Page 144 - Satellite Communications, Fourth Edition

124 Chapter Five

With the unit polarization vector at the earth station denoted by p,
the angle between it and f is obtained from the vector dot product as

p ? f
arccos (5.9)
Z Z
f
Since the angle between a normal and its plane is 90°, the angle
between p and the reference plane is j 90 h and
o
p ? f
arcsin (5.10)
Z Z
f
This is the desired angle. Keep in mind that the polarization vector is
always at right angles to the direction of propagation.
The next step is to relate the polarization vector p to the defined
polarization at the satellite. Let unit vector e represent the defined
polarization at the satellite. For vertical polarization, e lies parallel to
the earth’s N-S axis. For horizontal polarization, e lies in the equato-
rial plane at right angles to the geostationary radius a GSO to the satel-
lite. A cross-product vector can be formed,
g k e (5.11)

where g is normal to the plane containing e and k, as shown in Fig. 5.9.
The cross-product of g with k gives the direction of the polarization in
this plane. Denoting this cross-product by h gives

h g k (5.12)

The unit polarization vector at the earth station is therefore given by
h
p (5.13)
Z h Z

r f

p
k
g

a GSO e Figure 5.9 Vectors g k e and h g h.
h