Radar System Engineering

20 THE RADAR EQUATION [SEC.22
radiate a well-defined beam. This maximum value of G we shall denote
by GO. The narrow, concentrated beams which are characteristic of
microwave radar require, for their formation, antennas large compared
to a wavelength. In nearly every case the radiating system amounts to
an aperture of large area over which a substantially plane wave is excited.
For such a system, a fundamental relation connects the maximum gain
G,, the area of the aperture A, and the wavelength:
(1)
The dimensionless factor .f is equal to 1 if the excitation is uniform in
phase and intensity over the whole aperture; in actual antennas f is often
as large as 0.6 or 0.7 and is rarely less than 0.5. An antenna formed by a
paraboloidal mirror 100 cm in diameter, for a wavelength of 10 cm,
would have a gain of 986 according to Eq. (1) with ~ = 1, and in practice
might be designed to attain GO = 640.
The connection between gain and beamwidth is easily seen. Using
an aperture of dimensions d in both directions, a beam may be formed
whose angular width, 1 determined by diffraction, is about X/d radians.
The radiated power is then mainly concentrated in a solid angle of X2/dZ.
An isotropic radiator would spread the same power over a solid angle
of 4T. Therefore, we expect the gain to be approximately 4rd2/X’, which
is consistent with Eq. (1), since the area of the aperture is about d’. For
a more rigorous discussion of these questions the reader is referred to
Vol. 12, Chap. 5.
A complementary property of an antenna which is of importance
equal to that of the gain is the e.fective receiving cross section. This
quantity has the dimensions of an area, and when multiplied by the power
density (power per unit area) of an incident plane wave yields the total
signal power available at the terminals of the antenna. The effective
receiving cross section A, is related to the gain as follows:
A . G@2.
,
4rr
Note that G, not Go, has been written in Eq. (2), the applicability of
which is not restricted to the direction of maximum gain or to beams of
any special shape. Once the gain of the antenna in a particular direction
is specified, its effective receiving cross section for plane waves incident
jrom that direction is fixed. Equation (2) can be based rigorously on the
Reciprocity Theorem (see Vol. 12, Chap. 1). Comparing Eqs. (2) and
(1) we observe that, if the factor j is unity, the effective receiving cross
1Wherever a precise definition of beamwidth is intended, we shall mean the
angularintervalbetweentwo directionsfor whichG = G,/2.
(2)