15.
16.
17.
18.
19.
20.
tions times de 3-D Green’s function or its gradient on a plane triangle,
IEEE Trans Antennas Propagat 41 (1993), 1448-1455.
T.F. Eibert and V. Hansen, On the calculation of potential integrals for
linear source distributions on triangular domains, IEEE Trans Antennas Propagat 43 (1995), 1499-1502.
M.G. Duffy, Quadrature over a pyramid or cube of integrands with a
singularity at a vertex, SIAM J Numer Anal 19 (1982), 1260-1262.
L. Rossi and P.J. Cullen, On the fully numerical evaluation of the
linear-shape function times the 3-D Green’s function on a plane
triangle, IEEE Trans Microwave Theory Tech 47 (1999), 398-402.
M.A. Khayat and D.R. Wilton, Numerical evaluation of singular and
near-singular potential integrals, IEEE Trans Antennas Propag 53
(2005).
A.R. Krommer and C.W. Ueberhuber, Computational integration, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1998.
W.-H. Tang and S.D. Gedney, An efficient evaluation of near singular
surface integrals via the Khayat-Wilton transform, Microwave Opt
Technol Lett 48 (2006).
© 2007 Wiley Periodicals, Inc.
CAD FORMULAS FOR PATCH
DIMENSIONS OF RECTANGULAR
MICROSTRIP ANTENNAS WITH
VARIOUS SUBSTRATE THICKNESSES
Ali Akdagli
Faculty of Engineering, Department of Electrical and Electronics
Engineering, Mersin University, 33343 Ciftlikkoy, Mersin, Turkey
means that the patch length is approximately one-half of the
wavelength in the substrate material. Besides, the patch length
affects the resonant resistance and the E-plane radiation pattern.
The patch width affects the resonant frequency, the efficiency, the
bandwidth, the H-plane radiation pattern, and the cross polarization. To select the patch width properly, it must be considered that,
due to the excitation of surface waves, a small width results in a
large bandwidth, and low antenna efficiency and gain. However, a
large width results in the excitation of high-order modes, which
may distort the radiation pattern, decrease the bandwidth, and
increase the efficiency. To avoid exciting higher order modes, the
patch width must be less than a wavelength in the substrate [18].
Several formulas varying in computation effort and accuracy
for calculating the physical dimensions of rectangular MSAs are
available in the literature [1–17, 22]. However, they do not achieve
similar results, and the patch dimensions computed by using most
of these formulas do not agree well with the measured data. The
rectangular MSA designer needs the simple and accurate formulas
to calculate the physical dimensions. In this letter, therefore, the
simple and accurate formulas for computing the patch length and
patch width are successfully developed by utilizing a differential
evolution [23]. The differential evolution is a simple, fast, and
robust evolutionary algorithm that has proven effective in determining the global optimum for general type of numeric optimization problems. It does not require initial guesses, does not use
derivatives, and, it is also independent of the complexity of the
objective function considered. Since its introduction, differential
evolution has been rapidly gaining in acceptance, and has already
been applied to the solution of various challenging engineering
tasks [24 –27].
Received 5 February 2007
ABSTRACT: This letter presents the closed-from formulas for determining the patch dimensions of electrically thin and thick rectangular
microstrip antennas. The presented formulas constructed by utilizing a
differential evolution algorithm become very useful as computer-aided
design (CAD) models for patch antenna design. A comparison between
the calculated and experimental results derived from 33 patch antennas
demonstrates that the proposed CAD formulas provide ⬍0.16% and
0.22% errors on average for the patch length and patch width, respectively. The results obtained by using the formulas given here are found
to be closer to the experimental values, as compared with previous results presented by several researchers. © 2007 Wiley Periodicals, Inc.
Microwave Opt Technol Lett 49: 2197–2201, 2007; Published online in
Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.
