Modeling a magnifier

To: "'Tesla List'" <tesla-at-pupman-dot-com>
Subject: Modeling a magnifier
From: Tesla List <tesla-at-stic-dot-net>
Date: Sat, 21 Feb 1998 16:26:16 -0600
Approved: tesla-at-stic-dot-net

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From:  Antonio Carlos M. de Queiroz [SMTP:acmq-at-compuland-dot-com.br]
Sent:  Friday, February 20, 1998 4:41 PM
To:  Tesla List
Subject:  Modeling a magnifier

Hi:

I was studying how to model a Tesla magnifier, and would like to hear
comments on my reasoning:

A conventional capacitor-discharge Tesla coil is composed by two resonator
LC tanks tuned for the same frequency, with the coils magnetically coupled
with a low coupling coefficient.
The lumped model for a conventional Tesla coil after the firing of the
spark gap, and before any breakout of sparks in the secondary, ignoring
resistances, is:

+-----+     +-----+
|     | <k> |     |
C1    L1    L2    C2
|     |     |     |
+-----+     +-----+

k is the coupling coefficient, k=M12/sqrt(L1*L2).
This model works as well as a transmission-line model. The two tanks
resonate at the same frequency, and the effect of the coupling is to
produce an "oscillation" in the oscillation, that appears modulated
in amplitude (DSB) at both tanks, with the energy moving back and forth
between the two tanks. Periodically, all the energy is in the secondary tank,
and if the spark gap is quenched at one of these instants (better if at
the first), the energy is trapped in the secondary, in the resulting high-
voltage RF in C2 produces the effects that people like to see.

A Tesla magnifier has a transformer with higher coupling coefficient, and
a separate "third coil" resonator mounted some distance away.
The ideal lumped model for a magnifier would be:

+-----+      +--L3-+
|     | <k'> |     |
C1    L1     L2'   C3
|     |      |     |
+-----+      +-----+

k' is the coupling coefficient, k'=M12'/sqrt(L1*L2').

This model is exactly equivalent to the model of the conventional coil,
if M12=M12', k=M12'/sqrt(L1*(L2'+L3)), L2=L2'+L3, and C2=C3.

As the energy transference only occurs efficiently if both tanks in the
first circuit resonate at the same frequency, this must also happen
in the second circuit, when everything is connected together.

The resonance frequency is 1/(2*pi*sqrt(C1*L1)).
The relation C1*L1=C3*(L2'+L3) must hold.
The maximum output voltage is VC3max=VC1max*sqrt(C1/C3)=
                                    =VC1max*sqrt((L2'+L3)/L1)
The higher coupling coefficient in the magnifier transformer is only a
consequence of the splitting of the secondary coil. The actual coupling
coefficient, considering the transformer and the third coil, is as low
as in a conventional coil.

And this is more polemic:
The dynamic behavior of the magnifier system is practically identical to the
one of a conventional coil (with that model, exactly), and so there is no need
for different operating frequencies and special spark gaps (as I see commented 
in several places).
The advantages of the magnifier are that the high voltage terminal can
be moved away from the primary circuit, and that a more compact and predictable
primary circuit can be used.

What do you think?

Antonio Carlos M. de Queiroz
http://www.coe.ufrj.br/~acmq