resonant frequency calculation for circular microstrip antennas with a dielectric cover using ...

28 Pages·2007·0.52 MB·English

Progress In Electromagnetics Research, PIER 72, 279–306, 2007
RESONANT FREQUENCY CALCULATION FOR
CIRCULAR MICROSTRIP ANTENNAS WITH A
DIELECTRIC COVER USING ADAPTIVE
NETWORK-BASED FUZZY INFERENCE SYSTEM
OPTIMIZED BY VARIOUS ALGORITHMS
K. Guney
Department of Electrical and Electronics Engineering
Faculty of Engineering
Erciyes University
38039 Kayseri, Turkey
N. Sarikaya
Department of Aircraft Electrical and Electronics
Civil Aviation School
Erciyes University
38039 Kayseri, Turkey
Abstract—This paper presents a method based on adaptive-network-
based fuzzy inference system (ANFIS) to calculate the resonant
frequency of a circular microstrip antenna (MSA) with a dielectric
cover. The ANFIS is a class of adaptive networks which are
functionally equivalent to fuzzy inference systems (FISs). Six
optimization algorithms, hybrid learning, least-squares, nelder-mead,
genetic, differential evolution and particle swarm, are used to
determineoptimallythedesignparametersoftheANFIS.Theresonant
frequencyresultspredictedbyANFISareinverygoodagreementwith
the results reported elsewhere. When the performances of ANFIS
models are compared with each other, the best result is obtained from
the ANFIS model optimized by the LSQ algorithm.
280 Guney and Sarikaya
1. INTRODUCTION
Microstrip antennas (MSAs) have many attractive features such as
light weight, low production cost, low profile, integrability into arrays,
conformability to curved surfaces, simplicity of fabrication, and ease
of integration with microwave integrated circuit (MIC) or monolithic
microwave integrated circuit (MMIC) components [1–17]. These
attractive features have recently increased the application of MSAs
and stimulated greater effort to investigate their performance.
A dielectric cover or superstrate is usually employed in practice
to protect the MSAs from environmental hazards, or may be naturally
formed (e.g., ice) during flight or severe weather conditions. Due to
the limited bandwidth of MSAs, it is important to determine the effect
of a dielectric cover on the resonant frequency of MSAs in order to
introduce appropriate corrections in the design of MSAs.
Several methods [18–24], varying in accuracy and computational
effort,havebeenproposedandusedtocalculatetheresonantfrequency
of a circular MSA with a dielectric cover. The Hankel transform
analysis [18], the modified Wolff model [19], Galerkin’s method in
the Hankel transform domain [22], a formula [23] based on an
improvedcavitymodel[25]andBernhard’swork[26]werepresentedfor
computing the resonant frequency of a circular MSA with a dielectric
cover. The spectral-domain moment-method proposed by Fan and
Lee [20] can also be used to calculate the resonant frequency. Lee
and Fan [21] presented a Fortran program of CAD (Computer-Aided
Design) formula for the resonant frequency of a circular MSA with a
dielectric cover. This CAD formula was obtained by using a database
built by moment-method. Closed-form Hankel transforms for circular
disk basis modes involving Chebyshev polynomials and edge condition
closed-form expressions for two kinds of Hankel transform integrals,
which are encountered in the spectral moment method solution of a
circular patch, were obtained in [24]. These expressions were applied
in the formulation of the the full-wave moment method solution for
the resonance frequency of a circular microstrip patch antenna with
a dielectric cover. It is clear that the methods used in calculating
the resonant frequency can be broadly classified into two categories:
analytical and numerical methods. The analytical methods, based
on some fundamental simplifying physical assumptions regarding the
radiation mechanism of antennas, are the most useful for practical
designaswellasprovidingagoodintuitiveexplanationoftheoperation
ofMSAs. However,thesemethodsarenotsuitableformanystructures,
in particular, if the thickness of the substrate is not very thin. The
numerical techniques provide accurate results but usually require
Progress In Electromagnetics Research, PIER 72, 2007 281
considerable computational time and costs.