22679
2. PATCH DIMENSIONS OF RECTANGULAR MICROSTRIP
ANTENNAS
Figure 1 shows a typical rectangular MSA consisting of a patch of
length L and width W over a ground plane with a substrate of
thickness h and a relative dielectric constant ␧r. When a rectangular MSA design is concerned, a proper dielectric substrate material
is firstly selected, and then the values of patch length and patch
Key words: rectangular microstrip antennas; patch dimensions; differential evolution; CAD
1. INTRODUCTION
Microstrip antennas (MSAs) are becoming a popular choice for a
broad range of applications from portable wireless communication
systems to biomedical systems [1–10]. Although they have some
advantages, such as light weight, low profile, low cost, easy
manufacturing, high level of integration, the possibility of making
conformal antennas, they also have some disadvantages, such as
narrow bandwidth, low overall efficiency due to dielectric losses,
and the generation of surface waves at the air dielectric interface.
A rectangular patch antenna is the most widely used MSA, and is
characterized by its length and width [1–22].
The proper selection of patch dimensions of rectangular MSAs
is important in the design, because the antenna performance characteristics depend firmly on the physical dimensions. For instance,
the rectangular MSA radiates efficiently when it resonates, which
DOI 10.1002/mop
Figure 1
Geometry of a rectangular MSA
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007
2197
TABLE 1
Comparison of Measured and Computed Patch Lengths of Rectangular Microstrip Antennas
L (cm)
⑀I
h (cm)
(h/␭d)
fI (MHz)
Measured
[16, 7]
2.22
2.22
2.22
2.55
10.2
2.33
2.33
2.55
2.55
2.55
2.55
2.55
2.50
2.50
2.50
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
0.017
0.017
0.079
0.079
0.127
0.157
0.157
0.163
0.163
0.200
0.242
0.252
0.300
0.300
0.300
0.476
0.330
0.400
0.450
0.476
0.476
0.550
0.626
0.854
0.952
0.952
0.952
1.000
1.100
1.200
1.281
1.281
1.281
0.006535
0.007134
0.015577
0.032505
0.062193
0.040421
0.038384
0.056917
0.048587
0.066004
0.090814
0.077800
0.083326
0.126333
0.103881
0.129219
0.140525
0.151895
0.145395
0.147462
0.16165
0.175363
0.155278
0.209105
0.181413
0.201683
0.197629
0.211852
0.228353
0.221646
0.218197
0.203196
0.214787
7740
8450
3970
7730
4600
5060
4805
6560
5600
6200
7050
5800
5270
7990
6570
5100
8000
7134
6070
5820
6380
5990
4660
4600
3580
3980
3900
3980
3900
3470
3200
2980
3150
1.290
1.185
2.500
1.183
1.000
1.860
1.960
1.350
1.621
1.412
1.200
1.485
1.630
1.018
1.280
1.580
1.080
1.255
1.450
1.520
1.440
1.620
1.970
2.300
2.756
2.620
2.640
2.676
2.835
3.130
3.380
3.500
3.400
This letter
Eq. (5)
Eq. (7)
[7]
[12]
[1, 6, 15]
[16, 7]
[22]
1.294
1.185
2.500
1.179
1.000
1.862
1.966
1.349
1.596
1.411
1.202
1.486
1.639
1.021
1.280
1.579
1.080
1.261
1.448
1.521
1.457
1.619
1.953
2.301
2.756
2.612
2.640
2.676
2.835
3.139
3.377
3.502
3.404
1.290
1.185
2.500
1.182
1.000
1.862
1.960
1.346
1.599
1.413
1.199
1.489
1.632
1.017
1.281
1.580
1.080
