resonance happens when two coils have same frequency or harmonic. the output trafo primary resonates 50/60 hz but output secondary doesnt. it just gets same frequency induced by induction and its not air core.
when pulsed dc is used to feed primary the secondary gets induced ac of same frequency.
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when pulsed dc is used to feed primary the secondary gets induced ac of same frequency.
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I know they should be here parallel to primary coils and they should be turned on/off sequentially because that is how this circuit works I think (push-pull ?).
I have just problems with electronics - how they change current path. You posted important schematic ! He he ...it's nice how simple things are the most powerful. 
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