is based on the assumption of a perfectmagnetic wall and neglects the fringing ﬁelds atthe open-end edge of the microstrip patch. Toaccount for these fringing ﬁelds, there are a num-
ber of suggestions 1, 6
14 . The most common
Ž .
suggestion is that side length
a
in eq. 1 bereplaced by an effective value
a
. The same
eff
suggestion is also used in this study. The effective
Figure 1.
Geometry of equilateral triangular mi-crostrip antenna.
side length,
a
, which is slightly larger than the
eff
physical side length
a
, takes into account theinﬂuence of the fringing ﬁeld at the edges andthe dielectric inhomogeneity of the triangular mi-crostrip patch antenna. It is clear from all of the
formulae proposed 1, 6
14 that the effectiveside length of a triangular microstrip antenna isdetermined by the relative dielectric constant of the substrate,
, the physical side length,
a
, and
r
the thickness of the substrate,
h
. Therefore, theeffective side length expression,
a
, to be found
eff
must be larger than
a
and depend on
,
a
,
r
and
h
.The problem in the literature is that an expres-sion that is as simple as possible for the effectiveside length should be obtained, but the theoreti-cal results obtained by using the expression mustbe in good agreement with the experimental re-sults. In this work, a new technique based on thetabu search algorithm for solving this problemefﬁciently is presented. First, a model for theeffective side length expression is chosen, thenthe unknown coefﬁcients of the model are deter-mined by a modiﬁed tabu search algorithm.To ﬁnd the proper model for the effective sidelength expression, many experiments were carriedout in this work. After many trials, the followingmodel, depending on
,
a
, and
h
, which pro-
r
duces good results, was chosen:
2
Ž .
a
a
h
2
eff 1
3
ž /
r
where the unknown coefﬁcients
,
, and
1 2 3
are determined by a modiﬁed tabu search algo-
Karaboga et al.
ˇ
6
Ž .
rithm. It is evident from eq. 2 that the effectiveside length,
a
, is larger than the physical side
eff
length,
a
, provided
,
, and
are greater
1 2 3
than zero.The tabu search algorithm, which is based onintelligent problem solving tenets, is an optimiza-tion technique developed especially for combina-torial optimization problems, but it has also pro-duced efﬁcient solutions for numeric problems. Itis a form of iterative search and does not usederivative-based transition rules.The tabu search starts with an arbitrary solu-tion created by a random number generator. Inthis particular problem, it is equivalent to starting with randomly generated values for the effectiveside length expression coefﬁcients. A solution is
Ž
represented with a vector of real numbers coef-
.
ﬁcient values and an associated set of neighbors. A neighbor is reached directly from the presentsolution by an operation called ‘‘move.’’ A succes-sion of moves is carried out to transform thearbitrary solution to an optimal one. The newsolution is the highest evaluation move amongthe neighbors in terms of the performance valueand tabu restrictions which exist to avoid newmoves that were evaluated in earlier iterations.The tabu search used in this work employs anadaptive mechanism for producing neighbors. Theneighbors of a present solution are created by thefollowing procedure.
Ž . Ž .
If
a t
,
,
is the solution vector
eff 1 2 3
Ž Ž ..
at the
t
th iteration, two neighbors
a n
,
n
of
eff 1 2
this solution of which the element
is not in
k
the tabu list are produced by:
Ž .
t
for odd neighbors
k
Ž .
a n
,
n
eff 1 2
½
Ž .
t
for even neighbors
k
Ž .
3
Ž . Ž Ž . . Ž .
a n
,
n
Remain
a n
,
n
,
4
eff 1 2 eff 1 2 max
with
k
3
LatestImprovementIteration
Ž .
t
k
1
k
2
Iteration
LatestImprovementIteration
Ž .
5 where Iteration stands for the current iterationnumber and LatestImprovementIteration is theiteration number at which the latest improvement was obtained. The value of
, which is larger
max
than zero for each coefﬁcient, is determined afterseveral experiments by the designer, and which is
Ž .
taken as 5 in this work. The index,
t
, in
t
represents the iteration number. The Remain
Ž .
function in eq. 4 keeps the elements of thesolution within the desired range. While
k
in eq.
1
Ž . Ž .
5 determines the magnitude of
t
,
k
and
k
2 3
Ž .
control the change of
t
. The proper values for
Ž .
the parameter
k
,
k
, and
k
in eq. 5 are
1 2 3
determined by experience on the tabu search. Inthe present work, the values taken for
k
,
k
, and
1 2
k
are 10, 2, and 2, respectively.
