Psst, the "transformer" is called a speed coil or choke. It takes a while to learn all of the terminology.
The "slower running speed" and quieter operation of the hospital fans is due to the 6 pole stator and slower overall operating speeds. Not due to the speed coil.
Also, to some degree, the speed coil and stator are generally matched, and it's best practice not to mix and match stators and rotors, since normally this doesn't work to a good outcome. Keep in mind that the speed coil only affects med and low speed. In high speed, the speed coil is out of the circuit.
Hope that helps.