148 



SITING AND COVERAGE OF GROUND RADARS 



In the construction of a vertical coverage diagram 

 it is important to be able to draw the lines of constant 

 path difference. Of special interest are the lines of 

 maxima in the center of the lobes and the lines of 

 minima or nulls. These lines correspond to 



A = ?• — rd = constant . (52) 



For the case of a flat earth these lines are by defini- 

 tion confocal hyperbolae with T and T' as foci and 

 A as the major axis. This is shown in Figure 48 for 

 a target at short range. In a typical case ft will be 



TARGET 



Figure 48. Constant path difference hyperbola. 



positive and approximately equal to 7. From 

 geometry 



4/ii 2 - A 2 



Td = 



2A - 4/u sin 



(53) 



When r& is very large compared with hi the angle 

 7 is equal to /3, and the denominator of equation (53) 

 is practically zero, giving 



sin 7 = 



2/h ' 



(54) 



Using equation (51) with the (irA)/X = tt/2, the 

 lobe maxima are given by 



A = n 



(55) 



where n = 1, 3, 5 ■ 

 of n = 0, 2, 4, 6, ■ • • . 



Substituting in equation (54) 



Minima are given by values 



n X 

 4/h 



sin 7 

 or to a sufficient approximation 



7 



n X 

 4/Ti " 



(56) 



(57) 



Here 7 = the angle of elevation of the target referred 

 to the horizontal at the ground below the 

 antenna, in radians; 



n = number of half-wavelengths difference 

 between the direct and indirect paths. 

 n = 1, 3, 5, etc., for maxima of lobe 

 number 1, 2, 3, ■ • • (n + l)/2 counting 

 from the reflector up. n = 0,2, 4, 6, etc., 

 for minima of null numbers 1, 2, 3, 4, 

 • • • (n + 2)/2; 



hi = the height of the center of the antenna 

 above the reflector; 



X = wavelength; 

 with hi and X in the same units. 



From Figure 47 it follows that di, the distance to 

 the reflection point, is given by 



di = 



hi 



tan * ' 



or taking tan ^ = sin ty = 7 (which may be done 

 provided di is small enough compared to d 2 and large 

 compared to hi) and substituting in equation (57), 



rfi = 



4/1 r 2 



(58) 



Here d\, hi, and X must be expressed in the same units' 

 For high sites and distant targets the angle 7 

 becomes smaller, and the approximation involved 

 in equations (57) and (58) requiring d x to be small 

 compared to rf 2 becomes worse. In Table 4 are listed 

 the minimum values that n\ may have for an error 

 of 1 per cent or less in equation (57) at different 

 antenna heights. Also is given the minimum value 



Table 4. One per cent error in 7. 



of 7 corresponding to «X. Thus equation (57) when 

 used on a 100-ft site at 100 mc (X = 9.84 ft) will 

 give values which are in error by less than 1 per cent 

 for all lobes and for angles above 0.8 degree. If the 

 100-ft site operated at 1,000 mc (X = 0.98 ft) the 

 minimum value of n would be 6 corresponding to 

 the fourth null. The error in 7 is always positive 

 and increases rapidly with antenna height, and at 

 a height of 1,000 ft and a frequency of 100 mc the 

 formula is incorrect for all angles of interest. At 



