RE: Calculating secondary resonance of bipolar coils

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

Original poster: "by way of Terry Fritz <twftesla-at-uswest-dot-net>" <paul-at-abelian.demon.co.uk>

Darren -

You pose some good questions.

I wrote:

>> The 'half-wave' is the simplest possible mode of resonance of a
>> solenoid, and is the natural 'ground state' of an open circuit coil
>> in free space, or placed horizontally above a ground plane.

Darren asked:

> By this are you implying multiple modes of oscillation as in
> quantum mechanics? Would it then be possible to achieve multimode
> oscillation?

Not in the same way as in quantum mechanics - I was just borrowing a
term to use for the 'lowest possible self resonance' which didn't have
the word 'wave' in it, so that I could emphasise that the half wave
and quarter wave resonances are really just extreme shapes of the same
resonant mode, the shape being determined by the degree of asymmetry
of the distributed capacity. In that respect the quarter or half wave
is a convenient classification rather than something fundamental, and
you can operate a coil somewhere in between if you want.  When it
comes to calculating the resonant frequency, the cylindrical symmetry
of the vertical monopole aids calculation of the capacitance
distribution, and in free space the bipolar also has axial symmetry
too, so it's even easier. Horizontal above a ground plane leaves us
with only the axial symmetry so we have more work to do to calculate
the capacitance, but it's not fundamentally different to the vertical
monopole.

All real inductors display a spectrum of self resonant modes or over-
tones. Their frequencies are not harmonically related so I'm always
careful not to call them harmonics. They can of course be
independently excited.  BTW these are 'axial' modes in which the
E-field lies along the axis of the coil. At very much higher
frequencies there are transverse modes in which the solenoid begins to
look more like a waveguide than a coil. For TCs we use the lowest
axial mode and we call it quarter or half wave according to how
much of the electrical length is spanned by the coil itself.

I believe that there is no advantage to exciting a TC secondary at a
higher mode because the higher modes require a greater stored energy
to achieve the same end-end peak voltage.

> Can the system be considered to be linear or is the dielectric of
> the enabel going to saturate and result in some interesting coupling
> between various modes of oscillation?

With small signal levels the solenoid is linear and the modes are
not coupled together. Energy presented to the coil will be either
accepted or reflected according to the available modes and the
amplitude of each mode decays independently, each with its own time
constant.  The coil materials do not normally contribute any
non-linearity worth speaking about until a certain power level is
reached, indicated by emission of smoke or flame usually.
The major departure from linear behaviour at operating power levels is
due to the varying endload applied by the corona. tssp models the 
linear small signal behaviour, which has limited application to the
operating conditions of a real TC, but I believe a thorough
understanding of the small signal behaviour is a prerequisite for
tackling the more difficult problems. One obstacle to modeling the
coil under operating conditions is knowledge of the corona impedance,
which it is safe to assume is a function of both voltage and frequency
for any given topload. I don't know of any formula which estimates
this, and as far as I know the only attempt to tackle this in the TC
community was made by Terry, details to be found in a little gem
tucked away at

 http://hot-streamer-dot-com/TeslaCoils/MyCoils/CWCoil/CWImpedance.txt

The gist of it is that as you turn up the power, the size of the 
discharge increases, taking more surface area and thus more charge 
per volt to fill it. Thus greater capacitive load reactance as the
power goes up, and the resonant frequency comes down, etc. Terry 
proposes a model of the brush discharge as a resistive volume
surrounded by a charge storing surface - IMO an eminently reasonable
picture and a good candidate for both modeling and further experiments
- any takers for this crucial 'gateway' topic which could open the
way to coil and topload optimisation?

> I was quite convinced that the lumped inductor model was appropriate
> for explaining the profile he obtained, since the mutual inductance
> between turns is greater in the middle and he found a minor increase
> in volts/turns towards the middle but no '1/4 wave' effect, which
> would be a sinusoidal profile wouldn't it? 

I don't recall any profile measurements involved, just a phase compar-
ison between top and bottom, with the erroneous expectation that a
transmission line quarter wave would show a 90 degree difference - it
doesn't. The worrying part about it was that it was happily accepted
by the TC community. It's to the benefit of everyone if people's 
pronouncements are view critically before acceptance. It doesn't do 
any harm - if an idea is correct it will survive any amount of
scrutiny.

More recently Terry has made precision voltage profile measurements,
see

 http://hot-streamer-dot-com/TeslaCoils/MyPapers/NSVPI/NVSPI.htm

which are reviewed in 

 http://www.abelian.demon.co.uk/tssp/pn2510/

Hope I answered your questions Darren and thanks for your kind words
about tssp. In line with my comments above, please don't accept
anything I say about TC modes, etc. Scrutinise what's said and form
your own considered opinion.

Cheers
--
Paul Nicholson,
Manchester, UK.
--