1.257
1.446
1.522
1.455
1.620
1.955
2.298
2.756
2.620
2.640
2.680
2.834
3.130
3.381
3.500
3.400
1.300
1.190
2.512
1.184
0.987
1.867
1.971
1.349
1.599
1.407
1.189
1.477
1.625
0.992
1.258
1.534
0.960
1.054
1.254
1.303
1.158
1.199
1.603
1.452
1.982
1.706
1.757
1.668
1.637
1.871
2.045
2.272
2.094
1.275
1.168
2.485
1.191
1.001
1.903
2.004
1.403
1.644
1.485
1.306
1.587
1.764
1.164
1.415
1.805
1.151
1.290
1.517
1.582
1.443
1.537
1.975
2.001
2.571
2.313
2.360
2.313
2.360
2.653
2.877
3.089
2.922
1.299
1.189
2.508
1.183
1.010
1.864
1.968
1.351
1.600
1.412
1.200
1.485
1.636
1.107
1.275
1.577
0.997
1.109
1.309
1.365
1.232
1.300
1.691
1.672
2.169
1.933
1.976
1.932
1.970
2.217
2.404
2.585
2.442
1.298
1.189
2.508
1.183
1.009
1.864
1.968
1.351
1.600
1.412
1.200
1.485
1.636
1.016
1.275
1.577
1.085
1.264
1.454
1.527
1.459
1.618
1.957
2.301
2.754
2.612
2.638
2.677
2.836
3.140
3.379
3.501
3.405
1.291
1.182
2.502
1.181
1.002
1.865
1.969
1.350
1.598
1.411
1.202
1.486
1.639
1.020
1.279
1.577
1.080
1.262
1.449
1.523
1.458
1.619
1.955
2.301
2.756
2.613
2.640
2.676
2.831
3.137
3.376
3.502
3.404
width are determined for a given operating frequency. It is clear
from all of the formulas presented in the literature [1–17, 22] that
only three parameters, h, ␧r, and resonant frequency fr, are required
to describe the patch dimensions of rectangular MSAs. However,
it is difficult to accurately determine the patch dimensions, because
MSA is inhomogeneous and the radiation appears at the edges of
the patch. Many of the formulas available do not provide a ready
solution for finding the patch dimensions of a rectangular MSA for
a given operating frequency. The latest formulas proposed by
Guney [22] for computing the patch dimensions of the rectangular
MSAs with thin and thick substrates are given as
冦
冦
0.51 ␭ d共h/␭d兲0.0112 ⫺ 0.565h共h/␭d兲⫺0.117 ⫹ 0.014␭o,
if h/␭d ⱕ 0.13
L ⫽ 383.0445␭ 共h/␭ 兲0.00052 ⫹ 0.36h共h/␭ 兲0.014 ⫺ 239.37␭ , (1)
d
d
d
o
if h/␭d ⬎ 0.13
9.8636 ␭ d共h/␭d兲0.4561 ⫺ 10.9h共h/␭d兲⫺0.446 ⫹ 0.00204␭o,
if h/␭d ⱕ 0.13
W ⫽ 382.466␭ 共h/␭ 兲⫺0.000907 ⫺ 0.0003h共h/␭ 兲⫺0.04 ⫺ 239.73␭ ,
d
d
d
o
if h/␭d ⬎ 0.13
(2)
with
2198
␭d ⫽
␭o
冑␧r
⫽
c
fr 冑␧r
(3)
where ␭d is the wavelength in the dielectric substrate, ␭o is the
free-space wavelength, and c is the velocity of electromagnetic
waves in free space.
In this letter, based on the experimental data available in the
literature [16, 17], the simple and accurate CAD models are
provided for calculating the patch length and patch width of the
rectangular MSAs with thin and thick substrates. To find proper
models for the patch dimensions, several experiments were
carried out. The coefficient values of the models found are then
optimally determined by utilizing differential evolution so as to
minimize the following total absolute errors (TAE)
TAE ⫽ ⌺兩PDme ⫺ PDca兩
(4)
where PDme and PDca are, respectively, the measured and calculated patch dimension (L or W). Two methods given as follows are
proposed for the calculation of patch dimensions.
