Electrical Machines · Single-Phase Motors

Torque Produced in Single-Phase Induction Motors

Part 6 · Single-Phase and Special Machines — two opposing torque curves add to zero at standstill, and to a useful torque once the rotor moves.

Prof. Mithun Mondal Engineering Devotion Digital Textbook
i Learning Objectives

By the end of this chapter you should be able to:

  • Construct the torque–speed curve of a single-phase motor from its forward and backward components.

  • Derive the slips \(s_f = s\) and \(s_b = 2 - s\).

  • Explain why the forward field dominates when running, and the origin of double-frequency torque pulsations.

  • Explain how an auxiliary winding makes the motor self-starting.

Demonstrative Video

Section 1

Forward and Backward Torques

Each of the two rotating MMF waves produces induction-motor action on the rotor, and their torques oppose each other. The resultant torque–speed characteristic is the sum of the two component curves, with slip running from 0 to 2.

Forward, backward and resultant torque-speed curves
Forward torque, backward torque and their resultant.

  • At standstill, \(s = 1\) and \(2 - s = 1\). \(T_f\) and \(T_b\) are numerically equal and opposite, so \(T_{st} = 0\).

  • If the motor is started by auxiliary means, the net torque acts in the direction in which it was started.

Section 2

Slips with Respect to Each Field

Let \(n_s\) be the synchronous speed and \(n\) the rotor speed in the direction in which the motor was started (the forward direction).

\[s_f = s = \frac{n_s - n}{n_s} = 1 - \frac{n}{n_s}\]

The backward field rotates the other way, at \(-n_s\), so relative to it the rotor moves at \(n_s + n\):

\[s_b = \frac{n_s - (-n)}{n_s} = 1 + \frac{n}{n_s}\]
2
Slip Relation
\(s_f + s_b = 2 \quad\Rightarrow\quad s_b = 2 - s\)
Section 3

Why the Forward Field Dominates

The simple picture assumes the two flux waves stay equal while the rotor moves. In reality they do not.

  • As the rotor speeds up, the backward slip approaches 2. The backward rotor currents rise and, by their demagnetising effect, strongly reduce the backward flux.

  • The forward slip falls, so the forward rotor currents fall and the forward flux increases.

  • The total flux is roughly constant, so the stator counter-EMF stays nearly constant and the stator leakage drop has little effect.

Forward and backward flux waves when running
When running, the forward flux wave is much larger than the backward one.
Like a polyphase motor. In the normal running region the forward field is much larger than the backward field. The air-gap flux then resembles the constant-amplitude rotating field of a polyphase motor, and the torque–speed curve looks much like a polyphase motor's.

The two oppositely rotating waves still interact to give torque pulsations at twice the stator frequency. These produce no average torque but cause vibration and noise. Torque–speed curves show the time-averaged torque.

1 Worked Example — Slips at running speed

Problem. A 6-pole, 50 Hz single-phase motor runs at 960 rpm. Find \(s_f\), \(s_b\) and the frequency of the torque pulsation.

\[n_s = 1000\ \text{rpm}, \qquad s_f = 0.04, \qquad s_b = 1.96\]

The torque pulsates at 2 × 50 = 100 Hz.

Section 4

Making It Self-Starting

The motor is made self-starting by turning it temporarily into a two-phase motor. An additional starting winding is placed on the stator alongside the main winding.

Main and starting windings
Main and starting windings in parallel across the supply.

  • The two windings are spaced 90° electrical apart and connected in parallel to the single-phase supply.

  • Their impedances are made different, so their currents are phase-shifted. The ideal shift is 90°.

  • The two currents create a revolving flux, and the motor starts.

Section 5

Summary

  • The resultant torque equals the forward torque minus the backward torque. It is zero at standstill.

  • \(s_b = 2 - s\). When running, the backward flux is suppressed by large rotor currents.

  • Double-frequency torque pulsations are inherent in single-phase motors.

  • A space-displaced starting winding with a phase-shifted current provides starting torque.

Section 6

Chapter Review

Multiple-Choice Questions

  1. MCQ 1. The torque pulsation in a 50 Hz single-phase motor is at
    (a) 25 Hz (b) 50 Hz (c) 100 Hz (d) slip frequency

    Show answer
    (c) 100 Hz.

  2. MCQ 2. At synchronous speed in the forward direction, the backward slip is
    (a) 0 (b) 1 (c) 2 (d) −1

    Show answer
    (c) 2.
Conceptual Questions

  1. Why is the backward flux small when the motor runs near synchronous speed?

  2. Why are single-phase motors noisier than three-phase motors?