Chapter 6 
COVERAGE DIAGRAMS 
DEFINITIONS 
HE 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 structure 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 determined in part 
by contributions from the diffraction terms as well 
as by the contributions of the direct and reflected 
waves. 
PLANE EARTH 
Field Strength 
For horizontal polarization and a reflection coeffi- 
cient equal to —1 (ie., 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, 
d 
Figure 1. Coverage diagram for plane earth (heights 
he are exaggerated relative to distance d). n = 1,3,5 
. ... for the first, second, third... . lobes. d = dmax 
sin (7/2) and dmax = 2dp. 
which in turn determines the phase retardation, 
5iag, due to path difference. The angle 2/2 used in 
calculating H by equation (29) in Chapter 5 is a 
function of 6, and @, since 2 = dj, + o’. The 
effect of a reflection coefficient p less than unity is to 
reduce the length of the lobe maxima to values less 
than 2d) and to increase the minima above zero as 
indicated by the dotted lines of Figure 1. The angles 
433 
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 7 radians, while lobe minima 
(nulls) occur when the total phase shift is an odd 
multiple of 7 radians. If p = 1, the nulls are equal 
to zero. 
It follows that for horizontal polarization 
(@ = =z), maxima occur when 6 equals 7z, 37, 
5m, ete., and minima when 6 = 0, 272, 47, etc. 
This means that a path difference equal to an odd 
multiple of \/2 gives a lobe maximum while a path 
difference equal to an even multiple of \/2 gives a 
null. This holds only for horizontal polarization 
(6 = zm). Applying equation (52) in Chapter 5 
to the case when d; < < dk (i.e., d2 =), 
A a Dhahe 
d 
= 2h tan y => 2h tan Y,¥= 2hry, (1) 
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, y must be less than 
0.2. Hence for maxima, 
Qhry a n = odd integer, 
and for minima, (2) 
2hy = = n = even integer, 
or ce aL. 
4h, 
and m has a range of 0 to 2 for the first lobe, 2 to 4 
for the second lobe, ete. 
The limitations of equation (2) may be summarized 
as follows: 
1. The phase shift at reflection is 7 radians. 
This assumes horizontal polarization, or y © 0 for 
vertical 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 
