72 



CALCULATION OF RADIO GAIN 



Hence for a transmitter of height hi al)o^•e the 

 ground the height above the tangent plane at the 

 reflection point, the so-called equivalent height is, to a 

 first approximation, 



dr 



so that 



and for the receiA'er the equivalent height is 



d2- 



(5S) and 



h' = h, - 



2ka 



(59) 



The equivalent heights are shown in Figure 14, 

 which illustrates the geometry of the spherical earth 

 having an effective radius of ka. 



*^-^ Angles 



Referring to Figure 14 and remembering that the 

 angles are greatly exaggerated in the figure, it is 

 seen that 



hi' h' 



tan \j/ ^ 



di 



(60) 

 (61) 

 (62) 

 (63) 



Angle 4/ must be evaluated in order to determine the 

 reflection coefficient. Angles \pd and ;' determine the 

 antenna pattern factors Fi and F^, which are shown 

 in Figure 11. The angle y is significant in coverage 

 calculations and angular approximations. 



5.5.4 



Determination of Reflection 

 Point (di) 



Inasmuch as several equations of Sections 5.5.2 

 and 5.5.3 depend upon rfi, it is necessary to be able 

 to determine this distance when the transmitter 

 and receiver heights are given and the distance 

 between them is known. Let 



f/i = - (1 + h), 



(64) 



and 



h = ^^^) (1 + c), (65) 



d, = - (1 - b), 



, h + h-z ,. s 

 h = — (1 - c). 



where 



and 



c = 



di - d. 

 d, + d, 



hi - h. 

 hi + h, ' 



(66) 

 (67) 

 (68) 

 (69) 



Assume hi > ho and di > do, so that h and c ^\'ill 

 always be positive. This is always possible because 

 of the principle of reciprocity. From equation (60), 



hi' ho' 



= — or 

 di do 



h 

 di 



A 

 2ka 



ho 



do 



2ka' 



(70) 



Substituting for rfi and do from equations (64) and 

 (66), 



hi + ho 





hi+ho/l - c 



Simplifying, 



hi + ho 2(c - b) 

 d 1 -b- 



d(l + b) 

 ika 



d(l - b) 

 4ka 



bd 

 2ka' 



Solving for c, 



where 



b + bm(l - V), 

 d"- 



■ikaihi + ho) 



(71) 

 (72) 



To determine di, equation (71) must be solved 

 for 6. This is a cubic ec^uation, which is easily solved 

 when m is small in comparison to unity. However, 

 for )n values comparable to unity, or larger, it is 

 easier to plot a series of curves showing c as a func- 

 tion of ri) for assigned values of b ranging from 

 to 1. These are straight lines with a slope of 

 6(1 — b'-) and are given in Figure 15. 



The procedure for calculating di when hi, ho, and 

 d are given is as follows : 



1. Compute c = 



hi - ho_ 

 hi + ho ' 



