3-2] THE RADAR RANGE EQUATION 139 



Normally, the transmitter is not an isotropic radiator but possesses a 

 directivity or power gain due to the influence of an antenna. The power 

 gain on transmission is denoted by Gt and the resulting power density at the 

 range R is 



P G 



Power density with an antenna = D = - — ^- (3-2) 



This power is incident upon some sort of target which reflects a portion to 

 the receiver. The target will be characterized by an idealized or eff"ective 

 cross-sectional area a. This area is defined to reradiate isotropically all the 

 incident energy collected. The target cross section will be dependent on the 

 radar frequency being used and the aspect from which the target is viewed. 

 It is normally determined experimentally and often represents a large 

 unknown factor in radar system calculations. By definition, the power 

 collected by the target is Da. When this power is reradiated isotropically, 

 the power density at the receiver, which is assumed to be located near or at 

 the transmitter, is simply 



Power density at the receiver = - — ^r;; = , . ,„„. • (3-3) 



■iirK- (4x)-K* 



The effective area of the receiving aperture is denoted by Jr- The power 

 intercepted by the effective area of the receiving antenna is simply the 

 product of this area and the power density. The receiving area is related to 

 the receiving gain Gr and the wavelength X by the following relation.^ 



^r = ^- (3-4) 



We shall assume, as is normally the case, that the same antenna is used for 

 reception and for transmission. In this event, the receiving gain will equal 

 the transmission gain or Gr =" Gt — G. 



The power received by the receiver will be simply the product of the 

 power density at the receiver and the receiving area. Combining Equations 

 3-3 and 3-4, the received signal power will be 



Received signal power = ^S' = , yn^ (3-5) 



This expression represents one version of the radar range equation. It 

 shows how the received power varies with target range and size and with 

 the wavelength and power gain of the antenna. The received power can 

 represent either average power or peak power, depending upon what the 

 transmitted power Pt represents. 



'See Paragraph 10-1 for a further discussion and references. 



