That Gas core theory is not what Stan was talking about, I mean, what they are doing there is showing two different systems of Stans and mixing them together, and giving them incorrect names, we know exactly what that transformer looks like from the control and driver patent, and they show it as a gas core, which is wrong, Yes, the EPG can be used as a transformer and a VIC, but I wouldn't call it a Gas core, because the gas isn't acting like a core, a core concentrates the magnetic flux, but this EPG, this is different, the gas is itself magnetized, and it's just being moved around, so calling it a gas core is a misunderstanding, it's more like a gas-rotor alternator, where in an alternator the rotor has "field coils" to make a magnetic field, to induce voltage in the stator, where as in the EPG you have the magnetic gas acting like your "field coil" to produce the magnetic field, which is pumped around, to induce a voltage in the pickup coils (stator)
Anyway, good luck to them, it's nice to see them thinking for themselves and actually building things, through this they will learn a lot.
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The chokes are part of the secondary, as he says in the control and driver patent, and if you look at the transformer you can see it has 5 coils all on the same core, the meaning of a secondary is a coil on the core that picks up magnetic flux induced by the primary, so anything that is not a primary coil (or input coil) is a secondary coil (or output coil) ... and the "three secondaries" (secondary and two chokes) are in series... there are different ways to think of it, but they all have a voltage induced in them from the core, and they are connected together, electrically and/or magnetically
the diode is used to disconnect the circuit (open circuit) (non-conducting)(diodes only conduct in ONE direction) when the voltage goes negative, to keep it from discharging the capacitor, and to allow the magnetic field from the chokes to collapse and create another pulse, to double the frequency
the resonance is not AC resonance
the ground you are asking about is labeled J in your drawing, Isolated electrical ground, there is no ground symbol in the drawing, it just says it.