# Double Revolving Field Theory of Single-Phase Induction Motors

## Operating Principle of Single-phase Induction Motor

A single-phase induction motor consists of a squirrel cage rotor and a stator carrying a single-phase winding. When a single-phase AC supply is fed to the stator winding, a pulsating magnetic field (not the rotating) is produced. Under these conditions, the rotor does not rotate due to inertia. Hence, a single-phase induction motor is inherently not self-starting, but requires some auxiliary starting means.

If the stator winding of a single-phase induction motor is excited and the rotor is rotated by an auxiliary means and the starting device is then removed, the motor continues to rotate in the same direction in which it is started.

The double revolving field theory is suggested to analyse the performance of a single-phase induction motor. It explains why a torque is produced in the rotor once it is turning.

## Double Revolving Field Theory

According to the double revolving field theory of single-phase induction motor, a stationary pulsating magnetic field can be resolved into two rotating magnetic fields. Both the magnetic fields are of equal magnitude but rotating in opposite directions. The motor responds to each magnetic field separately and the net torque produced in the motor is equal to the sum of the torques due to each of the two magnetic fields.

Mathematically, an alternating magnetic field whose field axis is fixed in the space is given by,

$$\mathrm{𝐵(𝜃) = 𝐵_{max}\:sin\:\omega 𝑡\:cos\:𝜃 … (1)}$$

Where, Bmax is the maximum value of magnetic flux density which is sinusoidally distributed in the air-gap of the motor.

This magnetic field is produced by a properly distributed stator winding carrying a current of frequency ω and θ is the space displacement angle measured form the axis of the stator winding.

$$\mathrm{(∵\:sin\:𝑋\:sin\:𝑌 =\frac{1}{2}sin(𝑋 − 𝑌) +\frac{1}{2}sin(𝑋 + 𝑌)}$$

$$\mathrm{∴\:𝐵(𝜃) =\frac{1}{2}𝐵_{max}\: sin(\omega 𝑡 − 𝜃) +\frac{1}{2}𝐵_{max}\:sin(\omega 𝑡 + 𝜃) … (2)}$$

The first term of eqn. (2) represents a revolving field which is moving in the positive θ direction and has a maximum value equal to $\frac{1}{2}$ while the second term represents a revolving magnetic field which is moving in the negative θ direction and has a maximum value also equal to $\frac{1}{2}\:B_{max}$ .

The magnetic field rotating in positive θ direction is known as the forward rotating field, while the magnetic field rotating in the negative θ direction is known as the backward rotating field.

The positive direction is the direction in which the single-phase induction motor is started initially. Both the magnetic fields rotate at synchronous speed in opposite direction. Hence,

$$\mathrm{Forward\:rotating\:field =\frac{1}{2}𝐵_{max}\:sin(\omega 𝑡 − 𝜃)}$$

And

$$\mathrm{Backward\:rotating\:field =\frac{1}{2}𝐵_{max}\:sin(\omega 𝑡 + 𝜃)}$$

Therefore, it can be concluded that a stationary magnetic field can be resolved into two rotating magnetic fields, both of equal in magnitude and rotating at synchronous speed in opposite directions at the same frequency as the stationary magnetic field pulsates. Such a theory which is based on a resolution of a stationary pulsating magnetic field into two opposite rotating magnetic fields is known as double-revolving field theory of single-phase induction motors.

When the rotor is stationary, the two torques produced are equal and opposite. Hence, at standstill, the net torque is zero. However, if the rotor is given an initial rotation by some auxiliary means in either directions, then the torque due to the rotating magnetic field acting in the either direction of initial rotation will be more than the torque due to the other rotating magnetic field. Thus, the motor develops a net torque in the same direction as the initial rotation. Therefore, the motor will keep running in the same direction of the initial rotation.