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SIMPLIFIED DESIGN THEORY FOR A
CIRCULAR MICROSTRIP
PATCH ANTENNA
Jibendu Sekhar Roy and Bernard Jecko
IRCOM
Faculte des Sciences
URA CNRS No 356
123, Avenue A,-Thomas
87060 Limoges Cedex, France
KEY TERMS
Circular patch antenna, microstrip, antenna design
ABSTRACT
An accurate and simple design procedure for a circular microstrip
patch untenna io operate in the dominant mode is obtained, for
which it is not necessary to compute ihe conventional design theory,
involving the Bessel function. The design equations presented in this
article provide for the design of a circular microstrip antenna, in a
direct way, for specified resonant resistance and gain in the desired
,frequency. From ihis design theory, it is found that the effective gain
of u circular microsirip antenna in the dominant mode can be evaluated measuring resonant resistance only. This is also verified experimentaiiy. 0 199.1 John Wiley & Sons. Inc.
INTRODUCTION
Microstrip antennas have gained considerable popularity in
recent years due to their various advantages. There are several
theories like wire grid model, transmission-line model, cavity
model etc., for the analysis of microstrip elements with varying degrees of accuracy and complexity. The transmissionline model gives the best physical insight and, of course, the
simplest model. But the model is not very simple when patch
geometries differ from rectangular or square. The cavity
model gives a deeper insight into the operation of the microstrip patch antennas of all patch geometries and is more complex than the transmission-line model.
Circular microstrip antennas are being used in a variety of
low-profile antennas. A number of articles have appeared [l191 describing the characteristics of circular microstrip antennas. The methods of computation of the resonant frequency,
input impedance, and radiation characteristics of circular
microstrip antennas using different models have been described in the above articles.
The design procedure for circular microstrip antennas was
discussed thoroughly in [ l l ] for the dominant mode, which is
the normally used mode of circular microstrip antennas because for this mode the beam is in the broadside direction.
This design procedure shows that for specific gain and specified resonant resistance of a circular microstrip antenna in
the dominant mode, a designer should compute a lengthy
theory involving integrations of Bessel functions, in order to
determine the dielectric constant of the substrate and the
dimension of the patch. This is complicated and time-consuming.
In this article a simple and accurate design procedure for
a circular microstrip antenna is reported, by simplifying the
existing theory for the dominant TMll mode. Using this theory, for given gain and resonant resistance, a designer can
compute the dielectric constant and the patch dimension for
his desired resonant frequency using a hand calculator only,
without computing the theory of edge-fed circular microstrip
antennas. For good impedance matching, sometimes it is nec-
essary to feed inside the patch, and then the design equations
presented in this article reduce design complexity and computation time drastically. From the present design theory, the
following advantages can be achieved.
1. The dielectric constant of the substrate to be chosen
and the patch dimension of the antenna can be found
accurately within a moment.
2 . The effective gain of a circular microstrip antenna in
the dominant mode can be evaluated with accurate measurement of the resonant resistance only, if the dielectric constant and the substrate thickness are known.
The gain of an edge-fed circular microstrip antenna in the
dominant mode was evaluated measuring resonant resistance
only and using the present theory, and then the result was
compared with the measured gain of the antenna. Good
agreement between the two supports the present theory.
CONVENTIONAL DESIGN THEORY
Since the proposed design theory is a simplification of the
conventional design theory for a particular mode, it is necessary to present the conventional design theory.
Modeling the circular microstrip patch antenna (Figure 1)
as a cylindrical cavity, bounded at its top and bottom by
electric walls and on its sides by a magnetic wall, which can
be resonant height-independent T M modes, the electric field
inside the cavity can be written as
where J,(kp) is the Bessel function of order rz, k = ( m i
c ) G r ( W = circular frequency, c = velocity of light in free
space, E, = relative dielectric constant of the substrate) and
( p , +‘, z) are the cylindrical coordinates.
