RADAR CROSS SECTION AND GAIN 



21 



introducing a quantity Gn, called the gain of the 

 target. (Jr is the gain of a target in the direction 

 of the radar receiver relative to a shorted (dumnij^) 

 doublet. 



Bj^ writing formulas connecting the radar gain 

 mth 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 Gr instead of tr. 



Applying equations (15) and (6), 



Pe = Wi 



3X2 



(48) 



for the case where the target is a shorted doublet. 

 Pj is the total scattered power and TF,- is the power 

 per square meter incident on the target. For a 

 target with a radar gain Gr it follows that 



Ps = W.^^Gn. 



(49) 



In a similar way a formula for W,, the scattered 

 power per unit area at the receiver, can be developed. 

 A target which scattered equally in all directions 

 would scatter a,n amount 



P/ = 47rdW, . 



(50) 



But 



P/ 



GuPs 



where Pj is the amount scattered by an actual 

 target with gain Gr. [The factor 3/2 appears be- 

 cause the gain of the target relative to an isotropic 

 radiator is (3/2) Ga-] Hence 



43rdW, = - GrP, . 



(52) 



Eliminating Pj from equations (49) and (52), 



W^ ^ 9X2_(V 



Wi 167r2d2 ■ 

 Putting this value of W,/Wi in equation (41), 



9X- 



4Tr 



Gr', 



(53) 



(54) 



which is the required general formula connecting 

 target gain and radar cross section. It will be noted 

 that the factor 9X-/47r is just the radar cross section 

 of the shorted doublet. 



Inserting the value of a given by equation (54) 

 into equation (45), 



§ = AGiG,Gr' 



Pi 



XSirdJ ' 



(55) 



which is the radar gain formula for free space in 

 terms of the gain of the target relative to a dummy 

 doublet. 



The reasonableness of the factor 4 in the above 

 equation may be made apparent by the following 

 analogy. Compare the doublet antenna with a 

 generator whose internal resistance corresponds 

 to the radiation resistance of the antenna. ^^Tien the 

 generator is shorted all the power is dissipated in 

 the internal resistance. When the doublet is shorted 

 all the power is reradiated. The maximum power 

 that can be extracted from either the generator or 

 the antenna occurs when the load resistance equals 

 the internal generator, or antenna radiation, re- 

 (51) sistance. It is 3^ the above short-circuit power. 



This is the 4 that occurs in the above equation. 



Equation (55), in the nonfree-space case, takes the 

 form 



Pi 



P. 



= ^GiGoGr- 



\8wd/ 



(56) 



where Ap is the path gain factor defined by equation 

 (20). 



