(10.9a)
(10.9b)
This region is known as the far field [
Ott, 1988
;
Mardiguian, 1992
;
Cheng, 1989
]. Not
surprisingly, the wave impedance in this region can be seen [from the ratio of
equations
(10.9a)
and
(10.9b)
] to be the impedance of free space,
0
.
Nonideal Dipole Radiators.
As for the radiating loop, many references directly apply electric dipole equations for
frequencies and circuit dimensions that violate the initial assumption of
<< . The reader
should be aware that the solutions are not theoretically correct when this assumption is not
met. However,
equations (10.7a)
and
(10.9a)
can be modified to be usable with longer
dimensions, much like the length of the loop was modified for the loop radiator, by fixing the
length of the radiating wire to /2. Beyond this length, the radiating wire looks like a
transmission line and the current is not uniform. Thus the effective radiating area shrinks.
The fields from the other /2 segments cancel each other out [
Mardiquian, 1992
].
If only one side of the radiating wire has a strong capacitance or connection to ground, the
radiating wire will behave like a monopole rather then a dipole, and twice the length should
be used in
equation (10.9a)
. It should be noted that at resonant lengths (such as multiples of
/2) where standing waves will be present on the radiator, there will be directional lobes to
the radiation, and the maximum field intensity may be significantly higher.
Common-Mode Radiation in the Presence of a Reference Plane.
An other modification is required for a radiating wire if the radiating wire is close to a
reference plane (ac ground). A ground plane that extends sufficiently in the vicinity of the
radiator (i.e., a quasi-infinite plane) will act as a reflector. The reflected wave will be inverted
from the incident wave much like a wave on a transmission line reflecting from a short-circuit
termination. Depending on the distance of the reflector and the wavelength being considered,
the reflected wave can add constructively or destructively with the radiated wave.
To see the destructive and constructive effects of the reflecting plane, one may use
trigonometric identities as follows. Consider the radiated wave to be sin and the phase shift
angle of the reflected wave to be = 2 /h, where h is the distance from the constant-
voltage plane. Thus the resultant wave will be sin - sin( + ). Substituting the trigonometric
identity sin( + ) = sin cos + cos sin reveals that if the phase shift is small, such as in
the case of a PCB trace over a ground plane, then sin 0, cos 1, and the incident
wave will cancel with the reflected wave, causing very little radiation. This is why PCB board
traces are typically not considered for common-mode radiation except for very high
frequencies or traces with no local reference plane. In such a case, the loop radiator
equation can still be applied to the board trace. In a case where the phase shift of the
reflected wave is larger, say close to a value of , then sin 0, cos = -1, and the
reflected and radiated waves add constructively to double the radiated amplitude.
For a close ground plane (closer then /10), the reduction in radiation is sometimes given as
[
Mardiguian, 1992
]
(10.10)