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How could a DC oscillator produce AC after a capacitor?
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The standard way of looking at this is to consider a capacitor-resistor series combination going to ground. Connect a 10v (wrt ground) supply to the capacitor and the voltage across the resistor rises to +10v, then decays. Now connect that capacitor to ground and that same resistor gets -10v across it, which then decays. Whatever is connected to the capacitor "top" terminal has to be able to sink current as well as source it.
That's what generators in simulators do - they have zero internal impedance (usually). They sink currents as well as source them.