Single-Phase Induction Motors GATE Quick Notes - Electrical Engineering

Fundamentals

Single-Phase Induction Motor - Basics

Key Characteristics

  • Not self-starting - Zero starting torque

  • Single-phase AC supply creates pulsating magnetic field

  • Requires auxiliary starting arrangement

  • Power rating: Typically up to 1 kW

Construction:

  • Stator: Single-phase winding + auxiliary winding

  • Rotor: Squirrel cage type (similar to 3-phase IM)

  • Air gap: Uniform (unlike shaded pole motors)

Double Field Revolving Theory

Double Field Revolving Theory

A pulsating magnetic field can be resolved into two rotating magnetic fields of equal magnitude, rotating in opposite directions at synchronous speed.

Mathematical Expression:

\[\phi(t) = \phi_m \cos(\omega t) = \frac{\phi_m}{2}[\cos(\omega t) + \cos(-\omega t)]\]

Where:

  • \(\frac{\phi_m}{2}\cos(\omega t)\) = Forward rotating field

  • \(\frac{\phi_m}{2}\cos(-\omega t)\) = Backward rotating field

  • Each field has magnitude \(\frac{\phi_m}{2}\)

Slip Relations

Forward Field:

  • Synchronous speed: \(N_s\) (forward direction)

  • Slip: \(s_f = \frac{N_s - N}{N_s} = s\)

Backward Field:

  • Synchronous speed: \(N_s\) (backward direction)

  • Relative speed w.r.t. rotor: \(N_s + N\)

  • Slip: \(s_b = \frac{N_s + N}{N_s} = \frac{N_s + N}{N_s} = 2 - s\)

Important

At standstill: \(s_f = s_b = 1\), hence net torque = 0

Equivalent Circuit

Equivalent Circuit

Circuit Parameters:

  • \(R_1, X_1\) = Stator resistance and reactance

  • \(R_2', X_2'\) = Rotor parameters referred to stator

  • \(X_m\) = Magnetizing reactance

  • Forward branch impedance: \(Z_f = \frac{R_2'}{2s} + j\frac{X_2'}{2}\)

  • Backward branch impedance: \(Z_b = \frac{R_2'}{2(2-s)} + j\frac{X_2'}{2}\)

Torque Analysis

Torque Characteristics

Forward Torque:

\[T_f = \frac{K_1 \phi_m^2 R_2' s}{(R_2')^2 + (sX_2')^2}\]

Backward Torque:

\[T_b = \frac{K_2 \phi_m^2 R_2' (2-s)}{(R_2')^2 + ((2-s)X_2')^2}\]

Net Torque:

\[T_{net} = T_f - T_b\]

Key Points

  • At \(s = 1\): \(T_f = T_b\), hence \(T_{net} = 0\) (not self-starting)

  • At \(s = 0\): \(T_b \approx 0\), \(T_f\) small (low running torque)

  • Maximum torque occurs at intermediate slip values

Starting Methods

Starting Methods

Why Starting Methods Required?

Single-phase induction motors have zero starting torque due to equal and opposite torques from forward and backward rotating fields.

1. Split-Phase Starting:

  • Auxiliary winding with higher resistance

  • Phase difference: \(\phi_m - \phi_a = 20°\) to \(30°\)

  • Starting torque: 1.5 to 2 times full-load torque

2. Capacitor Starting:

  • Capacitor in series with auxiliary winding

  • Phase difference: \(\phi_m - \phi_a = 90°\) (ideal)

  • Starting torque: 3 to 4 times full-load torque

3. Capacitor Start-Capacitor Run:

  • Two capacitors: Starting and running

  • Better performance throughout speed range

Capacitor Starting - Details

Capacitor Value for Starting:

\[C_s = \frac{1}{2\pi f X_C}\]

Where \(X_C\) is chosen to get 90° phase shift.

