It seems like there might be a confusion in your question. A repulsion motor is not a type of single-phase induction motor; rather, it's a type of single-phase commutator motor. Repulsion motors are not considered induction motors, but rather fall under the category of commutator motors.
In a repulsion motor, the stator contains windings similar to those found in single-phase induction motors. However, the rotor is equipped with a commutator and brushes, similar to those used in DC motors. The interaction between the stator's magnetic field and the rotor's commutator and brushes allows the motor to operate.
The operation of a repulsion motor involves initial starting as a repulsion motor and then transitioning into an induction motor mode. The brushes and commutator play a crucial role in this transition. Initially, the brushes are set in a position where the commutator segments short-circuit the rotor windings, causing the rotor to behave like a squirrel-cage rotor of an induction motor. As the motor gains speed, the brushes are gradually shifted to a position where the commutator segments are no longer short-circuited, and the motor operates more like a standard induction motor.
Repulsion motors were used in various applications in the past, particularly where variable speed and high starting torque were required. However, due to their complexity and maintenance requirements (due to the presence of brushes and commutator), as well as advancements in motor technology, their usage has declined over time. Instead, single-phase induction motors or more efficient and reliable motor types are often used in modern applications.
If you're interested in learning about single-phase induction motors, I'd be happy to provide information about those as well!