Radar System Engineering

22 THE RADAR EQUATION [SEC.25
Again we call attention to the fact that Eq. (3b), like Eq. (2), contains G
rather than Go, and is not restricted to any particular direction or to
beams of any special shape. The sole restriction which has not yet been
made explicit is that G should not vary significantly over the angle which
the target subtends at the radar antenna.
Usually we shall be interested in the signal that is returned when the
target lies somewhere along the maximum of the radar beam, and we
should then replace G by G,. It is instructive to proceed then to eliminate
GOby means of Eq. (l), obtaining
S = P(TA ‘j2
47rR%2”
(4a)
Note that AZnow appears in the denominator, while the numerator contains
the square of the area of the antenna aperture. A further manipulation
of Eq. (4a) is of interest. Suppose the minimum power required
for satisfactory detection, Smin, is known; we may solve Eq. (4a) for the
maximum range of detection, R~..:
4 PuA ‘jz
R .s, = — (4b)
J 4~S~,.A2”
At this point it may be well to get an idea of the order of magnitude
of the quantities involved by inserting numbers not unusual in wartime
pulse-radar practice. If we choose X = 0.10 ft ( = 3.0 cm), P = 10’
watts, A = 10 ft 2, j = 0.6, u = 100 ftz (typical for small aircraft),
& = 5 X 10–12 watts, we obtain R~.. = 155,000 ft or 29 statute miles.
We observe that a 16-fold increase in transmitted power is required to
double the maximum range; on the other hand, it would appear that R~..
could be doubled by doubling the linear dimensions of the antenna. But
the latter step would at the same time reduce the beamwidth by a factor
of 2 and as we shall see in Sec. 2“11 that this indirectly affects Sti.. A
change in wavelength is even more difficult to discuss as it entails changes
in S~i., P, and possibly u as well.
2.5. Beams of Special Shapes.—In several applications of radar, use
is made of an antenna designed to spread the radiated energy out over a
considerable range in angle in one plane. The object usually is to
increase the angular region covered at one time. An example of such a
radiation pattern is the simple “fan beam” sketched in Fig. 2.1. It is
easy to produce such a beam by means of an antenna whose effective
aperture is wide in the direction in which the beam is to be narrow and
narrow in the direction in which the beam is to be broad. The connection
given by Eq. (1) between aperture and gain still holds, implying
that by reducing the vertical dimension of the antenna aperture (in
Fig. 2.1), gain has necessarily been sacrificed; of course this is inevitable,