The characteristic equation for the resonant frequency of
a circular microstrip patch is given by
JA(kU)
=
0,
a being the radius of the circular patch. For the dominant
mode (TMI1), the resonant frequency can be obtained using
the equation
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1.84118~
f r
=
(3)
~
2?ru,Gr’
where up is the effective radius of the circular patch and is
given by [7]
circular Patch
1
-
Substrate
Ground Plane
Figure 1 Circular microstrip antenna configuration
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 6, No. 3, March 5 1993
201
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The far fields in spherical polar coordinates (r, 0, 4) are
given by [ 111
PROPOSED DESIGN THEORY
The proposed design theory starts from the end of the conventional design theory described above.
Taking Eqs. (9) and (8), Eq. (7) can be written as
a,
711 = 1207~ the free-space impedance.
where V = hEoJn(kae)is the edge voltage at 4 = 0 and
ko = WJC.
The resonant resistance R can be calculated using the relation [ l l ]
From Eqs. (4) and ( 5 ) , for n
=
1, one can obtain
Then the expression for effective gain given by Eq. (10) becomes
1
-
If the disk is fed at an arbitrary point (po, 0) and if R0 is the
resonant resistance when the disk is fed at (po = a, 0), then
PT’
Again, from Eq. (6):
where PT = Pd + P, + P, is the total power loss in the cavity.
Pd is the dielectric loss, P, is the conductor loss, and P, is the
radiation loss, expressions for which can be found in [ll].
The effective gain of the antenna may be calculated using
the relation [ l l ]
Then from Eqs. (12) and (13),
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G, = 77D,
(7)
Then the patch radius is given by
where the efficiency 77 is given by
77 =
p,
PT
-3
Now, using Eqs. (3) and (15), eliminating a,, one can write
and the directivity of the antenna D may be expressed as [ l l ] ,
G,
R
- e,
(9)
n
= - (1.84118)2 =
2.825 x lo-? rnho.
770
Equations (16) and (3) are the proposed design equations,
from which one can design a circular microstrip antenna very
easily for required gain and resonant resistance. In order to
design a circular microstrip antenna in a desired frequency
for certain gain and resonant resistance, one should obtain e,
from Eq. (16) and then a from Eq. (3).
In order to design a circular microstrip antenna with specified gain and resonant resistance, one should compute Eqs.
(1)-(9) to find the appropriate substrate and dimension of the
patch.
TABLE 1
h
0.16 cm
a
=
2.2 crn
5.8
Resonant
Frequency
Resonant
Resistance
4.36
(GHz)
(W
Gain
Gain
Measured
1.896
184
1.300
Computed
1.866
205
1.329
Evaluated from
measured resonant
resistance and
using Eq. (16)
1.200
E, =
202
=
(16)
~
=
Sim
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 6, No. 3, March 5 1993
tan 6 = 0.02
The most important things which have been obtained from
this analysis are
uated accurately by measuring resonant resistance only. The
difference between measured and computed resonant resistance is due to the inadequacy of the simple cavity model,
where effects of other modes and the feed probe have not
been taken into account.
The variations of the factor G,E,IR with feed position for
different substrate thicknesses are plotted in Figures 2 and 3.
The variations of the factor with feed position for different
dielectric constants and for the same h are shown in Figure
4. From Figures 2 and 3, it is evident that for the same dielectric constant, the above factor is almost independent of
substrate thickness for high values of normalized feed position. This is also true for antennas with the same substrate
thickness and different dielectric constants, as shown in Figure 4.
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1. The effective gain of a circular microstrip antenna in
the dominant mode can be evaluated accurately by measuring resonant resistance only and using Eq. (16) for
a given substrate.
2. The factor G,&,IR, given by Eq. (16), is a function of
feed position, E,, and h , and independent of all types
of losses in the antenna. For the edge-fed case, for a
particular dielectric constant, the factor is independent
of h and all types of losses.
