20 



FUNDAMENTAL RELATIONS 



where 11'^ is the scattered p(jwer per unit area at the 

 receiver, (/ the distance from the target to the 

 receiver, and Fj the total scattered power. Tliis 

 gives, using ecjuation (39), 



,S = 4Tr<l- 



JVr 



Wi 



(40) 



as a formula for the scattering cross section of an 

 isotropic scatterer which involves scattered power 

 per unit area at the rpcei\'er W, instead of total 

 scattered power Pj. 



For targets other than isotropic scatterers, how- 

 ever, this procedure fails since t)ne cannot say that 

 Wr = Ps/iwd^. Nevertheless, it is useful to define a 

 parameter a which is called the radar cross section, by 



= iwd' 



, w. 



Wi 



(41) 



in analogy with ec^uation (40). Here W, is the actual 

 power per unit area at the receiver. From the pre- 

 ceding discussion it is apparent that <t 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 recei^'er. For a target 

 scattering isotropically, c = S, but for any other 

 type of target er does not, in general, eciual aS. 



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, 

 with the directly measurable quantities Pi (power 

 output) and P^ (received power). From ecjuation 

 (6), Wi = £'Vl20ir in which E is the field strength 

 incident on the target. Substituting this value of E 

 into equation (7) gives IF,- = 3Pi/Sird'- for a doublet 

 transmitter in free space. Including the gain of any 

 type of transmitting antenna, this takes the form 



3PiG'i 



W, = 



S-rrfP 



(42) 



Further, the p(jwer received by a doublet with a 

 matched load, equation (17), may be written 



3X- 



Pi = 



TF, 



(43) 



if £"-/1207r is replaced bj' \\\, where here E is the 

 field at the receiver. If the receiver is not a doublet, 

 equation (43) may be replaced by 





(44) 



values for TF; and W,, gi\'en by equations (42) and 

 (43), into equation (41) yields 



^ = fnG. -^- f-^Y (45) 



This is the radar gain for t\\o-\\'ay transmission in 

 free space. By means of it, a msiV be measured, or if 

 (7 and Po/Pi are known, it may be used to calculate 

 ranges. Generalizing equation (45) we have 



^ = r;/;, -^- (--\^^p^ (46) 



Pi iird- KSwdJ '^ 



where Ap is the path gain factor (see Section 2.2.1). 



It may be observed here that some writers call 

 (7.4/, not (T, the radar cross section. These writers 

 call their a, for the case .4^=1 (free space), the free- 

 space radar cross section o-q. Since, in this volimie, 

 the complicated terms appearing in .4^ are treated 

 separately and not as part of the cross section, this 

 distincti(_)n is not made here. 



For some simple targets, a may l)e calculated. 

 The following are a few of the values. 



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 ovev 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 cr bj' 



a = 47rL^ (47) 



2.4.2 



Radar Gain 



where C,'^ is the gain of the receiver. Substituting the 



It is possible to write equations for two-way 

 transmission which bear a formal resemblance to 

 corresponding eciuations for one-wa.v transmission by 



