I remember a long time ago, I was discussing my induction heater. I had used a figure of VAs, calling it energy (without justification). At the time, Win Hill agreed that it was energy (VARs, circulating power, not real power), but didn't grok in the moment exactly what the relation is. Well it turns out I created the formula on a whim.
Thought process: VAs are like watts, but like torque (which is N.m, but is like force, not like energy in J = N.m), they're sorta rotating. So there's a 2*pi in there, and there's frequency in there. So I wrote E = Er / (2*pi*F) ("Er" = resonant energy).
Example: Tank of 1uH, 10uF, fo = 50kHz, carrying something like V = 70V and I =
240A, so Er = 240 * 70 = 17kVAR. E = 17k / 2*pi*50k = 53mJ. Alternately, 1/2 * 10uF * (70 * sqrt(2))^2 = 49mJ (one sig fig rounding error).Algebraic proof: Vpk = Vrms * sqrt(2) Ec(pk) = 1/2 * C * Vpk^2 (using the capacitor; the inductor is equally valid) = C * Vrms^2
Er = Vrms * Irms Irms = Vrms / Xc = Vrms / (1 / 2*pi*F*C) = Vrms*2*pi*F*C Er = Vrms^2 * 2*pi*F*C Er / 2*pi*F = Vrms^2*C So by the transitive property, Ec(pk) = Er / 2*pi*F.
I'm sure this is *somewhere*, but I haven't ran across it in my textbooks so far. It's just as important as E = 1/2 * C*V^2, just for tanks. Of course you can use Vpk as I started, but you can also get it right away from the tank parameters.
Tim