Optimum toroid size

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

My previous posting on this subject was intended merely to illustrate
the difficulty of finding a simple formula for the optimum toroid
size. If we're going to look at this further I should highlight, for
the record, a couple of assumptions that were made.

First, I took it that the breakout voltage of the topload would
firmly clamp the output voltage, allowing no further rise. This may
be approximately true, but a more thorough analysis might introduce
the breakout effect in terms of a voltage dependent (ie non-linear)
load resistance applied to the coil.

Second, I introduced the parameters describing the top load as a set
of numbers S but then went on to treat it like a scalar quantity. I
carefully avoided drawing attention to the consequences of S having
more than one component. In fact, the formula for the point of
intersection,

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

removes one degree of freedom from S, leaving not a single solution
but a hyperplane cutting the space of S into two. A further step is
then required to explore this plane to find the single point with
maximum Fb( S).

Having just made the problem seem harder still, I'll just point
out that regardless of the complexity of the functions Fc and Fb,
or the dimensionality of S, a reasonable solution for the optimum S
is readily obtainable on a computer by the Monte Carlo method, which
conveniently allows the functions Fb and Fc to be represented by
table lookup if necessary.

John Couture makes a number of simplifying assumptions for the case
of a sphere, and succeeds in finding a simple formula which can be
summarised as

   R = cube_root( 169 * J) inches

where J is the net energy into the sphere per bang, and R is the
radius of the optimum sphere. This formula assumes that 30% of the
bang energy reaches the topload, and this 30% includes not only the
energy efficiency, but also the proportion by which the topload
capacitance contributes to the total capacitance. If we follow
John's approximations but separate out the top capacitance from the
coil capacitance, we have

   Eff * J = 0.5 * (Ccoil + Ctop) * V^2

where J is now the input bang size, Eff is the overall efficiency of
energy transfer into the capacitance, Ctop is the sphere capacitance,
and Ccoil is the remaining total equivalent self capacitance of the
coil.

If we follow John and intersect with the breakout voltage
V = 65x10^3 R, and substitute Ctop = 2.8x10^-12 R, we get the
relationship

   2 * Eff * J = [Ccoil + 2.8x10^-12 R] * 65^2 * 10^6 R^2

which tidies up to the cubic equation in R:

   R^3 + 0.36 Ccoil R^2 - 169 * Eff * J = 0

if we take the Ccoil to be given in pF.

We might then put in Medhurst's formula for Ccoil, and ignore the
effect on this due to the presence of the sphere, to end up with a
cubic equation for optimum sphere radius in terms of the coil
dimensions, bang size, and energy efficiency.

The solution by radicals of these cubics for R isn't too hard, but
it's not the sort of thing you want to work out by hand. So if you're
going to write a program to do it, consider a Monte Carlo method
instead, which is not a lot harder.

The value of a Monte Carlo solution lies in the ease with which the
approximations for breakdown voltage and sphere capacitance can be
arbitrarily refined, changes which would of course break the cubic.

"John H. Couture" <couturejh-at-worldnet.att-dot-net> wrote:
> It is interesting to note that this [the 20.29pF solution] is close
> to Tesla's method that gave 19.34 pf ...
> ...I believe this is only a coincidence

Yes, I think so too, in view of the arbitrary 30% topload efficiency.

> ...with a toroid the solution would be much more complicated.

Maybe, but perhaps not that much. Has anyone done any work on 
toroid breakout voltages? Formula proposals wanted! Again, the
advantage of using the Monte Carlo method is clear.

> What is the best way to find this efficiency?

The answer to this might be fairly regarded as the holy grail
of tesla physics! Given an answer to this, your Monte Carlo solver
could be extended slightly to optimise the whole system.

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