Torque-slip characteristic

Reference notes

Use Next → on the narrator above to step through the construction of the torque-slip curve and the role of the rotor resistance R₂.

The torque-slip equation

From the equivalent circuit, the electromagnetic torque developed by an induction motor (per phase, then × 3 for 3-phase) is:

where ω_s = 2π·n_s/60 is the synchronous mechanical angular speed in rad/s. E₂, R₂, X₂ are the standstill rotor EMF, resistance, and leakage reactance.

The curve's three regimes

1. Low-slip region (s << R₂/X₂): the (sX₂)² term in the denominator is negligible. T simplifies to T ≈ K·s — torque is linear in slip. This is the stable steady-state operating region.
2. Breakdown (maximum) torque: occurs at s = s_m = R₂/X₂. At this slip the torque peaks.
3. High-slip region (s >> R₂/X₂): now R₂² is negligible and T ≈ K'/s — torque falls as slip increases further.

The two famous identities

The second fact is surprising at first but powerful: changing the rotor resistance shifts WHERE the peak occurs but not how high it is. Add more rotor resistance → peak slides toward s = 1 (the standstill side); the peak value stays the same.

Starting torque and the rotor-resistance starting trick

Plug s = 1 into the torque equation:

For typical squirrel-cage motors R₂ << X₂, so T_start << T_max. To get high starting torque, you want the peak of the T-s curve to sit near s = 1 — which means making R₂ ≈ X₂. Two practical ways to do this:

• Slip-ring rotor with external resistance: insert resistance via the slip rings, get peak torque AT starting (s_m = 1). As the motor speeds up, gradually cut out the external resistance — you ride along the family of T-s curves, keeping near maximum torque the whole way. Used in cranes, hoists, and mills.
• Double-cage / deep-bar rotors: a squirrel-cage trick that effectively gives a higher rotor resistance at start (when rotor frequency is high) and a lower resistance at running speed (rotor frequency low). Same benefit without the slip rings.

Why does R₂ not change T_max?

Intuitively: at s_m, the rotor impedance is purely real (|R₂| = |jsX₂|), so power transfer is "matched". The peak torque depends on E₂ and X₂ but not R₂. Algebraically: substitute s = R₂/X₂ into the torque equation; the R₂'s cancel.

Stalling (s = 1) is dangerous

When a motor is stalled — locked rotor under load — slip is 1 and the rotor isn't moving. From the 1 : s : (1−s) power-flow ratio of the equivalent-circuit lesson, all of the air-gap power P_g becomes rotor copper loss (s·P_g = P_g), and mechanical power is zero. The rotor bars cook in seconds. This is why motor protection trips out on prolonged stalls, and why the torque-slip curve's high-slip region is a brief transient regime, not a sustained operating point.

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