Re: Magnifer vs. Tesla Coil

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

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

Antonio wrote:

 > Ok. You consider that there are two "wave modes" that add up.

No, we're accounting for three modes - the lowest three normal modes
of the system:  Two quarter wave modes, plus the 3/4 wave.  The
two 1/4 wave modes have similar voltage and current distributions,
but have opposite phases of primary current and voltage.

[3rd mode contributing to channel heating?]
 > The magnifiers that appear to work well have really a third mode
 > at high frequency (modes a:a+1:a+n>2).

Yes, and this is difficult to achieve with a single coil of uniform
h/d and uniform turns.  With a single solenoid of typical TC secondary
proportions, the 3rd mode is only some 2.5 to 2.9 times the frequency
of the 1/4 wave mode.   However, if we move to a secondary-tertiary
arrangement, we have the opportunity to choose coil sizes such that
the 3rd mode is raised higher in frequency, relative to the lower
two modes.

 > A possibility would be to design the system to maximize the
 > amplitude of the high-frequency mode.

Your example shows quite a high C2 to achieve this, and this agrees
with the distributed model.. adding C2 increases the mode 3 content,
but also brings down its frequency.

Wonder if anyone can devise a way to fire up a non-maggy TC to
breakout, while at the same time injecting a modest amount of higher
frequency CW power into the secondary base - tuned to one of the
higher secondary modes.  At the top of the coil, a low-ish impedance
HF will then be present on the pedestal of the high voltage
fundamental.  Does the additional HF injection contribute
to streamer brightness?  If so, do the extra hot streamers allow
longer streamers to develop?  Can such a hybrid outperform a maggy?

There seems to be two more questions to consider:

a) What proportion of bang energy should go into the 3rd mode? Should
    this be as much as possible?
b) Should the frequency of the 3rd mode be raised as high as possible?

Applying shunt C to the transmission line seems to allow us to control
the tradeoff between these two requirements.  I'd have to do a bit of
work to see if (a) and (b) can both be raised together by some
strategy.

The secondary and tertiary geometries, plus C2, determine the actual
values of 3rd mode energy proportion and 3rd mode frequency.  If we
knew what the ideal was, in terms of energy proportion and frequency,
we would be in a good position to suggest suitable geometries.

 > But why then not simply design the entire system to oscillate at
 > high frequencies?

Because we need to initiate breakout and make some streamers develop
by setting up a high enough E-field.  This is much easier to achieve
at lower frequencies.   But the heating would be more efficient at
higher frequencies - wouldn't it?   The topload of the maggie
simultaneously offers a high voltage, fairly high impedance low
frequency waveform, plus a somewhat lower impedance, higher frequency
signal voltage.   This is true of the 2-coil TC too, but to a much
lesser extent - just a few percent of the energy is in the higher
modes.   Could it be that the maggie can seem to work better than
the 2-coil because its output waveform is more favourable to streamer
production, and is little or nothing to do with the efficiency gain
due to faster transfer?

I wrote:
 >> ...the voltage zero of this [the 3rd] mode is not far below the
 >> topload, and the voltage peak is about half way.

Antonio wrote:
 > This would not then correspond to the ideal lumped design.

We need to resolve the issue of just where the voltage peaks and
zeroes of the 3rd mode sit with respect to the coils involved, and
I can try to come up with some animations to show what I think is
happening, based on a distributed model.  This must tie up exactly
with the lumped model too, and I don't think we're too far off.

At least we agree on

 >>    a) All three peak simultaneously, and at that instant...
 >>    b) sum to zero on the t-line,
 >>    c) zero on Cpri,
 >>    d) no current anywhere,
 >>    e) and a max volts on the topload.

 > Correct.

Let's just explore this a little more

 > The high-frequency oscillations of the third mode are clearly
 > observable only at the transmission line.

Yes, I'm saying this is close to the 3rd mode volts peak.  The higher
modes don't raise a lot of voltage at the topload - it's hard to
see them on the scope - this due to the severe truncation of the top
quarter wave - so quite a low impedance at the top.

The 3rd mode is at quite a high impedance at the location of its
volts peak, quite a bit higher than at the topload  - 2 or 3 times
as high.

 > For an optimized design this would be at the middle of the
 > secondary coil

I don't see how the lumped model can predict or require a volts
peak inside any of the coils.  Along the secondary-tertiary, you
only have three voltage 'nodes' to consider: ground, transmission
line, and topload.

Let me try this argument on you, which stays entirely within the
lumped model...

