Re: VA and stored energy in capacitors

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

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

Jolyon wrote:

> Since relationship between apparent power (VA), frequency (F) and
> stored energy (E) in a capacitor in an AC circuit is VA=4pFE, where
> it is assumed that F is the line frequency (50 to 60 Hz) during the
> period when the capacitor is being charged by the transformer
> prior to the firing of the spark-gap, ...

In general for a capacitor carrying a sinusoidal current, the stored
energy is 0.5 * C * Vp^2 where Vp is the peak voltage.  This peak
voltage is sqrt(2) * Vrms, so in terms of RMS voltage, E = C * Vrms^2. 
The associated RMS current is 2 * pi * F * C * Vrms, so
we have 
  VA(R) = Vrms * Irms 
        = V * 2 * pi * F * C * Vrms
        = 2 * pi * F * C * Vrms^2
        = 2 * pi * F * E 

I put the (R) in VA(R) to remind that we're talking reactive VA, not
real power - the energy delivered to the C in one quarter-cycle is
returned to the source during the next so the net energy transfer is
zero.  For example, a resonating LC behaves as if the stored energy E
shuffles back and forth 2 * pi * F times per second.

The above doesn't quite apply to an idealised TC charging circuit, 
because the discharge of the C is via another path.  The line-side
RMS ammeter is only registering the charging current, not the two-way
charge-discharge current.  To a first approximation the RMS ammeter
reading is halved to pi * F * C * Vrms. So now

  VA = Vrms * Irms 
     = V * pi * F * C * Vrms
     = pi * F * C * Vrms^2
     = pi * F * E 

where I've dropped the (R) because now some portion of the VA is real
and some reactive.  This wouldn't be expected to apply to a real
charging circuit as it stands because the currents and voltages are
no longer sinusoidal - form factors other than sqrt(2) relate RMS
readings to peak values, readings depend on just where in the circuit
you put the metering, and the effect of chokes and ballasts all come
into play.  Best thing to do is to model your particular circuit to
estimate how your actual meter readings relate to the peak voltages
and stored energy.

> doesn't the same equation apply when the gap fires when F becomes
> the resonant frequency of the Tesla coil primary?

After the gap fires the low resistance of the arc effectively 
decouples the charging circuit from the primary LC.  Things get a lot
simpler, the LC currents and voltages are sinusoidal, and RMS meter
readings in the LC circuit would obey VA(R) = 2 * pi * F * E during
each cycle of the ringdown. Of course, real meters report an RMS
value averaged over the response time of the meter's pointer which
is typically much longer than the whole firing event, so again some
circuit modeling would have to be applied to make sense of real
measurements.

> Does this not go some way into accounting for high peak powers
> observed in TC discharges?

It certainly describes the 'power compression' taking place in
the charging/firing/primary circuit.  Roughly speaking, energy stored
in C over a quarter-cycle of line frequency is released over a
quarter-cycle of the resonant frequency, so you're getting a
'compression ratio' in the order of F_res/F_line.  However, the 
discharges from the secondary are compressed further still, because
once the bulk of your stored energy has made its way to the topload,
it can all discharge to ground in nano-seconds.  Again, the formula
VA = 2 * pi * F * E could be said to apply during the arc discharge,
but this time the F is the frequency of oscillation of the discharge
itself, which involves the topload C and the self-inductance of the
arc and its ground return path - a very high frequency.  Therefore
ultimately, the peak power in the discharge could be compressed by
up to F_discharge/F_line.
--
Paul Nicholson
--