Wound-rotor induction motor: how rotor resistance shapes the torque curve

Slip rings and the wound rotor

An ordinary squirrel-cage rotor has bars shorted by end rings, so its resistance is fixed by design. A wound rotor instead has a three-phase winding, usually star-connected, whose three ends go to slip rings on the shaft. Carbon brushes press on the rings and connect the winding to an external three-phase resistor bank, or to a short circuit when the machine is running normally.

The stator and the rotating field are the same as in a cage motor, and the speed is still close to N_s = 120 × f / p. What changes is that the rotor circuit is accessible, so its resistance, and with it the torque–speed curve, can be changed from outside.

Why rotor resistance shifts the curve

In the usual approximate equivalent circuit, the torque is

T = (3 / ω_s) × V² × (R2 / s) / [(R1 + R2 / s)² + (X1 + X2)²]

where V is the phase voltage, ω_s the synchronous angular speed, R2 the rotor resistance referred to the stator and X1, X2 the leakage reactances. Setting dT/ds = 0 and neglecting the stator resistance R1 gives the slip at the maximum torque and the maximum torque itself:

s_max ≈ R2 / (X1 + X2)

T_max ≈ 3 × V² / (2 × ω_s × (X1 + X2))

The key point is that R2 appears in s_max but not in T_max. Adding external resistance R_ext makes R2 larger, which pushes the peak of the curve towards standstill without lowering it. When R2 = X1 + X2 the peak is at s = 1, so the maximum torque is available at the moment of starting.

The starting torque comes from putting s = 1 into the formula. For R2 below the reactance, adding resistance raises it, because the rotor current becomes better in phase with the rotor flux. The starting current falls, because the rotor impedance grows.

Starting: high torque, low current

A cage motor started directly draws five to eight times rated current while giving a modest starting torque. A wound-rotor motor can be started with the full resistance in the rotor circuit and the resistance cut out in steps as the speed rises. Each step moves the operating point onto a curve with a smaller peak slip, so the torque stays high through the whole acceleration, and the line current stays low.

Worked example. Treat the total leakage reactance as X = 1 Ω per phase referred to the stator, the rotor resistance as R2 = 0.2 Ω, and neglect the stator resistance for a simple comparison.

• Rings shorted: s_max = 0.2 / 1 = 0.2. Starting torque is proportional to R2 / (R2² + X²) = 0.2 / 1.04 ≈ 0.19, and starting current is proportional to 1 / √(R2² + X²) ≈ 0.98.
• With 0.8 Ω added: R = 1.0 Ω, s_max = 1.0. Starting torque is proportional to 1.0 / 2.0 = 0.50, which is about 2.6 times larger, and starting current to 1 / √2 ≈ 0.71, or 28% lower.
• The peak torque, 3V² / (2ω_s X), is the same in both cases.

This combination is why wound-rotor motors were used for cranes, hoists, crushers, mills and other loads that start under heavy torque from a weak supply.

Speed control and its cost

With the rings connected to a resistor, the motor can also run at a reduced speed. For a load that needs constant torque, the rotor current is almost fixed, so the slip rises in proportion to the total rotor resistance:

s2 / s1 = (R2 + R_ext) / R2

The speed falls as the slip increases, so doubling the rotor resistance doubles the slip. The price is paid in heat. The air-gap power P_ag crossing to the rotor splits into mechanical power (1 − s) × P_ag and rotor copper loss s × P_ag. The share of that loss that falls on the external resistor is R_ext divided by the total rotor resistance:

P_ext = s × P_ag × R_ext / (R2 + R_ext)

At 20% slip, one fifth of the air-gap power is dissipated as heat, so the efficiency cannot exceed about 80%. The speed also changes with load, since for a given resistance it depends on the torque. Resistor speed control is therefore used for short periods or small ratings, and has been replaced by power electronics (slip-power recovery and doubly-fed drives) for continuous duty.

Practical points

• Running with the rings shorted gives the same efficiency and speed regulation as a cage motor, and some motors lift the brushes at that point to reduce wear.
• Slip rings and brushes need maintenance and add cost and size compared with a cage rotor.
• Liquid or grid resistors must be rated for the starting heat: the energy of accelerating a load is dissipated in them.
• The resistance in the formulas is per phase and referred to the stator, which includes the turns ratio squared.
• Wound-rotor motors are seldom specified in new installations where a variable frequency drive can do the same job.

How the simulator models rotor resistance

The simulator uses the same equivalent circuit as the three-phase induction motor, calibrated to the nameplate with the rings shorted. The external resistance is added to the rotor resistance, which in the model is R_rotor(s) = r2 × (1 + 0.1 × s) + R_ext. It is treated as a plain resistor per phase, referred to the stator, so with 0 Ω the results equal the three-phase model.

Starting torque and current use the resistance at switch-on, while the running point and the breakdown torque use the final running resistance. In the stepped starter mode the resistance begins at the starter value and falls in equal steps until it reaches the running value. The slip-power loss is P_ext = s × P_ag × R_ext / R_rotor, and a warning appears when it exceeds 10% of input power.

Limits: the live loss readout uses the running resistance, not the stepped value during start-up, and the model is for teaching, not for selecting a starter.

Try it in the wound-rotor induction motor simulator

Open the wound-rotor induction motor simulator.

• Keep Rotor resistance mode on Fixed and set the External rotor resistance to 0 Ω. Note the Starting torque and the Slip at breakdown torque.
• Raise the external resistance. The hint shows the rotor's own resistance in ohms. The Starting torque rises, the starting current falls and the Slip at breakdown torque grows, while the torque chart shows the peak moving towards standstill at nearly the same height. The Starting torque (rings shorted) readout gives the comparison.
• Look at Rotor speed at a constant load. A larger resistance increases the slip, which reduces the speed. The Loss in rotor resistor readout shows the price of that.
• Switch to Stepped starter and set Starter resistance at switch-on, Number of steps and Time per step. Run the timed experiment and watch the resistance fall in steps as the speed rises.
• In the 3D scene, look for the slip rings, the brush gear and the external rotor resistor cabinet, whose grids glow with the dissipated power.

This shows the central trade: resistance buys starting torque and speed control, and you pay for it in heat.

Try it in the simulator