Re: 1/4 wave theory/cite the variance?

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Ed Phillips by way of Terry Fritz <twftesla-at-qwest-dot-net>" <evp-at-pacbell-dot-net>

> To flog the dead equine a bit... resonant (in the sense of having no
> reactive impedance component) lengths are a bit short of the multiple of
> 1/4 wavelength (and the amount short varies by diameter of conductor,
> relative to length, the medium, and which multiple you are talking a bit).
> However, if one is speaking of the frequency at which the current, voltage,
> charge, or whatever,  forms a nice sinusoid along the wire, that IS at the
> actual appropriate fraction of the free space wavelength. (and, the antenna
> has a fair amount of reactance at this point..)
> 
> The latter is what the open/closed tube or clamped bar, or string, etc.,
> would be comparable to.  I wonder what the equivalent impedances are in
> both cases (assuming that the amplitude is small enough that the
> nonlinearities are negligible)

	For an open-circuited transmission line (analagous to a TC with no top
loading) the minimum input impedance occurs when its electrical length
is a quarter wave or ODD multiples there of, and the maximum when its
length is a half wave or multiples there of.  As for the current
distribution, it is sinusoidal only when there is no top loading;
becoming constant with enough top loading.  (Realize I'm mixing
antennas, TC's, and transmission lines here, but basic principles are
similar.  Even the unloaded TC isn't quite a distributed transmission
line, but close to it as the Corums expound.)

	For the record, I ran some simple and approximate calculations of the
self-resonant frequency of an unloaded coil as a function of its length
to diameter ratio.  Here is what comes out:

L/D        Wire length

1:1      0.295 wavelength
2:1      0.374     "
3:1      0.412     "
5:1      0.448     "
10:1     0.477     "
100:1    0.498     "
1000:1   0.499     "

(This table may not look right if you're reader is not using mono-spaced
type, but should still be readable.

	In the limit, for an infinitely long coil the resonant frequency
approaches that corresponding to a wavelength of twice the wire length
in free space.  This is an experimental observation, and has been known
for a very long time - shows up in handbook's as far back as the 1930's
and probably before.  Tesla was usually dealing with coils of much
smaller L/D, where my approximations are not accurate enough to use, so
I can't comment on the length/wavelength correlation.  It may approach
1/4 wavelength in the limit; I can't say.

	I don't know any way to make the same calculations for a spiral
antenna, and don't know if any any empirical relations between geometry
and self-capacitance have been worked out and published.  Someone could
surely solve the problem with modern analysis techniques, but I'm not
man enough to do it.

Ed