The two important theories used for explaining the operating principle are:

Double revolving field

Cross-field

The Double Revolving Theory is a simplified explanation used to understand the operation of single-phase induction motors.

The field produced in a

**single-phase induction motor**can be explained by the**double revolving field theory**.This theory is based on the

**“Ferraris Principle”**, asserting that the pulsating field produced can be resolved into two components of half the magnitude and rotating in opposite directions at the same synchronous speed.The alternating flux across the air gap at standstill consists of a combination of two fields of the same strength, both revolving at the same speed.

One field rotates in the

**clockwise direction**, while the other rotates in the**anticlockwise direction**.The strength of each of these fields is equal to

**one-half of the maximum field strength**of the actual alternating field.

Let \({\phi_m}\) be the pulsating field, which has two components, each of magnitude \(\frac{\phi_m}{2}\).

Both components are rotating at the same angular speed \(\omega_s\) in opposite directions.

The resultant of the two fields is \(\phi_m \cos(\theta)\)

Resultant field varies according to the cosine of the angle \(\theta\).

At any instant t, the two fluxes have been rotated through angle \(\theta \left(\theta = \omega t \right)\)

Total value of flux along \(x\)-axis

\[\frac{\phi_m}{2} \cos \omega t+\frac{\phi_m}{2} \cos \omega t=\phi_m \cos \omega t\]

Total value of flux along \(y\)-axis

\[\frac{\phi_m}{2} \sin \omega t-\frac{\phi_m}{2} \sin \omega t=0\]

Resultant flux, \[\phi=\sqrt{\left(\phi_m \cos \omega t\right)^2+(0)^2}=\phi_m \cos \omega t\]

Thus an alternating field can be represented by the two fields each of half the magnitude rotating at same angular (synchronous) speed of \(\omega_s\) radians/sec but in opposite direction

The resultant air-gap mmf whose axis is fixed in space: \[\begin{aligned} \mathcal{F}_{ag} & = F_{\mathrm{max}}\cos(\theta)\cos(\omega t)\\ & = \dfrac{1}{2}F_{\mathrm{max}}\cos(\theta - \omega t) + \dfrac{1}{2}F_{\mathrm{max}}\cos(\theta + \omega t) \\ & = \mathcal{F}^{+}_{ag} + \mathcal{F}^{-}_{ag}\\ & = \text{Forward mmf} + \text{ Backward mmf} \end{aligned}\]

\[\begin{aligned} \dfrac{1}{2}F_{\mathrm{max}}: &~ \text{Max. value of mmf} \\ \omega : & ~ \text{Frequency of the stator current} \\ \theta : & ~ \text{Space displacement angle from stator winding axis}\\ +: &~ \text{the direction in which motor started initially} \end{aligned}\]

Conclusion:

A stationary pulsating magnetic field can generate two equal-magnitude rotating magnetic fields.

These fields move at synchronous speed in opposite directions, matching the frequency of the stationary magnetic field alternations.