Re: Magnifying Coil

To: tesla-at-pupman-dot-com
Subject: Re: Magnifying Coil
From: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> (by way of Terry Fritz <twftesla-at-uswest-dot-net>)
Date: Mon, 15 May 2000 18:29:17 -0600
Approved: twftesla-at-uswest-dot-net
Delivered-To: fixup-tesla-at-pupman-dot-com-at-fixme
Tesla List wrote:
> 
> Original Poster: "Dan Kunkel" <kunk77-at-juno-dot-com>

> I have read the page you refered to above and I have a few questions.
> 
> I don't understand when you say the primary and secondary have a coupling
> coefficient of 'k12.' Is k an undefined constant? I thought coils had a
> coupling coefficient of 0-1.0 (1.0 being max).

k12 = coupling coefficient (k) between primary (1) and secondary (2) 
coils. Just a name.
 
> You use several variables I am unfamiliar with...k,l,m,w. Can you tell me
> what each one is/represents/how it is applied to Tesla Coils? Well I
> guess 'l' is inductance (got that one down).

Just names too. k, l, and m are integer constants that multiply the base
frequency w (should be omega). It's the same notation used in the 
reference papers.
 
> Your page unfortunately leaves my head spinning in the technical realm.
> >From what I gather, the resonant frequencies (self C and topload factored
> in) must occur at certain integer multiples of each other. I would assume
> each coil then be "tuned" to its own harmonic on the sine wave...can you
> please explain to me in less technical terms please.

The article is really somewhat technical. Let me try to explain better:

Ignoring losses:
A circuit composed of one L and one C freely resonates at a single
frequency, and all the currents and voltages are sinusoidal.
A circuit composed of two Ls and two Cs resonates at two different
frequencies, producing waveforms that are sums of two sinusoids,
or sinusoids with "beats". A capacitor-discharge Tesla coil is one
of these. The transformer doesn't add more complexity.
A circuit composed of three Ls and three Cs resonates at three
different frequencies, producing the complex waveforms seen in
magnifiers, that are sums of three sinusoids. Again, a transformer
doesn't add more complexity.
When the charged primary capacitor is connected to the circuit,
all the coils and capacitors start to resonate simultaneously at
the three frequencies. With those design equations followed, there
is an instant when all the energy is stored in the terminal capacitance
(and in the self-C of the third coil). After the same time interval,
all the energy returns to the primary capacitor, and the cycle
repeats forever. 
Other designs may work reasonably too, but will always leave some 
energy in other parts of the circuit when the output voltage is maximum,
and this energy will not be available at the terminal.
In a practical circuit, there are resistive losses that progressively
drain the energy, but the waveforms are not changed significantly
from the ideal lossless case.
That design was derived to solve a problem that was occurring in
a high-energy device for physics experiments, where a three-coils
system was being used to allow very fast energy transfer. With
three coils, it's possible to have very high output voltage and
very tight coupling between the primary and secondary-third coil
system, essential for fast energy transfer. But in that case,
after the output energy was used, the remaining energy in the
system was still very high, and had to be dissipated by a special
device.
With that optimized design, almost no energy was left in the
system after the removal of the energy in the terminal, at the
right instant when the voltage was maximum.
 
Antonio Carlos M. de Queiroz