DIRECTIVE DIAGRAMS OF ANTENNA ARRAYS 



307 



Appendix 



Formulae for the directive diagrams of this paper are conveniently 

 expressed in terms of polar coordinates, as follows: For a linear array 

 of n antennae with the parameters (aX, bT), 



sin n (ira cos d-^-wb) 



(1) 



The area of this 



n sin (ra cos d-{-Tb) 



where d is measured from the axis of the array 

 diagram, relative to the unit circle, is 



(n — 1 V 



■|+ ^ (n-k)M2Trka) cos (27rkb)\ (2) 



For the special case n = 2, Fig. 1, formula (1) reduces to 



r = I cos (ira cos 6-\-Trb) \ , (3) 



and formula (2) for the area to 



5 = i(l + ^o(27ra) cos (27r&)) . (4) 



For the special case w= oo in a total interval (^X, BT), the limit of 

 formula (1) for a=^A/n and b^Bjn, as n becomes infinite, is 



sin {-kA cos Q-\--kB') 



(5) 



■kA cos Q^-kB 



For the array of Fig. 11, 



r = I cos (|x cos Q cos 0) cos (^tt sin Q cos </>) | , • (6) 



where is the angle which the radius vector makes with the plane 

 of the array, and Q the angle which the projection of the radius vector 

 in this plane makes with one side of the square. For the array of 

 Fig. 12, 



sin (Stt cos fl cos 0) sin (Stt sin Q cos </>) 



r = 



32 sin (^TT cos Q cos 0) 32 sin (|x sin Q cos 0) 



(7) 



where and Q are the same as for formula (6) except that the latter 

 is measured from one diagonal of the square. 



