I read a manual of how to connect a speed controller to a motor. I see there is a capacitor as shown below. My question is what the function of the capacitor is? Thanks
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I think the speed controller is built to control a three-phase motor from a single phase power line. In short a three phase motor creates a rotating magnetic field in its stator (the part that does not rotate, the housing), and the rotor is dragged along with this rotating field (at a slightly lower rate called 'slip').
To change the direction for a three phase motor, it is sufficient to exchange two of the phases (hence the CW [clockwise] and CCW [counter clockwise] terminals).
The real trick is to create three phases that are about 120° apart and that is where the capacitor comes in. By putting a capacitor in series with one of the windings, the phase angle will be sufficiently shifted to create a rotating magnetic field in the stator and as such you don't need a full three phase controller. Of course the size of the capacitor depends on the current drawn by the motor. Advantage: price of the controller is lower than a full three phase controller; disadvantage: output power of the motor is lower than its three-phase maximum.
The image and more background information is found here.