Torque-slip curve and rotor resistance

Torque-slip characteristic

For a 3-phase induction motor, the torque at slip s is

T = (3 / ω_s) × s E₂² R₂ / (R₂² + (sX₂)²) N·m

where E₂, R₂ and X₂ are the standstill rotor emf, resistance and reactance per phase, and ω_s is the synchronous speed in rad/s.

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

• At s = 0, T = 0. For small s, T ∝ s (stable working region, nearly linear).
• T rises to T_max at s = s_m, then falls as T ∝ 1/s (unstable region) down to the starting torque T_st at s = 1.
• Normal full-load slip is only 2 to 5 %, in the stable linear region.

Condition for maximum torque

Set dT/ds = 0:

(R₂² + s²X₂²) − 2s²X₂² = 0, so R₂² = s²X₂²

s_m = R₂ / X₂, and T_max = 3E₂² / (2ω_sX₂)

Maximum torque occurs when the rotor resistance per phase equals the rotor reactance at that slip.

Effect of rotor resistance

• T_max does not depend on R₂, but the slip at which it occurs (s_m = R₂/X₂) increases in proportion to R₂.
• Higher R₂ shifts the peak towards s = 1 (lower speed), and the starting torque increases.
• If R₂ = X₂, then s_m = 1, and the starting torque equals the maximum torque.
• If R₂ is increased beyond X₂, the starting torque begins to fall.

   R2 small             R2 larger            R2 = X2
   (low Tst)            (higher Tst)         (Tst = Tmax)
 T |  **              T |     *******      T |         ******
   | *  **              |   **       ***     |      ***
   |*     **            |  *                 |    **
   |        ******      | *                  |  **
   |              *     |*                   |**
   +---------------> s  +---------------> s  +---------------> s
   0              1     0              1     0              1

This is why a slip-ring induction motor, with external resistance added to the rotor circuit, gives a high starting torque with a low starting current. The extra resistance is cut out gradually as the motor speeds up, to keep good efficiency and low slip.