I'll do a post on resonant charging then, later.
Perhaps for your Grandpa's oscillator it could save a lot of iron if you start out with a high frequency transformer such as is often used with halogen lighting. What they do is, the first create an AC of about 40 KHz and then they use a small step-down transformer to get 12V. This 40 KHz AC will make resonant charging a lot easier (read: less iron required, compact design)