THE CALCULATION OF VERTICAL COVUKAGK 



163 



'•'"'' " Antenna I'atlerns 



To obtain/.,, the relative amplitude of the radia- 

 tion from the antenna, as a function of the vertical 

 angle 7 it is only necessary to take into account, the 

 path differences of the elements of the array. The 

 absolute field intensity and time phase will not be 

 considered. In Figure 61 is shown an array of four 



Figure 61. The four-element array. 



horizontal half -wave dipoles spaced a half wavelength 

 apart. The radiation from A in the direction 7 may 

 be taken as proportional to cos cot. The path differ- 

 ence of radiation from B is A/2 ■ sin 7. The corre- 

 sponding phase difference is 



2x X . 



T x 2 sm7 = 



-tv sin 7 



For C and D the phase is — 2x sin 7 and — 37r sin 7 

 respectively. The total field intensity pattern is 

 f A = cos cot + cos (cot — x sin 7) 



+ cos (at — 2x sin 7) + cos (at — 3x sin 7) , 

 grouping 



[cos at + cos (at — 3x sin 7)] 



+ [cos (at — x sin 7) + cos (wt — 2x sin 7)] . (89) 

 From the identity 



cos A + cos B = 2 cos f (A + B) cos \ (A - B) 

 equation (89) may be written 



f A = 2 cos ( cot — sin 7 J cos ( — sin 7 



-r- 2 cos ( Lit — -=- sin 7 j cos (^ sin 7 ) 



f A = 2 cos ( at ■— K — sin 7 



C0S ( Y sin 7 ) + cos " (1 sin T ) 



i A = 4 cos ( at 



3tt 



sin 7 ) cos (ir sin 7) 



sin 7 



Since only the rms value of this equation is signifi- 

 cant, the terms containing ul may be dropped, and 

 the result for the four-element array is 



f A = cos (it sin 7) cos I ^ sin 7 



(i sir 



(90) 



It is easily verified that this is a special case of the 

 general expression for an iV element array spaced at 

 intervals of nX and excited in phase (not derived here) 



sin (Nn x sin 7) 



h = 



N sin (n ir sin 7) 



(91) 



The effect of a reflecting screen may be computed 

 by treating it as though it were X/4 from the dipole 

 as in Figure 62. In practice the spacing may be 



image * 4 



■SCREEN 



Figure 62. The reflecting screen. 



more nearly X/8 but for 7 less than 30° the method 

 given here is satisfactory and avoids a complicated 

 analysis. The path difference QR is (X/2) cos 7, and 

 the phase difference is x cos 7. Then 



f A = COS Lit — cos (cot — x cos 7) . (92) 



From the relation 



cos A — cos B = —2 sin \ (A + B) sin § (A — B), 

 it follows that equation (92) may be written in the 

 form 



cot 



COS 7 



f A =- 2 sin L.. 

 Dealing only with the rms value, 



) Sin {l 



cos 7 



f A = sin ( - cos 7 



(93) 



For small angles this factor is usually unimportant. 

 Factors are given in Table 10 for some typical arrays 

 with horizontal radiators in a vertical column and 

 a reflector screen. 



