Three-Phase Induction Motor Torque-Slip Calculator: Plot the Torque-Speed Curve

This calculator plots the torque-speed characteristic of a three-phase induction motor from its rotor parameters. It shows all three regions of operation: braking, motoring and generating. Change the rotor resistance, supply voltage or poles and watch the curve move. It is built to go with the article on the torque equation of a three-phase induction motor. Valid for three-phase induction motors only, not single-phase motors.

Torque-speed calculator

Supply frequency f (Hz)

Number of poles P

Supply voltage (% of rated)

Rotor emf E₂ (V per phase)At standstill, rated voltage

Rotor resistance R₂ (Ω per phase)

Rotor reactance X₂ (Ω per phase)At standstill

External resistance (Ω per phase)Slip-ring motor only

Operating point given as Slip (%)Rotor speed (rpm)

Operating slip (%)

Negative slip = generating, above 100% = braking

Graph x-axis Rotor speed (rpm)Slip

Graph range Braking + motoring + generatingMotoring only

Also show the curve for R₂ = X₂ (maximum starting torque)

Hover or tap the graph to read values.

Machine results

Operating point

Torque is positive along the direction of the rotating field. Negative P₂ means power flows from the rotor back to the supply. Negative mechanical power means the shaft is supplying power.

Braking, motoring and generating regions

The same torque equation covers all three regions. Only the slip changes. Slip is s = (N_s − N) / N_s, where N_s is the speed of the rotating field and N is the rotor speed.

• Motoring (0 < s ≤ 1): the rotor turns in the direction of the field, slower than synchronous speed. Torque drives the load. Normal running is at small slip, to the right of the peak.
• Braking, also called plugging (1 < s ≤ 2): the rotor turns against the field. This happens, for example, if two supply leads are swapped while the motor is running. The field reverses, but the rotor is still turning the old way. Torque now opposes rotation, so the motor slows down. Rotor currents and losses are high.
• Generating (s < 0): a prime mover, such as a wind turbine, drives the rotor above synchronous speed. The rotor conductors now cut the field in the opposite sense, so torque reverses and the machine delivers real power to the supply. It still draws reactive power from the supply for its magnetic field.

Also Read: Induction Generator Working Theory

With stator resistance and reactance neglected, the curve is symmetric: T(−s) = −T(s). The peak generating torque has the same size as T_max. In a real machine, the stator resistance makes the generating peak somewhat larger. Check this against your textbook before quoting it in an exam answer.

Formulas used

All rotor values are per phase. Stator resistance and leakage reactance are neglected, so rotor emf is taken as proportional to the supply voltage. N_s is in revolutions per second inside the formulas.

N_s = 120f / P (rpm),   s = (N_s − N) / N_s,   N = N_s(1 − s)
T = [3 / (2πN_s)] × sE₂²R₂ / (R₂² + (sX₂)²) (valid for any s)
s_m = R₂ / X₂,   T_max = 3E₂² / (4πN_sX₂)
P₂ = T × 2πN_s,   rotor copper loss = sP₂,   gross mechanical power = (1 − s)P₂

The power figures ignore stator losses and friction and windage. The calculator treats R₂ + external resistance as the total rotor resistance.

Things to try

Start with the worked example (4 poles, 50 Hz, E₂ = 120 V, R₂ = 0.1 Ω, X₂ = 1 Ω). It should show T_st ≈ 27.2 N·m, T_max ≈ 137.5 N·m at 1350 rpm, and about 94.8 N·m at 4% slip.

1. Add 0.9 Ω external resistance. Total R₂ becomes 1.0 Ω = X₂. The peak moves to standstill and T_st equals T_max. The peak height does not change.
2. Double R₂ to 0.2 Ω. T_max stays at 137.5 N·m. Only the slip at the peak changes (0.1 to 0.2).
3. Set the supply to 90%. T_max falls to about 111.4 N·m, a drop of 19%, because torque varies as V².
4. Enter a slip of −4%. The rotor runs at 1560 rpm, above synchronous speed. Torque is about −94.8 N·m and P₂ is negative. The machine is generating.
5. Enter a slip of 150%. The rotor speed is −750 rpm. Torque is positive (about 18.3 N·m) and acts to slow the rotor. This is the braking region.
6. Move the operating slip above s_m (but below 100%). The result panel switches to the unstable motoring region.

Common mistakes

• Entering line values for E₂. The formulas need the per-phase value.
• Entering the running rotor reactance. Enter X₂ at standstill; the calculator applies the factor s itself.
• Using the calculator for a cage motor and adding external resistance. A cage rotor has no accessible rotor circuit.
• Thinking negative torque in the generating region means something is wrong. It only means the torque acts against the direction of rotation.
• Treating the output as a nameplate value. It is only as accurate as the rotor values you enter and the assumptions above.

FAQ

Why does the curve rise to a peak after standstill?

At standstill the rotor reactance is large, so the rotor power factor is poor. As the motor speeds up, rotor reactance falls and torque rises until the peak at s_m.

What does external resistance do to the curve?

It moves the peak toward standstill without changing its height. That raises the starting torque.

Which part of the curve is the normal running region?

The nearly straight section between the peak and synchronous speed, at small positive slip.

Why is torque zero at synchronous speed?

At s = 0 the rotor and field move together. No emf is induced in the rotor, so there is no rotor current and no torque.

Can an induction machine generate without a supply?

A grid-connected machine needs the supply to provide its magnetizing current. Stand-alone operation needs extra capacitors and is a separate topic.

Also Read: Starting methods of three phase induction motors and Speed control methods of induction motor.