Page 173 - Satellite Communications, Fourth Edition

Antennas 153

which is the xz plane, for which 0, and the E plane, which is the xy
plane, for which 90°. It simplifies matters to examine the radiation
pattern in these two planes. Consider first the H plane. With 0, it
is seen that Y 0, E 0, and

a
X sin
(6.21)
l
and with C set equal to unity,

sin X
E (
) cos
(6.22)

X
The radiation pattern is given by:
(
) ZE (
)Z 2
g H
2
2 sin X (6.23)
cos
P P
X
A similar analysis may be applied to the E plane resulting in X 0, E

0, and
b
Y sin
(6.24)
l
sinY
E (
) (6.25)

Y
g (
) ZE (
)Z 2
E

sinY 2 (6.26)
P P
Y
A function that occurs frequently in communications engineering is
the sampling function defined as Sa(x) sinx/x. This function is avail-
able in tabular form in many handbooks, and may also be available in
programmable calculators. A point to bear in mind when evaluating
this function is that the denominator x must be in radians. Sa(x) 1
for x 0, and Sa(x) 0 for x n , where n is the integer. It is seen that
the H plane pattern contains the function Sa(Y) and the E plane, the
function Sa(X). These radiation patterns are illustrated in Example 6.1.

Example 6.1 Plot the E-plane and H-plane radiation patterns for the uniformly
illuminated aperture for which a 3l, and b 2l.
Solution Looking first at the H plane, for a 3l, X 3 sin
. As noted here, the
sampling function has well-defined zeros, occurring in this case when sin
1/3,
2/3, or 1. The g H (
) function will have correspondingly zeros or nulls. (The cos
term
will also have zeros for
= n /2, where n is any odd integer.
For the E plane and b 2l, Y 2 sin
. Again, the sampling function has well-
defined zeros occurring in this case when sin
1/2 or 1, and the g E (
) function