There are not many ways that a single phase (split phase) motor can be controlled over a range of speed. It is possible at least in some cases through using a Variable Frequency Drive (VFD). This varies the frequency and the voltage so that the volts/hz characteristic is satisfied. The frequency determines the speed, and the volts is adjusted to suit the frequency. There are various problems, such as the centrifugal switch in some split phase motors. It is more common to use a VFD with single phase input and 3 phase output. As the VFD can control the speed, this can be incorporated into a closed loop control, so it is much like controlling a 3 phase induction motor with a VFD. For circuitry (topology anyway) the 3 phase concept and the single phase output concept are similar. The same ideas apply. Very simply, a speed control loop consists of a set point (desired speed) a speed sensor (measured speed). An error comparator determines the difference between the two and adjusts to control