But on MANY single phase induction motors, there is a centrifugal switch for the starting capacitor and if you lower the voltage to reduce the torque to where the fan slows down below that speed to keep the centrifugal switch closed, it opens and put the start winding and capacitor on-line again. Since those are not designed for continuous use, they burn up.
You can only use voltage control on Shaded Pole motors and some SPECIFICALLY DESIGNED types of PSC (Permanent Split Capacitor) type single phase motors, i.e. those used in some ceiling paddle fans. CSIR (Cap Start - Induction Run) and CSCR (Cap Start - Cap Run) and most standard PSC type single phase motors cannot be speed controlled with voltage only.
PSC* motors can be controlled via special VFDs, and there are a few small companies that sell VFDs for these single phase MOTORS, but you really have to want to do it; those drives are expensive.
*Technically, Shaded Pole motors can be controlled by these VFDs as well, but since they can be controlled with cheaper voltage control, there is no point.
