Reluctance motor
A single-phase induction motor built with a variable air gap and without DC excitation is called a reluctance motor. It operates based on the principle of magnetic reluctance, which is the opposition to the magnetic flux in a magnetic circuit.

Operating principle
The reluctance motor consists of a stator and a rotor. The stator has a laminated iron core with windings that are connected to an AC power supply. The rotor is made of laminated iron and has salient poles. The air gap between the stator and rotor can be adjusted to control the motor's performance.

When the stator windings are energized with AC current, a rotating magnetic field is produced. The magnetic field induces magnetic poles of opposite polarity in the rotor. As the magnetic field rotates, the rotor poles try to align with the stator poles due to magnetic attraction.

Variable air gap
The variable air gap in a reluctance motor allows for control over the motor's torque and speed characteristics. By adjusting the air gap, the magnetic reluctance of the motor can be changed. The magnetic reluctance is the opposition to the magnetic flux and is determined by the material properties and dimensions of the magnetic circuit.

No DC excitation
Unlike other types of induction motors, the reluctance motor does not require DC excitation. This means that there is no need for external brushes or slip rings to supply DC current to the rotor. The rotor poles are only magnetized by the rotating magnetic field produced by the stator windings.

Advantages and applications
Reluctance motors have several advantages, including simplicity, low cost, and high efficiency. They are commonly used in applications where variable speed control is required, such as washing machines, refrigerators, and air conditioners. The variable air gap allows for smooth speed control and efficient operation.

In conclusion, a single-phase induction motor built with a variable air gap and without DC excitation is called a reluctance motor. It operates based on the principle of magnetic reluctance and can be used in various applications where variable speed control is required.