RE: 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:
 
> Paul -
> I did not intend to belittle your approach to finding a solution to
> the optimum toroid problem. I was only trying to find some TC
> parameter that could simplify the problem.

No, on the contrary, I was pleasantly surprised that you were able to
pick up on the suggestions and respond so quickly with such
straightforward approximations for those potentialy messy Fc and Fb
functions. You certainly illustrated the idea nicely.

Forgive me for feeling the need to point out (in my previous post)
that the energy J which appears in your formula is itself very
dependent on the radius R, since the topload efficiency which you took
as a fixed constant, contains a factor Csphere/(Csphere + Ccoil). Thus
I would say that the extra little refinement which takes this into
account, leading to the cubic 

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

is justifiable to achieve a genuine optimisation of radius R.

[Note that Eff * J in this formula is the net bang energy delivered
to the total secondary capacitance, whereas in John's simpler formula
the J is the net bang energy delivered to the top capacitance only.]

My ulterior motive was to raise a plug for the Monte Carlo method of
tackling optimisation problems such as this. For example, a direct
solution of the above for R took 20 lines of code, and a Monte Carlo
took 11.

> The toroid does not have this [Fc and Fb linear in R] advantage so
> the solution for optimum size will be more complicated.
> Any suggestions?

Here's my guess: Lets take Fb to be dependent on the radius of the
toroid tube in the same way as a sphere, ie

   Fb = 65 * Rtube kV.

Then for the capacitance, we can use one of the regular formulae -
I'll borrow one from Matt Behrend's neat web page 
http://home.earthlink-dot-net/~electronxlc/tccalc.html

   Ctor = 1.4(1.2781 - Rtube/Rtor) sqrt( 4 * PI * Rtube(Rtor-Rtube))

Ctor in pF, Rtube is the toroid tube radius, Rtor is the overall
toroid radius, both in inches.

>From the energy equation, we have
 
   Eff * J = 0.5 * 10^-12 * (Ccoil + Ctor) * V^2

where the caps are in pF, J is input bang energy (Joules) and
Eff the energy efficiency (factor) into the secondary capacitance.

Then Fc = 10^3 * sqrt( 2 * Eff * J/(Ccoil + Ctor(Rtube, Rtor))) kV

So we need to find Rtube and Rtor, such that

     Fc( Rtube, Rtor) = Fb( Rtube)
and 
     Fb( Rtube) is a maximum.

with an additional proviso that Rtube and Rtor are physically
realizable! Here's a little C code to solve the above problem,

 http://www.abelian.demon.co.uk/mc/optimum_toroid1.c

Interestingly, the optimum toroid always comes out to be as
fat as possible, ie the tube radius wants to be half the toroid
radius, so that the hole in the middle just disappears - the
toroid wants to be a sphere.

> In the past coilers have found that increasing the toroid size
> in tests would increase the output spark.

This is often found. Does this imply that topload sizes as commonly
used are, on the whole, generally below our proposed optimum size?

If you run the above program, you'll see that the optimum toroid
does come out rather bigger than what you might normally use,
eg John's example, 900 watts, 120 bps, and say, 50% efficiency and
40pF coil capacitance, the optimisation gives 66 inch radius as best
toroid radius. Thats one big fat 200pF toroid!

(I suspect that Fb, borrowed from the sphere, is too conservative for
 the toroid, since its curvature in one dimension is at most half
 that of the other dimension - thus favouring the larger toroids too 
 much. Fixing this would I'm sure bring the optimum toroid size down
 some. However, unless we can take the coil dimensions into account,
 I think we'll always come out with 'zero-middle' toroids.)

Regards,
--
Paul Nicholson,
Manchester, UK.
--