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007
DOI 10.1002/mop
TABLE 2
Comparison of Measured and Computed Patch Widths of Rectangular Microstrip Antennas
L (cm)
⑀I
h (cm)
(h/␭d)
fI (MHz)
Measured
[16, 7]
2.22
2.22
2.22
2.55
10.2
2.33
2.33
2.55
2.55
2.55
2.55
2.55
2.50
2.50
2.50
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
2.55
0.017
0.017
0.079
0.079
0.127
0.157
0.157
0.163
0.163
0.200
0.242
0.252
0.300
0.300
0.300
0.476
0.330
0.400
0.450
0.476
0.476
0.550
0.626
0.854
0.952
0.952
0.952
1.000
1.100
1.200
1.281
1.281
1.281
0.006535
0.007134
0.015577
0.032505
0.062193
0.040421
0.038384
0.056917
0.048587
0.066004
0.090814
0.077800
0.083326
0.126333
0.103881
0.129219
0.140525
0.151895
0.145395
0.147462
0.16165
0.175363
0.155278
0.209105
0.181413
0.201683
0.197629
0.211852
0.228353
0.221646
0.218197
0.203196
0.214787
7740
8450
3970
7730
4600
5060
4805
6560
5600
6200
7050
5800
5270
7990
6570
5100
8000
7134
6070
5820
6380
5990
4660
4600
3580
3980
3900
3980
3900
3470
3200
2980
3150
0.850
0.790
2.000
1.063
0.910
1.720
1.810
1.270
1.500
1.337
1.120
1.403
1.530
0.905
1.170
1.375
0.776
0.790
0.987
1.000
0.814
0.790
1.200
0.783
1.256
0.974
1.020
0.883
0.777
0.920
1.030
1.265
1.080
This letter
Eq. (6)
Eq. (8)
[13]
[14]
[1, 11, 15]
[16, 7]
[22]
0.850
0.795
2.000
1.063
0.903
1.723
1.810
1.272
1.495
1.334
1.119
1.399
1.538
0.896
1.171
1.375
0.776
0.790
0.980
1.004
0.818
0.786
1.177
0.782
1.256
0.963
1.016
0.881
0.763
0.920
1.032
1.270
1.084
0.850
0.790
2.000
1.065
0.908
1.719
1.810
1.271
1.498
1.337
1.120
1.405
1.535
0.901
1.169
1.379
0.776
0.790
0.981
1.001
0.828
0.784
1.190
0.782
1.256
0.974
1.020
0.879
0.776
0.920
1.031
1.261
1.080
1.163
1.065
2.267
1.164
1.957
1.779
1.873
1.372
1.607
1.452
1.277
1.552
1.708
1.126
1.370
1.765
1.125
1.262
1.483
1.546
1.411
1.503
1.931
1.957
2.514
2.261
2.308
2.261
2.308
2.594
2.813
3.020
2.857
2.341
2.145
4.565
2.187
1.838
3.496
3.681
2.577
3.019
2.727
2.398
2.915
3.240
2.137
2.599
3.315
2.114
2.370
2.786
2.905
2.650
2.823
3.628
3.676
4.723
4.248
4.335
4.248
4.335
4.873
5.284
5.674
5.368
1.527
1.399
2.978
1.457
1.378
2.297
2.419
1.716
2.010
1.816
1.597
1.941
2.152
1.419
1.726
2.208
1.407
1.578
1.855
1.935
1.765
1.880
2.416
2.448
3.145
2.829
2.887
2.829
2.887
3.245
3.518
3.778
3.574
0.848
0.794
2.001
1.063
0.905
1.725
1.811
1.275
1.497
1.337
1.123
1.404
1.153
0.902
1.177
1.385
0.777
0.798
0.987
1.013
0.828
0.793
1.192
0.783
1.265
0.964
1.018
0.884
0.777
0.929
1.039
1.271
1.088
0.848
0.794
2.000
1.062
0.910
1.722
1.808
1.273
1.495
1.335
1.120
1.400
1.539
0.897
1.172
1.377
0.773
0.796
0.982
1.009
0.826
0.791
1.187
0.780
1.262
0.961
1.015
0.881
0.773
0.925
1.035
1.267
1.084
The First Method
W
冦
0.483 ␭ d ⫺ 0.5957h ⫹ 0.0281␭ocos共共h/␭d兲⫺0.02142 兲,
if h/␭d ⱕ 0.13
(5)
L ⫽ 0.265␭ ⫹ 1.4097h ⫹ 0.0029␭ cos共共h/␭ 兲⫺1.099 兲,
d
o
d
if h/␭d ⬎ 0.13
冦
0.5913 ␭ d ⫺ 2h ⫹ 0.3211␭dcos共共h/␭d兲⫺0.1798 兲,
if h/␭d ⱕ 0.13
W ⫽ 1.2848␭ ⫺ 3.636h ⫹ 0.4469␭ cos共共h/␭ 兲⫺0.5626 兲, (6)
d
d
d
if h/␭d ⬎ 0.13
冦
0.5909␭d ⫺ 2h ⫹ 0.3241␭dcos共共h/␭d兲⫺0.179 兲
⫹ 0.0009␭ocos共共h/␭d兲⫺1.75565 兲, if h/␭d ⱕ 0.13
