Page 167 - Satellite Communications, Fourth Edition

Antennas 147

R R R
1 radian
radian R
R

R

(a)

Surface Surface area
2
area R 2 A = R
R R

R
R
Unit
steradian Steradians

(b)
Figure 6.7 (a) Defining the radian. (b) Defining the steradian.

This is illustrated in Fig. 6.7a. The circumference of a circle is given by
2 R, and hence the total angle subtended at the center of a circle is 2
rad. All this should be familiar to the student. What may not be so
2
familiar is the concept of solid angle. A surface area of R on the surface
of a sphere of radius R subtends unit solid angle at the center of the
sphere. This is shown in Fig. 6.7b. The unit for the solid angle is the stera-
dian. A solid angle of steradians defines a surface area on the sphere
2
(a spherical cap) of R . Looking at this another way, a surface area A
2
subtends a solid angle A/R at the center of the sphere. Since the total
2
surface area of a sphere of radius R is 4 R , the total solid angle sub-
tended at the center of the sphere is 4 sr.
The radiation intensity is the power radiated per unit solid angle.
For a power P rad radiated, the average radiation intensity (which is also
the isotropic value) taken over a sphere is

P rad
U W/sr (6.7)
i
4