# RE: Optimum toroid size

```
Paul, All -

Thank you for your post regarding the optimum toroid for Tesla coils. You
gave me some ideas and I decided to dig deeper into the matter. I found that
if a sphere is used instead of a toroid for the secondary top terminal the
optimum sphere solution was much simpler. The ascending and descending
curves were much easier to develop. The secret is in properly applying some
standard Tesla coil equations.

The graph consists of the following.
x axis - sphere capacitance - pf
y axis - sphere volts - KV

The ascending curve -
sphere capacitance Cs (pf) = 2.8 R   R = radius in inches
sphere breakout voltage KV = 65 KV per inch radius R
Ascending curve  KVs = 65/2.8 = 23.2 x Cs

The descending curve -
Example NST 15 KV 60 ma - 120 breaks per second
input watts 900/120 = 7.5 watt seconds per break
It is assumed that 30% of this energy reaches the sphere.
energy at sphere J = 7.5 x .30 = 2.25 joules
The energy equation  J = .5 Cs x Vs^2
Descending curve  KVs =1000 sqrt(2J/Cs)  Cs in pf

This will give a graph with two curves that will intersect at about 20.29 pf
or a sphere of about 15 inch diameter. It is interesting to note that this
is close to Tesla's method that gave 19.34 pf for the optimum secondary top
load (my earlier post). However, I believe this is only a coincidence
because the two methods use completely different variables. I will leave it
up to the reader to find the sphere voltage.

To the left of the intersection of the two curves the sphere voltage is
limited by the breakout of the smaller spheres. To the right of the
intersection the sphere voltage is limited by insufficient energy to charge
the sphere. This gives you the optimum sphere size for the given conditions.
I agree that with a toroid the solution would be much more complicated.

It is interesting to note that the optimum top load of a Tesla coil is
dependent on only two parameters, the input energy and the efficiency at the
top load. What is the best way to find this efficiency?

Tesla coilers usually accomplish the same optimization by trial and error
methods. But does a single random spark length indicate optimum conditions?

I would be interested in any comments.

John Couture

-----------------------

-----Original Message-----
From: Tesla list [mailto:tesla-at-pupman-dot-com]
Sent: Sunday, October 01, 2000 7:28 PM
To: tesla-at-pupman-dot-com
Subject: Optimum toroid size

Original poster: paul-at-abelian.demon.co.uk

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
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/
--

```