3.
1 linear Arrays
and ^ 119 118 Pattern Synthesis for Linear and Planar Arrays
N7-/2-I
lrL /,„ = T M {zo cos(sn/N T)} cos[(2w - l)sir/N T] (3.17b)
0= A (3.21) NT
(r -l)(X/L)Bh r+2
where the
( _.,ii broadening factor B/, for a large Chebyshev J
M» 1 , 2 , 3 , . . . , ( N T / 2- 1)
array is B b = 1 + 0.636 {(llr) cosh [(cosher) 2 - n 2] m)
for Nj even, where Nj is the number of elements; 2N + 1 is Nj odd; and 2N is Nj
(3.22) even; and again M = Ny - 1.
Other authors have given formulas valid for d xl\ < 0.5 for arrays with odd
and the beamwidth is, as in Chapter 1, #3 = 0.886(A/L)B/,. These two equations are
numbers of elements. Stegen's formulas are obtained by expanding the Chebyshev
used in Figure 1.9.
radiation pattern in a Fourier series and are more convenient and stable to compute
Figure 3.5(a) compares directivity as computed by Drane [17], using Elliott's
than the original equation of Dolph or those derived prior to Stegen's work. The
formulas [18] with the exact calculation. Good agreement is shown over a wide
Chebyshev pattern synthesis procedure has received much attention in the literature.
range of array lengths. The figure also shows that the directivity does not increase
Brown and Scharp [14] give extensive tabulations of current distributions computed
indefinitely with L, but reaches a maximum value 2r% or 3 dB greater than the
from the above formulas (although Hansen [15] has pointed out that the numerical
numerical value of the specified sidelobe level. This effect is demonstrated in Figure
accuracy of the tabulated data does not meet current standards). Stegen and others
3.5(b), due to Elliott [18], which shows the computed
1000 give equations for beamwidth, and there are several convenient expressions for
10 array gain valid for large arrays.
Stegen [16] gives the following expression for directivity
R = 100
(3.18) D = ^
directivity versus array length
N7
where
- 10
Nj
W = —z --- 1 for Nj even
= —j— for Nj- odd
For spacings greater than A/2, Drane [17] gives the following equation for the
directivity of a large array:
U/L')r2[In(2r)/ir] 1/2
D= (3.19)
C
,1/2
100 Relative
length (LA) and the beamwidth in radians:
(a) = O.18|A/L')(SLdB (3.20)
C
Figure 3.5 Characteristics of Chebyshev patterns: (a) Array directivity versus length for -20-dB (fi
= 10) and -40-dB (R = 100) sidelobe arrays: comparison between approximation of In these expressions, 1/ is the physical array length L' = (Nj - l)d x. Drane also
Drane (....), Elliott (xxxx), and exact values, (from: [17]. © 1968 IEEE. Reprinted with gives similar relations for arrays with spacing less than A/2.
permission.) (b) Array directivity versus length. Note: text uses L in place of Elliott's Lz;
Elliott [18] gives the following approximate expression of the directivity in
sidelobe levels are-IS to-60 dB. {From: [18]. © 1966 Academic Press, Inc. Reprinted
with permission.) (c) Array beamwidth versus length for -20- and -40-dB sidelobe terms of the beam broadening factor. This expression is valid for large arrays: r
arrays: comparison between approximation of Drane, Elliott, and exact values. (From:
[17]. © 1968 IEEE. Reprinted with permission.) (d) Taper efficiency e j = D/Nj versus
sidelobe level.
€
I
C
