--- title: "Torque-slip curve; linear operating region" subject: "Electrical Machine" chapter: 5 key: "semester_4/electrical_machine:torque-slip-linear-portion" source: https://www.bctengineer.com/q/semester_4/electrical_machine/torque-slip-linear-portion site: BCT Engineering --- # Torque-slip curve; linear operating region Electrical Machine · Chapter 5: Three Phase Induction Machines - Asked in: 2071 Bhadra, 8 marks - Video: [IM 12 Torque Slip Characteristics](https://www.youtube.com/watch?v=qoeOtMZzhkc) — Lectures in Electrical Engineering - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machine/torque-slip-linear-portion - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machine/chapter-5 ## Question Explain torque slip characteristics of 3-phase induction motor. Why the induction motor operates only in linear portion of torque-slip characteristics? ## Short answer The motor operates in the low-slip linear region (0 to s_m) because that region is stable: dT/ds > 0, so an increase in load increases slip and torque. Efficiency is high due to small rotor copper loss, speed is nearly constant, and current and power factor remain acceptable. Beyond s_m, torque falls and the motor stalls. ## Answer The torque-slip characteristic of a three-phase induction motor is the curve of developed torque against slip (or rotor speed) at constant supply voltage and frequency. #### Torque equation T = (3/ω_s) · s E₂² R₂ / (R₂² + s²X₂²) where E₂ and X₂ are the rotor emf and reactance per phase at standstill, R₂ is the rotor resistance and ω_s is the synchronous speed in rad/s. #### Shape of the curve ```text T | Tmax (s = R2/X2) | ****** | * **** | * ***** | * ******* | ** Tst | * |* +-------+--------------------+--> s 0 s_m 1 (N = Ns) (N = 0) stable unstable ``` | Region | Slip | Torque behaviour | |---|---|---| | Low slip | 0 to about 5 % | s²X₂² << R₂², so T ≈ k·s/R₂, a straight line through the origin | | Medium slip | about 5 % to s_m | torque rises more slowly and reaches T_max | | High slip | s_m to 1 | s²X₂² >> R₂², T ≈ k·R₂/(sX₂²), torque falls as slip rises | - The starting torque T_st is found at s = 1. - A typical squirrel-cage motor has T_max about 2 to 3 times full-load torque and T_st about 1 to 2 times full-load torque. #### Why the motor operates only in the linear (low-slip) portion 1. **Stability.** In the region 0 < s < s_m, dT/ds is positive. If the load increases, the speed drops, slip rises, torque rises and a new balance is found. In the region s > s_m, dT/ds is negative. A rise in load slows the motor, the torque drops further, and the motor stalls. So steady running is possible only on the stable part. 2. **Efficiency.** The rotor efficiency is (1 − s). At low slip the rotor copper loss s·P_g is small, so the efficiency is high (85 to 95 %). In the high-slip region the rotor loss is very large. 3. **Almost constant speed.** From no load to full load, the slip changes only by about 0.5 to 5 %, so the speed is almost constant, which suits most drives. 4. **Power factor and current.** Near rated slip, the rotor reactance sX₂ is small, so the rotor power factor is near unity and the current is moderate. At high slip the current is very large with a poor power factor, which causes overheating. 5. **Loading.** The full-load point lies on this portion, and the margin up to T_max (the overload capacity) protects the motor against temporary overloads. Hence the motor is always designed and operated in the low-slip, linear, stable region. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is the region 0 < s < s_m stable? A. dT/ds is positive so extra load slows the motor and torque rises to balance B. Slip is constant C. Torque is zero D. dT/ds is negative Answer: A. Torque rises with slip, giving a new balance. (Key point: Stability) 2. What happens in s > s_m if load increases? A. Torque rises B. Torque falls further and the motor stalls C. Nothing D. Speed increases Answer: B. dT/ds is negative there. (Key point: Unstable region) 3. Why is efficiency high at low slip? A. Flux is large B. Friction is zero C. Efficiency is about (1 - s) and rotor copper loss s·P_g is small D. Stator loss is zero Answer: C. Rotor copper loss is a fraction s of air-gap power. (Key point: Efficiency) 4. How much does slip change from no load to full load? A. None B. About 100 % C. About 50 % D. Only about 0.5 to 5 % Answer: D. Speed is almost constant. 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