LOBE-ANGLE METHOD 



143 



The path difference A is given by 



A = A + B - ra=— (0 = 180°). 



Hence 



A + B - yU= + B- + 2AB cos 2.^ 



n\ 



Squaring, 



B{2A - «X - 2A cos 24^) = n\A - f — V . 

 Therefore, 



B 



A _ 1 /nXV- 



r 2\2/ 



A(l - C0S2.A) --- 



For small angles, 



cos 2^ = 1 - ^ = 1 - 2ip\ 



Hence 



7?X.4 _ [ //iX\' 

 Y 2 



//^XV- 4 _ "^ 



V2/ /,X ' 4 



2\I/-A — — 2A\1/- — — 



2 2 



Since the angle \p is of the order of a few degrees 

 only, it is permissible to write A = di in the above 

 equation , 



Neglecting further the term ?!X/4 in comparison 

 with di, which is permissible for short waves and 

 small values of ??, the above expression for B reduces 

 to equation (44g). 



The method of determining the locus of points 

 having a path difference of X/2 (i.e., n = 1) is as 

 follows. Assume a value of rfi and calculate the 

 corresponding values of 



/(/ by equation (44d), 

 xp by equation (44a), 

 B by equation (44g), 

 r^ by equation (44e), 

 \l/d by equation (44f), 

 8 by equation (44b), 

 7i bj" eciuation (44c). 



The assumed values of rfi are limited to those -whicli 

 will give positive values of B in equation (44g). 

 Select as manj^ values of di as are necessary to plot a 

 smooth path-difference locus. Repeat for n = 3, etc. 



This method of determining the angles of lobe 

 maxima is of particular value in constructing the 

 first few lobes, since the approximations of Sections 

 6.6.2 to 6.6.6 may cause considerable error for low 

 angles. These path-difference loci vfiW intersect a 

 vertical line dra\\Ti from the antenna to the ground 

 below at heights equal to nX/4. For short waves, 

 this height is negligilDle for the lower lobes. 



'^ LOBE-ANGLE METHOD 



(VERTICAL POLARIZATION) 



(.7.1 Angles of Lobe Maxima 



The values of ?i corresponding to the angles of 

 lobe maxima are determined exactly as in Section 

 6.2.3 for the case of a plane earth. The values of n 

 in the expression 7' = iiK/^hi are increased above 

 those for horizontal polarization by an amount {An) 

 to compensate for the reduced phase shift on reflec- 

 tion. In other words, the path difference must be 

 greater than integral multiples of X/2 to compensate 

 for the reduced phase shift. The expression for this 

 compensation, from equation (5), is 



Hence 



An 



7' = (h + An) 



4/(i' 



(45) 



(46) 



[See Figure 8 and eciuation (32). 



6.7.2 



Construction of Lobes 



As a first approximation, the angles of lobe maxima 

 are calculated on the basis of horizontal polarization. 

 A table is constructed giving values of n and 7' 

 for maxima and minima. The next approximation 

 is to let i/ = 7'. This assumes that rfi < < ch. The 

 values of (f)' and p may then be found from reflection 

 curves, and (An) calculated from equation (45). 

 The corrected values of 7' may be determined from 

 equation (46). It is simpler to find 7' by interpolat- 

 ing between integral values of n in the n versus 7' 

 table previously constructed. The values of rfnia:: 

 and rfmin are 



dmax = ^Gldoil + A'), (47) 



rfmin = V(ud„(l - A'), (48) 



