A MATHEMATICAL THEORY OF LINEAR ARRAYS 107 



where \f/ = ^f (cos 6 — \). In this case D — 1 and 



r^ • « ^« 2 r ... sin 2{n - IW , 2 sin 2(« - 2)^^ 



/ $ sni ddd = -—- n^l + + — ^— 



Jo fir^^ L w — 1 n — 2 



^ (63) 

 ^...^ 3sin2(.-3W ^...^^^_^^^.^ n^ 



w — o J 



When the separation between the elements is exactly an integral number 

 of quarter wavelengths, (63) becomes 



I 



2 

 ^sinddd = - (64) 



n 



and consequently the gain is 



G = 10 Logio n. (65) 



Figure 21 contrasts the directive gain of a pair of sources of equal strength 

 with the phase delay ItC/X (Curve A) with a directive gain of another pair 

 of sources of equal strength but with the phase delay r — 2irf/\ (Curve B). 

 In one case the directive gain diminishes with separation between the ele- 

 ments and in the other it increases. Figure 22 shows the directive gain of 

 three-element and four-element end-on arrays with nulls equispaced in the 

 range of z. 



As the separation between the elements decreases, the directive gain of an 

 end-on array with nulls equispaced in the range of z increases but the radia- 

 tion intensity per ampere-meter decreases. This circumstance would be of 

 •no importance if we had perfect conductors at our disposal to make trans- 

 mitting and receiving antennas; but in reality parasitic losses in themselves 

 cannot be removed and the efficiency of an array decreases, therefore, with 

 the separation between the elements. This decrease in efficiency will 

 impose an upper limit on the overall gain that can be obtained with small 

 antenna arrays in spite of the fact that the directive gain could be made very 

 large. 



Likewise the band width diminishes as the distance between the elements 

 decreases. This imposes another limitation on arrays of this type. 



