Theoretically, Cltr/Cres = pi/2

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Original poster: "Terrell W. Fritz by way of Terry Fritz <twftesla-at-qwest-dot-net>" <terrellf-at-qwest-dot-net>


Hi All,

Ralph wanted to know how one derives the LTR primary capacitor size for a
static-gap system.  The usual way is to look at the tables at:

http://hot-streamer-dot-com/TeslaCoils/Misc/NSTStudy/NSTStudy.htm

One can look for the highest real power or 120 BPS to find the static-gap LTR
primary cap value.  In the case of a 15/60 NST, this value is about 15 to
16nF.  These tables are based on a large number of computer simulations to
arrive at the numbers.  Sort of empirical data gathered from computer
simulations.  The tables are fairly accurate since they take some important
losses into account.  However, the "exact value" is a little "fuzzy".  The NST
currents peak at 16nF, the real power peaks at 15nF, and the 120 BPS point is
also at 15nF.  In general, we simply say that the static-gap LTR value is 1.5
times the resonant value and that gives a pretty close number right off.  The
"exact" value is not super critical, so it all works. :-)

But I think there really is a "perfect world" solution as well.

For a resonant system:

Cres = Inst / (Vnst x 2 x pi x Fline)           ...Eq1

for a 15/60 NST this is:

10.61nF = 0.06 / (15000 x 2 x pi x 60) 

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For the static-gap LTR case, we extract the maximum power from the NST.  For a
15/60 NST, the VA rating is 900VA but the maximum real power is only half that
due to the current limiting at 450 watts real.   Thus, the available power is
450 watts or 450 Joules per second of energy.  

E = Vnst x Inst / 2

450 = 15000 x 0.06 / 2

That is what we must process in the LTR case.  Since we run LTR coils at 2X the
line frequency (120 or 100 BPS), each bang's energy is:

Ebang = Vnst x Inst / (2 x BPS) ...Eq2

3.75 = 15000 x 0.06 / (2 x 120)

However, also we have a direct relationship for capacitor size, energy, and
voltage.

Ebang = 1/2 Cltr x Vfire^2  but...

Vfire = Vnst x SQRT(2) so...

Ebang = Cltr / Vnst^2

Equating this into Eq2 gives:

Cltr / Vnst^2 = Vnst x Inst / (2 x BPS)

Solving for Cltr...

Cltr = Inst / (2 x Vnst x BPS)  ...Eq3

So with Eq1 and Eq3 we can find the ratio:

Cltr / Cres = Eq3 / Eq1 = (Inst / (2 x Vnst x BPS)) / (Inst / (Vnst x 2 x pi x
Fline))

or:

Cltr / Cres = (pi x Fline) / BPS

However, for the LTR case Fline / BPS = 1/2 so:

Cltr = 1/2 x pi x Cres  ...Eq4

So, in a perfect lossless world, the static-gap LTR value is not 1.5 times the
resonant value but actually pi/2 the resonant value or:

Cltr = 1.57079632... x Cres     ...Eq5
===================

For our 15/60 case we get:

10.61nF x 1.57 = 16.66nF which is correct for the lossless case.

A similar analysis for the sync-gap LTR coil case fails since the gap timing
and inductive kick effects ruin the analysis.  In that case, the added factor
of dwell time is not obvious to derive and losses start to really take a big
toll.  It gives a value of 2X or 33.33nF which is too far off the "real" value
to be very useful.  The factor there is a consistent 2.6 (assuming optimal
dwell) which is independent of line frequency and NST type...

One should also consider the great work of Tero Ranta in this area:

http://tesla.tr-labs-dot-com/design/param/

http://tesla.tr-labs-dot-com/design/resocap/

BTW - I just now made all this up, so take it with a grain of salt :-)) 

Cheers,

        Terry