A MULTIPLE UNIT STEER ABLE ANTENNA 345 



This exponential series may be evaluated with the aid of the 



identity ^^ 



. nd 



sin-y (n-l)d 

 1 + e'« + e^-^" + • • • + e'C-i^" = j e^ ^ ^"^ • 



sin- 



Using this summation we ha"ve 



N 

 sin— [0 — Iwaiv — cos 8)j 



J^ _ J ___Z gy[u«+(iV-l)/2(0-27ra(u-pos «)])_ ^3) 



sin- \jj) — 27ra(i^ — cos 5)] 



The amplitude of A in (3) is the array directional pattern or array 

 factor. It is zero when the numerator alone is zero, i.e., when 



-[</)- liraiy - cos 5)] ?^ 0, ± tt, ± 27r • • • 



and simultaneously 



N 



Y [0 - 27ra(u - cos 5)] = 0, ± TT, ± 27r • • • . 



It attains its maximum value of NI when the denominator and 

 numerator are zero simultaneously, i.e., when 



- [0 — 27ra(y — cos b)~\ = 0, ± tt, ± 27r • ■ • 



and 



A'' 



y [(/) - liraiy - cos 5)] = 0, ± TT, ± 27r • • •. 



Plots of (3) for ten unit antennas (iV — 10) spaced five wave-lengths 

 (a = 5) are shown in Fig. 4 for two arbitrary values of ^. The same 

 array used at twice the frequency (iV = 10, a = 10) has the directional 

 patterns shown in Fig. 5. The abscissas are labeled earth angle 

 although nothing has been said thus far concerning the disposition of 

 the N antennas with respect to the earth. In order that the simple 

 multiple phase shifts of Fig. 1 shall suffice to steer the array, reflection 

 from the ground must affect the phase of all antenna outputs identi- 

 cally. This is assured by constructing the array over, and parallel to, 

 a flat expanse of ground. Since the angle b measures the direction of 



'2 This identity may be deduced by substituting e'* for r in the well-known for- 

 mula for the sum of a geometrical progression 



]. -\- r -\- r'^ + r^ -\- ■■ ■ + /-"-i = '"' ~ ^ . 



r — 1 



