101.3
PARTIAL NONUNIFORM SPACING OF ARRAY ELEMENTS
Randy L. Haupt
Department of E l a t r * d Enginering
Unitsd S t a b hh Form Academy 80840
INTRODUCTION.
The most common methods for producing low sidelobe arrays are amplitude
tapering the elements and nonuniformly spacing the elements. These methods require
a complex feed network that is more expensive and difficult to test than a uniform feed
network. If only a portion of the elements are amplitude tapered or nonuniformly
spaced, the feed network is greatly simplified. In general, low sidelobe distributions
have the inner elements weighted higher than the edge elements. Therefore, the edge
elements are the most sensible to amplitude taper or nonuniformly space to produce
low sidelobes.
This paper describes a method in which all the elements are equally weighted
and only the center elements are equally spaced. Elements towards the edge of the
array are nonuniformly spaced to produce a low sidelobe array pattern. The spacing of
the edge elements is computed using numerical optimization.
LOW SIDELOBES FROM NONUNIFORM SPACING.
The linear array consists of 3K uniformly spaced, uniformly weighted elements
at the center. These elements provide the bulk of the antenna gain and require a
simple feed network. There are M elements on either side of these elements that are
not uniformly spaced. These elements contribute to the antenna gain and lower the
sidelobe levels a t the expense of making the feed network more complicated.
The array factor for a uniformly weighted array is the vector sum of the fields
of all the individual elements.
K+ni
AF =
?cos(~xnc~)
n=l
where
AF = array factor
xn = element spacing
4 = angle with respect to array axis
X = wavelength of electromagnetic radiation
If the 2X center elements are uniformly spaced and the 2M edge elements are
nonuniformly spaced (Figure l), then the array factor is written as
E=
where
d = uniform spacing of the elements in the center of the array
U.S. Government work not protected by US.
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2(N+M) = total number of elements
The spacing of the outer elements that produces the minimum peak sidelobe
level is determined using numerical optimization. The advantage of nonuniformly
spacing only part of the array is that low sidelobes are possible while maintaining a
standard feed network for the center elements of the array. Figure 2 is a flow chart of
the optimization procedure to find the optimum spacing of the outer elements.
Initially, the 2M+SN elements of the array are spaced a t intervals of A/?. The center
2N elements remain at this spacing while the smallest value of the maximum sidelobe
level is found by moving the outer M edge elements of the array. The Nelder Meade
minimization algorithm was used to minimize the maximum sidelobe level with respect
to the outer element spacing.
RESULTS.
The following two examples demonstrate this concept. The dashed line in the
far field patterns represents the far field pattern of a uniformly spaced array (d=X/2)
of 2N+2M elements and the solid line represents the optimized array with 2M
nonuniformly spaced edge elements.
1) Figure 3. Total number of elements = 44 with N=20 and M=2 and lcosdl element
pattern.
initial spacing of outer edge elements: 10.25X, 10.75X
optimized spacing of outer edge elements: 12.63X, 14.64X
maximum sidelobe level: -18.52 dB
% of elements nonuniformly spaced: 9%
observations: The maximum sidelobe level is about 5 dB lower than the
maximum sidelobe level of the uniform array.
2) Figure 4. Total number of elements = 48 with N=20 and M = 4 and I C O S ~ element
~
pattern.
initial spacing of outer edge elements: 10.25X, 10.75X, 11.25X, 11.75X
optimized spacing of outer edge elements: 10.99X, 12.88X, 13.93X, 16.47X
maximum sidelobe level: -21.57 dB
% of elements nonuniformly spaced: l i %
observations: The maximum sidelobe level is about 8 dB lower than the
maximum sidelobe level of the uniform array.
Adding more elements increases the ability to lower the maximum sidelobe
level but raises some of the sidelobe levels beyond 60'.
CONCLUSIONS.
The above examples demonstrate the ability to lower sidelobe levels through
optimally spacing only some of the edge elements. Keeping as many elements equally
spaced as possible, makes the design and manufacture of the array cheaper.
0
0
I-
Figure 1. Diagram of an array with 2N uniformly spaced center elements and and 2M
nonunifomily spaced edge elements.
PICK ARRAY SIZE: M,NI
1
\CALCULATE PATTERN OF CENTER 2N ELEMENTS1
1
CALCULATE PATTERN OF 2M EDGE ELEMENTSJ
1
ISUM THE TWO PATTERNq
1
FIND THE LARGEST SIDELOBE LEVEL]
1
~
y
e
S
+
i
X
1no
OPTIMIZE EDGE ELEMENT SPACING1
Figure 2. Flow chart of the MATLAB computer algorithm.
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R
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I
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0
f
-15 -
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-25-30-
-35 -40
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0
Figure 3. An optimized 44 element array a*ith N=20, M=2, andl CO$ I element
patterns (-)
and a 48 element uniform array with I cos4 I element patterns (-).
0
D
Figure 4. An optimized 48 element array with N=20, hk4, andl cos4 I element
patterns (-)
and a 48 element uniform array with 1 cos$ I element patterns (-).
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