Genetic Algorithm Optimization of a Corrugated Conical Horn Antenna
You Chung Chung*, youchung@ieee.org
Randy Haupt, haupt@ieee.org
Electrical and Computer Eng.
Utah State University
Logan, UT 84322
Dooyeong Yang
yeongyd@cheju.cheju.ac. kr
Telecom. & Computer. Eng.
Cheju National Univ.
Cheju, Korea
1. Introduction
The corrugated conical hom has a rotationally symmetrical radiation pattem and a very
low level of cross-polarization [I], making it ideal for a feed of a reflector antenna. The
theoretical hom antenna analysis has been studied by Clarricoats and Saha [2, 31.
reference [4] introduced a corrugated hom antenna design using the mode-matching
method. The genetic optimizer has been used to optimize a muti-choked hom to generate
a broad sectoral hom antenna pattem [SI.The least squares optimization method has been
used to design a horn antenna in [6].
This paper presents a design procedure for a wideband corrugated multi-sectional conical
hom using a genetic algorithm (GA). The GA optimizes the length, number of sections,
flare angles, and depth of the first slot In each horn section in order to match the
admittance, reduce the voltage standing wave ratio (VSWR), and generate the desired copolarization and cross-polarization pattems. A VSWR < 1.6 is obtained from llGHz to
18GHz.
11. Spherical Mode AnaIysis of a Multi-Sectional Corrugate Conical Horn
The geometry of a multi-sectional corrugated conical horn and slot are shown in Figure I .
The ithsection of the horn begins at y,.i and ends at y,, and has a flare angle of 9,'. Some
= hom admittance of ith
other parameters include h, =radius of ith section of the horn,
section, Y , s=slot admittance of ith section,
and t =tooth width.
r, = reflection coeficient at y,, w =slot width,
The equation for hom admittance at the boundary is
j x m h , W, I]
sin 9,'
v,(v,+ 1)
h,!"(ky)
=a
spherical Hankel functions of the second kind of v order
(cos$) = an associated Lcngendre polynomials of the 1st kind of v order.
k and y o are the wave number and wave admittance of free space.
-
A : coefficient depends on the boundary conditions.
The equation for slot admittance at the boundary is
0-7803-733@8/02$17.0002002 IEEE
342
-jy,
ly =-
kyi
[ev:(cosS,')P:'(cos
9,') - P," (cos S,')Q;'(cos
[Q,: (cos $,')Pc(cos 91) - P," (cos $,')er
9; )]
(cos 9; )]
(2)
where
QVm
(cos$) = associated Lengendre polynomials of the second kind, order v,
(cos 9) = the first order of differential function of Q: (cos 9)
QV"
Pv"(cos 9)= the first order of differential function of Pum(cos 9) ,
9;
When
ky, (9,'
= the
(2)
angle of slot depth in the ithsection of the horn.
has
- 91) =
conditions,
kyi >> 1
and
kw<< 1,
then
v, z k y , ,
n(l + 1/ 2), and the open-circuit condition (Yt=O) is satisfied. Thus,
$1
) = i l / 4(21+ 1) , where I and j are integer order. And
h is the wavelength in the corrugated conical hom. When w < W2, only the TM spherical
mode exists in the corrugation. Its depth is 1/ 4 when E, = 0 at 9 = 9,' .
the slot depth is d / = y , ( 9,2-
111. CA Optimization
The parameters of the GA are sectional length ( y,), flare angle of hom (9,')and order of
Hankel and Legendre function at the conducting boundary of horn ( v ,). When the hom
has M sections, the chromosome is written as: [7]
Chromosome=[(y,9: v , ) , ( y 2 9; v 2 ) , . . . . . ( y u 91 v,)]
The GA evaluates the admittance difference and fitness function of population. The
admittance difference function is represented by admittance difference between the horninside (1) and the circumferential slots (2). The admittance difference function at the
conducting boundary (9 = 9,')with the open circuit condition is given by
M
Admittance difference=
21y u - ys I
(3)
,=I
The fitness function is given by
Fitness = K,G,
where
- K2GBc
- K,G,<
(4)
Go,,= the antenna gain at the degree of B = 0",
Go,= the taper gain at the sub-reflector edge at B = e,, and
G, = the maximum side lobe level.