For each mode to be a distinct normal mode, there must be a
qualitatively different electrical motion of the three voltage nodes
of the secondary-tertiary part of the network.  Fixing ground as the
reference, you then only have two choices for normal mode
oscillations:

a) t-line and topload oscillate up and down together.
b) t-line and topload oscillate up and down in antiphase.

(a) obviously describes the 1/4 wave modes, which are distinguished
from each other by primary voltage polarity. This leaves (b) as the
only remaining option for the 3/4 wave.  Thus in the lumped model,
the t-line and topload can only be in antiphase in mode 3.  Indeed,
it seems you can't force the t-line to be a zero of any mode, without
introducing another lumped voltage node between the t-line and
ground, ie by splitting L2.   There seems to be no alternative for
the lumped model, and the t-line voltage of the 3rd mode must be in
antiphase to the topload, ie a 1/2 wave apart, and is therefore a
voltage max rather than a zero.

I'm sure this can be confirmed by looking at the complex amplitudes
of mode 3 in the lumped model, comparing the phase of the t-line
volts with the top volts.  If I'm wrong, the amplitude of mode 3
on the t-line will either be identically zero (ie a zero of the mode,
which I think is impossible in the lumped model), or if I'm still
wrong, they will be in phase.  But if it agrees with the distributed
model they will be exactly in antiphase in a lossless model.  This
condition, applying in isolation to the 3rd mode, should be true
independently of tuning.

Look at another argument based on electrical length...

To place the voltage zero of the 3rd mode on the transmission line
would require a very short electrical length for the tertiary.
Yet for the 1/4 wave modes we want plenty of electrical length in
the tertiary, otherwise most of the voltage rise occurs across the
secondary.  Your scenario demands an extraordinary amount of
dispersion from the top coil.  Likewise for the bottom coil, you'd
need an entire half wave in there at the 3rd mode, but not too many
degrees at the lower two modes.

[modelling]
 > It's possible to model even a two coils system as a magnifier,
 > by considering the resonator coil as composed of two coils with
 > a grounded capacitor at the center and another at the top. The
 > magnetic coupling adds some complications, but it's probably
 > possible to find an equivalent without magnetic coupling to the
 > top part (I didn't try).

Yes, that's how I deal with magnifiers in what is essentially a
2-coil simulation.   Fortunately I can put into tssp some fairly
arbitrary piecewise descriptions of the 'secondary', so I can model
a 'secondary' which consists of a few hundred turns, then say, a 100
metre gap, then a few more hundred turns.  Thus the two halves of the
secondary are essentially decoupled in the capacitance and inductance
matrices and are only joined through the coil current.  The result is
effectively the magnifier.

And of course, lumped capacitive loading of the transmission line is
easy to put in, just by adding lumps to the relevant elements of the
capacitance matrix.

Here's a set of graphs showing the voltage distributions along the
two coils of a magnifier,

  http://www.abelian.demon.co.uk/tmp/mag1.gif

I'll have to work on animating this set, meanwhile you'll just have
to imagine them waggling up and down!  The two coils here have same
h/d, the tertiary is twice as long, and has twice the turns, of the
secondary.  The relative voltages between the two modes is not
significant in this set of graphs.   There's no additional C2.

Someone will ask what precisely happens at the instant of peak
topvolts, at which the t-line is momentarily at zero volts.  The
distributed model says this:  All three modes reach their voltage
peak with no current anywhere.  The two 1/4 wave signals are of the
same polarity and add to some value on the t-line, but the 3/4 wave
is peaking at the opposite polarity at that instant, and so all three
together sum, momentarily, to zero (with the correct tuning, that is).

 >> The eigenvectors of the 'round-trip' operator are the normal modes
 >> of the resonator.

 > Seems complicated...

Not really.  Define [V] and [I] to be complex column vectors
representing the coil volts and current distributions.  Then
construct a square matrix [L] such that [V] = [L][I], where the
elements of [L] come from the inductance matrix of the coil and will
contain factors of j*omega.  Then use the capacitance matrix (the one
we've been talking about in the other thread) to produce another
equation [I] = [C][V].   Put the two together to get [V] = [L][C][V].
Then you see that the [V] which satisfy this equation are the
eigenvectors of the square matric [L][C].   Look for the omegas which
make the determinant |[L][C] - [1]| = zero, and find the corresponding
[V] for each one. If [V] is N dimensions, there are N modes.  If you
run through this with the N=3 lumped model to illustrate the idea, then
your magnifier equations should pop out.  The distributed model is
just large N, and the distributed real world turns the matrices [L] and
[C] into integral operators.
--
Paul Nicholson
--