In this paper, a method for computing the resonant frequency
of the circular MSA with a dielectric cover, based on the adaptive-
network-based fuzzy inference system (ANFIS) [27,28], is presented.
First, the antenna parameters related to the resonant frequency are
determined, and then the resonant frequency depending on these
parameters is calculated by using the ANFIS. The ANFIS is a class of
adaptive networks which are functionally equivalent to fuzzy inference
systems (FISs). The FIS is a popular computing framework based
on the concepts of fuzzy set theory, fuzzy if-then rules, and fuzzy
reasoning. The ANFIS can simulate and analysis the mapping relation
between the input and output data through a learning to determine
optimal parameters of a given FIS. It can be trained with no need
for the expert knowledge usually required for the standard fuzzy logic
design. A prominent advantage of the ANFIS is that, after proper
training, ANFIS completely bypasses the repeated use of complex
iterative processes for new cases presented to it. Even if training takes
afewminutes, thetestprocesstakesonlyafewmicroseconds. Because
oftheseattractivefeatures,theANFISinthispaperisusedtocalculate
the resonant frequency of a circular MSA with a dielectric cover.
In previous works [29–38], we successfully utilized ANFIS for
computing accurately the various parameters of the rectangular,
triangular, and circular MSAs, and for tracking multiple targets and
estimating the phase inductance of the switched reluctance motors.
In reference [32], the resonant frequency of circular MSA without a
dielectric cover has been computed by using ANFIS. In this paper,
ANFISisemployedtocomputetheresonantfrequencyofcircularMSA
with a dielectric cover. In previous works [29–38], only the hybrid
learning (HL) algorithm [27,28] was used to determine the optimum
design parameters of the ANFIS. However, in this study, six different
optimization algorithms, HL algorithm, least-squares (LSQ) algorithm
[39–41], nelder-mead (NM) algorithm [42,43], genetic algorithm (GA)
[44,45], differential evolution algorithm (DEA) [46–48], and particle
swarm optimization (PSO) [49,50], are used to determine optimally
the design parameters of the ANFIS. These optimization algorithms
areemployedtoobtainbetterperformanceandfasterconvergencewith
simpler structure.
In this paper, the next section briefly describes the spectral-
domain moment-method [20,21] for computing the resonant frequency
of the circular MSA with a dielectric cover. The application of the
ANFIStotheresonantfrequencycomputationisgiveninthefollowing
section. The results are then presented and conclusion is made.
282 Guney and Sarikaya
2. RESONANT FREQUENCY OF A CIRCULAR
MICROSTRIP ANTENNA WITH A DIELECTRIC
COVER
The basic geometry of a circular MSA covered by a dielectric layer and
the co-ordinate system under consideration is depicted in Figure 1.
The patch is circular with radius a and is fed through coaxial probe.
r
a
φ P
d
z
a
ε d
r0
ε I h
r1 s 1
I
ε p h
r2 2
z=0
2a
p
Figure 1. Geometry of the circular microstrip antenna with a
dielectric cover.
The coaxial probe is modeled by a uniform cylindrical current sheet
of radius a . The substrate-dielectric cover thicknesses and relative
p
dielectric constants are h , h and, ε , ε , respectively. The resonant
2 1 r2 r1
frequency of the circular MSA with a dielectric cover can be obtained
by using the spectral-domain moment-method proposed by [20,21].
In this method, the problem is formulated in the Hankel transform
domain. First, the integral equations are obtained for the surface
current density on the patch. Galerkin’s method is then used to solve
forthesurfacecurrentdensity,andtheresonantfrequencyiscalculated
by using the input impedance formula. The surface current density I
p
is given by the following equation for 1A total current in the coaxial
Progress In Electromagnetics Research, PIER 72, 2007 283
probe [20]:
1 δ[|r−xˆd|−a ]
p
I = zˆI (r,φ) = zˆ (1)
p p 2π |r−xˆd|
where d is the feed location. The problem can be solved either with
vector Hankel transforms or Hankel transforms with the r and φ
components in quadrature [51]. The following vector F is defined by
F± = Fr ±jFφ (2a)
The Hankel transform quantities F˜± are given by
(cid:2)∞
F˜±(α) = F±(r)Jn±1(αr)rdr (2b)
0
The following integral equations are obtained for the unknown patch
current densities [20].
