Dr. Slack wrote ...

There's not enough in the description in the first link to tell what else the guy has got in the circuit other than the pig, the capacitor, and a 'homebuilt ballast', which could be anything.

In the second link, the description specifies two Lincoln welder transformers in parallel as the ballast. The point about a welding transformer is that it has a large, and controllable (to set the current), leakage inductance, to provide series current limiting to stabilise the negative resistance of the welding arc.

So yes, the series capacitor is resonating out some of the ballast's series inductance. If you have an inductive ballast that cannot be set low enough, then resonating out some of it is a way to set the effective inductance even lower, that would be one valid reason for using a cap.

A problem with using a cap in series to resonate a series inductance is that the overall series impedance drops, so you'll get more current for any given voltage. However, the capacitor and inductor still have their original impedances, so what this means is that the voltage across each increases (the 'resonant rise') to force more current through each. This increase in voltage can be very profound if you do hit actual resonance, and can over-volt the cap. I notice they're using a 100kV cap in both videos, so there should be some margin to allow it to survive.

Just because there's a cap in there doesn't mean it's resonated out all of the ballast inductance. Indeed, if it had, the pig would blow the supply breakers, so this is not equivalent to 'running unballasted'. A useful experiment would be to measure the input current to the pig (you know, measurement rather than guesswork, the thing that scientists do when they want to actually understand something) for different settings of the ballast (one, two in parallel, series) with and without series caps, then we would have a correspondance between current consumption and arc appearance that could be judged.

So why didn't he simply ballast to the current he wanted? Who knows, you ask him. Perhaps he thought (like some people do) that resonance always leads to MOAR POWA, and tried it, without understanding what it was doing. Perhaps he had a 100kV Maxwell and wanted to show it off in a video, I would if I had!

Putting the resonant components either side of a transformer can be self-defeating because transformer saturation will prevent any significant resonant rise.

In any case, fewest series components carrying the current will result in the best efficiency.

Dr. Slack wrote ...

Scotth, that's a good suggestion from johnf.

There's nothing like playing with a few ideal components and tracing out the waveforms to get to understand what's going on.

There are a couple of problems with simulating this sort of thing in Spice however. Ideal transformers are easy to do, transformers with saturation are possible, but trickier. For the moment, stick to ideal transformers. Secondly, modelling an arc in full, with the temperature modulated conductivity, is not really possible. Just model it as a resistor, and use a wide range of values to see the range of things that can happen. But you don't need to model either transformers or arcs to see how resonance with Ls and Cs works.