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Torque-slip curve; linear operating region

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.

Explain torque slip characteristics of 3-phase induction motor. Why the induction motor operates only in linear portion of torque-slip characteristics?

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

 T
 |      Tmax (s = R2/X2)
 |     ******
 |    *      ****
 |   *           *****
 |  *                 *******
 |                           ** Tst
 | *
 |*
 +-------+--------------------+--> s
 0      s_m                   1
(N = Ns)                   (N = 0)
  stable        unstable
RegionSlipTorque behaviour
Low slip0 to about 5 %s²X₂² << R₂², so T ≈ k·s/R₂, a straight line through the origin
Medium slipabout 5 % to s_mtorque rises more slowly and reaches T_max
High slips_m to 1s²X₂² >> 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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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