The constants Kt, K2 and K, are arbitrary constants. Therefore, the GA maximizes the
fitness value of a horn while it optimizes the admittance difference to zero.
IV. Results
The corrugated hom has been optimized at the center frequency (14.47GHz) using a GA.
The optimum number of sections is 4, which was determined from numerous runs. Table
1 shows the optimum data for the four section horn optimized by the GA. The patterns
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are dependent on the flare angle, order v, and hom length of each section.
Since hom (1) and slot admittance (2) values between the cascaded hom sections have
different values, the depth of the first slot (d;) of each section has been optiniized to
reduce the reflections. The optimized depths of the first slot of each section have been
shown in Table 1. All slots except the first one have a depth (d;) of a quarter wavelength.
The optimized corrugation dimensions are t=l.3mm = U16, w=3.89mm zs 3U16, and
d! = 5.18mm=h/4 at the center frequency. These are composed of four corrugati.ons per
wavelength at the center frequency to generate the transverse magnetic modes in the slots
of hom.
Fig. 2 shows the mean of the admittance difference and fitness values calculated with (3)
and (4) each generation. The admittance difference values converge to almost zero after
five iterations. The fitness value converges to 28.7dB after 30 generations. The variables
are encoded with 8-bits. The genetic parameters used in this paper have 384 members in
the initial population, 0.5 discard rate, and 5% mutation rate.
Fig. 3 shows the normalized radiation pattern of the corrugated conical horn. For
comparison, the radiation pattems calculated by the GA have been analyzed at the center
of transmit and receive frequency bands (11.94/17.54GHz). It has better results i:han the
non-sectional horn. Co-polar and cross-polar pattems shown in the in the E-plane and
diagonal plane (4 = 45”)of the hom. The co-polar pattems are circularly syinmetric
pattems with very low sidelobe levels. Also, the peak cross-polar levels are less than 70dB for transmit and -60dB for receive bands.
Fig. 4 shows the magnitude of the VSWR for the optimum hom antenna when TE,,
modes are incident from the circular waveguide (WC69). As shown in Fig. 4, this
antenna has a VSWR below 1.6 over the frequency range from llGHz to l8GHz. Also,
retum loss has a good value below -13dB.
References
1. K. S. Rao, “A simple dual-band corrugated horn with low cross polarization.” IEEE
Trans. Antennas and Propagation, vol. 38, No. 6, pp. 946-950, June 1990.
2. P. J. B. Clarricoats and P. K. Saha, “Propagation and radiation behaviour of
corrugated feeds,” Part 1 & 2, Proc. IEE, vol. 118, pp. 1177 - 1186, Sept. 1971.
3. P. J. B. Clarricoats and A, D. Olver, Corrugated Horns for Microwave Antennas, IEE
Electromagnetics Waves Series 18, Peter Peregrinus Ltd., pp. 58-96, 1984.
4. R. R. Collmann and F. M. Landstorfer, “Calculation of the field radiated by homantennas using the mode-matching methods,” IEEE Trans. Antennas and
Propagation, vol. 43, No. 8, pp. 876-880, Aug. 1995.
5. G. Muller, “Optimisation of compact horn with broad sectoral radiation pattern,” Electronics
Letters, vol. 37, no. 6, pp. 337-338, March 2001.
6. C. Hartwanger, K. Schittkowski and H. Wolf, “Computer aided design of hom
radiators for satellite communication by least squares optimization,” Engineering
Optimization, vol. 33, no. 2, pp. 221-244,2000,
7. R. L. Haupt and Sue E. Haupt, Practical Genetic Algorithms, John Wiley arid Sons
Inc., Chapter 2, pp. 25 - 48, 1998.
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Yn
Y,,"
Yl
r;,
h
J-l
Y,
Y;'
U:'
Y,"
r,
r,
Fig. 1. Geometry of the corrugated conical horn with multi-section
Fig. 2. Convergence of GA.
Fig. 4.Optimized VSWR and return loss
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