88 BELL SYSTEM TECHNICAL JOURNAL 



For free space \E\ = |//| so 



s=^E^. (19) 



Now the total power radiated through a sphere enclosing an array 

 of sources is 



Pj = Csda =^ r r^ E,"^ sin ed4>dd. (20) 



A second system would give 



t-Jo Jo 



p. =^ j I £2' sin ed<i>dd. (21; 



The radiated powers of these two systems might be so adjusted 

 at the source as to give equal fields at any point along a preferred 

 direction. A ratio of these powers, therefore, would be a convenient 

 measure of the relative directional properties of the two arrays. This 

 "test ratio" may conveniently be set up in terms of the equations of 

 the diagrams derived above. In which case 



ri2 sin ed(i>d6 



Jo Jo 



^2" sm 



If we assume all comparisons are to be made with respect to a single 

 linear oscillator the denominator reduces to 87r/3, so 



r = A r f"" ri' sin ed(f>dd. (23) 



>^Jo Jo 



This ratio may conveniently be expressed in decibels. In which 

 case G = 10 logio l/T is sometimes called the gain of an array. 



If we are interested in the solid array shown in Fig. 21, where 

 n-N'N linear oscillators, each having respective space and phase 

 separations of a\, bT; A\, BT\ and A\, BT, are arranged progressively 

 along the three principal coordinate axes, this becomes 



_ 3 r^" r^' sin^ [t?7r(a cos </> sin + Z>)] 

 8^ Jo Jo ^^ ^^"^ ^'^^^ ^°^ (i> sin d -\- b)2 

 sin'' iNxjA sin sin g + Jg)] 

 * N^ sin2 liriA sin sin + 5)] 



. ^f^^^^f/'^'+^ll ^sin^ed^de. (24) 

 N^ sm^ \_Tr{A cos 6 + B)j 