EVALUATING GAIN, MEASURING RESONANT
RESISTANCE ONLY
Equation (16) can be used to evaluate the gain of a circular
microstrip antenna in the dominant mode, measuring resonant resistance only. In order to verify this, a circular microstrip antenna was fabricated on epoxy substrate and then
resonant resistance and gain were measured by a WILTRON
360 network analyzer. The antenna was excited in the dominant mode by a 50-R coaxial SMA connector, feeding at the
edge of the patch. The maximum of the real part of input
impedance, plotted on Smith chart, was taken as resonant
resistance. Now from this measured resonant resistance, the
gain of the antenna was evaluated using Eq. (16) and then
compared with the measured gain. The results are tabulated
in Table 1.
Good agreement in Table 1 confirms that the gain of a
circular microstrip antenna in the dominant mode can be eval-
cr
CONCLUSION
An accurate and simple design theory for circular microstrip
antennas, operating in the dominant mode with specified gain
and resonant resistance, is presented in this communication,
using a simple cavity model. Using this theory, a designer can
determine the dielectric constant of the required substrate
and the dimension of the circular patch without the lengthy
theory of circular microstrip antennas.
Since the proposed design theory is not an approximation
of the conventional design theory, the accuracy of the first is
same as that of the second.
Also presented in this article is a method of evaluating the
gain of a circular microstrip antenna in the dominant mode
by measuring resonant resistance only.
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=2.32 a=6.8 cm
tan8 =0.0005
0.~1
k0.318 cm
k0.080 cm
--------------.
0.2
0.3
0.4
0.5
0.6
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0-7
0.8
0.9
1.0
Normalized Feed Position
Figure 2 Variation of the factor G,E,IR with normalized feed position (fils)
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 6, No. 3, March 5 1993
203
0.4
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&r
=2.55
a=2.0 cm
tan6 =0.0022
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0.3
L
E
W
0.2
0.1
0.0
I
.2
I
0.3
I
0.4
I
0.5
I
0.6
I
0.7
1
I
0.8
0.9
1.0
Normalized Feed Position
Figure 3 Variation of the factor G , E , / Rwith normalized feed position
h=0.15 cm
a=1.0 cm
0.4
0.3
0.2
0 . 4
0.c
I
.2
0.3
I
0.4
I
0.5
I
0.6
I
I
0.7
0.8
Normalized Feed Position
Figure 4 Variation of the factor G,E,/Rwith normalized feed position
204
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I
0.9
1
1.0
REFERENCES
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I . J . Watkins, “Circular Resonant Structures in Microstrip,” Electron. Lett., Vol. 5, 1969, pp. 524-525.
2. P. K. Agrawal and M. C. Bailey, “An Analysis Technique for
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AP-25, 1977, pp. 756-759.
COMBINED EFFECT OF HARMONIC
DISTORTION AND PULSE DISPERSION
IN OPTICAL FIBER TRANSMISSION
F. Javler Fraile-Pelhez and Fernando Gil-Vhzquez
Dept. Tecnologias de las Cornunicaciones
Universidad de Vigo
E.T.S.I.Telecomunicaci6n
Campus Universitario
36200 Vigo, Spain
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zyxwvutsrqponmlk
zyxwv
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zyxwv
zyxwvutsrqpon
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3. Y. T. Lo et al., “Study of Microstrip Antennas, Microstrip Phased
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“Resonant Frequency of a Circular-Disc, Printed-Circuit Antenna,” IEEE Trans. Antennas Propagat., Vol. AP-25, No. 4,
July 1977, pp. 595-596.
8. Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and Ex-
periment on Microstrip Antennas,” IEEE Trans. Antennas Propagat., Vol. AP-27, No. 2, March 1979, pp. 137-145.
9. W. F. Richards, Y. T. Lo, and D. D. Harrison, “An Improved
Theory of Microstrip Antennas with Applications,” IEEE Trans.
Antennas Propagat., Vol. AP-29, No. 1, Jan. 1981, pp. 38-46.
