Re: Magnifer vs. Tesla Coil

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

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

Hi Antonio, All,

I don't know if anyone else is following this most enjoyable
thread, but I hope there are a few determined old hands still
with us!   They'll find some interesting details and animations
in this post - I hope!

Many thanks for the figures from mrn6.  Things are beginning to
become much clearer with regard to how the lumped and distributed
models coincide.   And coincide they must, if they are both correctly
implemented (which I think they both are).

In particular, the qualitative behaviour of the three normal modes
involved must come out the same, and we can look at each mode in
isolation to determine that - so the issue is quite independent of
relative tuning.

I now have some waveform plots and animations of each mode, so we can
illustrate exactly what is happening in the magnifier.  I will show
how the lumped and distributed models tie together.

Let me work through each of the three modes, starting with the lowest
mode:

 > Mode 1:2:3
 > v3: A31=   0.3125000000
 > v2: A21=   0.2343750000
 > v1: A11=  -0.3125000000

Here we see the transmission line volts A21 and the topload volts A31
in phase, and we would all be happy to label this as a 1/4 wave mode.

An animation of the voltage distribution of this mode:
  http://www.abelian.demon.co.uk/tmp/mag1.mode1.anim.gif
and the predicted terminal waveforms:
  http://www.abelian.demon.co.uk/tmp/mag1.mode1.plot.gif

In these animation files, I've displayed the secondary and tertiary
together as a single moving graph.  You'll have to imagine the
transmission line fitted in at a position of 30% on the x-axis.
So positions 0-30% correspond to the secondary, L2, and positions
30-100% correspond to the tertiary, L3.    You can see a distinct
'glitch' in the profiles at the interface - this is where the
transmission line shunt capacitance is applied.

You will also notice a softer glitch in the current profile, at
around 20%, which is 2/3rds of the way up the secondary coil.  This
corresponds with the top of the primary - which in this modelled
dummy system is a 20 turn helical fairly closely wound around the
secondary.  The capacitance between the primary and secondary has
quite an impact on the secondary current distribution in this region.

I now want to jump from the lowest mode to the highest mode, aka the
3rd mode:

 > The high-frequency component is always negative at the transmission
 > line, to cancel the first component there, and positive at the top.
 > (3/4 wave too, with a zero somewhere at the third coil).

Yes, that description agrees with the normal mode that I label as
'3/4 wave', which is the argument I've been making.  And your
printout says the same thing:

 > Mode 1:2:3
 > v3: A33=   0.1875000000
 > v2: A23=  -0.2343750000
 > v1: A13=  -0.1875000000

Here we see the topvolts A33 having the opposite phase to the
transmission line voltage A23 of this 3rd mode.  The lumped model
implies a zero of this mode occurs inside L3, and the distributed
model agrees.

Have a look at the animation for this mode,
  http://www.abelian.demon.co.uk/tmp/mag1.mode3.anim.gif
and the predicted terminal waveforms:
  http://www.abelian.demon.co.uk/tmp/mag1.mode3.plot.gif

It becomes clear at this stage that the dummy system that I'm
modelling is not very well tuned.  In fact to create this simulation
I've done what many coilers would in practice end up doing - I took
some convenient coils that looked about 'right' and bolted them
together!   I added a primary and tuned it to about the right
resonant frequency, then numerically 'fired' it up.

What we've ended up with, is the voltage max of mode 3 sitting part
way up the tertiary.  Perhaps we should really add on some extra
C2 capacitance to shift that max down to the transmission line.

Notice that the voltage zero of this mode is not far below the topload
(this model is fitted with quite a large toroid) which allows mode 3
current to be drawn off at a fairly low impedance from the top.

That seems to resolve the issue of mode 3, the highest frequency
mode - we would both agree that it's a 3/4 wave mode, with a rather
truncated top quartwave, and having a voltage max somewhere near
the transmission line.

Now lets look at mode 2, the middle frequency mode, which resolves
where my confusion with the lumped model comes from.

For Antonio's node voltages, we have
 > Mode 1:2:3
 > v1: A32=   0.5000000000
 > v2: A22=   0.0000000000
 > v3: A12=   0.5000000000

Clearly, the transmission line is a voltage zero, with a max at the
top. All is revealed when we look at the distributed animation for
mode 2:
  http://www.abelian.demon.co.uk/tmp/mag1.mode2.anim.gif
and the terminal waveforms:
  http://www.abelian.demon.co.uk/tmp/mag1.mode2.plot.gif

Here we have a voltage zero close to the transmission line - only a
short distance up into the tertiary.  Extra applied C2 would shift
this down to the transmission line if required.  The secondary
then spans a half wave with a voltage peak somewhere within it.

