86 BELL SYSTEM TECHNICAL JOURNAL 



which is readily summed giving 



so 



(10) 



(11) 



Reducing to common voltage level and including a term sin d to cover 

 the case of radiation from Hnear oscillators we have for the equation 

 of the directional diagram 



_ sin mr{a cos <^ • sin + ^) 

 n sin 7r(a cos ^ • sin + 6) 



sin Nir{A sin (/> • sin g + .B) sin iVx(i4 cos 6 -{- B) ^.^ ^ .^2) 

 * iVsin Tr{A sin </> • sin + 5) ' iVsin t:{A cos 6 + B) ' 



It will be recognized that this equation is made up of four factors. 

 The first three account for the effects of the disposition of elements 

 along the x, y, and z axes, respectively, while the fourth, of course, 

 accounts for the direction of radiation from a linear oscillator. This 

 is an equation giving magnitudes only. In plotting polar diagrams 

 from this equation negative signs have no physical significance, and are 

 plotted in a positive sense. 



An examination of this equation shows that there are many possi- 

 bilities which allow radiation in preferred directions, and at the same 

 time limit it in others. Some of these are discussed below. 



Special Cases 

 If we assume n = 2, a = I, b = — j and B = B = 



sin (NttA sin <^ • sin 6) sin (NttA cos d) 

 N sin {tA sin </> • sin d) N sin {ttA cos 6) 



• cos- (cos • sin — 1)- sin 6. (13) 



This corresponds to the practical case of transmission along the 

 x axis from an antenna curtain and reflector made up of N vertical 

 columns of N elements each. 



