Page 194 - Satellite Communications, Fourth Edition

174 Chapter Six

(Imaginary)
j

E ψ
E R
E
E ψ
Figure 6.27 Phasor diagram for the in-
0 (Real) line array of dipoles.

The Argand diagram for the phasors is shown in Fig. 6.27. The magni-
tude of the resultant phasor can be found by first resolving the individual
phasors into horizontal (real axis) and vertical (imaginary axis) compo-
nents, adding these, and finding the resultant. The contribution from the
first element is E, and from the second element, E cos jE sin . The
third element contributes E cos 2 jE sin 2 , and in general the Nth
element contributes E cos(N 1) jE sin(N 1) . These contribu-
tions can be added to get:

E E cos jE sin E cos 2 jE sin 2 c
E R
N 1
a E cos n jE sin n
n 0 (6.37)
N 1
jn
E a e
n 0
Here, N is the total number of elements in the array. A single element
would have resulted in a field E, and the array is seen to modify this by
the summation factor. The magnitude of summation factor is termed the
array factor (AF):
N 1
AF 2 a e jn 2 (6.38)
n 0
The AF has a maximum value of N when 0 , and hence the max-

imum value of E is E R max NE . Recalling that as given by Eq. (6.36)
R
is a function of the current phase angle, , and the angular coordinate,
, it is possible to choose the current phase to make the AF show a peak
in some desired direction . The required relationship is, from Eq. (6.36),
0
2
s cos 0 (6.39)
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