A MATHEMATICAL TIIKORY OF LINEAR ARRAYS 93 



at Q is comparatively nondirective. Substituting for this couplet another 

 couplet with a null at B should improve the directivity of the array. This 

 is indeed the case: In Fig. 12, Curve A depicts directive properties of the 

 uniform array and Curve B depicts those of an array with its nulls at P and 

 B. The field strength of the second array is 



V* = \{z - e '^){z + 1)1 = U^ + (1 - e~^)z - e~ ^ \ 



I _ _ii -ill TT ^^^^ 



= 1 1 + 2 Vs e ^ -{- z e ^ I , z = X (cos d — \); 



hence the amplitudes of the elements are proportional to 1, -x/S, 1 and the 



total progressive phase delay in the direction of maximum radiation is 



7r , TT 2ir ,. 

 ;r + 7 = -5- radians, 

 I b 6 



The minor lobe of the second array is substantially smaller than that of 



the first array. The major lobes, however, are equally "wide"* although 



one lobe is somewhat sharper than the other. The width of the major lobe 



can be reduced at the expense of increasing the minor lobe by moving the 



null from P io M (Fig. 11). The effect of this change is shown by Curve 



C (Fig. 12). The corresponding field strength is 



\/¥ = I (s + i)(z + 1) I = I 2' + (1 + i)z + i I 



j iv IT . \^^ ) 



= I 1 - i(l + i)z - iz^\ =\l -\- V2e~^z + T^z' |; 



hence the amplitudes are proportional to 1, v^2, 1 and the total progressive 



phase delay is - + - = -— . 

 2 4 4 



For arrays of six elements, one-quarter wavelength apart and with t} = 

 7r/2, we have Fig. 13. Curve A represents the directive characteristic of a 

 uniform array, with its nulls as shown in Fig. 14A, and Curve B shows the 

 directive properties of an array with its nulls equispaced on the lower half 

 of the unit circle as shown in Fig. 14B. 



If the spacing between the elements is ^ = X/8 and if the phase delay i? = 

 7r/2, then the effect of distribution of the null points is even more pronounced 

 (Figs. 15 and 16). This time z is confined to the fourth quadrant of the 

 unit circle. In Fig. 15, n — 3; Curve A corresponds to an array with equal 

 amplitudes in which case the nulls are equispaced on the complete unit 

 circle (Fig. 17 A) and Curve B corresponds to an array with its nulls equi- 



^ If the "width" of a lobe is measured by the angle of the cone of silence enclosing the 

 lobe. _ 



^ When transforming the expressions for V*, it is well to remember that the absolute 

 value of a complex quantitj' does not change if this quantity is multipUed by a unit complex 

 number. 