3
Tabu restrictions used here are based on therecency and frequency memory storing the infor-mation about the past steps of the search. Therecency-based memory prevents cycles of lengthless than or equal to a predetermined number of iterations from occurring in the trajectory. Thefrequency-based memory keeps the number of change of solution vector elements. If an elementof the solution vector does not satisfy the follow-ing tabu restrictions, then it is accepted as tabu:Tabu Restrictions
Ž .
recency
k
restriction period
Ž .
6or
Ž .
frequency
k
frequency limitTo select the new solution from the neighbors,evaluation values of the neighbors are calculatedusing their recency, frequency, and performance values.The formula used for the evaluation of a solu-tion is:
Ž .
evaluation
i
Ž . Ž .
a
improvement
i
b
recency
i
Ž . Ž .
c
frequency
i
7 where
a
,
b
, and
c
are the improvement, recency,and frequency factors, and equal to 4, 2, and 1,
Ž .
respectively, in this study. In eq. 7 the improve-ment is the difference between the performanceof the best solution found so far and that of the
i
th neighbor. The performance of a neighbor canbe computed using various formulas. In our work,the following is employed:
N
Ž . Ž . Ž . Ž .
P i
A
f j
f j
8
Ý
me ca
j
1
where
A
is a positive constant selected to be
Ž .
large enough so that
P i
values are positive for
Effecti
e Side Length Expression
7
TABLE I. Comparison of Measured and Calculated Resonant Frequencies of the First Five Modes of anEquilateral Triangular Microstrip Antenna with
a
10 cm,
2.32, and
h
0.159 cm
r
f f
da cl
1
f
present
f f f f f
moment
f f f f
me bb hj gl ga sd cl
2
gu
1
kk gu
2
7 method 1 6 8 9 10 method 11 11 12 14 13
Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .
Mode MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHzTM 1280 1281 1413 1299 1273 1340 1273 1288 1296 1280 1289 1280
10
TM 2242 2218 2447 2251 2206 2320 2206 2259 2244 2217 2233 2218
11
TM 2550 2562 2826 2599 2547 2679 2547 2610 2591 2560 2579 2561
20
TM 3400 3389 3738 3438 3369 3544 3369 3454 3428 3387 3411 3387
21
TM 3824 3842 4239 3898 3820 4019 3820 3875 3887 3840 3868 3841
30
all possible solutions, which is taken as 1000 inthe present work, and
f
and
f
represent,
me ca
respectively, the measured resonant frequency values and the calculated resonant frequency val-ues by using effective side length expression con-structed by the modiﬁed tabu search algorithm.The measured data sets used for the optimizationand evaluation process have been obtained fromthe previous works, which are given in TablesI
III. The fourth entries in these tables are usedfor the evaluation process to demonstrate theaccuracy of the model and the remainder 12 data
Ž .
sets
N
12 in eq. 8 are used for the optimiza-tion process. Only three measured data sets areused for the evaluation process because of thelimited measured data available in the literature.The unknown coefﬁcient values of the model
Ž .
given in eq. 2 are optimized by the modiﬁedtabu search algorithm just described. The opti-mum values found are:
Ž .
0.1,
8,
2 9
1 2 3
The following effective side length expression,
a
, is obtained by substituting the coefﬁcient
eff
Ž . Ž .
values given by eq. 9 into eq. 2 .8
Ž .
a
a
h
0.1
10
eff 2
ž /
r
The resonant frequencies are then calculatedby the formula2
c
1
22 2
Ž .
f
m
mn
n
11
mn
1
2
Ž .
3
a
eff
r
RESULTS AND DISCUSSION
To determine the most appropriate suggestiongiven in the literature, we compared our com-puted values of the resonant frequencies for theﬁrst ﬁve modes of the different equilateral trian-gular patch antennas with the theoretical andexperimental results reported by other scientists, which are all given in Tables I
III. The entries of
f
,
f
,
f
,
f
,
f
,
f
,
f
,
f
,
f
,
f
,
f
,
me da bb hj gl ga sd cl
1
cl
2
gu
1
kk
and
f
represent, respectively, the values mea-
gu
2
sured 7, 11 , calculated by this method, calcu-
lated by Bahl and Bhartia 1 , calculated by Hel-
szajn and James 6 , calculated by Garg and Long
8 , calculated by Gang 9 , calculated by Singh
et al. 10 , calculated by using moment method
11 , calculated by using the curve-ﬁtting formula
proposed by Chen et al. 11 , calculated by Guney
¨
12 , calculated by Kumprasert and Kiranon 14 ,
and calculated by Guney 13 . In Table I, the
¨
resonant frequencies were measured by Dahele
and Lee 7 . In Tables II and III, the resonant
frequencies were measured by Chen et al. 11 .The total absolute errors between the theoreticaland experimental results in Tables I
III for everysuggestion are also listed in Table IV.The theoretical results predicted by Garg and
Long 8 , and Singh et al. 10 are the same,because the analytical formulas proposed by thesescientists are the same.In ref. 11, the moment method full-wave analy-sis and also the curve-ﬁtting formula based on thedata set obtained from this moment method full- wave analysis were presented for the resonantfrequency of a triangular patch antenna. How-ever, it is apparent from Tables I
IV that thetheoretical resonant frequency results calculatedfrom this curve-ﬁtting formula and the momentmethod full-wave analysis are not in very goodagreement with the experimental results. It is alsoevident from Tables I
IV that the theoreticalresonant frequency results calculated from the
theories available in the literature 1, 6
14 arealso not in very good agreement with the experi-mental results. For these reasons, the data set
Karaboga et al.
