Slip and the torque–speed curve of a three-phase induction motor

Synchronous speed

The stator windings of a three-phase motor create a magnetic field that rotates at the synchronous speed

N_s = 120 × f / p

where f is the supply frequency in hertz and p is the number of poles. The result is in revolutions per minute.

• 2 poles at 50 Hz: 3000 rpm
• 4 poles at 50 Hz: 1500 rpm
• 4 poles at 60 Hz: 1800 rpm
• 6 poles at 50 Hz: 1000 rpm

The rotor of an induction motor never reaches this speed while producing torque, which is why the motor is also called asynchronous.

What slip is and why it exists

Slip, s = (N_s − N) / N_s, measures how far the rotor lags the field. It is often quoted as a percentage.

The lag is essential. Torque comes from currents induced in the rotor bars, and currents are induced only when the rotor conductors move relative to the field. If the rotor ran exactly at N_s, there would be no relative motion, no induced current and no torque. As the load increases, the rotor slows slightly, the relative motion grows, more current is induced and more torque is produced.

At standstill s = 1 (100%), and at synchronous speed s = 0. The frequency of the rotor currents is s × f, which is why it is small at normal operating speeds, for example 2 Hz on a 50 Hz motor with 4% slip.

Worked example

Take a 4 kW, 400 V, four-pole motor on a 50 Hz supply with a nameplate speed of 1440 rpm. This is the default motor in the induction motor simulator.

1. Synchronous speed: N_s = 120 × 50 / 4 = 1500 rpm. 2. Rated slip: s = (1500 − 1440) / 1500 = 0.04, or 4%. 3. Shaft torque at rated output: T = P / ω = 4000 / (2π × 1440 / 60) ≈ 26.5 N·m.

If the supply frequency changes, the synchronous speed changes in proportion. At 60 Hz the same four-pole motor has N_s = 1800 rpm, and the same 4% slip gives 1728 rpm.

The torque–speed curve

The torque–speed curve of an induction motor has a characteristic shape. At standstill, the motor produces its starting torque. As speed rises, torque increases to a maximum, called the pull-out or breakdown torque, then falls to zero at synchronous speed.

Near synchronous speed, torque is roughly proportional to slip, so the curve is close to a straight line. This is the stable operating region, and a loaded motor runs in it, a few percent below N_s. Beyond the pull-out point, torque falls as slip grows, so a motor loaded past that point slows down further and stalls.

Torque also varies roughly with the square of the supply voltage. A drop to 80% voltage cuts the torque available at any speed to about 64%, which matters at start-up and under heavy load.

Curvo Studio's induction motor simulator does not plot this curve. It reports slip, rotor speed and stator current, and works through the stable region and the stall condition described below.

How the simulator models load and slip

The simulator uses a deliberately small steady-state lesson model, not a full equivalent circuit. With the load as a percentage of rated load and the voltage ratio V_r = line voltage / rated voltage:

• Rated slip is taken from the nameplate speed, limited to 0.5%–15%.
• Slip = no-load slip + (rated slip − no-load slip) × load / V_r², where the no-load slip is 20% of rated slip, capped at 1%. The V_r² divisor has a floor of 0.25. Slip therefore rises linearly from the no-load value to the nameplate slip at 100% load and rated voltage, and rises faster when the voltage is reduced.
• The motor stalls when the load exceeds 1.1 × V_r², and then slip is 100%, speed is zero and the current jumps to 4.2 times rated current times V_r.
• Stator current combines a magnetizing part (32% of rated current × V_r) and a load part, which are added as the square root of the sum of squares.

For the default motor at 400 V this gives a speed of 1488 rpm at no load (slip 0.8%), 1457 rpm at 65% load (slip 2.88%, the default setting), and exactly the 1440 rpm nameplate speed at 100% load (slip 4%).

Try it in the induction motor simulator

Open the three-phase induction motor simulator and keep the default 4 kW, 400 V, four-pole, 50 Hz motor.

• Set the load to 0%, 25%, 50%, 75% and 100% and note the rotor speed. You should read about 1488, 1476, 1464, 1452 and 1440 rpm (the last is the nameplate speed). Slip rises in a straight line with load.
• Raise the load to 111%. At rated voltage this passes the stall limit, so the speed falls to 0 and the current jumps to about 34 A.
• Return to 65% load and reduce the line voltage to 320 V (80%). Slip rises to about 4.05%, 1439 rpm, and the stall limit drops to about 70% load, so setting 75% stalls the motor.
• Change the frequency to 60 Hz and see the synchronous speed move to 1800 rpm.

The last two experiments show the same lesson: slip is the price of torque, and lower voltage means more slip for the same load.