Starting Torque with Capacitor:

\[T_s = \frac{K V^2}{2} \left[\frac{R_2'}{s} - \frac{R_2'}{2-s}\right]\]

GATE Important

  • Capacitor starting gives maximum starting torque

  • Optimal capacitor value depends on motor parameters

  • Centrifugal switch disconnects starting capacitor

Power Relations

Power Analysis

Input Power:

\[P_{in} = V I \cos \phi\]

Stator Copper Loss:

\[P_{cu1} = I^2 R_1\]

Core Loss:

\[P_{core} = \text{Hysteresis loss + Eddy current loss}\]

Air Gap Power:

\[P_{ag} = P_{in} - P_{cu1} - P_{core}\]

Rotor Copper Loss:

\[P_{cu2} = s \times P_{ag}\]

Mechanical Power Developed:

\[P_{mech} = (1-s) \times P_{ag}\]

Output Power:

\[P_{out} = P_{mech} - P_{friction} - P_{windage}\]

Efficiency and Power Factor

Efficiency and Power Factor

Efficiency:

\[\eta = \frac{P_{out}}{P_{in}} = \frac{P_{mech} - P_{mechanical\ losses}}{P_{in}}\]

Power Factor:

\[\cos \phi = \frac{P_{in}}{V I}\]

Typical Values:

  • Efficiency: 60-80% (lower than 3-phase motors)

  • Power factor: 0.6-0.8 (lagging)

  • Starting current: 5-7 times full-load current

GATE Focus

Single-phase motors have lower efficiency and power factor compared to 3-phase motors of same rating due to backward rotating field losses.

Special Types

Shaded Pole Motor

Construction:

  • Salient pole stator with shading coils

  • Copper ring around part of each pole

  • Squirrel cage rotor

Working Principle:

  • Shading coil delays flux in shaded portion

  • Creates rotating magnetic field effect

  • Low starting torque but self-starting

Characteristics:

  • Starting torque: 50-100% of full-load torque

  • Efficiency: Very low (30-40%)

  • Power factor: Poor (0.3-0.5)

  • Applications: Small fans, toys, timers

GATE Problem Types

GATE Problem Categories

1. Equivalent Circuit Problems:

  • Calculate input current and power factor

  • Determine torque at given slip

  • Find efficiency at rated conditions

2. Starting Method Analysis:

  • Compare starting torques of different methods

  • Calculate capacitor value for starting

  • Determine phase relationships

3. Performance Calculations:

  • Power flow analysis

  • Slip calculations

  • Torque-speed characteristics

4. Conceptual Questions:

  • Double field revolving theory

  • Reasons for not being self-starting

  • Comparison with 3-phase motors

Key Formulas

Important Formulas for GATE

Must Remember

  • Forward slip: \(s_f = s\)

  • Backward slip: \(s_b = 2 - s\)

  • Net torque: \(T = T_f - T_b\)

  • Rotor copper loss: \(P_{cu2} = s \times P_{ag}\)

  • Mechanical power: \(P_{mech} = (1-s) \times P_{ag}\)

  • Synchronous speed: \(N_s = \frac{120f}{P}\)

  • Slip: \(s = \frac{N_s - N}{N_s}\)

Efficiency:

\[\eta = \frac{P_{out}}{P_{in}} = \frac{(1-s)P_{ag} - P_{mech\ losses}}{P_{in}}\]

Summary

Quick Revision Points

  1. Single-phase motors are NOT self-starting

  2. Double field revolving theory explains operation

  3. Starting methods create phase difference between windings

  4. Capacitor starting gives highest starting torque

  5. Efficiency and power factor are lower than 3-phase motors

  6. Equivalent circuit has forward and backward components

  7. Slip relations: \(s_f = s\), \(s_b = 2-s\)

  8. Applications: Fans, washing machines, small pumps

GATE Tip

Focus on torque analysis, starting methods, and equivalent circuit problems. Understand the concept of forward and backward rotating fields.