⫽ 0.5811␭ ⫺ 1.83h ⫺ 0.0092␭ cos共共h/␭ 兲⫺1.0825 兲
d
d
d
⫺ 0.0013␭ocos共共h/␭d兲⫺3.1727 兲, if h/␭d ⬎ 0.13
(8)
The formulas for calculation of patch dimensions suggested in
the first method are simpler than those suggested in the second
method, however, the formulas of the second method achieve more
accurate results. Thus, the antenna designer can make a trade-off
between the simplicity and the accuracy.
3. RESULTS AND CONCLUSIONS
The Second Method
L
冦
0.4826 ␭ d ⫺ 0.6115h ⫹ 0.0015␭dcos共共h/␭d兲⫺1.9133 兲
⫹ 0.0305␭ocos共共h/␭d兲⫺0.0275 兲,
⫽ 0.2716␭ ⫹ 1.3711h ⫹ 0.0063␭ cos共共h/␭ 兲⫺1.1116 兲
d
d
d
⫺ 0.00065␭ocos共共h/␭d兲⫺3.7 兲,
if h/␭d ⱕ 0.13
if h/␭d ⬎ 0.13
(7)
DOI 10.1002/mop
To appraise the validity and accuracy of the new formulas
suggested for patch dimensions, the calculations were carried
out on 33 rectangular patch antennas built on various substrates.
The antennas, which consist in measurement data [16, 17] given
in Tables 1 and 2, vary in electrical thickness, defined as h/␭d,
from 0.006535 to 0.228353, and in physical thickness from
0.017 to 1.281 cm, and operate over the frequency range 2980 –
8450 MHz. The results calculated by using the formulas given
in the first and second methods for the patch length L and patch
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007
2199
TABLE 3 Total Absolute Errors Between the Measured and
Computed Patch Lengths (L)
can be very useful for many engineering applications because of
their simplicity and high accuracy.
This letter
Eq. (5)
Eq. (7)
[7]
[12]
[1, 6, 15]
[16, 7]
[22]
REFERENCES
(TAE)
(cm)
0.130
0.085
12.762
5.244
9.349
0.170
0.140
TABLE 4 Total Absolute Errors Between the Measured and
Computed Patch Widths (W)
This letter
Eq. (6) Eq. (8)
(TAE)
(cm)
0.133
0.072
[13]
[14]
22.553 74.122
[1, 11, 15] [16, 7]
36.918
0.164
[22]
0.143
width W are, respectively, tabulated in Tables 1 and 2. To make
a comparison, these tables also contain the patch dimensions
measured [16, 17] and those computed by other scientists [1, 6,
7, 11–17, 22]. Total absolute errors between the measured and
calculated by every suggestion for the patch length and patch
width are listed in Tables 3 and 4, respectively.
As it can be seen from Tables 1 and 2, the methods previously published in the literature for the patch dimensions give
comparable results—some cases are in good agreement with
measured data, and others are far off. From the Tables 3 and 4,
the patch dimensions obtained by using the formulas given in
the first method and the second method proposed in this work
provide the more accurate results, as compared with those
reported by the other researchers. The formulas given by Eqs.
(7) and (8) in the second method achieve the best results, and
the average percentage deviations for the patch length and patch
width are found to be 0.16% and 0.22%, respectively, for 33
rectangular MSAs considered in this work. The results reported
by Kara [16, 17] and Guney [22] are also close to the experimental results, however, the proposed formulas given by Eqs.
