236 BELL S YSTEM TECH NIC A L JOURNA L 



where A is a negligibly small fraction of a wavelength for every point on .9 

 then we see from (17) that the electric vector at Q is gi\en by 



Js \r Xd 



The power flow per unit area at Q is therefore 



1 £^5' PtS 



P = 



UOir \W \H' 



Po the power flow per unit area at Q when power is radiated isotropically 

 from is found by assuming that Pt is spread evenly over the surface of a 

 sphere of radius d. 



The gain of a lossless, uniphase, uniamplitude, linearly polarized antenna 

 is, by the definition of equation 1, the ratio of 19 and 20. 



It follows from 12 that the effective area of the ideal antenna is 



A ^ S (22) 



In other words in this ideal antenna the effective area is equal to the actual 

 area. This is a result which might have been obtained by more direct 

 arguments. 



3.3 Gain and Efeclive Area of an A ntenna with Aperture in a Plane and with 

 Arbitrary Phase and Amplitude 



Let us consider an antenna with a wave front in the XY plane which has 

 a known phase and amplitude variation. Let the electric intensity in the 

 wave front be 



E{x, y) = Eoaix, y)e'*^''''^ (23) 



polarized parallel to the x axis. The radiated power is equal to the power 

 flow through 5 and is given by 



_ E'o I a'{x, y) dS 



P... = " J " " (24) 



1207r 



The input power to the antenna is 



Pt = PradA (25) 