(cid:2)∞(cid:3) (cid:4)
G11I˜s++G12I˜s−+H1 Jn+1(αr)αdα = 0, on the patch (3a)
0
(cid:2)∞(cid:3) (cid:4)
G21I˜s++G22I˜s−+H2 Jn−1(αr)αdα = 0, on the patch (3b)
0
with
(cid:5) (cid:6)
ζ k2S R
G = G = 0 0 2 + 2 (4a)
11 22
2k Q D
0 (cid:5) (cid:6)
ζ k2S R
G = G = 0 0 2 − 2 (4b)
12 21
2k Q D
0
ζ R 2αI˜
H = −H = 0 2 p (4c)
1 2 2k D γ2
0 2
S = jγ tanγ h +γ (4d)
2 0 1 1 1
R = γ γ (jγ tanγ h +ε γ ) (4e)
2 (cid:7)1 2 1 1 1 r1 0
µ
ζ = 0, k2 = w2µ ε (4f)
0 ε 0 0 0
0
D = ε γ (−γ +jε γ cotγ h )
r1(cid:8) 1 2 r2 0 2 2(cid:9)
− jε2 γ γ +ε γ2cotγ h tanγ h (4g)
r1 0 2 r2 1 2 2 1 1
284 Guney and Sarikaya
Q = γ (−γ +jγ cotγ h )
1(cid:8) 0 2 2 2(cid:9)
− jγ2+γ γ cotγ h tanγ h (4h)
1 0 2 2 2 1 1
1
I˜ = J (αd) J (αa ) (4i)
p n 0 p
(cid:10)2π
γ = ε k2−α2, i = 0, 1, 2. (4j)
i ri 0
Galerkin’s method can be used to solve Eq. (3). The surface patch
current density Is± is expanded in terms of basis functions:
(cid:11)∞ (cid:11)M
Is±(r,φ) = ejnφ Cm±(n)Ism±(r,n) (5)
n=−∞ m=1
The surface patch current density Is± in the Hankel transform
domain can be written as
(cid:11)M
I˜s±(α,n) = Cm±(n)I˜sm±(α,n) (6)
m=1
The following equations are obtained by substituting Eq. (6) into
Eq. (3) and multiplying the resulting equation by rIsi±(r,n) and
integrating from r = 0 to ∞,
(cid:11)M (cid:8) (cid:9)
Zi1m1Cm++Zi1m2Cm− = Vi1, i = 1,... ,M (7a)
m=1
(cid:11)M (cid:8) (cid:9)
Zi2m1Cm+ +Zi2m2Cm− = Vi2, i = 1,... ,M (7b)
m=1
with
(cid:2)∞
Z11(n) = I˜ G I˜ αdα = Z11(n) (8a)
im si+ 11 sm+ mi
0
(cid:2)∞
Zi1m2(n) = I˜si+G12I˜sm−αdα = Zm21i(n) (8b)
0
(cid:2)∞
Zi2m2(n) = I˜si−G22I˜sm−αdα = Zm22i(n) (8c)
0
Progress In Electromagnetics Research, PIER 72, 2007 285
(cid:2)∞
V1(n) = − J˜ H αdα (9a)
i si+ 1
0
(cid:2)∞
Vi2(n) = − J˜si−H2αdα (9b)
0
The input impedance can be expressed as [20,21,52]
(cid:12) (cid:13)
(cid:11)∞ (cid:11)M (cid:11)M
Zin = π Cm+Vm1 + Cm−Vm2 +jxp (10a)
n=−∞ m=1 m=1
with
ζ
x = √0 tan(k h ) (10b)
p 0 2
ε
r2
The resonant frequency is the frequency at which the input resistance
is maximum.
