CALCULATION OF RADIO GAIN 413 
Graphs for the Case 
of the Dielectric Earth (8>>1) 
1. Fundamental formulafor gain factor. Thisformula 
18 
A = 32 (ah)s (@h)s, (172) 
where q = 1 when h < 300°”. 
For the dielectric earth, 5 > > 1. See equation (193) 
Tf both antennas are low (h < 30*/*) equation (172) 
and the accompanying figures (Figures 31 to 41) are 
valid for all distances d such that 
Ass a (173) 
Tf one or both antennas are elevated, equation (172) 
is valid only well within the diffraction region of the 
transmitter, i.e., for 
d>> dy. (174) 
The following quantities required to find 20 log A 
are given in Figures 31 to 41: 
sas a function of ) is given in Figure 31. 
20 log F versus sd in Figures 32 and 33. 
20 log d can be found by using Figure 34. 
eas a function of A by Figure 35. 
20 log g versus eh is given by Figures 36 and 41. 
When one antenna is low, h < 307°, and the other 
quite elevated, h > 1,200d7”*, a result valid for h, near 
the line of sight can be found from the formula and 
graphs on page 419 , obtained by summing several 
modes. 
A more general method of finding the gain near 
the line of sight is to use equation (172) well below 
the line of sight to obtain a curve of A versus hz and 
by constructing a similar curve for the optical region 
d eh 20log g h 20logh 
NOTE 
h scale is 
being multiplied 
by to, therefore 
Value given by 
= ehp must be 
DSI Multiplied by 10 
— 20 log h=443 
20 log g=1.5 
Figure 42. Illustrating use of Figure 41. Note: (eh 
represents ehg(6)); [eh2 = 12.3] represents [ehog(5) = 
12.3]. 
by the method of Chapter 6, ‘“Coverage Diagrams.” 
By joining the two curves into a smooth overall 
curve, it is possible to estimate A in the transition 
region near the line of sight. 
For the case of short distances and receiver below 
the interference region, see page 380. 
2. For h<4/l. Vertical potarization. A more 
accurate result can be obtained by replacing the 
height-gain (gh) by H,/I or in decibels by 20 log Hz, 
— 20 log 1. Hy, is given by Figure 47 and 1 by 
Figure 46 (see Table 3). 
3. Graphical aids (continued). 
A. Definition of A. Figures 31 to 36 can be com- 
bined into a form more convenient for numerical 
computation. In Figure 37, a curve parameter A is 
introduced, defined by 
ek, 
higi 
where g; is a function of eh;. This may also be ex- 
pressed in the form 
(175) 
20log A = 20log A — 20 log hig. (176) 
(For hy < 4/l, 20 log A = 20 log A — 20 log H;, + 20 
log 1.) j 
Equation (172) can be written as 
7 F 
A, = iia 3< Wr’ 177 
ae (17) 
or 
20 log A = — 135 + 20 log = + 20logy,  (177a) 
s 
where 
vy = (che) J2; 
with gz a function of eh. Note that F,/(sd)? is a 
function of sd only and is independent of height. 
While h, usually represents the transmitter antenna 
height and hz that of the receiver, the role of h; and 
hg in equations (176) and (177) may be interchanged. 
To facilitate the use of Figure 37, three nomograms 
have been added (Figure 40 gives sd when ) and d 
are given. Figure 41 gives ef, 20 log h and 20 log g 
when ) and h are known. Figure 43 gives the modi- 
fied height h’ and distance d’ for given h, d, and k.) 
To find sd for a value of d which is not on the nomo- 
gram, say 120 km, find sd corresponding to a distance 
100 times smaller (i.e., 1.2 km) and multiply result- 
ing sd by 100. Proceed similarly for eh. 
B. Both antennas low. In the case of both an- 
tennas low, hf; and h: < 30 d”/, the contours 20 log A 
are given by Figure 39. If both antennas are so low, 
say, fi and hz < 4/l (see Table 4) that it is desired 
to use H,, for greater accuracy for vertical polariza- 
tion (Figure 47), then for 20 log A we take 20 log A 
+ 20 log 1 — 20 log Hz, (see text above ) and 
instead of ehs in Figure 39, we use as ordinate eHz/1. 
Then for given sd, Figure 39 would give the value 
of eHz/l. If the frequency is given, e/l is known 
