Page 261 - Maxwell House

Chapter 5 241

−1
⁄ ⁄
be reached if ⁄ ≤ cos (− 1 ) . For example, when = 1.293 we need to keep
≤ 0.7814 at any frequency. Evidently, the any drop in SLL increases the parameter b value
⁄
and narrows the frequency range without grating lobes. Now we have everything to design a
linear array with all sidelobes at some prescribed level. The plots in Figure 5.4.5 demonstrate
that the parameter b defines SLL
= −20 log� ()� [dB]

⁄
cosh −1 (10 − 20 � (5.81)
)
= cosh � �

as well the pattern beamwidth: higher b drives down the level of sidelobes (compare blue, green
and red line graphs) but broadens the main beam. That is a consequence of the never proved
but despite this well working Murphy’s Law: “In nature, nothing is ever right. Therefore, if
everything is going right ... something must be wrong.” Note that the parameter b was defined
inverting the last equity in (5.79). From the above discussion and drawing in Figure 5.4.5
follows that the array factor (/), ∈[-1, 1], oscillates equally in the interval ∈
[ (−1/), (1/)] (blue, green and red points). Therefore, the desired SLL of 0dB (blue
dotted line) never exceeds if the antenna factor behaves like Chebyshev’s polynomial ()

with the same 0dB oscillations (solid blue line) or

(/) = + 2 ∑ (/) = () , ∈ [−1, 1] (5.82)

=1

0
It is well-known that two polynomials of the same degree are equal if and only if the coefficients
of the corresponding powers of the variable x are identical. If so, we need to equate coefficients
on both sides of (5.82) and derive all excitation coefficients from the resulting set of linear

equations. Let us illustrate this procedure with a simple example assuming N = 4. According to
(5.78) and (5.82)
4 2 2
⁄
⁄
8 − 8 + 1 = + 2� ( ) + �2 ( ) − 1�
0 1 2 (5.83)
3
2
4
⁄
⁄
⁄
⁄
+ (4 ( ) − 3 ( )) + �8 ( ) − 8 ( ) + 1��
4
3
Therefore, associating the coefficients belonging the same power we obtain the simple set of
four linear equations and its solution as
4
8 = 16 / 4 ⎫ = ⁄ 2
4
4
0 = 8 / 3 ⎪ ⎧ = 0
⎪ 3
3
−8 = 2(2 − 8 / ) ⇒ = 2 − 2 (5.84)
2
2
2
4
⁄
2 4 2
⎬ ⎨
0 = 2( / − 3 /) = 0
1 3 ⎪ ⎪ 1
1 = − 2 + 2 ⎭ ⎩ = 1 + 2 − 2
0 2 4 0 2 4
The more detailed expressions for the excitation coefficients’ calculation can be found in [6, 8].
Putting (5.84) into (5.78) and (5.80) we obtain

() = 0 + 2[ cos(2cos) + cos(4cos)] (5.85)
2
4

It is worthwhile to point out that the absence of the odd elements in array ( = = 0) doubles
1
3
the separation between the adjacent active radiators. But as we have earlier accepted this
distance should not exceed = . To fulfil it we have to halve the parameter in (5.85)
putting = /2. The array factor () for such array is normalized to the main beam peak
as shown in Figure 5.4.6 (first and second plot from the left). The array of 5 radiators is grouped