Supply a three-phase motor with AC mono-phase current.

Asynchronous three-phase motor needs three sinusoids shifted by 120 °.
When you do not require the 100% of the power it is possible to use a phase-shifting capacitor to supply the motor.

The power and the efficiency will not be high because the phase-shifting obtained is not optimized but is enough for a lot of applications like fans, pumps, drills, grinders, small lathes and any other applications which do not require high current at starting point.

Calculation of the phase-shifting capacitor

Supply voltage

V

Supply frequency

Hz

Rating power of the motor

W kW CV

Result.

Capacitance of the capacitor

µF

Connections of the capacitor.
Most cases concern the delta connection, useful for a three-phase motor 220-380 V supplied with 220 V mono-phase.
The next picture shows the connections for three-phase asynchronous motors supplied in mono-phase with wye and delta connections and with clockwise and counterclockwise rotation.

What capacitor does.
The capacitor delays the electric signal reproducing the phase shift required to obtain a rotating magnetic field inside the motor.
The phase shift will not be optimal because its entity depends on capacitance, current and load applied. It will be always a compromise.

Motor performances.
The results reported here are compromises, a motor which will supplied in this way can not give the rated power but only a 60-70% of that.
In way to obtain an higher power it is possible to increase the capacitance of the capacitor but be careful because in that case the motor must not work without loads or with small ones, you might burn out it.

Starting.
The worst phase using this kind of power source is when you start the motor. The available torque is the 30-40% of the obtainable if normally supplied.
To fix this problem you could connect a second capacitor of the same capacitance in parallel and commuting it with an amperometric relay.
The second capacitor must not be connected after the starting but it has to work only to give an higher torque until current exceeds the established value.

Warnings.
Please take note that in this particular situation the capacitor is subjected to high currents and continuous polarity reversals, if it is not adapt to this kind of job it may explode.
Use only not polarized capacitors with a maximum operating voltage of 15-20% higher than the supply voltage and realized for AC.