Chapter 2 
FUNDAMENTAL RELATIONS 
THE ELECTRIC DOUBLET 
IN FREE SPACE 
Radiation of an Electric Doublet 
T IS CONVENIENT to present the basic relation- 
I ships of radiation and reception by antennas in 
their simplest form, that of the radiation and recep- 
tion of electric doublets in free space. The resulting 
formulas will later be generalized to include other 
types of antennas and their positions relative to the 
earth. 
An electric doublet is a rectilinear antenna, which 
is symmetrical about the point or points of connec- 
tion thereto and is so short that its directive proper- 
ties are independent of its length. The field of such 
an antenna does not depend on the distribution of 
current along the wire, because the wire is so short 
Figure 1. Polar coordinate system. 
that there is no phase difference between waves 
reaching a point in space from different portions of 
the wire. In symbols, 1 << \ where 1 is the length 
of the antenna and ) is the wavelength of the radia- 
tion. F 
To facilitate the analysis of the field of the doublet 
antenna, the spherical polar coordinate system 
shown. in Figure 1 is introduced. The upper half 
of the doublet is shown in the figure. The distance 
from the center of the antenna to a point in space 
is here denoted by 7. Elsewhere in this volume 
this quantity is written d. 
Let dl be an infinitesimal portion of 1, the length of 
the doublet, and let the current in this portion be 
the real part of Ie?" Let dE,, dE,, dE, and dH,, 
dH,, dH, be the components of the electric and 
magnetic field strengths at any point P(r, 6, ) due 
to the current in dl. A straightforward solution of 
the fundamental equations of electromagnetic theory 
336 
gives the following values for these components, 
valid at distances large compared with the length of 
the doublet: 
[+ au al dl. cos 6 ei 2"/*)(e-") 
2 Oars 
volts per meter, 
eg al dl sin @ ef 27/*)et-") 
4q?73 
volts per meter, (1) 
dE, = 60 fz 
Cart ayaa iva) ae 
dE. = 0, dH, S 0, dH, = 0, 
I [ j 1 : On 
dH, = —| + + —— | dlsing e727’ t-) 
OT lie” =| 
amperes per meter, 
where 
c = velocity of light = 3 X 10® meters per second 
jolt, 
and all distances are measured in meters. Electric 
field strengths are in volts per meter and magnetic 
field strengths are in amperes per meter. Unless 
otherwise explicitly stated, the mks rationalized 
units are used throughout this volume. 
Equation (1) can be simplified at once. Since the 
time variation of the field is assumed sinusoidal, 
e @rct/*) may be omitted. The term e?°"” gives the - 
phase, and it too can be omitted when only the 
amplitude is required. From here on, unless other- 
wise stated, it is understood that root-mean-square 
(rms) values will be used for dH,, dE», dH,, and I. 
The field in the neighborhood of the doublet -is 
called the induction field and is given by the terms 
in equation (1) which include the highest powers of 
r in the denominators. This field is important when 
mutual effects between closely spaced antennas, or 
antennas and reflectors or directors, are involved. 
The radiation field, of greater interest for most of 
the purposes of this volume and the only important 
field at large distances (r>>)), is given by the 
terms in equation (1) containing 7}. ‘Thus the 
radiation field for the element dl of the doublet may 
be written: 
dE, = OOalalsin() volts per meter, 
Ar (2) 
Idlsind dE, 
dH, = — = — 
¢ nr 120_ amperes per meter. 
The other components are relatively negligible 
except near the antenna or near ground for low 
antennas. The electric field dH, is perpendicular 
to the radius vector r and lies in the 7,2 plane, and 
the magnetic field dH, is perpendicular to r and to 
dE,. It will be noted that H/H = 1207 = 376.7 
