Overtones and velocity factors

["Tesla list" <tesla@xxxxxxxxxx>]

Original poster: Paul Nicholson <paul@xxxxxxxxxxxxxxxxxxx>

Gerry wrote (in another thread):

> about tfss270501 and md110701 spectral graphs. It appears
> the first overtone is close to a 3rd harmonic frequency
> (but not quite) and higher overtones seem to be noticably
> lower in frequency than the 5th and 7th harmonics.  Is it
> true that the spectral graphs are not harmonic in nature??

True.  For example here are the resonant frequencies of an
unloaded coil (one of Terry's, h/d=2.92)

    Freq        Mode       Ratio
    148.4kHz    1/4 wave   1.0
    353.4kHz    3/4 wave   2.38
    513.8kHz    5/4 wave   3.46
    666.4kHz    7/4 wave   4.49
    819.8kHz    9/4 wave   5.52
    977.4kHz   11/4 wave   6.59
   1133.1kHz   13/4 wave   7.64

As you can see, the mode frequency ratios are less than the
quarter wave count.   When the coil is loaded with a typical
topload things change a bit,

    Freq        Mode       Ratio
    97.9kHz     1/4 wave   1.0
   321.4kHz     3/4 wave   3.28
   490.2kHz     5/4 wave   5.0

The 1/4 wave is pulled down by a greater factor than the higher
modes.  There's a particular value of top C that will make the
3/4 wave exactly 3 times the fundamental - that may or may not
be a good thing for a square wave drive signal. Just by luck,
this example has just the right topload to make the 5/4 overtone
match the 5th harmonic.

I'll just say a bit about propagation velocity with respect
to wire length.   For the above coil, unloaded, 817 metres
of wire, we have

    Freq        Mode      Overall wire velocity factor
    148.4kHz    1/4 wave         1.62
    353.4kHz    3/4 wave         1.28
    513.8kHz    5/4 wave         1.12
    666.4kHz    7/4 wave         1.04
    819.8kHz    9/4 wave         0.99
    977.4kHz   11/4 wave         0.97
   1133.1kHz   13/4 wave         0.95

in which the velocity factor is given by 817/n * 4 * Fres/c,
where n is the number of 1/4 waves in the resonance and c
is the fundamental constant 300e6.

Here are the figures for a coil h/d = 4.66, with 2078 metres
of wire.

    Freq        Mode      Overall wire velocity factor
    61.9kHz    1/4 wave        1.72
   157.9kHz    3/4 wave        1.46
   229.7kHz    5/4 wave        1.27
   294.4kHz    7/4 wave        1.17
   355.6kHz    9/4 wave        1.09

These are typical patterns, with the velocity tending towards
something below c as the in-coil wavelength becomes shorter.
If the velocity factor was the same for all frequencies, you
would get harmonic overtones, but you can see that instead we
have a lot of dispersion.   This dispersion originates in the
long range 'longitudinal' coupling, ie mutual L and C between
remote regions of the coil.

Long range coupling becomes less effective at high frequencies. The
distributed mutual reactances are the same of course,  but because
the coil is now carrying several or many quarter wave sections,
the L & C coupling to a point on the coil from some remote turn is,
on average, balanced by an equal but opposite coupling from another
turn at a similar range somewhere else.  The longitudinal coupling
becomes dominated by reactance just with neighbouring turns, h/d
has less of an influence on velocity, and the winding pitch becomes
more of a significant factor,  eventually being dominated by the
direct turn-turn coupling. I've no idea what the limiting velocity
factor is or how to calculate it, but I've seen research on the
topic which involves using tridiagonal matrices to represent the
periodic coupling between adjacent turns.

At even higher frequencies the voltage around a turn is no
longer uniform and we start to get an E-field component acting
across the diameter of the coil - a component which can rotate at
the signal frequency.  The result is circularly polarised EM
waves travelling along the coil and being radiated with reasonable
efficiency, beaming along the axis.  In this realm the pitch and
circumference become the main determining factors of the resonant
modes of the structure.

Bart wrote (in another thread):
> I started building a Seibt Coil myself, but I have yet to
> finish it.

Those Seibt coils might be good ones to use for examining
propagation of the various modes.

> I have everything except the high voltages.

High voltages - who needs 'em :)  Measurements are more rewarding
in the long run.  Making sparks - Bah!   Hope you can measure
some resonant frequencies and the associated in-coil wavelengths
to demonstrate the above.
--
Paul Nicholson
--