I want to control the speed of a ceiling fan. This fan originally came with a wall mounted controller containing an autotransformer with several outputs. Speed is controlled by simply switching coils and thus varying the supply voltage to the fan. 20 years ago I made a PIC-based controller with 5 relays, to select the different transformer taps. But this is bulky since it needs a big transformer "hidden" up there.
Other alternatives seem to use switched capacitors to change the phase angle between motor coils, for example: Calculating the capacitor values to control ceiling fan speed
If simply changing the capacitance between coils is enough to control the speed of the fan, why did it come with such a complex (and bulky) transformer?
Is there an advantage to supplying the fan with a lower voltage instead of changing the motor capacitance?
The only reason I can think they did it this way is so the fan can run in a "predictable" way: supplying it full line voltage will make it run full speed, and the speed controller can be omitted (good for maybe commercial installs with a single switch controling multiple fans).
I'd prefer to switch capacitors instead, since capacitors are much smaller and the circuit will be much simpler.