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How many joules --> e.m. waves?



MEA CULPA again. I should have looked into the derivation of the R_rad
formula to realize that what was presented in my previous message as the
area was actually the radius of the loop, r. The dimensional analysis
was clearly pointing to that conclusion. If you do save posted messages
please delete my previous EURECA message. This is the modified edition.
........................................................................
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.

R_rad = (4*PI/3)*sqrt(mu/epsilon)*(2*PI*A/lmd)^4 = 2.46e6*(r/lambda)^4

where r is the radius of the loop and lambda is the wavelength (SI units).
For C=30 pF and L=0.083 mH we have f=10 MHz and lambda=c/f=29.8 meters.
The radius of our antenna, r is 3 m and consequently R_rad=252 ohms. This
is much larger than the ohmic resistance R=0.1 ohms. The efficiency of
the antenna must thus be 252/252.1 or nearly 100%. (it wouldl be 96%
if the ohmic resistance were 10 ohms rather than 0.1 ohms.)

The results of computations for other radii are shown in the table
below. Note that both L and Rohmic decrease linearly with r, as it would
be if the wire thickness were costant. Also note that for a thinner wire
all Rohmic would be higher and the efficiencies would be lower than those
shown below. For examle, a wire which is 3 times thinner (Rohmic is 9 times
larger) would result, for r=0.1 m, to the efficiency of 90.3 %.

r(m) L(mH) Rohmic f(MHz) lambda (m) R_rad(ohms) effic (%)
------------------------------------------------------------------
3 0.0083 0.10 10 29.8 252 99.96
1 0.0028 0.033 17.4 17.2 28.0 99.88
0.3 0.00083 0.010 31.8 9.42 2.52 99.61
0.1 0.00028 0.0033 55.1 5.44 0.28 98.82
0.03 0.00008 0.001 100 2.98 0.0252 96.19
------------------------------------------------------------------

I am still puzzled by high efficiencies of antennas whose diameters are
about 5 times smaller than the wavelengths. (For a released guitar string
the energy of sound is only a small fraction of 1%, even when the "antenna"
is as long as 1/2 * wavelength. Perhaps something is wrong with my
understanding of the radiation resistance; it is a new concept to me.
Please help find what is wrong. I realize that my calculations refer
to resonance frequencies but this can not be a source of a very big
error because I am matching the natural f for each L. The w*L are
large in comparison with Rohmic and resonances are very narrow. The
LCR circuit through which C is being discharged oscillates for many
cycles at nearly constant f.

Yes, 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 the oscillating circuit)?
Ludwik Kowalski