Chapter 6 

 COVERAGE DIAGRAMS 



61 DEFINITIONS 



THE LOCUS of points in space having a constant 

 field strength is called a coverage diagram. In 

 the optical region this is also called a lobe diagram. 

 The construction of these diagrams is an important 

 part of the predetermination of the performance of 

 radar and communication sets. The basic concepts 

 and formulas will be developed first for the case of 

 the plane earth and then applied with necessary 

 modifications to propagation over a spherical earth. 

 The method outlined in this chapter is applicable 

 only to the lobe stiiicture lying above the tip of the 

 first lobe. In this region the field is given almost 

 entirely by the vector sum of the direct and reflected 

 waves. The lower portion of the first lobe is dis- 

 torted from the regular lobe structure because, in 

 this region, the field strength is detennined in part 

 by contributions from the diffraction terms as well 

 as by the contributions of the direct and reflected 

 waves. 



6.2 

 6.2.1 



PLANE EARTH 

 Field Strength 



For horizontal polarization and a reflection coeffi- 

 cient equal to —1 (i.e., p = 1, </> = 180°), the re- 

 ceived field intensity oscillates from zero to twice 

 the free-space value, depending on the position of 

 the point in space, as shown in Figure 1. The posi- 

 tion in space determines the path difference A, 



Figure 1. Coverage diagram for plane earth (heights 

 hi are exaggerated relative to distance d). n = 1,3,5 

 .... for the first, second, third .... lobes, d = dmax 

 sin (7rre/2) and dmax = 2do. 



which in turn determines the phase retardation, 

 Slag, due to path difference. The angle Q/2 used in 



calculating E by equation (29) m Chapter 5 is a 

 function of Sjag and (j), since fl = 5iag + <^'. The 

 effect of a reflection coefficient p less than unity is to 

 reduce the length of the lobe maxima to values less 

 than 2rfo and to increase the minima above zero as 

 indicated by the dotted lines of Figure 1 . The angles 

 at which the maxima and minima occur depend 

 upon the phase shift at reflection, as will be explained 

 in the following section. 



*^^ Angles of Lobe Maxima 



(Horizontal Polarization) 



Lobe maxima occur whenever the sum of the phase 

 shifts caused by reflection and path difference equals 

 an even multiple of -k radians, while lobe minima 

 (nulls) occur when the total phase shift is an odd 

 multiple of tt radians. If p = 1, the nulls are equal 

 to zero. 



It follows that for horizontal polarization 

 {4> = tt), maxima occur when 5 equals w, Sir, 

 Sir, etc., and minima when 5=0, 27r, iir, etc. 

 This means that a path difference equal to an odd 

 multiple of X/2 gives a lobe maximum while a path 

 difference equal to an even multiple of X/2 gives a 

 null. This holds only for horizontal polarization 

 {(f> = it). Applying equation (52) in Chapter 5 

 to the case when di < < ^2 (i.e., ^2 = d), 



A = ^= 

 d 



where y is the angle in radians between the hori- 

 zontal and the line joining the receiver or target 

 to the base of the transmitter (see Figure 8 in Chap- 

 ter 5). For equation (1) to hold, 4/ must be less than 

 0.2. Hence for maxima, 



= 2ih tan i/- ^ 2hi tan y ^ 2hiy, (1) 



2/ii7 = — , n = odd integer, 



and for minima. 



(2) 



n\ 



or 



2/ii7 = — , ■" = even integer. 



ri\ 



y = — ' 

 4/ii 



129 



