I guess what I should have asked was how the motor speed is controlled by changing the voltage.
No problem. Hopefully, I can answer these questions concisely.
Let's take the normal stabs at the possibilities at hand.
In an AC motor, the motor's "synchronous" speed is a function of the AC frequency.
This is the common method of speed determination with an Induction Motor.
The Rotor (Secondary) will respond to the Frequency of AC Induced into it from the Stator Windings (Primary).
For example, a 2 Pole Motor connected to 60 Hz spins the Rotor at 3600 RPMs. A 4 Pole Motor at 60 Hz spins the Rotor at 1800 RPMs.
I've worked with VFD's quite a bit.
Same thing here, only the VFD alters the Hz delivered to the Motor to obtain a given Rotor Speed - across a Fixed
But how do you vary an AC motor's speed by changing the voltage?
Normally, this will only limit the Volt-Amps reaching the Motor, and when applied to a Motor with a Fixed Load - such as a Surfacer, Lathe, Grinder, etc., the Rotor stalls under heavy Load and no real reduction in Speed is observed.
In the case of a Split Phase Single Phase Motor with a Start switch, the reduction in Speed results in Aux. Circuit being activated, with an end result of smoking the Aux. Winding or tripping the Overload Protection.
Now substitute a PSC (Permanent Split Capacitor) type Motor, or a Shaded Pole Motor, in place of the other types.
Along with this, make the Load type very Linear - meaning the higher the Speed, the higher Power requirement will be needed, with a corresponding higher amount of work done (or larger volume moved by the Motor).
In simpler terms, apply a PSC Motor for Fan Duty.
Attach a simple Axial Fan Blade assemblage to the Shaft of the PSC Motor, so the Motor moves Air.
This Linear type of Load (Environmental Air, or similar compressible Gases pushed by an Axial Blower), will respond to the Speed of the Blower.
For a given Fan size, if the Speed is fast, the CFM throughput is large. If the Speed is slow, the CFMs throughput is low.
The more volume - in CFM - that is required, the more True Power is required at the Rotor - delivered to the Motor from the Power Supply.
If the input restricts the level of True Power reaching the Rotor, then the Rotor's Speed is affected - and therefore spins slower (along with moving less Air).
This is achieved by reducing the input Voltage at the PSC Motor's Stator windings.
The Aux. Circuit is always in connection, so there's no Overload threat for it.
Basically, reducing the input Voltage results in a reduction of Rotor Speed and a corresponding reduction in Air flow.
The Motor draws "X" amount of True Power, and once the Air flow matches the drawn Power level, the Rotor's Speed becomes stable.
This is the same for a Shaded Pole Motor, and most Brush type Communtator Motors.
This would also apply to Cap Start, Resistance Start, and Cap. Start/Run type 1 Phase Motors, only that they will slip into the point where the Start Switch is activated.
The key reason this works is the Load Characteristics vs the Motor design.
Just wish I could explain this stuff better! I can think it, but not say it!
Hopefully another Member will come along and wrap this up in a simple, easy to understand answer.