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Single phase 120 and 240 V motors just do not produce as smooth a cut.
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I've spent many hours turning with a single phase motored lathe. No offense intended, there are plenty of good reasons to want three phase, all 6 of my lathes now are three phase, but surface finish isn't a reason I'd think of.
If you have single phase power then a single phase lathe motor is superior, and the only reason I know of to get a phase converter is if you have a three phase motor and no three phase power. Even then the output power is not true balanced three phase. Bad gear geometry or lack of rigidity would be a reasonable worry for surface finish, not the phase of the motor.
