FUNDAMENTAL RELATIONS 339 
transfer is 
P, ( 3 y i 
—= = Aj. 18 
P, 0 (18) 
The ratio P:/P, = Ao? (as used here) is called the 
free-space radio gain for matched doublets or for 
short the free-space gain, since all objects, including 
the earth, are supposed remote from both doublets. 
This free-space gain, 49 = 3A/8zd. On the decibel 
scale, it takes the form 
10 logio i = 20 log Ao 
1 
= — 18.46 — 20 login < decibels (19) 
The nomogram. Figure 3, gives a convenient 
means of calculating the radio gain for doublets 
adjusted for maximum power transfer. 
POWER TRANSMISSION. RECIPROCITY 
Radio Gain 
Preceding formulas (16- 19) apply to doublets in 
free space. This section considers the modifications 
that must be made in the formulas when the re- 
striction of free space is removed. In actual trans- 
mission problems, ground reflection, reflection from 
elevated layers of the atmosphere, diffraction by 
earth curvature and by obstacles, and refraction by 
the atmosphere must be considered. In Chapters 5, 
6, and 7 special forms of gain are discussed and 
separate gain factors are introduced to take care of 
each effect. For the present a factor that will be 
called the path-gain factor, A», representing the 
product of all these special factors will be used. 
A, is defined by 
E = E,A;, (20) 
where F is the absolute value of the actual field 
strength and Hp is the absolute value of the free- 
space field strength that would exist at the same 
distance d from the doublet transmitter in free space. 
Replacing Ho with HoA,, equation (17) for the 
received power, becomes 
_ EvA, 3” 
1207 87 
whiie the power output as given by equation (16) 
remains unchanged, so that 
P, (2) 
(21) 
2 
P. 1 Sad. 
replaces equation (18) as the ratio of received 
power to output power for maximum power transfer 
between doublets. The quantity defined by (22) 
is the free-space gain and A is the gain factor. 
The general relation between the input voltage at 
the receiver and the received power is V; = VP2R), 
where R; is the resistance of the receiver load circuit 
(which is equal to the radiation resistance for maxi- 
= |——) A? = (A.A,)? = A? (22) 
mum power transfer) and V; is the input voltage. 
Hence, using equation (14), 
V; = 0.0178E,AVR, volts (23) 
1 Ss EyL 
= ——= E,\VR; = 
Sa RRNA Ty au. Mae 
Antenna Gain. Polarization 
The equations given before (1-19)may be further 
generalized to apply to any type of antenna through 
the introduction of a quantity called the antenna 
gain. The term gain, as applied to an antenna, isa 
measure of the efficiency of the antenna as a radiator 
or receiver as compared with that of a doublet 
antenna, with all antennas located in free space. 
Quantitatively, the gain, Gi, of a directive trans- 
mitting antenna is the ratio of the power P;’ radiated 
by a doublet antenna to the power P; radiated by the 
antenna in question to give the same response in a 
distant receiver, with both transmitting antennas 
adjusted for maximum transfer of power. Hence 
, 
Gua (24) 
1 
The gain ( of a directive receiving antenna is the 
ratio of the power P,” radiated by a transmitting 
antenna, which produces a certain response in the 
matched load circuit of a distant doublet receiving 
antenna, to the power P; radiated by the same 
transmitting antenna to produce the same response 
in the matched load circuit of the receiving antenna 
in question, with both receiving antennas adjusted 
for maximum transfer of power. Hence 
‘ in IR 
Ge rae (25) 
From the definitions given above it follows that 
for a transmitting and receiving antenna combina- 
tion in free space, with gains G, and Gz and adjusted 
for maximum power transfer, the power ratio is 
equal to 
P, (2 2 
Pica Nead 
= GiG.A 0°, (26) 
where P;, G; are the power output and gain of the 
transmitter and P, is the power delivered to the 
matched load of a receiving antenna of gain Go. 
If the antennas are not in free space, equation (26) 
becomes 
GiGa(AoA,)?, (27) 
= (1G2A2, 
sp) 
2 
2 |¥ 
— 
> 
She) 
Il 
where A is the gain factor and A, is the path-gain 
factor. Note that for highly directive antennas A, 
may depend upon the directivity characteristic of 
the antennas, e.g. when the antenna discriminates 
between the direct and reflected waves. 
Since power is proportional to the square of field 