10. K. R. Carver and J. W. Mink, “Microstrip AntennaTechnology,”
IEEE Trans. Antennas Propagat., Vol. AP-29, No. 1, Jan. 1981,
pp. 2-24.
11. I. J . Bahl and P. Bhartia, Microstrip Antennas, Artech House,
Dedham, MA, 1980.
12. W. F. Richards, “Microstrip Antennas,” in Antenna Handbook,
Van Nostrand Reinhold, New York, 1988, Chap. 10.
13. W. C. Chew, J . A. Kong, and L. C. Shen, “Radiation Characteristics of a Circular Microstrip Antenna,” J . A p p l . Phys., Vol.
51, NO. 7, July 1980, pp. 3907-3915.
14. A. G. Derneryd, “The Circular Microstrip Antenna Element,”
IEEE Conf. on Antennas and Propagation, 1978, pp. 307-311.
15. S. A. Long, L. C. Shen, M. D . Walton, and M. R. Allerding,
“Impedance of a Circular-Disc Printed Circuit Antenna,” Electron. Lett., Vol. 14, 1978, pp. 684-686.
16. J. R. James, P. S. Hall, and C. Wood, Microstrip Antenna Theory
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Disk Antenna with a Thick Dielectric Substrate,” IEEE Trans.
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18 S. Yano and A. Ishimaru, “A Theoretical Study of the Input
Impedance of a Circular Microstrip Disk Antenna,” IEEE Trans.
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Peter Peregrinus, London, 1989, Vol. 1.
ABSTRACT
We study the transmission of Gaussian pulses b y subcarrier modulation of the laser power and obtain an analytical expression of the
detected photocurrent. This result provides insight into the effects of
the fiber dispersion on the transmission of AM-modulated subcarriers. 0 1993 John Wiley & Sons. Inc.
INTRODUCTION
The technique of subcarrier multiplexing (SCM) for transmitting microwave signals over optical links (1, 21 permits us
to take advantage of the large bandwidth available with semiconductor lasers and photodetectors, which currently exceeds
the speed of the baseband digital circuits used to generate
digital data. With SCM, several rf or microwave carriers are
amplitude- or frequency modulated by either analog or digital
signals. The composite signal, in turn, modulates the optical
intensity of the transmitter laser. The most common application of SCM is the transmission of video channels in CATV
systems.
The main impairment of SCM systems, originating from
the nonlinearity of the laser, is the harmonic and interharmonk distortion experienced by the subcarriers. An additional, more fundamental source of distortion is the dispersive
character of the fiber propagation. In [3], a numerical study
of the fiber-induced distortion was carried out. In this work,
the optical electric field, intensity modulated by one or two
subcarriers, was obtained at a propagation distance z by Fourier series expansion of the field taking into account the frequency dependence of the propagation constant. In the present work, we shall first consider an unmodulated subcarrier
and obtain an expression of the detected photocurrent
explicitly showing the periodicities of the fundamental, second-, and third-order harmonics. In order to obtain an analyFical expression in a relatively simply manner, a pure, chirpless intensity modulation is considered. This would be the
case with an ideal chirp-free semiconductor laser or an external modulator (see, for example, [4]).Next we shall extend
the calculations to the case of a subcarrier modulated by a
Gaussian pulse, also obtaining an analytical result. This will
allow us to discuss the similarities between the unmodulated
and modulated cases and illustrate the fiber-induced distortion
in SCM when the carriers are modulated, which obviously is
always the case in practice. The example of Gaussian pulses
is particularly illustrative for subcarriers modulated by digital
signals.
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Received 9-28-92
Microwave and Optical Technology Letters, 613, 201-205
0 1993 John Wiley & Sons, Inc.
ccc 0895-2477193
KEY TERMS
Analog optical transmission, subcurrier, fiber dispersion, pulse distortion
PURE SUBCARRIER PROPAGATION
We assume a single-tone intensity modulation of the optical
carrier of angular frequency q).
If fl is the subcarrier fre-
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205