Induction Motors Torque Equation; Torque equation

Torque Equation of Induction Motor: Explained with Factors Affecting Speed-Torque Characteristics

Induction motors are widely used in industries due to their robustness, low cost, and simple construction. One of the most important aspects of analyzing an induction motor is its torque equation, which helps in understanding how the motor develops torque under different operating conditions.

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Torque Equation of an Induction Motor

The general torque equation of an induction motor is:

$$T = \frac{3}{\omega_s} \cdot \frac{V^2 \cdot \frac{R_2'}{s}}{(R_1 + \frac{R_2'}{s})^2 + (X_1 + X_2')^2}$$

Where:

• T = Torque developed

• V = Applied stator voltage per phase

• R1 = Stator resistance

• R2’ = Rotor resistance (referred to stator side)

• X1 = Stator reactance

• X2’ = Rotor reactance (referred to stator side)

• s = Slip

• ωs = Synchronous angular speed

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Key Relation: Torque is Proportional to Voltage Squared

From the above equation, we see:

$$T \propto V^2$$

This means the torque developed is directly proportional to the square of the applied voltage. Even a small reduction in supply voltage leads to a significant drop in torque, which is crucial for applications requiring high starting torque.

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Factors Affecting Speed-Torque Characteristics of Induction Motor

The speed-torque characteristics of an induction motor are influenced by several parameters:

(a) Applied Voltage

• Torque is proportional to $V^2$.

• Reduction in voltage decreases both starting torque and running torque.

• Motors may fail to start under reduced voltage conditions in heavy-load applications.

(b) Supply Frequency

• Increase in supply frequency reduces starting torque.

• Higher frequency increases synchronous speed ($N_s = 120f/P$), shifting the torque-speed curve.

• Motors designed for fixed frequency (50 Hz/60 Hz) must not be operated at significantly higher frequencies.

(c) Rotor Resistance ($R_2’$)

• The maximum torque (Tmax) is independent of rotor resistance.

• Increasing rotor resistance:

  • Improves starting torque.

  • Shifts the slip at which Tmax occurs to a higher value (lower motor speed).

• Useful in wound rotor induction motors for controlled starting.

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Practical Implications

• Industrial Applications: Motors with high starting torque (e.g., cranes, hoists) may use external rotor resistance for better performance.

• Energy Efficiency: Voltage drops in long transmission lines can drastically affect induction motor torque.

• Variable Frequency Drives (VFDs): Allow frequency and voltage control to optimize speed-torque characteristics.

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✅ Summary:

• Torque in induction motors is directly proportional to the square of voltage.

• Supply voltage, frequency, and rotor resistance significantly affect the speed-torque curve.

• Understanding these factors helps in motor selection, efficiency improvement, and troubleshooting in industrial applications.

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