Resonance, and now magnifiers

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

Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk> 

 >From another thread:
 > Using a large top load capacitance ... helps force the
 > best resonant point near the 1/4 lamda point.

This doesn't make any sense, although something along these
lines is often heard from experienced coilers.  It sounds as
if we're supposed to choose the topload C so as to make the
resonance a 1/4 wave.  Larges toploads are certainly good,
but the justification in terms of tuning to some quarter-wave
point is faulty.

It's a 1/4 wave resonance regardless of how much extra C,
if any, is added. We can list some of the changes that occur
as top C is added:-

* All the mode frequencies move down.  The fundamental (1/4
wave) moves down more than the overtones.

The remaining points all refer to the 1/4 wave mode...

* The coil current becomes more uniform.
* As a consequence, the effective inductance becomes closer
to the low-frequency inductance, Ldc.   For coils with h/d
more than about 1.5, this means the effective inductance
increases.
* Because Les moves closer to Ldc, the coil's effective
capacitance becomes closer to the Medhurst value, so the
calculation Fres = 1/(2*pi*sqrt(Ldc *(C_med + C_top)) gives
a more accurate prediction of Fres as top C is increased.

The benefits to the coiler of the heavily toploaded coil
are, possibly:-

* Now the fundamental is much lower than the overtones, so
we may find less of the bang energy is going into these higher
modes.
* The large top C makes plenty of charge and energy available
for streamer formation.
* The frequency is more accurately predicted by a simple
calculation involving C_medhurst.
* The topload controls the E-field gradient around the coil,
protecting the top turns from breakout, but also shapes the
wider field to that breakout tends to develop horizontally
outwards from the toroid, as opposed to taking the shortest
distance (vertically downwards) to earth, or looping back
into the coil itself.

The secondary resonance itself offers no meaningful target by
way of a specific value for top loading.  It will happily
display quarter-wave behaviour with any top C.   I think the
choice of a large topload is made purely to obtain a favourable,
or even optimum, coupling between the secondary and the
discharge/streamer load.

Perhaps I can throw in a few words about magnifiers?
For these comments I'll treat the joint secondary-tertiary
as a single resonator.

I've described how end-loading pulls down the fundamental much
more than the higher modes.  With the magnifier, we exploit
the next higher mode of the resonator - the 3/4 wave, as well
as the fundamental.  The idea is to tune the 3/4 wave mode so
that it will reach the voltage peak of its alternating cycle
at the same instant as does the fundamental. This will increase
the output voltage beyond that of a similar coil whose 3/4 wave
mode is just left to some random tuning.

To tune the 3/4 mode, we must pull its frequency down from
the unloaded value so that it has the correct frequency
relationship to the fundamental.   Applying end loading pulls
both modes down and gets us part of the way.  But obviously
we need to control the 3/4 wave frequency independantly of
the 1/4 wave.  We do this by applying more external capacitance,
but this time attaching it near the other voltage maxima of
the 3/4 wave mode - say about 1/3rd the way up the resonator.

This 'middle' capacitance affects the 3/4 wave mode more than
the 1/4 wave mode, so we now have the means of tuning each mode
with some independence of the other.  This tends to be done by
splitting the coil at the appropriate point and maintaining the
connection with a piece of wire (the 'transmission line' of
magnifier terminology).  The resonator now finds extra
capacitive loading near its 3/4 wave voltage maxima: the top
end-effect C of the secondary, plus the transmission line C,
plus the lower end-effect C of the tertiary.

If done correctly, the 3/4 wave mode is now timed to reach
a voltage peak simultaneously with the fundamental after a
certain (design choice) number of RF cycles have elapsed.

Why bother going to all this trouble?  Well the 3/4 wave
mode is excited anyway, to some extent, whether we like it
or not.  So rather than waste that energy, we might as well
try to use it.   The extent to which higher modes are excited
in any coil depends on how the primary induction is distributed
along the coil.  If we want to achieve high coupling (for what-
ever reason) we cannot do so by spreading the primary along
the secondary, for reasons of voltage breakdown.  So we have
to apply strong coupling to just a short region of the
secondary at its cold end.  It's this highly end-concentrated
coupling which tends to put a greater proportion of the bang
energy into the higher resonances of the secondary.  Therefore
it's a natural evolution of the TC to try to tame and exploit
these.[*]

