Single phase motor

Most home and business appliances operate on single phase AC power. For this reason, single-phase AC motors are in widespread use. A single-phase induction motor is larger in size, for the same horsepower, than a three-phase motor. When running, the torque produced by a single phase motor is pulsating and irregular, contributing to a much lower power factor and efficiency than that of a polyphase motor. Single-phase AC motors are generally available in the fractional to 10-hp range and all use a solid squirrel-cage rotor.

The single-phase induction motor operates on the principle of induction, just as does a three-phase motor. Unlike three phase motors, they are not self-starting. Whereas a three-phase induction motor sets up a rotating field that can start the motor, a single-phase motor needs an auxiliary means of starting. Once a single-phase induction motor is running, it develops a rotating magnetic field.
An induction motor rotor can be either wound rotor or a squirrel cage rotor. The majority of commercial and industrial applications usually involve the use of a three-phase squirrel-cage induction motor. A typical squirrel-cage induction motor is shown. The rotor is constructed using a number of single bars short-circuited by end rings and arranged in a hamster-wheel or squirrel-cage configuration. When voltage is applied to the stator winding, a rotating magnetic field is established. This rotating magnetic field causes a voltage to be induced in the rotor, which, because the rotor bars are essentially single-turn coils, causes currents to flow in the rotor bars. These rotor currents establish their own magnetic field, which interacts with the stator magnetic field to produce a torque. The resultant production of torque spins the rotor in the same direction as the rotation of the magnetic field produced by the stator. In modern induction motors, the most common type of rotor has cast-aluminum conductors and short-circuiting end rings.
In theory, it is possible to control the rotor flux vector in any n-phase machine. People play with five and six phase machines all the time but the single phase machine seems to be the singularity because of the bidirectional air gap flux. I think mathematically we can work out some way of controlling that flux but as others have said here, why bother?

One more assumption that I see is that the single phase machine is incapable of higher powers. The single phase machine can be designed to very high powers, but we don't do it because the machines would be much larger than a 3 phase machine and take up more material. In fact there are relatively large single phase machines in the integral power range, designed mainly for rural use, where three phase power is not available.

Capacitor start single phase motors have a stationary switch & centrifugal switch that switch the motor out of the start winding to the run winding. The centrifugal switch requires the rotor to turn at a certain speed (apprx. 3/4 speed) to engage the contacts on the stationary switch. Its possible the motor would never get out of the start winding and quickly fry.
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