(5) and (6) in the first method not only are simpler and but
provide the more accurate results. The better agreement between the measured and our computed patch dimension results
supports the validity and accuracy of the new formulas.
The formulas proposed here are suitable for CAD and are
directly applicable for the integration of rectangular MSAs.
These CAD formulas are very useful to antenna engineers since
they require no complicated mathematical functions and no
iterative techniques. By using a hand calculator, one can accurately compute the patch dimensions without possessing any
background knowledge of the rectangular MSAs.
4. CONCLUSION
In this letter, the simple and accurate CAD formulas for the
calculation of patch length and width of the rectangular MSAs
with various substrate thicknesses, which can be readily used by
a MSA designer practically without any background in sophisticated mathematical techniques, are presented. They are obtained by utilizing the differential evolution algorithm. Two
methods for the patch dimension formulation, which give a
trade-off between the simplicity and the accuracy, are suggested. The validity of the new formulas has been verified by
comparing the achieved patch dimension results with the experimental results. The CAD formulas proposed in this work
2200
1. I.J. Bahl and P. Bhartia, Microstrip antennas, Artech House, Dedham,
MA, 1980.
2. J.R. James, P.S. Hall, and C. Wood, Microstrip antennas-theory and
design, Peter Peregrinus, London, 1981.
3. J.R. Mosig and F.E. Gardiol, A dynamic radiation model for microstrip
structures, In: Advances in electronics and electron physics, Vol. 59,
Academic Press, New York, 1982, pp. 139 –227.
4. R.E. Munson, Microstrip antennas, In: R.C. Johnson (Ed.), Antenna
engineering handbook, 3rd ed., McGraw-Hill, New York, 1983.
5. K.C. Gupta and A. Benalla (Eds.), Microstrip antenna design, Artech
House, Canton, MA, 1988.
6. Y.T. Lo, S.M. Wright, and M. Davidovitz, Microstrip Antennas, In: K.
Chang (Ed.), Handbook of microwave and optical components, Vol. 1,
Wiley, New York, 1989, pp. 764 – 889.
7. D.M. Pozar and D.H. Schaubert (Eds.), Microstrip antennas-the analysis and design of microstrip antennas and arrays, IEEE Press, New
York, 1995.
8. R.A. Sainati, CAD of microstrip antennas for wireless applications,
Artech House, Norwood, MA, 1996.
9. K.F. Lee and W. Chen, Advances in microstrip and printed antennas,
Wiley, New York, 1997.
10. R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip antenna
design handbook, Artech House, Canton, MA, 2001.
11. K.G. Schroeder, Miniature slotted cylinder antennas, Microwaves
(1964), 28 –37.
12. R.E. Munson, Conformal microstrip arrays and microstrip phased
arrays, IEEE Trans Antennas Propagat AP-22 (1974), 74 –78.
13. A.G. Derneryd, A theoretical investigation of the rectangular microstrip patch antenna element, IEEE Trans Antennas Propagat AP-26
(1978), 532–535.
14. H.D. Weinschel, Measurements of various microstrip parameters, Proceedings of Workshop on Printed Circuit Antenna Technology, New
Mexico State University, Las Cruces, Oct. 1979, pp. 2.1–2.15.
15. I.J. Bahl, Build microstrip antennas with paper-thin dimensions, Microwaves (1979), 50 –53.
16. M. Kara, Formulas for the computation of the physical properties of
rectangular microstrip antenna elements with various substrate thicknesses, Microwave Opt Technol Lett 12 (1996), 234 –239.
17. M. Kara, Empirical formulas for the computation of the physical
properties of rectangular microstrip antenna elements with thick substrates, Microwave Opt Technol Lett 14 (1997), 115–121.
18. M. Kara, Design considerations for rectangular microstrip antenna
elements with various substrate thicknesses, Microwave Opt Technol
Lett 19 (1998), 111–121.
