130 



COVERAGE DIAGRAMS 



;iiitl ii> has a range of to 2 

 for the second lobe, etc. 



Tlie limitations of e(iuati(jn(2)maybe summarized 

 as follows: 



1 . The phase shift at reflection is tt radians. 

 This assumes horizontal polarization, or ^ ~0 for 

 \'ertical polarization. 



2. The reflection point is (relatively) close to the 

 transmitter. 



3. The grazing angles corresponding to lobe 

 maxima are less than 0.2 radians. In connection with 

 limitation (2), it should be noted that the angles for 

 which the approximation y — \p holds depend upon 

 the wavelength and transmitter height. The follow- 

 ing table shows the minimum angle for various 

 transmitter heights and wavelengths at which the 

 error in the path difference A introduced by this 

 assumption is less than 1 per cent. To satisfy equa- 

 tion (2), and have an error less than 1 per cent, 



Table 1 



the first lobe, 2 to 4 than tt radians (i. e., (f)' = (j) — w is negative). 

 It follows that the path difference for lobe maxima 

 must be greater than X/2 and greater than X for the 

 nulls. In other words, the value of n in equation (2) 

 must be increa.sed by (A -|- n) to compensate for the 

 flccreased pha.se shift of reflection, so that 



y should lie between the minimum \'alue and 0.2 

 radians. Table 1 shows, for increasmg transmitter 

 antenna height, how the angle for which eciuation (2) 

 is yiiYid "within 1 per cent also increases. 



If n is set equal to 2m — 1, integral values of m 

 corr(\spond to lobe maxima and half-odd integers to 

 lobe minima. The advantage of this notation is that 

 the value of m is the number of the maxima or lobe 

 number. Thus for the fifth lobe, 7n = 5. The gen- 

 eral expression for the grazing angles corresponding 

 to lolie maxima, for a plane earth and for horizontal 

 polarization, is 



7 = 



(2 m- 1) . 

 4/ii 



(3) 



«here integral value."^ of ni gi\'e maxima and half- 

 odd integers give minima. 



*^ ' yVngles of Lobe Maxima 



(Vertical Polarization) 



With \-ertically polarized radiation the reflection 

 phase shift (j) [equation (27) in Chapter 5] is less 



(A/0 



m - ■> 



<t>) = 't. (4) 



III 



(A.) = '^ - 1 = + i' 



Tor \eitical polarization, equation (2) becomes 

 n + (All) 



7 = 



iJh 



(5) 



(6) 



h.2.4 



Lobe Eqviation 



When p = 1, and (^ = tt, cj)' = 0, and = 8, 

 e( I nation (46) in Chapter 5 may be written as 



r/ = 2VGir/o sin - = 2ylGido sin - . 



(7) 



Sul)stituting 



gives 



d = 2VGirf„sin \-^)' 



(8) 



where do is the free-space range which may be com- 

 puted from the gain corresponding to the given 

 co\'erage diagram by use of the nomogram given in 

 Figure 3 in Chapter 2. Equation (S) may be written 

 as 



d = 2VGir/o sin f 27r -^ • sin \y\ . 



(9) 



]*](luation (9) shows that for fixed values of free- 

 space range do, transmitter height hi, and wa\'e- 

 length, the coverage lobe may be represented bj' a 

 polar sine function of the angle y at the base of the 

 antenna. This assumes that the slant range measured 

 to any point on the lobe may be considered equal 

 to the distance d measured along the surface of the 

 earth. 



