A MATHEMATICAL THEORY OF LINEAR ARRAYS 101 



The angle of the cone of silence enclosing the major radiation lobe is 

 determined from 



^S^cos^i - 1) = -— ^; (39) 



11 — 1 



thus 



If the arc ^ is equal to the range of z, then (40) becomes 



1 — cos ^1 = -, sin ~ = , (41) 



n - 1 2 Vw - 1 



In this case, the si2e of the first cone of silence is determined solely by the 

 number of elements. On the other hand, if ^ = 27r —Iw/n, the nulls are 

 equispaced on the unit circle and we have an ordinary uniform array; then 



1 - cos ^1 = £^. = f>' sin 2 = l/^r ^^2) 



^ - A 



n^t~ lie ""' 2 



This time the size of the first cone of silence depends upon the total length 

 L — {n — \)t of the array measured in wavelengths. 



When the number of elements in the first case and the total length of the 

 array in the second are large, then we have approximately 



y 2nr 



= U/irr (43) 



■\/n — 1 

 For a large n the ratio of the two cone angles is approximately 





(44) 



For example, if i— X/8, the angle of the major lobe in the first case is one-half 

 of that in the second case or one-quarter if we are to compare the solid angles. 

 Equispacing the null points in the range of z not only makes the major 

 lobe narrower but it also makes it sharper. Thus at the point lying halfway 

 between the point of maximum radiation and the first null point, the field 

 strength relative to the principal maximum is 



^ . 3iA . SiA . {In - 3)4' 



sin T. tn sm 77 T-, sm — • • • sm ^— {- 



_ ^n — 1) 4(w — 1) 4(w — 1) 4(w — 1) 



^ ~ . iA . 2^ . 3^ . (« - 1))A 



sm — sm —. — - — — sm —. — - — -, • • • sm ~ -^ 



2(n - 1) 2(w - 1) 2(w - 1) 2(w - 1) 



