Re: Magnifer vs. Tesla Coil

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

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

In this post we can try to compare a magnifier with the same
coils arranged as a traditional 2-coil TC.

In my last post I supplied animations of the first three resonant
modes of a model magnifier

  http://www.abelian.demon.co.uk/tmp/mag1.mode1.anim.gif
  http://www.abelian.demon.co.uk/tmp/mag1.mode2.anim.gif
  http://www.abelian.demon.co.uk/tmp/mag1.mode3.anim.gif

and the composite animation
   http://www.abelian.demon.co.uk/tmp/mag1.anim.gif

I took the model secondary and tertiary and joined them together to
make a single coil, kept the same toroid and helical primary, and
fired it up 'numerically' as a regular TC, albeit with much higher
coupling than is conventionally used. Here are the first three modes

  http://www.abelian.demon.co.uk/tmp/dual1.mode1.anim.gif
  http://www.abelian.demon.co.uk/tmp/dual1.mode2.anim.gif
  http://www.abelian.demon.co.uk/tmp/dual1.mode3.anim.gif

with the composite
   http://www.abelian.demon.co.uk/tmp/dual1.anim.gif

The helical primary extends 15% of the way up the secondary,
and its influence a little higher.

This dual coil model also uses 30 modes, ie up to around 3Mhz.
The mode frequencies come out much the same, a little lower but not
a vast change.  Peak output voltage is also about the same.

It's clear that the modes are qualitatively the same as the
magnifier, with mode 2 demonstrating the same half wave behaviour
associated with the primary coupling.  The dual coil model shows much
less HF content than the magnifier, due to the smoother distribution
of capacitance along the secondary, compared with that along the
secondary-tertiary of the magnifier.

Antonio wrote:
 > Note that the middle frequency is always the resonance frequency of
 > the L3-C3 system alone. So, if we consider this to be "the"
 > resonance frequency of the magnifier,

You make a good case for that view...although I think most of
the energy is still going into mode 1.  Perhaps that's just an
accident of this 'random' choice of coils and tuning.

 > ... then the third coil operates in 1/4 wave mode, and the
 > secondary coil operates in 1/2 wave mode.

Yes, given the correct tuning.  My randomly selected coils demonstrate
that the zero of mode 2 and the maximum of mode 3 won't automatically
position themselves at the junction of L2 and L3, and careful
selection of coils and tuning would be required.  I should think it
is unlikely to come out right by 'accident'.

 > No relation to the secondary being a voltage source driving the
 > third coil, as sometimes mentioned (the voltage there is always
 > zero at the central frequency).

Agreed.  There's little to be gained by trying to interpret this
system in terms of impedance matching at the transmission line.
We often see explanations along these lines, but they obscure much
of the essential physics that's going on.  If anything, mode 2 in
the tertiary could be described as 'current' driven.

We can summarise the mode behaviour for the secondary + tertiary

  mode 1: secondary and tertiary combined is 1/4 wave;
  mode 2: secondary is 1/2 wave, tertiary is 1/4 wave, total 3/4 wave;
  mode 3: secondary is 1/4 wave, tertiary is 1/2 wave, total 3/4 wave;

The 1/2 wave behaviour of mode 2 in the region influenced by the
primary coil becomes much less noticeable when we switch to
a normal 'flat' primary with a more modest coupling coefficient.  We
can see this for example in mode 2 from Marco's Thor,

  http://www.abelian.demon.co.uk/tmp/thor.mode2.anim.gif

which is almost a quarter wave, but if you look carefully there is
still a small voltage max on the lower few percent of the coil.
But you can probably see why, up until now, I have regarded this mode
as a 2nd quarter wave mode.  It is the fierce coupling of the helical
primary which is raising this extra 180 degrees of phase delay for
the 2nd mode.  This mode would like to be a 1/4 wave, just like mode
1, but the primary is 'pulling' on it the opposite way!   For this
reason I like to think of mode 2 as a degeneration of mode 1 - is
that a reasonable view?  I would argue yes, because as you reduce
the primary coupling, the shapes and frequencies of modes 1 and 2
converge.

