The radiation resistance of our antenna, which is an oscillating magnetic
dipole, can be calculated from the following formula. The derivation is
not trivial (I skipped it) but the formula is simple.
where A is the area of the loop and lmd is the wavelength. For C=30 pF
and L=0.083 mH we have f=3.19e6 Hz and lmb=c/f=94 meters. The area of
our antenna (r=9 m) is 28.3 m^2 and R_rad=20100 ohms.
This is a very large resistance in comparison with R=0.1 ohms. Does this
mean that our antenna is nearly 100% efficient? If true then the antenna
whose area is 10 times smaller (radius=0.94 cm = lmb/10000) would have
R_rad=2 ohms and its efficiency would be 20/20.1 or 99.5%. Strange for
an antenna whose radius is so small in comparison with the wavelength.
Even 50% efficiency (assuming R=R_rad=2 ohms) would be strange in this
case.
Going down by another factor of 10 in area (r=29.7 cm) we find that
R_rad=0.0002 ohms. For that antenna (assuming R=0.1 ohms) the efficiency
is 0.0002/0.10002 or 0.2%. This is not much but may be enough to hear
a click when a capacitor is discharged near a radio set.
But I am puzzled by all this; nearly 100% efficiency is suspecious. For
a released guitar string the energy of sound is only a smallfraction of 1%,
even when the "antenna" is as long as 1/2 * wavelength.
I know that my "definition" of efficiency does not include losses
in power supplies, etc., but I am still puzzled. Does anybody know how
efficient a typical radio station is (preferably in terms of energy of
waves over the energy received by its oscillating circuit)?
Just before posting I noticed something I do not like in the formula.
The last factor is not dimensionless unless A a length rather than the
area. I suspect a typographical error (however, A was clearly defined
on the previous page as Pi*r^2). The dimensional analysis (see below)
shows that R_rad is not in ohms when A is an area. Can somebody verify
the formula for R_rad of a magnetic dipole?
One ohm is Volt/Amp = (kg*m^2)/(A^2*s^3) while sqrt(mu/epsilon) is also
in ohms (kg*m^2)/(A^2*s^3).
Ludwik Kowalski