It's apparent from this that mode 2 should also be labelled as a 3/4
wave, and part of the problem is that I've been mislabelling it as a
second 1/4 wave mode - hence some of the confusion.

So then the modes are (along with my model frequencies)

   mode 1:   1/4 wave;   60.6 kHz
   mode 2:   3/4 wave;   99.5 kHz
   mode 3:   3/4 wave;  217.4 kHz

And if you want to know what the next higher mode looks like, see the
mode 4 animation in
  http://www.abelian.demon.co.uk/tmp/mag1.mode4.anim.gif

I'd like to use the terminal waveform plots of these three modes to
illustrate how the mode amplitudes are determined for a given firing.

 >From above, these are
  http://www.abelian.demon.co.uk/tmp/mag1.mode1.plot.gif
  http://www.abelian.demon.co.uk/tmp/mag1.mode2.plot.gif
  http://www.abelian.demon.co.uk/tmp/mag1.mode3.plot.gif

I'll just summarise the primary voltage of each mode at t=0:

  mode 1: 16 kV
  mode 2:  7 kV
  mode 3:  6 kV

These add up to 29kV which is close to the firing voltage of 30kV.
In fact, it should add up exactly to 30kV, and the difference is made
up from all the other higher modes which are excited by the firing,
modes 4, 5, 6... etc.

Similary, note the top voltage at t=0:

  mode 1: -200kV
  mode 2:  220kV
  mode 3:  -30kV

which together add to -10kV. Again the precise value is zero and
introducing further higher modes makes up the difference.

The crucial point is that in any firing, the modes are excited in
just the right mix of amplitudes to ensure that Vpri equals the
firing voltage; that Vtop equals zero; and also for everywhere else
on the resonator - that the initial conditions are matched by the sum
of all the modes.  This is a strict enough condition to determine all
the mode amplitudes.

If I set the model to use 30 modes, rather than just 3, we get a
very precise match throughout the resonator, producing the following
combined animation:

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

which accounts for all resonant modes up to around 3Mhz.

The significant HF content is quite visible as the ripple, and indeed
the modes come together to generate quite a vicious initial response
to the firing.  The overall voltage max is not quite were we'd like
it to be, and we're getting about half the output voltage appearing
on the transmission line!  Suffice it to say that this modelled coil
would need some redesign before it was actually built!

We can see from the initial primary voltages at t=0 how the energy is
shared between the modes:

  mode 1: 16 kV = 75% of the bang energy
  mode 2:  7 kV = 14%
  mode 3:  6 kV = 11%

I wonder if the lumped model predicts similar proportions for the
mode energies.  I suspect things would be a little different if the
model was tuned properly.

At this stage it would be nice to apply the distributed model to
reproduce the behaviour of some particular real magnifier.  It would
be great to examine a system which was known to produce good or
exceptional spark performance.

 > I was thinking that if for some mode (the central) there is zero
 > voltage at the ground, zero at the transmission line, and something
 > at the top, in a real system there is some voltage around the
 > middle of the secondary (caused by the middle frequency swinging to
 > negative before crossing zero at the transmission line). A question
 > of complicating the model to see if this really happens.

Yes, we do indeed see that in the mode 2 animation.

 > Ok, but the analysis shows one 1/4 wave mode (low), one 3/4 wave
 > mode (high), and a "degenerate" mode (middle).

Yes, I need to understand more about this middle mode, which I've
been happily interpreting, apparently incorrectly, as 1/4 wave.

 > What is missing is that one of the 1/4 wave components is
 > degenerate, as it is zero at the transmission line.

 > What happens with the middle-frequency component?

You were right to highlight that one as the crux of the issue.

 > A problem (changing the notation a bit): in the usual equations for
 > a group of inductors, V'=jw[L]I', V' are the voltages over the
 > inductors, and I' are currents in parallel with them, but in
 > I=jw[C]V, V are nodal voltages and I are currents between the
 > ground and the nodes.  So, I and V are not I' and V', although
 > it's easy to obtain ones from the others. So I suppose that you
 > are arranging the matrix [L] so V=jw[L]I.

Almost, in fact I rearrange so everything is in terms of V and I',
ie the node voltages and the series currents.  But yes, it does take
a bit of manipulation to put everything into these terms.

 > Ok. you are computing the natural frequencies of the network.

Yes, we get the normal modes without having to put in a forcing
function, which helps quite a bit.

--
Paul Nicholson
--