Radar System Engineering

26 THE RADAR EQUATION [SEC.2.5
small except for small values of O, and since G vanishes for values of o
outside the interval 00to ~/2, it is permissible to write and evaluate the
integral as follows;
()
H
=/2 - sin2 7raij —
1 G,
i
CSC2e
4UCSCZ00 ~. o
7ra4 2
(-) A
Hence
d@dO=~cos O,sin OO= 1. (10)
GO = ~sec
OOCSC00,
Returning nowto Eq. (6b), if Gisreplaced bytheright-hand side of
Eq. (n), andareplaced byk/a, weobtain,forS,
S = PLah tanz 0,
4uh3
(12)
When O,issmall, asisusually thecase, itispermissible toreplace h/tan 00
by Rti, the maximum range, which leads tothefinalr elation
(13)
The appearance of Xin this formula, which isto be contrasted with
theresult obtained forthesimple fan beam and point target, Eq. (5), can
be traced to the influence of the horizontal beamwidth upon the effective
cross section of the extended target. It will be observed that once the
other system parameters, P, L!&, and a, ale specified, the quantity
hR2 is fixed. That is to say, the maximum range obtainable is inversely
proportional to the square root of the height of the aircraft, keeping
everything about the radar set constant but the vertical radiation
pattern of the antenna, which we assume to be adjusted to optimum
shape for each height.
A problem related to the preceding one is met in the design of groundbased
air-search radar, which may be required to provide uniform coverage
at all ranges for point targets (aircraft) flying at some limiting
altitude h. Here, however, u is generally assumed to be constant: the
reader will easily verify that this assumption leads again to the requirement
that the gain vary as CSC20, but with a diflerent final result for the
dependence of S upon h, R, and a. It turns out in fact that for the point
target with u independent of angle, the quantity hR~,. is constant, rather
than hR5.,, and S is proportional to az rather than to ah.