I've described all this to show how a bit of 'distributed'
theory can be applied to understand the motivation for
constructing magnifiers. (I must point out that when coilers
think about magnifiers, they don't usually think about them in
the kind of terms that I've used above).  Antonio has
thoroughly documented this tuning of multiple resonance
networks using the lumped model,

  http://www.coe.ufrj.br/~acmq/tesla/magnifier.html

(There are of course 3 modes in operation in the magnifier,
not just the two I mentioned above.  The primary coil adds
adds another relevant degree of freedom and when this is
coupled to the secondary we find an additional 1/2 wave mode
at work along the secondary-tertiary.   This too is tuned to
reach a top-volts peak simultaneously with the other two.) [+]

You can see some modelled results for the mode spectrum of
Thor, in the last two graphs in

  http://www.abelian.demon.co.uk/tssp/tmod.html

This isn't a magnifier, just a heavily toploaded TC, but it
shows the predicted levels and current distributions of each
of the resonances.

Coiler's don't tend to take much notice of the higher order
resonances of either primary or secondary.  Many perhaps don't
even realise they are there - mistaking the behaviour of the LC
model for that of the real coil.  Some coilers will admit that
the secondary has lots of resonances, but will insist that
the primary has only one, thinking perhaps that because it is
more 'lumped' it is working in some basically different way.

I should think most of the time this lack of awareness isn't
any problem, after all the coils work.  But I think it's
useful to build up a mental picture of the distributed
resonance, if only for the coiler's satisfaction of having a
deeper understanding of the coil's behaviour.

And there's always the hypothesis that some of this HF
activity may contribute to the racing arc phenomena.
If for example, one of the primary resonances happens to
be similar in frequency to one of the secondary resonances.
Say the first primary overtone collided with, say, the
5th overtone of the secondary, there's nothing to prevent
the two coils transferring energy back and forth at these
high frequencies.

Admittedly the various overtone amplitudes are small compared
with fundamental.  But set against this is the fact that
the peak-trough distance of the overtone standing waves
spans fewer turns along the coil than does the fundamental -
fewer in inverse proportion to the number of quarter-waves.

Plus they sit atop the pedestal of the large 1/4 wave voltage,
so at the very least they will eat into the coil's breakdown
'headroom'.

Racing arcs tend to be cured by small adjustments to topload
height or coupling, eg raising the secondary or lowering the
primary.   When we model these kind of changes, we don't see
any great change in the voltage gradients reached.  Nor do
models show any real gradient problems under out of tune
conditions.  One wonders then why the real coils seem to show
such sensitivity to coupling and topload height with respect
to racing arcs.   One thing our models *dont* account for is
the effect of primary overtones.  It may be conceivable that
the small adjustments mentioned are having the side effect
of shifting the mode spectra of one or both coils with the
unwitting consequence of removing a nasty collision of HF
resonances.

Finally, returning to the issue of large toploads.  A point
not mentioned earlier is that the higher overtones of the
secondary are available at the top of the coil at a rather
low impedance, thanks largely to the top-C.  It might be the
case that this low impedance HF energy contributes to streamer
heating during breakout, helping to keep things cooking nicely
in between the rather lower frequency ebbs and flows of the
quarter-wave current.  Magnifiers, for example, are likely to
have higher overtone amplitudes than regular TCs, and so might
be predicted to give brighter streamers for a given streamer
length.

Anyway, there must be some food for thought and experiment in
there somewhere.

[*]  This phenomena of 'concentrated induction near at the end
of the coil' leading to 'greater proportion of energy in the
higher overtones' is a fairly general phenomena.  I like to
demonstrate it with an effect familiar to guitar players. As
the string is plucked closer to the bridge, the tone becomes
brighter as the overtones (almost harmonics in this case) take
up a greater proportion in the mix of string vibrations that
give the guitar its tone.  The guitarist is changing the shape
of the initial (triangular) displacement of the string. The
essential physics is that the mix of mode amplitudes (and phases)
must be such that the superposition of all the modes at t=0 along
the string (or coil) equals the initial displacement distribution.
This is a Fourier-like synthesis in the spatial domain, with the
normal modes of the string as basis functions.

[+] As Antonio shows, the idea extends to include many overtones.
As more and more overtones are tuned into coherence at the
voltage peak, the TC operation becomes more pulse-like compared
with the leisurely sinusoidal energy transfer of the normal TC.
--
Paul Nicholson
--