Page 544 - Engineering Electromagnetics, 8th Edition

526 ENGINEERING ELECTROMAGNETICS

On a short antenna, where 2 is significantly less than a half-wavelength, we see
only the first portion of the sine wave; the amplitude of the current increases in an
approximately linear manner, from zero at the ends to a maximum value at the feed, as
indicated in Figure 14.6. The gap at the feed point is small and has negligible effects.
The short antenna approximation (in which a linear current variation along the length
can be assumed) is reasonable for antennas having an overall length that is less than
about one-tenth of a wavelength.
A simple extension of the Hertzian dipole results can be performed in the short
antenna regime ( <λ/20). If this is the case, then retardation effects may be ne-
glected. That is, signals arriving at any field point P from the two ends of the antenna
are approximately in phase. The average current along the antenna is I 0 /2, where
I 0 is the input current at the feed. The electric and magnetic field intensities will
thus be one-half the values given in (22) and (23), and there are no changes in the
vertical and horizontal patterns. The power will be one-quarter of its previous value,
and thus the radiation resistance will also be one-quarter of the value given by (30).
Matters improve as the antenna length is increased, but retardation effects must then
be included.
For longer lengths, the current distribution is treated the same as that of an open-
ended transmission line that propagates a TEM wave. This will be a standing wave
in which the current phasor is given by

.
I s (z) = I 0 sin(kz) (51)
where the open end is located at z = 0. Also, for a TEM wave on a transmission line,
√
the phase constant will be β = k = ω µ . When the line is unfolded to form the
antenna, the z axis is rotated to the vertical orientation with z = 0 occurring at the
feed point. The current in (51) is then modified to be

. I 0 sin k( − z) (z > 0)
I s (z) = = I 0 sin k( −|z|) (52)
I 0 sin k( + z) (z < 0)
The strategy from here is to consider the antenna as made up of a stack of
Hertzian dipoles, each having length dz (Figure 14.7). The current amplitude in each
Hertzian is determined according to its position z along the length, and is given by
(52). The far-zone field from each Hertzian can then be written using Eq. (22), suitably
modified. We write this as a differential field contribution at a far point at distance r
and spherical coordinate angle, θ :

I s (z) kdz
− jkr
dE θs = j η sin θ e (53)
4πr
The coordinates r and θ are, of course, referenced from the center of the

Hertzian, which itself is at a position z along the antenna length. We need to ref-
erence these local coordinates to the origin, which occurs at the antenna feed point.
To do this, we borrow from the methods used to analyze the static electric dipole
as presented in Sec. 4.7. Referring to Fig. 14.7, we can write the relation between