Re: Improved Model for a Primary Charging CKT

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

Original poster: "Gerry Reynolds" <gerryreynolds-at-earthlink-dot-net> 



 > Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br>
 > >
 > Is someone using reactance as resistance?

I can't say for sure, but it seems that way if you read the recent posts on
charging times for capacitance.  There has been frequent mention of RC time
constants in the context of Vc = Vs(1-e^(-t/RC)).
 >
 >  > This model has always bothered me because the charging is not an RC
circuit
 >  > and the equation Vc = Vs(1-e^(-t/RC)) doesn't seem appropriate for this
 >  > purpose.  Also, the excitation is AC and not DC.  Maybe for STR
designs,
 >  > this works OK.  But for us NST users, this approach seems to break
down.
 >
 > Surely. But things are somewhat more complex. See below.

Yeh, that's why this bothers me.
 >
 >
 > This is steady state sinusoidal analysis. Valid only if you assume that
 > the spark gap never fires and wait several cycles until all the
 > transients decay.

Yes, the derivation is for steady state sinusoidal waveforms.  I know this
isn't the whole answer.  But I'm somewhat frustrated by the inability of
today's tools to handle resonant rise and I was hoping this approach would
help bridge the gap and set us on the path to the correct answer.  For
completeness, one would need to compute the transient response (natural
response) introduced by the sparkgap firing.  This would involve a damped
sine wave (natural frequency of the LRC circuit) superimposed onto the
forced response.  This alone wouldn't be too difficult except that before
this transient dies out, there would probably be another SG firing.  I don't
think I'm smart enough to solve this analytically.  Anyway, I just want
something better than what we currently got short of a full blown computer
simulation.

For example, using a value of Cp = Cres * 1.6 according to derivations of
others and my measurements gives a BPS of 120 when the static spark gap is
set to Vs_peak.  One TC designer tool that doesn't take resonant rise into
account shows 50 BPS when the spark gap is set this way.  This tool
indicates that I would need to close the gap substantially to get 120 BPS
and it greatly underestimates the processed power.  Measurements and
prediction for this LTR case seem vastly different.  Again, I'm only talking
static gaps here.



 > The actual maximum is a bit larger than Vs*Q = Vs*sqrt(L/C)/R, at a
 > frequency slightly smaller than 1/sqrt(L*C) rad/s.

Yes, a root locus will clearly show how the resistance will lower the
natural frequency.

 >
 >  > as w > [1/LC] the voltage starts decreasing from peak at resonance and
 >  > eventually becomes smaller than Vs.
 >
 > w>1/sqrt(L*C)

Oops, forgot the sqrt  as 1/LC has units of w^2.  The main equation should
be correct.


 > Your derivation is essentially correct, unless for the errors pointed.
 > Some problems are:
 > R would have little effect in a practical transformer, but it's not a
 > big problem to include it in the calculations.

I agree with one possible exception and that is when operating close to
resonance.  Without R  the voltage gain, if allowed, would go to infinity.
Since it is easy to measure, at least for NST's, I saw no harm in including
it.  One could plot the Vc equation and see how far from resonance one need
to be for it to be insignificant.

 > The presence of a spark gap complicates the analysis substantially.

Agreed
 > The circuit would never reach the steady state assumed in the
 > calculation
 > between "bangs".

This may be true, but I have to mention that the AC voltage I measured at
1.6 * Cres stayed fairly close to the steady state values between bangs.
But then again it seems like it would need to to get 120BPS.

 >
 > A more precise analysis must consider a transient response caused by
 > the AC source Vs and the current in the transformer inductance at
 > the end of the last gap firing. This solution would include the forced
 > response due to Vs as calculated above, added to a decaying oscillatory
 > waveform caused by both IL and Vs. The obtained solution would be valid
 > until the next gap firing, where the capacitor voltage would return to
 > zero (if you don't complicate considering the fast energy transfer
 > transient to the secondary circuit too) instantaneously, and the
 > process would repeat. The gap firings can be (almost) periodical in
 > a rotary spark gap, that can be syncronous with the power line or
 > not, or can be determined by Vc in a static gap.
 > In the general case, the obtained output voltage Vc(t) would be not
 > periodical, and very difficult to predict exactly without a simulation.
 > It's not difficult, however, to write a specific simulator for this
 > problem. (If you are interested in details, contact-me directly.)
 >
 > Antonio Carlos M. de Queiroz

The challenge is to do this short of a simulation (as I see it).  I'm
wondering about short cuts like computing the steady state RMS current at a
particular C and line frequency and using that instead of the nameplate
current to charge the C.

Great reply Antonio, and thanks.

Gerry R
Ft Collins, CO