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These are actually single-phase induction motors. There is no electrical connection to the armature itself. To make the armature turn within the windings an electric current has to be induced in the it, and this is done by making the magnetic field move in relation to the armature. (The same principle is involved as moving a magnet over a wire to get an electric current.) With an electric current flowing in the armature there is a magnetic field set up around it which pushes against the magnetic field in the windings to make the armature turn.

OK, but how to make the magnetic field move? That's the purpose of the capacitor.

Capacitors have the property of advancing alternating current by (if I remember correctly) 90 degrees. In the simplest case, there are two windings in one of these motors. Unaltered electric current is sent through one of the windings. Some electric current is sent through the other winding and a capacitor. Now, because the two windings are being energised by currents that are out of phase by 90 degrees it is as if the magnetic field is moving. The armature responds by moving.

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