676 BELL SYSTEM TECHNICAL JOURNAL 



approximations even where imperfect ground conditions are encoun- 

 tered. 



Vertical Plane Directivity 



The vertical plane directivity of the horizontal diamond-shaped an- 

 tenna is determined by three factors, i.e., the length of each leg, the 

 "tilt angle" and the height above ground. 



For the cases where the element length is an integral multiple of a 

 half wave-length and where the far end termination is the characteristic 

 impedance multiplied by the sine of (see Fig. 14), the equation for the 

 vertical plane directivity over perfect ground has been calculated to be, 



k[_\ - e-J4^^«inA/X-j 



-\- cos A 



* 



rj^ _L g—j2wl sin 4> cos A/X~|2 



sin^ cos^ A 

 where, as shown in Fig. 14, 



H = height above perfect ground in wave-lengths. 



A — wave angle from horizontal in the vertical plane 



4> — tilt angle of elements. 



/ — element length in wave-lengths. 



k — proportionality factor. 

 Ir — receiver current. 



It will be noted that neither the length nor the tilt angle appears in 

 the first bracketed term. It can be shown that this factor appears as a 

 multiplier for nearly any type of horizontal antenna, accordingly the 

 location of nulls and maxima for this factor are separately plotted in 

 Fig. 15. 



In the same manner the nulls and maxima of the product of the sec- 

 ond and third bracketed terms have been plotted in Fig. 16 for an 

 element length of four wave-lengths. 



The curves of Figs. 15 and 16 are design curves and their use can be 

 illustrated by the following example: Measurements on the directions 

 of wave arrival have indicated that the most usual directions are from 

 10 to 15 degrees above the horizontal. It is desired to construct a 

 horizontal diamond-shaped antenna for this reception, employing four- 

 wave-length elements. Fig. 15 indicates that the most economical pole 

 height for 15 degrees is approximately one wave-length. Now referring 

 to Fig. 16, we see that the largest tilt angle, to accomplish this, is about 

 65 degrees. It is always desirable to use the largest possible angle of 

 tilt to obtain the use of the largest lobe of the directive diagram. 



* In the third bracketed quantity use, in the ± sign, — when / is an even integral 

 multiple of X/2 and -|- when / is an odd integral multiple of \,2. 



