netted 
FUNDAMENTAL RELATIONS 343 
receiver, d the distance from the target to the 
receiver, and P, the totsl scattered power. This 
gives, using equation (39), 
S = 4rd? Ws (40) 
W; 
as a formula for the scattering cross section of an 
isotropic scatterer which involves scattered power 
per unit area at the receiver W, instead of total 
scattered power P,. 
For targets other than isotropic scatterers, how- 
ever, this procedure fails since one cannot say that 
W, = P,/4rd?. Nevertheless, it is useful to define a 
parameter o which is called the radar cross section, by 
7 
c= Anal 
’ 
in analogy with equation (40). Here W, is the actual 
power per unit area at the receiver. From the pre- 
ceding discussion it is apparent that o may be 
thought of as the scattering cross section which the 
target in question would have if it scattered as much 
energy in all directions as it actually does scatter 
in the direction of the radar receiver. For a target 
scattering isotropically, o = S, but for any other 
type of target o does not, in general, equal S. 
A radar gain formula analogous to the radio gain 
but applicable to two-way transmission can be de- 
veloped from equation (41) by replacing W, and W; 
witb the directly measurable: quantities P; (power 
output) and P» (received power). From equation 
(6), W; = E?/1207 in which £ is the field strength 
incident on the target. Substituting this value of # 
into equation (7) gives W; = 3P,/87d? for a doublet 
transmitter in free space. Including the gain of any 
type of transmitting antenna, this takes the form 
a 3PiGi 
8rd? 
Further, the power received by a doublet with a 
matched load, equation (17), may be written 
(41) 
(42) 
P, = — W,, (43) 
8a 
if #?/1207 is replaced by W,, where here FH is the 
field at the receiver. If the receiver is not a doublet, 
equation (43) may be replaced by 
pas 3n° 
82 
where (2 is the gain of the receiver. Substituting the 
values for W; and W,, given by equations (42) and 
(43), into equation (41) yields 
P; 2 
Bod (P(e (2) (45) 
TT 
W,Gs (44) 
This is the radar gain for two-way transmission in 
free space. By means of it, ¢ may be measured, or if 
« and P;/P, are known, it may be used to calculate 
ranges. Generalizing equation (45) we have 
P, Heed: (2) E 
So eae ee ( S| net 46 
P, a cay (46) 
where A, is the path gain factor (see page 339). 
It may be observed here that some writers call 
oA,', not oc, the radar cross section. These writers 
call their c, for the case A, = 1 (free space), the free- 
space radar cross section oo. Since, in this volume, 
the complicated terms appearing in A, are treated 
separately and not as part of the cross section, this 
distinction is not made here. 
For some simple targets, « may be calculated. 
The following are a few of the values. 
Radar 
cross 
Targets Condition section ¢ 
Conducting sphere, radiusa | a >> ma? 
Metallic plate, area = ab a>>X,b >> | 470%b2/d2 
Cylinder, diameter = d, Axis of cylinder mdl?/X 
length =] parallel to field 
andd >>), 
l>>X 
Matched load doublet Oriented parallel 92/1677 
to field 
Shorted doublet (dummy) Oriented parallel 9)2/47r 
to field 
Objects of tactical interest (ships, airplanes) have 
very complicated radar cross sections. In particular, 
a strong dependence on the aspect of these unsym- 
metrical targets is observed. For ships the situation 
is still further complicated by the variability of the 
incident field over the target area. 
Some writers on the subject of targets use a 
characteristic length L (sometimes also called a 
scattering coefficient) which is related to « by 
o = 4rL?. (47) 
Radar Gain 
It is possible to write equations for two-way 
transmission which bear a formal resemblance to 
corresponding equations for one-way transmission by 
introducing a quantity Gp, called the gazn of the 
target. Gp is the gain of a target in the direction 
of the radar receiver relative to a shorted (dummy) 
doublet. 
By writing formulas connecting the radar gain 
with the power per square meter incident on the 
target and the power per square meter scattered 
back to the receiver, it is possible to establish a 
connection between radar gain and the radar cross 
section defined in the last paragraph, and from this 
to calculate a gain formula involving Gp instead of c. 
Applying equations (15) and (6), 
Ee ey (48) 
Qn 