 > A lossless model, I imagine.

No it's a lossy model.  We account for some sources of loss, eg
wire resistance, a lumped ground resistance, and arbitrary load
resistances applied to the topload.   But because not all losses are
being realistically modelled, the predicted Q factors of each mode
are usually 2 or 3 times their real values.  The lossy model is
also easier to model numerically!

 > There is my magnifier:
 > http://www.coe.ufrj.br/~acmq/tesla/mag345.html
 > It didn't work very well due to the lossy C2, but I have the
 > experimental waveforms at low power listed and all dimensions.

This could be a tricky one to model.   Could you give the dimensions
of the tertiary and its disc+rod topload?  How high is this
coil raised up?  And what was the primary tank capacitance?

Robert Jones wrote:

 > I don't see how the two models can coincide. ...

 > The lumped model can only be accurate (<10%error) for a
 > limited region.

Yes, the lumped model with N LCR elements can be made exact only
at the spot frequencies of its N poles and N zeros.  In between these
frequencies it will remain quite accurate, but will not be exact.
But - this is also true of the distributed model!

Remember, the distributed model can only be implemented as a
lumped model - it just has a very large number of LCR elements, ie
a large number of poles and zeros.  It's behaviour approximates
the real world distributed physics arbitrarily closely, but it
remains a large lumped model, nevertheless.

As a result, we expect a lumped model with say N=3 to exactly match
a lumped model with N > 3, at the poles and zeroes produced by
the N LCR elements shared by the two - of course with the proviso
that these N LCR elements are given equivalent values in both models.

As an example, see

    http://www.abelian.demon.co.uk/tmp/acmq-zin-2.gif

which shows input impedance plots of a system, modelled by a lumped
LCR model with small N (acmq LC model), and a large N (tssp model).

The lumped network involved here for the red trace is

  o--+--L1---+---L3---+---L4---o Vtop
     |       |        |
     C0      C2       C4
     |       |        |
     =       =        =

so there are 3 LC elements, giving 3 poles and 3 zeros.

As you can see, the response agrees within the common domain of
poles and zeros.  The tssp model, because it has more of these to
draw on (around 200 or so), is able to represent the distributed
physics up to a higher frequency - ie higher resonant modes.

So much for the frequency domain, what about the spatial domain?

In terms of physical distributions of currents and voltages, a
model consisting of N LCR segments can tell you the voltage at
N voltage nodes and the current in N meshes.  But it says nothing
about the distribution in between these points.  For example the
N=3 magnifier model can describe Vpri, V_tline, and Vtop, and
Ipri, I_sec, I_tertiary.    As you increase N, not only do you
allow the model to represent the physics over a wider frequency
range, you also gain more spatial resolution.   So the tssp model
with the secondary in 200 elements can supply 200 poles, 200 zeros,
and 200 voltage and current points.  Hence the result is a fairly
detailed 'impersonation' of the real distributed physics.

In fact the animations presented here are using 363 elements,
that is 362 for the secondary (or secondary-tertiary) plus one
extra for the primary.   We don't need more than one element
for the primary because all but one of its natural resonant modes
are at frequencies higher than we are looking at...so one element
is enough, and that accounts for the single low frequency primary
mode - the LC resonance we all think of as the 'lumped' primary
resonator.

Even without setting the two models to the same N sets of LCR
parameters, they should a priori agree on the qualitative
relationships between the phases of each mode at each node - as
Antonio and I between us have now demonstrated.

It all boils down to the number of 'degrees of freedom' which we wish
to model.  The real system has practically infinite number of these,
but our computer models must make do with a small finite number. It
is entirely up to us which degrees of freedom we wish to account for
in the models.

Tssp software uses a large number of elements not because we're
looking at very high mode frequencies, instead we're after spatial
resolution so we can 'see' the voltages and currents sloshing around
in the coils - and we get high accuracy too, around 1%.  But for
practical design 2 or 3 modes is enough, because these contain 95% or
more of the energy.  That means the essential behaviour of the TC can
be modelled for practical purposes with a circuit of just 2 or 3 LC
lumped resonators.
--
Paul Nicholson
--