456 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
FREE SPACE — HIGH-ANGLE 
COVERAGE 
Maximum Range Formulas 
In free space, the gain factor A has the value 
3\/8rd (for maximum power transfer between 
doublets). Equations (3) and (5) in Chapter 5 
then take the simple forms: 
0 P. 3d \? ; 
One-way, radio gan: — = GG i : 4 
aie d Pui Nema 4) 
4 
Two-way, radar gain: Pa G2. 1670 | ( =) . (5) 
P, 9? 8rd 
If Pz is replaced by the minimum detectable power 
of the receiver, Pmin, and P; by the peak power IZ 
of the transmitter, the maximum range is given by: 
3 
“way, dmax = 
One-way 3 
S| 2p. ERE (6) 
T 
4 
Two-way, Gmnax = ae 
in Ma 
TT 
SCANNING LOSS IN DB 
: \ — T 1 
j 
1 ° 
' 2 3 5 10 20 30 
SCANNING RATE RPM 
Ficure 4. Scanning loss as a function of scanning 
speed and beamwidth. 
Deviation from Maximum of Beam 
For an antenna whose direction is fixed, the 
equations reported above apply only to points on 
the axis of the beam. Denoting by f(z) the ratio of 
gain in a direction at an angle 7 from the axis of the 
beam to the gain at the axis and by 27 the beam 
width between half-power points, then 
T(x) = exp — 0.692(7/ r0)?. (8) 
Accordingly, for points off the axis, G must be 
multiplied by f(z) before substitution in the formulas 
mentioned 
Performance Figure 
Equations (6) and (7) for dmax depend on the 
performance figure defined in the last section. The 
quantities which appear in the performance figure 
can be measured. The one which offers most diffi- 
culty is Prin, the minimum detectable power, which 
has been discussed in Chapter 2 . In Tables3 and 4 at 
the end of this chapter, noise figures and bandwidths 
of various sets are given. An important correction to 
Pni,, a8 determined from the noise figure and band- 
width is the scanning loss. This loss for various 
scanning speeds and beamwidths is represented in 
Figure 4. Another source of loss is deviation of the 
product of bandwidth B (mc) and the pulse width 
t (microseconds) from the optimum value of. 1.2. 
The losses for various values of the product are tabu- 
lated in Table 2. 
TaBLeE 2. Loss resulting from band- and pulse widths. 
Bt* Loss (db) 
0.1 5.0 
0.3 1.5 
0.7 0.5 
1.2 0.0 
2.5 0.8 
-5.0 3.0 
10.0 ~ 5.0 
20.0 8.0 
*B = i-f bandwidth (mc); t= pulse width (microseconds). 
A field measure of the performance figure of a 
radar can be determined by the use of a target of 
known radar cross section, such as a silvered balloon 
and equation (7). A check on variability of per- 
formance can be made by finding the maximum 
range on a plane (using a constant aspect, such as 
nose or tail). 
Radar Cross Section 
An important but troublesome factor in calculat- 
ing dmax of a radar from a knowledge of the perform- 
ance figure is a, the radar cross section (see Chap - 
ter 2 and Chapter 9). The value of o can be found by: 
1. Laboratory measurement of the factors which 
constitute the performance figure and field determi- 
nation of the maximum range d,,x. The* value of 
c is then given by equation (7). 
2. Measuring the signal returned by the target 
at a convenient distance on a calibrated A scope or 
by direct comparison with a pulsed signal generator. 
In this method neither P,,;, nor dmax enter. 
The equations involving o assume a point target. 
Since an airplane intercepts a small solid angle over 
which the beam strength varies little, the assumption 
of a point target is adequate for aircraft. 
LOW-ANGLE AND SURFACE 
COVERAGE 
Maximum Range 
Since the gain factor A for this case is more 
complicated than for free space, the relation be- 
tween d,,,x and the performance figure cannot be 
given in general by a simple expression, as can be 
seen from equations (172) and (184) in Chapter 5. 
For ranges such that the shadow factor F; ~ 1, Le., 
