# Building a VFD for a split phase (capacitor run) motor?

I'd like to get variable speed on a very tiny lathe (Taig, 4" swing). I've got a dishwasher motor I'm planning on using for it, which is rated for 3600rpm at 60Hz (it's about 1/5HP). But it's a capacitor-run motor (no centrifugal switch) and I think that means I have direct access to both windings. Any recommendations for building a VFD to drive this? I am fairly well versed in basic circuit analysis, just never designed much to deal with 110V.

I understand the general operating principle of a VFD, I'm just not sure how to make the low-voltage sinusoids I can easily generate control the 110V I'd get by putting a rectifier on mains voltage.