The space-domain basis functions can be chosen as [20]
Ism±(r,n) = Jn±1(kmnr), r < a (11)
where k is the mth root of J(cid:4) (k a) = 0. The prime sign denotes
mn n mn
the derivative of Bessel function J (x) with respect to the argument
n
x. The Hankel transform is given by
(cid:14)
I˜sm±(α,n) = a αJn±1(kmna)Jn(cid:4)±1(αa) (cid:15) (cid:8) (cid:9)
−kmnJn±1(αa)Jn(cid:4)±1(kmna) / km2n−α2 (12)
From the formulations given above and in the literature [18–24]
we see that only five parameters, a, h , ε , h , and ε , are needed to
1 r1 2 r2
describe the resonant frequency of the circular MSA with a dielectric
cover. It is clear that the method described above is mathematically
complex and requires high performance large-scale computer resources
andaverylargenumberofcomputations. Forthisreason,inthiswork,
the resonant frequency of the circular MSA with a dielectric cover is
computed by using a method based on ANFIS. Only five parameters,
a, h , ε , h , and ε , are used in calculating the resonant frequency.
1 r1 2 r2
3. APPLICATION OF ANFIS TO THE COMPUTATION
OF RESONANT FREQUENCY
The ANFIS [27,28] is a class of adaptive networks which are
functionally equivalent to FISs. It is a very powerful approach for
286 Guney and Sarikaya
building complex and nonlinear relationship between a set of input
and output data. The ANFIS used in this paper implements a
first-order Sugeno fuzzy model [27,28]. Among many FIS models,
the Sugeno fuzzy model is the most widely applied one for its high
interpretability and computational efficiency, and built-in optimal and
adaptive techniques.
The ANFIS architecture used in this work for the resonant
frequency calculation of circular MSAs with a dielectric cover is
illustrated in Figure 2, in which a circle indicates a fixed note, whereas
a rectangular indicates an adaptive note. For the ANFIS, the inputs
are h /a, h /a, ε , andε , and the output is the normalized resonant
1 2 r1 r2
frequency f /f . f and f represent, respectively, the resonant
r no r no
frequencies of the TM mode (dominant mode) of the circular MSA
11
with and without dielectric cover.
ε ε
h/a h/a
1 2 r1 r2
w w
1 1
h/a M11
1
M12 w1f1
M21
h2/a M22
M23
Σ f / f
r no
ε M31
r1
M32
M41
εr2 M42 w36f36
M43
w w
36 36 ε ε
h/a h/a
1 2 r1 r2
Fuzzy Product Normalized De-fuzzy Summation
Layer Layer Layer Layer Layer
Figure 2. Architecture of ANFIS.
The accuracy of a properly trained ANFIS depends on the
accuracy and the effective representation of the data used for its
training. A good collection of the training data, i.e., data which
is well-distributed, sufficient, and accurately simulated, is the basic
requirement to obtain an accurate model. There are two types of
data generators for antenna applications. These data generators are
Progress In Electromagnetics Research, PIER 72, 2007 287
the measurement and simulation. The selection of a data generator
depends on the application and the availability of the data generator.
The training and test data sets used in this paper have been obtained
from the Fortran program [21] developed by using the results of
spectral-domain moment method described in Section 2. 1000 data
sets are used to train the ANFIS. Training data sets are in the range
of 0 < h /a < 0.3, 0 < h /a < 0.2, 1 < ε < 11, and 1 < ε < 11.
1 2 r1 r2
177 data sets, which are completely different from training data sets,
are used to test the ANFIS. The input and output data sets are scaled
between 0 and 1 before training.
In the design of ANFISs, it is very important to determine the
typesandtheparametersofmembershipfunctions(MFs). Thenumber
of the MFs for the input variables h /a, h /a, ε , and ε are 2, 3, 2
1 2 r1 r2
and 3, respectively. The number of rules is then 36 (2×3×2×3 = 36).
ThetypesoftheinputMFsfortheinputvariablesh /a, h /a, ε , and
1 2 r1
ε aretheGaussian, Generalizedbell, Gaussian, andGeneralizedbell,
r2
respectively. The Gaussian and the Generalized bell MFs are specified
by two and three parameters, respectively. Therefore, the ANFIS used
in this work contains a total of 206 fitting parameters, of which 26
(2×2+3×3+2×2+3×3 = 26) are the premise parameters and
180 (5×36 = 180) are the consequent parameters.
