Single phase motor control

Mathematically, the single phase motor is actually a step ahead in the vector control game. If the main and aux windings are wound orthogonal to each other, this, in effect is what the Park and Clark transforms do for the three phase machine: take the degrees of freedom down from three to two and to decouple the two, i.e. make the two axis orthogonal to one another.

The three phase motor is a summation of three sinusoids spaced 120 degrees apart in displacement. If you do the math, assuming a sinusoidal in time current, you will see that the resulting MMF is uni-directional. The single phase machine doesn't have that trick up its sleeve, it has two windings spaced 90 degrees apart and when you do the math there, the result is essentially two fields, one going forward and one going backward. The capacitor is used to give the motor a phase shift so that a preferred direction is given and the motor can start.

In theory, it is possible to control the rotor flux vector in any n-phase machine. People play with five and six phase machines all the time but the single phase machine seems to be the singularity because of the bidirectional air gap flux. I think mathematically we can work out some way of controlling that flux but as others have said here, why bother?

One more assumption that I see is that the single phase machine is incapable of higher powers. The single phase machine can be designed to very high powers, but we don't do it because the machines would be much larger than a 3 phase machine and take up more material. In fact there are relatively large single phase machines in the integral power range, designed mainly for rural use, where three phase power is not available.

Capacitor start single phase motors have a stationary switch & centrifugal switch that switch the motor out of the start winding to the run winding. The centrifugal switch requires the rotor to turn at a certain speed (apprx. 3/4 speed) to engage the contacts on the stationary switch. Its possible the motor would never get out of the start winding and quickly fry.
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