ˇ
8
TABLE II. Comparison of Measured and Calculated Resonant Frequencies of the First Five Modes of anEquilateral Triangular Microstrip Antenna with
a
8.7 cm,
2.32, and
h
0.078 cm
r
f f
da cl
1
f
present
f f f f f
moment
f f f f
me bb hj gl ga sd cl
2
gu
1
kk gu
2
11 method 1 6 8 9 10 method 11 11 12 14 13
Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .
Mode MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHzTM 1489 1488 1627 1500 1480 1532 1480 1498 1498 1486 1493 1481
10
TM 2596 2577 2818 2599 2564 2654 2564 2608 2595 2573 2585 2565
11
TM 2969 2976 3254 3001 2961 3065 2961 2990 2996 2971 2985 2962
20
TM 3968 3937 4304 3970 3917 4054 3917 3977 3963 3931 3949 3918
21
TM 4443 4464 4880 4501 4441 4597 4441 4480 4494 4457 4478 4443
30
obtained from the moment method and the exist-ing theories are not used in this work. The mea-sured data set is used only for the optimizationprocess.We observe that our results calculated by using
a
presented here are better than those pre-
eff
dicted by other scientists. This is clear from Ta-bles I
IV. The very good agreement between themeasured values and our computed resonant fre-quency values supports the validity of the simplecurve-ﬁtting effective side length expression ob-tained using the modiﬁed tabu search algorithm,even with the limited data set. We expect that themodiﬁed tabu search algorithm will ﬁnd wide
Ž .
application in computer-aided design CAD of microstrip antennas and microwave-integratedcircuits. The results obtained demonstrate the versatility, robustness, and computational efﬁ-ciency of the algorithm.The effective side length expression,
a
, pro-
eff
posed in this study has good accuracy in the range
Ž .
of 2.3
10.6 and 0.005
h
0.034,
r d
where
is the wavelength in the substrate.
d
It is seen from Tables I
IV that the theoreti-
cal results reported by Garg and Long 8 ,
Kumprasert and Kiranon 14 , and Guney 12 are
¨
also close to the experimental results. However,the formulae given in this work is simpler than
the formulae given elsewhere 8, 12, 14 and alsoprovides the best results.Because the formula presented in this workhas good accuracy and requires no complicatedmathematical functions, it can be very useful forthe development of fast CAD algorithms. ThisCAD formula, capable of accurately predictingthe resonant frequencies of triangular microstripantennas, is also very useful to antenna engineers.Using this formula, one can calculate accurately,using a hand calculator, the resonant frequencyof triangular patch antennas, without possessingany background knowledge of microstrip anten-nas. It takes only a few milliseconds to producethe resonant frequencies on a 486 personal com-puter. Results predicted by the curve-ﬁtting for-mula obtained using the modiﬁed tabu searchalgorithm agree well with the measured results.The advantages of the formula given here aresimplicity and accuracy.It needs to be emphasized that better andmore robust results can be obtained by using the
TABLE III. Comparison of Measured and Calculated Resonant Frequencies of the First Five Modes of anEquilateral Triangular Microstrip Antenna with
a
4.1 cm,
10.5, and
h
0.07 cm
r
f f
da cl
1
f
present
f f f f f
moment
f f f f
me bb hj gl ga sd cl
2
gu
1
kk gu
2
11 method 1 6 8 9 10 method 11 11 12 14 13
Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .
Mode MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHz MHzTM 1519 1501 1725 1498 1494 1577 1494 1522 1509 1511 1490 1541
10
TM 2637 2600 2988 2594 2588 2731 2588 2654 2614 2617 2581 2669
11
TM 2995 3002 3450 2995 2989 3153 2989 3025 3018 3021 2980 3082
20
TM 3973 3971 4564 3962 3954 4172 3954 4038 3993 3997 3942 4077
21
TM 4439 4503 5175 4493 4483 4730 4483 4518 4528 4532 4470 4623
30

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