Q17Electrical Machines and Drives

Single-Phase Induction Motor - Double-Field Revolving Theory and Equivalent Circuit

A single-phase induction motor's stator winding, carrying a pulsating (single-phase) alternating current, produces a purely pulsating (stationary, alternating-in-magnitude) magnetic field along the winding's axis rather than a rotating field. The double-field revolving theory resolves this single pulsating field, mathematically, into two counter-rotating fields of equal magnitude (each half the peak amplitude of the original pulsating field) rotating at synchronous speed in opposite directions - one forward (in the direction the motor is running or will run) and one backward.

Each of these two revolving fields interacts with the rotor exactly as the single rotating field of a balanced polyphase motor would, inducing its own rotor currents and developing its own torque - the forward field develops a forward (positive) torque component and the backward field develops an equal-magnitude but oppositely-directed (negative) torque component at standstill, so the two torques exactly cancel at zero speed, which is precisely why a single-phase induction motor develops no net starting torque and cannot self-start from rest without auxiliary starting means (an auxiliary winding, shaded pole, or other starting technique).

Once the rotor is brought up to some speed N (in either direction) by an external starting mechanism, the slip of the rotor differs with respect to the two fields: with respect to the forward field, slip is sf = (Ns-N)/Ns = s (the ordinary slip), but with respect to the backward field (which is effectively rotating at -Ns relative to the same stator reference), the rotor's slip is sb = (-Ns-N)/(-Ns) = 2-s. As the motor speeds up (s decreases from 1 toward 0), the forward-field torque increases (approaching its normal induction-motor torque-speed behavior) while the backward-field slip (2-s) approaches 2, at which the backward-field rotor impedance becomes very large and its developed torque becomes small - the net result is a non-zero, forward-direction resultant torque once the rotor is set into motion, which is what sustains single-phase induction motor running operation despite its zero starting torque.

The resulting per-phase equivalent circuit therefore consists of the stator winding's own resistance R1 and leakage reactance X1 in series with two parallel-combination rotor branches connected in series with each other: the forward-field branch, representing half the rotor impedance referred with slip s (i.e., 0.5(R2/s + jX2) combined appropriately with the magnetizing branch for that field), and the backward-field branch, representing half the rotor impedance referred with slip (2-s) (i.e., 0.5(R2/(2-s) + jX2)), each with its own associated magnetizing reactance representing the respective forward or backward rotating flux component.

The net electromagnetic torque developed is the difference between the forward-field torque and backward-field torque (each individually computable from the power dissipated in the corresponding branch's rotor resistance term, divided by synchronous speed, exactly as in the three-phase torque analysis), and this equivalent circuit, though roughly twice as complex as the balanced three-phase motor's equivalent circuit (owing to the need to separately represent both forward and backward field branches), provides the complete basis for computing single-phase induction motor performance - torque, current, power factor, and efficiency - at any given slip, and clearly explains both the motor's zero starting torque and its satisfactory, though somewhat inferior compared to an equivalent three-phase motor, running performance and efficiency, the latter inferiority arising directly from the continued small backward-field torque and associated loss that persists throughout the normal running range.