19. T. Gunel, A fuzzy hybrid approach for the synthesis of rectangular
microstrip antenna elements with thick substrates, Microwave Opt
Technol Lett 26 (2000), 351–355.
20. R.K. Mishra and A. Patnaik, Designing rectangular patch antenna
using the neurospectral method, IEEE Trans Antennas Propagat AP-51
(2003), 1914 –1921.
21. B. Khuntia, S.S. Pattnaik, D.C. Panda, D.K. Neog, S. Devi, and M.
Dutta, Genetic algorithm with artificial neural networks as its fitness
function to design rectangular microstrip antenna on thick substrate,
Microwave Opt Technol Lett 44 (2005), 144 –146.
22. K. Guney, Simple and accurate formulas for the physical dimensions
of rectangular microstrip antennas with thin and thick substrates,
Microwave Opt Technol Lett 44 (2005), 257–259.
23. R. Storn and K. Price, Differential evolution—A simple and efficient
heuristic for global optimization over continuous spaces, J Global
Optimizat 11 (1997), 341–359.
24. K.A. Michalski, Electromagnetic imaging of elliptical cylindrical conductors and tunnels using a differential evolution algorithm, Microwave Opt Technol Lett 28 (2001), 164 –169.
25. X.F. Luo, A. Qing, and C.K. Lee, Application of the differential-
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007
DOI 10.1002/mop
evolution strategy to the design of frequency-selective surfaces, Int J
RF Microwave Computer Aided Eng 15 (2005), 173–180.
26. A. Akdagli and M.E. Yuksel, Application of differential evolution
algorithm to the modeling of laser diode nonlinearity in a radiooverfiber network, Microwave Opt Technol Lett 48 (2006), 1130 –
1133.
27. C. Yildiz, A. Akdagli, and M. Turkmen, Simple and accurate synthesis
formulas obtained by using a differential evolution algorithm for
coplanar striplines, Microwave Opt Technol Lett 48 (2006), 1133–
1137.
© 2007 Wiley Periodicals, Inc.
ANALYSIS OF ELECTROMAGNETIC
BAND-GAP WAVEGUIDE STRUCTURES
USING BODY-OF-REVOLUTION FINITEDIFFERENCE TIME-DOMAIN METHOD
Ming-Sze Tong,1 Ronan Sauleau,2 Anthony Rolland,2 and
Tae-Gyu Chang1
1
School of Electrical and Electronics Engineering, Chung-Ang
University, 221 Heukseok-dong, Dongjak-gu, Seoul 156 –756, Korea
2
Institute of Electronics and Telecommunications of Rennes,
University of Rennes I, Campus de Beaulieu, 263 Avenue du Général
Leclerc, 35042 Rennes, France
Received 12 February 2007
ABSTRACT: Study of electromagnetic band-gap (EBG) structures
has become a hot topic in computational electromagnetics. In this
article, some EBG structures integrated inside a circular waveguide
are studied. They are formed by a series of air-gaps within a circular dielectric-filled waveguide. A body-of-revolution finite-difference
time-domain (BOR-FDTD) method is adopted for analysis of such
waveguide structures, due to their axial symmetric properties. The
opening ends of the waveguide are treated as a matched load using
an unsplit perfectly matched layer technique. Excitations on a
waveguide in BOR-FDTD are demonstrated. Numerical results of
various air-gap lengths with respect to the period of separation are
given, showing an interesting tendency of EBG behavior. A chirpingand-tapering technique is applied on the EBG pattern to improve the
overall performance. The proposed EBG structures may be applied
into antenna structures or other system for unwanted signal suppression. Results show that the BOR-FDTD offers a good alternative in
analyzing axial symmetric configurations, as it offers enormous savings in computational time and memory comparing with a general
3D-FDTD algorithm. © 2007 Wiley Periodicals, Inc. Microwave Opt
Technol Lett 49: 2201–2206, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22668
Key words: EBG structures; circular dielectric-filled waveguide; BORFDTD method
candidate in guided wave structures, since the traveling EM waves are
well confined inside the dielectrically filled substrate of the device
during transmission. Their axial symmetric nature ensures an even
field distributions along the azimuthal (␾) direction at a cross section.
