Controlling speed of PSC induction motor (Questions about operating at high slip)
You have to understand an induction motor is just a transformer with an odd rotating, shorted secondary.


Those additional primary windings are magnetically coupled with the main winding and thus, the arrangement works like an autotransformer (plus the main transformer with the odd rotating, shorted secondary).
You get a finer speed control at slightly less efficiency. But the main problem with a triac dimmer is the voltage and in result the currents become non-sinusoidal, which makes the motor rattle audibly. That's why ceiling fans typically don't use this method.
The more slip, the more losses in the rotor. As the fan on the motor axle is less efficient at lower speeds, that's a double penalty.
The electrical power going into an induction motor is pretty independent from the speed. (Of course, at lower voltage, the electrical power is also lower.) All what's not turned into mechanical power is turned into heat. So you don't want to operate an induction motor continously at more than 10% slip. Ceiling fans seem to be the one exception, because people demand it.