Re: New formula for secondary resonant frequency

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "by way of Terry Fritz <twftesla-at-uswest-dot-net>" <paul-at-abelian.demon.co.uk>

Harvey wrote:

> I do not understand this here, or the formulas.

Hi Harvey, All,

I hope I can clarify things a little.

> ln([b/h]/0.2) or ln(b/[h/0.2])

My apologies, I assumed conventional left-to-right application of
equal precedence operators - familiar to programmers, but ambiguous to
the general population. You have

 turns;
 h = length of secondary winding, metres;
 d = diameter of secondary - metres;
 b = height of winding start above ground - metres;
 awg = wire gauge, AWG;

 (metres = inches * 0.0254)

Compute:
 
 x = h/d                                  (form factor)
 wd = 7.348e-3/pow(1.122932, awg-1)       (wire diameter - metres)
 sr = turns * wd/h                        (spacing ratio)

 fa = -94.6683*awg*awg*awg + 9000.55*awg*awg - 301175*awg + 3.64056e+6
 fs = 3.50662*sr*sr - 7.90171*sr + 5.83019
 fx = -0.000211179*x*x*x + 0.00557568*x*x + 0.0664809*x - 0.0153254
 t = fa * fs * (fx/h)/h
 s = -3.85188e-15*t*t*t + 1.17176e-8*t*t + 0.631829*t + 482.463

 fb = log( (b/h)/0.2)                     (use the natural logarithm)
 Fres = s * (1.02 + fb/98.9065);        (Hertz)

> Also the finished equation demanding another variable s
> Fres = s * (1.02 + fb/98.9065);    Can you be more specific as to
> what the quantity s is?

Yes, start with this last equation first. To calculate Fres you need
to know the numbers s and fb. To obtain s and fb, you use the formulas
 s = -3.85188e-15*t*t*t + 1.17176e-8*t*t + 0.631829*t + 482.463
 fb = log( (b/h)/0.2)
but in order to use these, you need to know t, which you obtain
from 
  t = fa * fs * (fx/h)/h
which in turn requires the numbers fa, fs, and fx, which you get from
 fa = -94.6683*awg*awg*awg + 9000.55*awg*awg - 301175*awg + 3.64056e+6
 fs = 3.50662*sr*sr - 7.90171*sr + 5.83019
 fx = -0.000211179*x*x*x + 0.00557568*x*x + 0.0664809*x - 0.0153254
in which sr and x come from 
 x = h/d
 sr = turns * wd/h                        (spacing ratio)
and
 wd = 7.348e-3/pow(1.122932, awg-1)       (wire diameter - metres)

So, given the five numbers: turns, h, b, d, and awg, which together
describe your coil, first work out sr, wd, and x. Then use these to work
out fa, fs, and fx. Use these results in turn to work out t, then s,
and fb, at which point you have the numbers you need to apply the last
equation. Hope that's a bit clearer - the fa,fs,fx,fb,s,t are just
intermediate values of no physical significance in themselves. The
reason for introducing them is simply for clarity - they could be
eliminated by substitution but the resulting formula for Fres
directly in terms of the five starting numbers would be horrendous.

> The wire diameter divided by the height of the coil is already
> a small number but when we in turn multiply that by
> the turns we are only reaching the value of h again? 

It does not quite reach h again, since the turns inevitably have some
separation. The quantity sr is the spacing ratio - a number smaller
than 1.0 which describes the proportion of the winding length actually
taken up by copper. Typical sr for tesla coils varies from 0.55 to
0.95, higher than 0.95 being hard to achieve.

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