Each possible combination of inputs and their associated MFs is
represented by a rule in the rule base of the ANFIS in the following
equation:
1. if (h /a is M ) and (h /a is M ) and (ε is M ) and (ε is
1 11 2 21 r1 31 r2
M ) then R = f (h /a, h /a, ε , ε )
41 1 1 1 2 r1 r2
2. if (h /a is M ) and (h /a is M ) and (ε is M ) and (ε is
1 11 2 21 r1 31 r2
M ) then R = f (h /a, h /a, ε , ε )
42 2 2 1 2 r1 r2
3. if (h /a is M ) and (h /a is M ) and (ε is M ) and (ε is
1 11 2 21 r1 31 r2
M ) then R = f (h /a, h /a, ε , ε )
43 3 3 1 2 r1 r2
4. if (h /a is M ) and (h /a is M ) and (ε is M ) and (ε is
1 11 2 21 r1 32 r2
M ) then R = f (h /a, h /a, ε , ε )
41 4 4 1 2 r1 r2
. . .
. . .
. . .
(13)
. . .
. . .
. . .
35. if (h /a is M ) and (h /a is M ) and (ε is M ) and (ε is
1 12 2 23 r1 32 r2
M ) then R = f (h /a, h /a, ε , ε )
42 35 35 1 2 r1 r2
36. if (h /a is M ) and (h /a is M ) and (ε is M ) and (ε is
1 12 2 23 r1 32 r2
M ) then R = f (h /a, h /a, ε , ε )
43 36 36 1 2 r1 r2
where M , R , and f represent the jth MF of the ith input, the
ij k k
output of the kth rule, and the kth output MF, respectively.
288 Guney and Sarikaya
In this work, the input MFs are Gaussian and Generalized bell
types and the output MFs are a linear type as given bellow:
(cid:16) (cid:17)
−1 x−cij 2
M (x) = Gauss(x;c ,σ ) = e 2 σij
ij ij ij
i = (1,3), j = (1,2), (x = h /aorε )
1 r1
1
M (x) = Gbell(x; a , b , c ) = (cid:18) (cid:18) (14)
ij ij ij ij 1+(cid:18)(cid:18)x−cij(cid:18)(cid:18)2bij
aij
i = (2,4), j = (1,2,3), (x = h /aorε )
2 r2
R = f (h /a, h /a, ε , ε )
k k 1 2 r1 r2
= d h /a+d h /a+d ε +d ε +d k = (1,... ,36)
k1 1 k2 2 k3 r1 k4 r2 k5
Here a , b , c , σ and d are the parameters that characterize
ij ij ij ij k
the shapes of the MFs. The optimal values of these parameters are
determined by training.
TheoutputoftheANFISistheweightedaverageoftheindividual
rule outputs. The weighting factor of each rule, which is expressed
as w , is calculated by evaluating the membership expressions in the
k
antecedent of the rule. This is accomplished by first converting the
inputvaluestofuzzymembershipvaluesbyutilizingtheinputMFsand
then applying the “and” operator to these membership values. The
“and”operatorcorrespondstothemultiplicationofinputmembership
values. Hence, the weighting factors of the rules are calculated as
follows:
w = M (h /a)·M (h /a)·M (ε )·M (ε )
1 11 1 21 2 31 r1 41 r2
w = M (h /a)·M (h /a)·M (ε )·M (ε )
2 11 1 21 2 31 r1 42 r2
w = M (h /a)·M (h /a)·M (ε )·M (ε )
3 11 1 21 2 31 r1 43 r2
w = M (h /a)·M (h /a)·M (ε )·M (ε )
4 11 1 21 2 32 r1 41 r2
. (15)
.
.
.
.
.
w = M (h /a)·M (h /a)·M (ε )·M (ε )
35 12 1 23 2 32 r1 42 r2
w = M (h /a)·M (h /a)·M (ε )·M (ε )
36 12 1 23 2 32 r1 43 r2
Once the weighting factors are obtained, the output of the ANFIS can
be found by calculating the weighted average of the individual rule