CAIN OF DIRECTIVE ANTENNAS 91 



This equation was used in calculating the data given in Figs. 18 and 19. 

 (6) The test ratio for the case of the rectangular array of nN 

 elements discussed in connection with Fig. 16 may be calculated by 

 placing N=l,a = i, b=— I and B = 0. In which case 



^« = 4r + -4r2''^' (^ - k)-q(2tKA, 0) 



+ -4r e' L {n - k){N - K) • cos (^) 



7l~I\~ K=l k=l \ ^ / 



Q(^2r^^+KU\oy (31) 



Areas of Directional Diagrams 



In general, the areas of directional diagrams may be calculated 

 from their equations by the usual integration methods. The special 

 case of N couplets in horizontal array, such as used rather generally 

 in practice and shown in Fig. 5 above, is of sufficient importance to be 

 given here. The area of the diagram in the (XY) plane is 



S = -^,\^ + 'z\n-K)' Jo{2tKA) ' cos ItKB 1 



(32) 



This equation was used in calculating the data given in Fig. 5. 



The area of diagrams in the horizontal plane due to a single array 

 of N oscillators is given by the equation: 



S = ^ 



^ N' 



N 



+ "Z {N - K) • Jo{2tKA) • cos 2tKB 1 .* (33) 



K=l J 



This differs from equation (32) by a factor of two and indicates that 

 regardless of whether the gain is reckoned by an integration over a 

 unit sphere or in terms of the area of the horizontal diagram the effect 

 of the reflector is to double the radiated field in the preferred direction. 

 Placing -tV = 1 in equation (32) 



S = h (34) 



This is analogous to equation (29) above. 



* R. M. Foster, "Directive diagrams of antenna arrays," Bell Sys. Tech. Jour., 5, 

 307; 1926. 