Because of the nonexistence of a secondary conducting medium other
than the outer metallic shell, waveguides support generally non-TEM
(transverse electromagnetic) mode operations, such as transverse electric (TE) or transverse magnetic (TM) modes.
In terms of analyzing tool, a body-of-revolution finite-difference time-domain (BOR-FDTD) method is adopted for analysis [7, 8], due to the axial symmetric properties existing in
circular waveguide structures. The BOR-FDTD method has
been demonstrated as a robust and versatile numerical tool for
solving axial symmetric problems as found in [9 –11], though
they can also be solved using other frequency domain methods
[12–14]. BOR-FDTD is a cylindrically based algorithm, and it
expands the angular dependence using Fourier series. The computational domain is thus compressed from a three-dimensional
(3-D) volume into a two-dimensional (2-D) ␳–z plane. On the
other hand, an unsplit perfectly matched layer (U-PML) technique [15] is used to model the two end-walls of the waveguide
as a matched load.
In this article, some EBG structures integrated inside circular
waveguides are taken for studies. They are constructed by aligning
a series of periodic air-gaps longitudinally inside a dielectrically
filled circular waveguide. Two operational modes, viz., TE11 and
TM01, are used for excitations. Frequency characteristics in terms
of scattering parameters are extracted for analysis. It is observed
that the given EBG structures exhibit a good band-gap behavior.
Additionally, to further improve the performance of bandwidth in
stop-band and the low frequency side-lobes, a chirping-and-tapering technique [16] is applied on the EBG air-gap pattern.
2. THEORY
2.1. BOR-FDTD Method
Derivation of BOR-FDTD starts with the general time-dependent
Maxwell’s curl equations:
ជ ⫽ ␧ 0关␧ r兴⭸Eជ /⭸t ⫹ 关␴e兴Eជ
ⵜ⫻H
ជ /⭸t ⫺ 关␴m 兴H
ជ
and ⵜ ⫻ Eជ ⫽ ⫺ ␮0 关␮r兴⭸H
(1)
This assumes linear and anisotropic media inside the computational domain; where [␧r], [␮r], [␴e], [␴h] are the diagonal tensors
of the relative permittivity, relative permeability, electric conductivity, and magnetic conductivity, respectively. The Maxwell’s
equations are expanded in cylindrical coordinates to handle cylindrical structures more efficiently. For axial symmetric structures,
viz., BOR structures, the electric and magnetic fields can be
expanded using infinite Fourier series [17]:
1. INTRODUCTION
One of the interesting areas that have recently been broadly conducted in computational electromagnetics (CEM) is the study of
electromagnetic band-gap (EBG) structures. Typically, an EBG
material acts as a band-stop filtering device to suppress electromagnetic (EM) waves inside a certain frequency range. This is
conventionally done through a periodic pattern of distributed elements. The concept originated in optics [1, 2], and thereafter
successfully transformed into microwave areas by proper frequency downscaling (e.g. [3, 4]).
Circular waveguides have been commonly used in microwave
engineering due to their high power capability and low power loss
during transmission [5, 6]. Dielectric waveguide is deemed as a good
DOI 10.1002/mop
冘
⬁
Eជ 共 ␳ , ␾ ,z;t兲 ⫽
关Eជ 共 ␳ ,z;t兲 evencos共m␾兲 ⫹ Eជ 共␳,z;t兲oddsin共m␾兲兴
(2a)
m⫽0
冘
⬁
ជ 共␳,␾,z;t兲 ⫽
H
ជ 共␳,z;t兲oddsin共m␾兲兴
ជ 共␳,z;t兲evencos共m␾兲 ⫹ H
关H
(2b)
m⫽0
where m is the mode number in the ␾-direction, and the field terms
with subscripts even and odd are the coefficients of cos(m␾) and
sin(m␾), respectively. Solutions for Maxwell’s curl equations are
obtainable by a proper selection of angular field variations, e.g.,
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 9, September 2007
2201