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*To*: tesla-at-pupman-dot-com*Subject*: Re: VA and stored energy in capacitors*From*: "Tesla list" <tesla-at-pupman-dot-com>*Date*: Thu, 30 May 2002 17:27:49 -0600*Resent-Date*: Thu, 30 May 2002 17:42:32 -0600*Resent-From*: tesla-at-pupman-dot-com*Resent-Message-ID*: <1tngxB.A.cZD.mjr98-at-poodle>*Resent-Sender*: tesla-request-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 --

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