348 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
length d/2 lies between 1.5 and 0.05 meters. In this 
section it is assumed that the current distribution 
is sinusoidal. 
1. Radiation field. The radiation field at point P, 
Figure 7, where d >> \, is obtained by dividing 
the half-wave current distribution into an infinite 
Ficure 7. Half-wave dipole. 
number of infinitesimal doublets, using equation (2) 
in Chapter 2 and taking into account the differences 
in phase at P introduced by the differences in the 
distances which the radiation from the various 
doublets must travel. Thc net result, using d in 
place of r, is 
ap) OF povsil G/2)eteost9)| 
E, : volts per meter, (3) 
d sin 6 
ll, = sue amperes per meter. (4) 
1207 
The normal part of the field, H, (difference), pre- 
scribes the antenna pattern factor (measured in 
relative field strength) and is plotted in Figure 11. 
The corresponding pattern for a doublet is H, ~ sin 8, 
which is a circle in polar coordinates. These patterns 
are circularly symmetric about the antenna axis. 
Squaring the radial lengths in the above patterns 
gives the pattern in terms of relative power.per unit 
area in the same angular direction. 
The radial component of radiated power per square 
meter (Poynting’s vector) is given by 
wv 2 
cos € cos 6 ) 
EP _ 301; 2 (5) 
1207 ma? sin 0 
watts per square meter. 
W, = EoH, = 
In the equatorial plane, 
Ey = os (6) 
2. Gain of half-wave dipole. The gain of the dipole 
relative to a doublet is the ratio of the power supplied 
to the doublet to the power supplied to the dipole to 
produce the same field strength at the same distance 
in the direction of maximum radiation (here the 
equatorial plane, 9 = 90 degrees). 
For equal maximum fields, comparing equations 
(3) in Chapter 2 and (6) in this chapter, 
I Idl = *h. (7) 
The power per unit area for the doublet, using 
equation (3), in Chapter 2, is 
EP _ 3017 sin? 6 
1207 ad? 
and for the dipole the power per unit area is given 
by equation (5). 
The dipole gain is then 
) (8) 
Ww doublet = 
hs i Waoubier dA _ Power radiated by doublet 
a Power radiated by dipole 
Waipole dA 
where the integration is carried out over spheres 
surrounding the antennas. Carrying out this opera- 
tion, 
Gaipole = 1.09 (or 0.4 db) . (9) 
3. Radiation Resistance. The radiation resistance 
of the half-wave dipole is 
ra 2 i WeipoedA = 73.1ohms. (10) 
4. Impedance of an Infinitely Thin Dipole. The 
formulas given here are valid only for a half-wave 
Ez 
CYLINDRICAL COORDINATES 
Figure 8. Half-wave dipole field components. 
dipole composed of wire of vanishing thickness. 
For wire of finite dimensions, see pages 351- 353. 
Here (type A in Figure 6) it is necessary to caleu- 
late the voltage V; required at the input to establish 
a current distribution J; cos [(27/\)z], as shown in 
Figure 8. To do this, the total field of the dipole 
must be known, including the induction field which is 
significantly large at short distances as well as the 
radiation field. In cylindrical coordinates, the total 
field is given by 
E, = + j301; a Ce) pene 5), 
Ue br 
(11) 
