Max torque independent of rotor resistance

A three-phase induction motor develops torque because of the interaction of the air-gap flux with the rotor current. Its torque changes with slip, and the plot of torque against slip (or speed) is the torque-slip characteristic.

Torque-slip characteristics

 T
 |        Tmax
 |         ***
 |       *     *
 |     *         *
 |   *             *
 | *                  *
 |*Tst                    *  T_fl
 +--|--------|-----------|---|--> s
   s=1      s_m        0.05  0
 (standstill)        (near sync)

• s = 0 (synchronous speed): rotor emf and current are zero, so T = 0.
• Small slip (0 to about 5 %): the rotor reactance sX₂ is negligible, so I₂ ∝ s and T ∝ s. The curve is nearly a straight line. This is the stable, normal running region.
• Slip s_m (about 10 to 25 %): torque reaches its maximum value (breakdown or pull-out torque).
• Beyond s_m up to s = 1: sX₂ dominates, T ∝ 1/s, and the torque falls. This is the unstable region. At s = 1 the torque is the starting torque.

Torque expression

The rotor emf per phase at slip s is E₂ᵣ = sE₂ and the reactance is sX₂, where E₂ and X₂ are standstill values.

I₂ = sE₂ / √(R₂² + s²X₂²)

cos φ₂ = R₂ / √(R₂² + s²X₂²)

Torque T = P₂ / ω_s, where the rotor air-gap power is P₂ = 3 E₂ I₂ cos φ₂ (= 3I₂²R₂/s). This gives

T = (3 / ω_s) · s E₂² R₂ / (R₂² + s²X₂²), with ω_s = 2πN_s/60.

Condition for maximum torque

Differentiate and put dT/ds = 0:

 d/ds [ s R2 / (R2² + s²X2²) ] = 0
 (R2² + s²X2²) - s(2 s X2²) = 0
 R2² - s²X2² = 0
 s_m = R2 / X2

Put s = s_m = R₂/X₂ in the torque equation:

 Tmax = (3/ws) · E2² · (R2/X2) · R2 / (R2² + R2²)
      = (3/ws) · E2² · (R2²/X2) / (2 R2²)
 Tmax = 3 E2² / (2 ws X2)

Result: T_max depends only on the supply voltage (through E₂, since E₂ ∝ V), the standstill reactance X₂ and the synchronous speed. R₂ does not appear, so the maximum torque is independent of rotor resistance. R₂ only decides the slip s_m = R₂/X₂ at which T_max occurs. Raising R₂ (external resistance in a slip-ring motor) moves the peak toward s = 1 without changing its height, and when R₂ = X₂ the peak occurs at starting (s_m = 1).