Torque-slip curve and max torque condition

Torque equation

In a 3-phase induction motor, the rotor current at slip s is I₂ = sE₂ / √(R₂² + (sX₂)²), where E₂ and X₂ are the rotor emf and reactance per phase at standstill. The torque is proportional to Φ I₂ cos φ₂, which gives

T = (3 / ω_s) × s E₂² R₂ / (R₂² + (sX₂)²) newton-metre

where ω_s = 2πN_s/60 is the synchronous angular speed.

Torque-slip characteristic

 T |
   |       Tmax
   |      *******
   |     *       *****
   |    *             ******
   |   *                    ***** Tst
   |  *
   | *
   |*
   +---------+------------------+--> s
   0        s_m                 1
 (N = Ns)                    (N = 0)

1. Low slip (s small): sX₂ << R₂, so T ∝ s. The curve is a straight line; this is the normal working region (stable).
2. Maximum torque at s = s_m.
3. High slip (s large): R₂ << sX₂, so T ∝ 1/s. The curve is a rectangular hyperbola, and the motor is in the unstable region.
4. At s = 1 (standstill) the torque is the starting torque T_st = (3/ω_s) E₂² R₂ / (R₂² + X₂²).
5. At s = 0 (synchronous speed), the torque is zero.

Condition for maximum torque

Put dT/ds = 0. Writing T = K s R₂ / (R₂² + s²X₂²):

d/ds [s / (R₂² + s²X₂²)] = 0 gives (R₂² + s²X₂²) − 2s²X₂² = 0

R₂² = s²X₂²

s_m = R₂ / X₂

That is, the torque is maximum when the rotor resistance equals the rotor reactance at that slip (R₂ = s_mX₂).

Substituting:

T_max = 3E₂² / (2 ω_s X₂)

T_max is independent of the rotor resistance, and proportional to the square of the supply voltage. Also T / T_max = 2 s s_m / (s² + s_m²).