344 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
for the case where the target is a shorted doublet. 
P, is the total scattered power and W; is the power 
per square meter incident on the target. For a 
target with a radar gain Gp it follows that 
2 
2, Ss Wes Cae (49) 
20 
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 an amount 
PJ, = 4r@W,. (50) 
But 
Pe 5 GaP, (51) 
where P, is the amount scattered by an actual 
target with gain Gp. [The factor 3/2 appears be- 
cause the gain of the target relative to an isotropic 
radiator is (3/2)Gp.] Hence 
4n@W, = > GaP. (52) 
Eliminating P, from equations (49) and (52), 
W, _ 9NGR (63) 
W; 167°d? 
Putting this value of W,/W; in equation (41), 
2 
= Ge, (54) 
4a 
which is the required general formula connecting 
target gain and radar cross section. It will be noted 
that the factor 9\?/47 is just the radar cross section 
of the shorted doublet. 
Inserting the value of o given by equation (54) 
into equation (45), 
PS ( 3X r 
= = 466.4? |} , 55 
P. Bde 82d. (55) 
1 
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. When 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- 
sistance. It is 14 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 
Pe (2) 
28 He@@o(=2\\ Ae 56 
Py SHE aay) e 
where A, is the path gain factor defined by equation 
(20). 
