Thinned Concentric Ring Arrays
Randy L. Haupt, rlh45@psu.edu
The Pennsylvania State University, Applied Research Laboratory,
State College, PA 16801
Introduction
A ring array is a planar array with elements lying on a circle. If several of these
arrays with different radii share a common center, then the resulting planar array
is known as a concentric ring array. Figure 1 is a diagram of a concentric ring
array with ring n having N n elements and a radius of rn . The physical distance
between elements on ring n is constant and given by d n .
Figure 1. Diagram of a concentric ring array.
A number of papers have presented results for reducing the sidelobe level of
concentric ring arrays by either amplitude weighting the elements or
nonuniformly spacing the rings [1], [2], [3], [4], [5], [6], [7]. Milligan
demonstrates statistical thinning of concentric ring arrays [8]. This paper presents
results from numerically optimizing the thinning of concentric ring arrays.
Uniform concentric ring arrays
The array factor for the concentric ring array with a single element at the center
(Figure 1) is given by
Nr
Nn
n =1
m =1
AF = 1 + ∑ wn ∑ e
jkrn [ cos φm sin θ cos φ + sin φm sin θ sin φ ]
where
N n = number of elements in ring n
N r = number of rings
k = wavenumber
wn = element weights for ring n
rn = radius of ring n
978-1-4244-2042-1/08/$25.00 ©2008 IEEE
(1)
( xn , yn ) == ( rn cos φm , rn sin φm ) location of element n
φm = 2π ( m − 1) / N n
Figure 2 is a diagram of a 279 element concentric ring array. There are nine rings
with rn = nλ / 2 and d n = λ / 2 . The number of equally spaced elements in ring n is
given by
N n = 2π rn / d n = 2π n
(2)
Since the number of elements must be an integer, the value in (2) must be rounded
up or down. To keep d ≥ λ / 2 , the digits to the right of the decimal point are
dropped. Table 1 lists the ring spacing and number of elements in each ring for a
uniform concentric ring array with nine rings as shown in Figure 2.
A uniform array has equal element spacing and weighting. For a uniform
concentric ring array, the ring spacing, rn , is a constant times the ring number and
the spacing between elements within a ring, d n , is approximately constant for all
rings. A nine ring concentric ring array has the array factor shown in Figure 3.
The array factor has a directivity of 29.35 dB, a peak sidelobe level of -17.4 dB,
and is symmetric in φ . As long as the array factor is predominantly a function
of θ , the maximum can be found from a slice of the array factor for a single value
of φ .
Table 1. Ring radius and number of elements per ring for a 9-ring uniform
concentric ring array.
n
1
2
3
4
5
6
7
8
9
rn ( λ ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Nn
6 12 18 25 31 37 43 50 56
Figure 2. Concentric ring array with
nine rings spaced λ / 2 apart and
dn λ / 2 .
Figure 3. Array factor due to the array
with nine concentric uniform rings.
Thinned concentric ring arrays
As with other types of arrays, it is possible to lower sidelobes by turning off
selected elements in a uniform array. The starting point is the nine ring uniform
array shown in Figure 2. Some elements within a ring are turned off or effectively
removed from the ring in order to modify the current density on the aperture. The
goal is to minimize the maximum sidelobe level by creating a low sidelobe
density taper on the array aperture.
A binary genetic algorithm is used to perform the thinning. The thinned aperture
is shown in Figure 4. It only has 66.3% of the 279 elements in the fully populated
array. Turning off 94 of the elements reduced the directivity to 24.17 dB and the
maximum relative sidelobe level to 22.44 dB. (In this case, directivity was
calculated by integrating over all angular space.) Since the thinning is not
symmetric, the array factor is not symmetric either (Figure 5). The number of
elements in each ring is shown in Table 2. The inner five rings have between 0
and 13% of the element removed, while the outer four rings have between 30 and
60% of the elements removed. A single chromosome in the genetic algorithm has
279 variables, so convergence to an acceptable solution is slow.
The thinning configuration found for the point sources was then applied to an
array of dipoles (Figure 6). Using the method of moments [9], the gain pattern in
Figure 7 was calculated. The peak gain is 24.4 dB with a maximum sidelobe level
18.4 dB below the main beam. A better result can be found for the dipole array by
further optimizing the thinning with a cost function based upon the method of
moments solution rather than point sources.
Table 2. Ring radius and number of elements per ring for a nine ring uniform
thinned concentric ring array.
n
1
2
3
4
5
6
7
8
9
rn ( λ ) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Nn
6
11 18 22 27 26 24 20 30
Figure 4. Thinned uniform
concentric ring array having 9 rings.
Figure 5. Array factor of the thinned
uniform concentric ring array.
Conclusions
This paper demonstrates the optimized thinning of concentric ring arrays using
point sources and dipoles.
Figure 6. Concentric ring array of
dipoles.
Figure 7. Gain pattern of thinned
concentric ring array of dipoles.
References
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low sidelobes,", IEEE AP-S Trans., Vol. 13, No. 6, Nov 1965, pp. 856- 863.
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frequency Invariant property for concentric ring," IEEE Signal Processing Trans.,
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1966, pp. 398-400.
[4] N. Goto and D.K. Cheng, "On the synthesis of concentric-ring," IEEE Proc.,
Vol. 58, No. 5, May 1970, pp. 839- 840.
[5] L. Biller and G. Friedman, "Optimization of radiation patterns for an array of
concentric ring sources," IEEE Trans. Audio and Electroacoustics, Vol. 21, No. 1,
Feb 1973, pp. 57- 61.
[6] D.A. Huebner, "Design and optimization of small concentric ring arrays,"
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[7] G Holtrup, A. Margulnaud, and J. Citerns, "Synthesis of electronically
steerable antenna arrays with element on concentric rings with reduced
sidelobes," IEEE AP-S Symposium, 2001, pp. 800-803.
[8] T.A. Milligan, "Space-tapered circular (ring) array," IEEE AP Mag., Vol. 46,
No. 3, Jun 2004, pp. 70-73.FEKO Suite 5.3, EM Software and Systems
(www.feko.info), 2007.
[9] FEKO Suite 5.3, EM Software and Systems (www.feko.info), 2007.