FUNDAMENTAL RELATIONS 337 
ohms. This is the impedance of free space in the 
mks rationalized system of units, the ohms of the 
electrical engineer. 
Equation (2) describes the radiation field of a 
differential element of the doublet. To get the 
radiation field of the whole doublet, these equations 
must be integrated over the length l. This gives 
1/2 
607 sin 6 Tdl 
E, = -1/2 
ar 
H, = E,/1207 amperes per meter. 
volts per meter, (3) 
Equation (3) may be written in exactly the form 
of equation (2) by introducing the effective length, L, 
of an antenna, which is defined as the length that a 
straight wire carrying current constant over its 
length would have if it produced the same field as 
the antenna in question. Calling the current meas- 
ured at the input point /;, 
i 
Idl 
L=+"___ meters, (4) 
and hence 
6071 ,L sin 6 
Ey = mae FS volts per meter, 
(5) 
Ay = lie meter 
* = Too, amperes per ; 
so that equations (5) are the same as equations (2) 
with J;L replacing | Idl. For a short dipole or 
doublet the current varies linearly from J; at the 
midpoint to zero at each end so that from equation (4) 
L = 1/2 for a doublet. 
The power per unit area, W (that is, the power 
flowing through a unit area normal to the direction 
of propagation), is represented by Poynting’s 
vector and is given by the product E,H, times the 
sine of the angle between Hy and Hy. This angle is 
90 degrees. Consequently, 
W = EH watts per square meter, 
ER? 
W= 1207 watts per square meter, (6) 
E = V1207W volts per meter. 
To find P, the power output of the doublet, W is 
integrated over a large sphere concentric with the 
source. Using equations (5), 
E*d? 
= 45 watts 
and PRE (7) 
Re 3V5 we 
d 
where d is written in place of r. The subscripts @ and 
@ have been dropped at this point because the 
E and Z referred to in equations (7) are the fields 
in the equatorial plane, where sin @ = 1. 
As the antenna is part of a circuit, it is often con- 
venient to think of the radiated power as being 
dissipated in a fictitious resistance called the radia- 
tion resistance, defined by 
ohms, (8) 
where P is the radiated power and J; the rms input 
current. For the doublet, 
2 
R, = 807° e ohms, (9) 
nN 
where L is the effective length given by equation (4). 
Reception by an Electric Doublet 
When an electromagnetic wave falls upon an 
antenna, a current is induced in the antenna and 
power is abstracted from the wave. If the antenna 
is connected to a load, the power abstracted is 
dissipated in two ways: (1) by absorption in the 
load (reception), and (2) by reradiation from the 
antenna (scattering). 
In this classification, the power dissipated by the 
antenna itself (due to its ohmic resistance) is ignored 
because this loss is likely to be negligible compared 
with the power dissipated through reradiation. 
Hereafter, power absorbed by the load will be called 
received power and power reradiated by the antenna 
will be called scattered power. The sum of these is 
equal to the power abstracted from the wave. 
The calculation of the received and scattered 
power may be carried out by means of the equivalent 
circuit of Figure 2. In this figure, Z, is the impedance 
VOLTAGE GENERATED IN ANTENNA 
Figure 2. Equivalent circuit of antenna and load. 
of the doublet and Z; is the impedance of the load, 
that is, the impedance connected across the terminals 
of the antenna when it is acting as a receiver. V is 
the voltage generated in the antenna. 
The load is supposed to be tuned, which means that 
the reactance part of Z, is set equal and opposite 
to the reactance part of Z,, so that Z, + Z, = Ra 
+ R, that is, the total impedance is simply the 
sum of the resistance parts of the impedances of the 
antenna and the load. Hence 
V 
Ra + Ry 
But P, = R,J* is the power ab- 
I= (10) 
gives the current. 
