SITING AND COVERAGE OF GROUND RADARS 101 
unequal. This gap filling is secured at the price of 
shorter lobes. Above 3 degrees the lobes cannot be 
distinguished from the nulls. 
Only lobes due to the main free space lobe of the 
antenna pattern are ordinarily plotted, as targets 
higher than about 10 degrees are of little interest 
to an early warning radar. Because most detection 
occurs at angles under 2 or 3 degrees, no distinction 
will be made between slant range and horizontal 
range. 
The calculation of the coverage diagram will be 
approached in successive steps. The first step will 
consist of calculation of the angular position of the 
lobe maxima and minima. This will be done in three 
different degrees of approximation corresponding to 
different situations encountered in practice. The next 
step is the calculation of the length and shape of 
the lobes themselves, which is given in a later section. 
Flat Earth Lobe Angle Calculations 
When the reflection point is so close that earth 
curvature may be ignored, the rays may be drawn 
as in Figure 47. The transmitter 7’ has the center 
of the antenna at height hi above the horizontal 
reflecting surface. The antenna is assumed to have 
horizontal polarization; that is, the dipoles are 
parallel to the reflecting plane and perpendicular 
to the direet ray rg. The target height is he. Both hi 
and he are several wavelengths or more, and rq is so 
large that the field at the target falls off as i/ra. 
The image of the antenna is at 7’ at a distance hy 
below the reflector. The length of the ray from 
T’ isr. 
The coefficient of reflection is p, and the phase lag 
at reflection is ¢. The electric field strength due to 
the combined direct and reflected waves (ra ~ r) 
may be written as 
BH = * Ji + p+ 2p cos @ + 8)" (50) 
where 6 an = phase lag due to the path differ- 
ence, . 
A = r — rq = path difference of the direct 
and reflected rays, 
E, = the field strength at unit distance. 
For horizontal polarization and small angles p is 
unity and ¢ is 180 degrees and equation (50) reduces 
to 
E= on od 6, 
r 2 
_ 2B, 7A 
= 7 tS) NG (51) 
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 = 7 — ra = constant . (52) 
For the case of a flat earth these lines are by defini- 
tion confocal hyperbolae with 7 and 7” as foci and 
A as the major axis. This is shown in Figure 48 for 
a target at short range. In a typical case 6 will be 
TARGET 
LINE OF CONSTANT 
he 
PATH DIFFERENCE 
a 
Figure 48. Constant path difference hyperbola. 
positive and approximately equal to y. From 
geometry 
4h? — A? 
(eo One 4h, sin B © (53) 
When rz is very large compared with h; the angle 
¥ is equal to 8, and the denominator of equation (53) 
is practically zero, giving 
sin y & = 5 (54) 
1 
2 
Using equation (51) with the (rA)/A = 7/2, the 
lobe maxima are given by 
d 
N= 7 3 , (55) 
where n = 1, 3,5 --- . Minima are given by values 
of n = 0, 2,4,6,---. 
Substituting in equation (54) 
é md 
sin y = Tihs (56) 
or to a sufficient approximation 
nX 
ASS ah, . (57) 
Here y:= the angle of elevation of the target referred 
to the horizontal at the ground below the 
antenna, in radians; 
nm = number of half-wavelengths difference 
between the direct and indirect paths. 
n = 1, 3, 5, etce., for maxima of lobe 
number 1, 2, 3, --- (n + 1)/2 counting 
from the reflector up. n = 0, 2, 4, 6, etc., 
for minima of null numbers 1, 2, 3, 4, 
--- (n + 2)/2; 
hy = the height of the center of the antenna 
above the reflector; 
\ = wavelength; 
with hf; and ) in the same units. 
