82 



CALCULATION OF RADIO GAIN 



Some useful approximations : 



For V large, g — » 1 and i? = - (1 - f-)- ap- 



P 



proaches the value 



R 2p, 



V 



■which, solved for p, becomes 



(122) 



liR>2, 

 liR<2, 

 For R>Z, 



p^Z^±i^l±-8. ^^23) 



p ~ - . (124) 



R 



3-R 



D~ 1. 



(125) 

 (126) 



The polarization is assumed horizontal, the 

 effective earth's radius 4a/3. 



The foUo-fting data are assumed. 

 Transmitter height : hi = 50 meters 

 Receiver height: h = 1,500 meters 

 Distance apart: d = 100 kilometers 



Wavelength: X = 1 meter (/ = 300 mc) 

 Gains (over doublet) : G'l = Ck = 100 (20 db) 



Oxe-Way Transmission 



1. rfi = 188 kilometers (Figure 2). d < d^, so 

 that the receiver is in the optical region. 



2. The u,v coordinates of the receiver are 



h 1,500 „„ 



M = — = = 30, 



h 50 



The calculations will he divided into four types. 



Type I. The direct calculation of the radio gain 

 (or field) when the heights and distance apart of the 

 antennas and the wavelength are given. 



Type II. The calculation of the radio gam as a 

 function of the receiver height /)2 when the trans- 

 mitter antenna height Ih, distance d, and wavelength 

 X are given. 



Type III. The calculation of the radio gain as a 

 function of the distance d when the transmitter 

 antenna height Ih, receiver height h, and \\avelength 

 are given. 



Type IV. The calculation of the possible positions 

 of the receiver in space ih,d) w'hen the radio gain 

 will have the given value for given values of the 

 gain factor A, the transmitter antenna height hi, and 

 the wavelength \. Special cases, such as the re- 

 ceiver antenna height, /!2 for given d, or d for given /(2, 

 can be solved by use of the curves in Type II and 

 Type III, in Sections 5.6.3 and 5.6.4. 



This type of problem is of importance in estimat- 

 ing the range of a set when the minimum detectable 

 power of the receiver and power output of the trans- 

 mitter are known. 



5.6.2 



Problem of Type I. Radio Gain 

 for Fixed Heights and Distance 



The radio gain at a given receiver is to be found, 

 their heights, as well as the wavelength being given. 



dr = V2/a(/ii = 29,100 meters 



d 100 „ ,„ 

 V = — = = 6.i6. 



dr 29.1 



3. Referring to Figure 19, s (= p/v) is estimated to 

 be 0.05. Since the result is very sensitive to slight 

 changes in s, it is desirable to improve the value of s. 

 In Newton's method, the next approximation, using 

 equations (101) or (119), is 



s = s — 



f(s) 





2s3 - ? s2 - — 



2 2v- 



[^'-'] 



(127) 



s' = 0.04794. 



The next approximation gives the same result. 



4. Using the above value of -s' and the relation 

 p = sv (« = 3.43), p is equal to 



p = 0.1645. 



With the value of p = 0.1645, equation (117) and 

 Figure 22 give the value 



D = 0.95. 



5. The path difference variable R is obtained from 

 equation (112) or Figures 21, 22, 23, as 



R = 5.477. 



6. The number r, from equation (115), is 2.91, so 

 that the lobe number n is 



„ = ^ = 1.88. 

 r 



Hence the receiver is on the upper part of the first 



lobe close to a null. 



