Such motors are used whenever a single phase power supply is more practical than a three phase power supply such as in domestic appliances. It is used to get high initial torque in initial start of the induction motor in single phase.

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Three phase motors are simpler in design and more energy efficient than single phase motors but require a three phase power source to operate.

The purpose of the capacitor used in single phase motor is to. Motor start and run capacitors are used in single phase ac induction motors. Another method is the capacitor start induction run motors. The run capacitor is wired in series with the start winding of the motor and stays in the circuit all of the time.
We know about the activity of a capacitor in a pure ac. General purpose ac electrical motors vary in design and power source. The electrolytic start capacitor helps the motor achieve the most beneficial phase angles between start and main windings for the most locked rotor torque per locked rotor ampere.
One often used method is the split phase motors. Capacitor is used for starting single phase induction motors. It produces the torque that makes a90 degrees out of phase between the starting auxiliary winding and the running main winding.
They are not as efficient as three phase ac induction motors however. Hope to give you an idea. Without the capacitor the motor would run up in either direction after an initial spin of the shaft.
The single phase induction motor can be made to be self starting in numerous ways. Also called induction motors the most common ac electrical motors are three phase motors and single phase motors. They are designed to dissipate heat associated with continuous operation of the motor.
The whole purpose of the run capacitor is to bring the start winding back in phase with the run winding. The capacitor is used to make a single phase motor rotate in one direction only by supplying a phase shifted voltage to a starting winding on the motor.

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