Radar System Engineering

CHAPTER 2
THE lU.DAR EQUATION
BY E. M.
PURCELL
The operation of a radar set depends on the detection of a weak
signal returned from a distant reflecting object. The factors which
control the strength of the signal so received are clearly of first importance
in determining the maximum range of detection of a given target by a
specified radar set. In Sees. 2.1 to 26 we shall formulate and examine
the basic relation between these quantities, which is commonly known
as the “radar equation. ” Specifically, we want to derive an expression
for the peak radio-frequency signal power S, available at the terminals
of the radar antenna, which will involve measurable properties of the
transmitting and receiving antenna system, the transmission path
through space, and the target itself. Now this relation will not suffice
to fix the maximum range of detection unless the minimum power
required for detection, Sk, is known. This important quantity S~ti we
prefer to discuss separately, beginning in Sec. 2.7 below. It will be found
to depend on many other factors, not all readily accessible to measurement,
ranging from thermal noise in a resistor to the integrating property
of the eye of the radar observer. Thus we choose to divide the problem
into two parts, by a fictitious boundary, as it were, between the radar
antenna and the rest of the set. The relations which we shall develop
in Sees. 2.1 to 2.6 are wholly geometrical ones in the sense that the factors
upon which the received power S depends are all lengths, apart from the
transmitted power P, to which, of course, S is always proportional.
THE RADAR EQUATION FOR FREE-SPACE PROPAGATION
2.1. The Meaning of Free-space Propagation.-Fortunately, the
quasi-optical nature of microwave propagation permits us to concentrate
our attention at the outset on a very simple case, which we shall call
“free-space propagation.” The circumstances implied would be realized
exactly if radar set and target were isolated in unbounded empty space.
They are realized well enough for practical purposes if the following
conditions are fulfilled:
1, No large obstacles intervene between antenna and target, along
an optical line of sight.
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