Optimum toroid size

[paul-at-abelian.demon.co.uk (by way of Terry Fritz <twftesla-at-uswest-dot-net>)]

John H. Couture <couturejh-at-worldnet.att-dot-net> wrote:
(in RE: Resonant wire length issues)

> What are your recommendations for finding the optimum toroid
> size for any secondary coil other than using the 1/4 wavelength
> as Tesla did? I thought Tesla had a clever answer for coilers
> asking how to find the optimum toroid.

I'm afraid I don't know of any theoretical formula for
selecting an optimum toroid size for a given coil.

Very generally, if you wish to optimise for maximum output voltage,
you would seek to use the minimum possible topload capacitance
consistent with achieving sufficient control over the field strength
to prevent breakout. I don't see a direct way to calculate that.

Lets explore the difficulties.

The tesla coil will try to attain an RMS voltage of around

  Vt = sqrt( Eff * Q * Pin * sqrt( L/C) / D)

where Q, L, C have their usual definitions, Pin is the mean RMS
input power, D the effective duty cycle, and Eff the overall power
efficiency of the exciter. (Those using capacitor discharge
excitation will prefer a different formula).

This voltage rise will be frustrated if the toroid size is smaller
than optimum, due to breakout. I'm unable to comment on how the
breakout potential for a given topload and coil geometry is
predicted, so I'll just pretend that a function 

   Vb = Fb( S...)

exists which gives the maximum sustainable potential Vb for a
given coil as a function of a set S of numbers which describe
the toroid geometry options. More certainly, a function

   C = Fc( S...)

also exists which gives the total equivalent capacitance of
the given resonator as a function of the same set of toroid
options.

We can reasonably suppose that both Fb and Fc increase with
the overall toroid dimensions S. If so then Vt will decrease
with toroid size while Vb will increase, and thus there is
a point of intersection. We get something like

      |  Vt
      |    -_            . Vb
volts |       - _    .
      |           X _ 
      |        .      - _
      |     .              -  _
      |  .                       -   _
      ------------------------------------------
       small       Dimensions S          large

The point of intersection X represents the optimum toroid
size for maximum voltage from the given coil. To the
left of X, the voltage is limited to the dotted Vb curve
by breakout. To the right of X the voltage is limited to
the dashed Vt curve by the reduced frequency and the need
to charge up the higher capacitance. 

Some applications might trade output voltage for higher stored
energy to achieve better regulation and would therefore use a
larger than optimum topload. A smaller topload may be appropriate
if spectacular corona displays are the aim.

The difficulty of calculating the position of X is seen
by setting Vt = Vb, to get

  sqrt( Eff * Q * Pin * sqrt( L/Fc( S...)) / D) = Fb( S...)

which would need to be solved for S. Neither Fc nor Fb are
likely to be trivial functions, and moreover, it can be seen
that the solution depends not only on the coil parameters 
L and Q, but also on the exciter efficiency power level, and
mode of operation.

Thus, most coilers will try to experiment with a range of toroid 
sizes and heights to find the best performance, which is
(apparently) a lot more fun than solving equations.

Regards,
--
Paul Nicholson,
Manchester, UK.
Secondary modeling project http://www.abelian.demon.co.uk/tssp/
--