MC-C4 · Torque-speed curve

The Torque-Speed Curve and Proportional Shift

The torque of an induction machine traces a hill against slip. Vary the rotor resistance, see how that hill moves, and sort out for yourself which of the size and position of the maximum torque is tied to the resistance.

What happens to the maximum torque if you raise the rotor resistance?

On this graph, whose horizontal axis is speed, the torque curve rises almost in proportion to slip on the right, where slip is small (near synchronous speed), and comes back down on the left, where slip is large (toward standstill): a hill. Moving the rotor-resistance slider shifts the hill sideways; which changes, the height of the hill (maximum torque) or its position (slip at maximum torque), is the question of this lesson.

Rotor resistance R2R2 = 0.15

Slide for continuous rotor resistance. The x-axis is speed; the dashes mark the maximum-torque height.

The slip at which the maximum torque occurs

smax = R2 / X2 = 0.30

Raising the rotor resistance does not raise the maximum torque. The height of the hill stays put and only its position shifts sideways; in the formula for the size of the maximum torque, R2 cancels out and disappears.

Raise R2 — the maximum torque is?

Misaligned

Torque is a hill of slip

Turning the mechanical output of the equivalent circuit into torque gives the form T ∝ sE2²R2 / (R2² + (sX2)²). At small slip, R2² dominates the denominator and the torque rises nearly in proportion to slip (the running region); at large slip, (sX2)² dominates and the torque falls in inverse proportion to slip (the region just after starting). So the torque curve becomes a hill that peaks somewhere in between. Because the numerator holds the square of the emf, torque is proportional to the square of the voltage, so even a small voltage dip cuts the torque by more.

ObserveT ∝ sE2²R2(R2² + (sX2)²)

Torque is a hill-shaped formula with slip and rotor resistance in the numerator.

R2 vanishes at the peak

To find the slip at which torque is maximum, differentiate with respect to slip and set the result to 0, and the clean condition R2 = sX2 comes out, so the slip at maximum torque is s = R2/X2. Put this slip back into the torque formula and the R2 in the numerator and denominator cancel, giving the maximum torque Tmax ∝ E2²/(2X2), with R2 gone. The size of the maximum torque is set only by the voltage and the leakage reactance, regardless of the rotor resistance, and the conclusion stays the same even when the stator's resistance and reactance are included in the calculation.

Choosemax T: R2 = ?

Maximum torque is where the rotor resistance equals slip times reactance.

Fill insmax = R2 / ?

The slip at maximum torque is the resistance divided by the reactance.

Buy starting torque with the proportional shift

That the slip at maximum torque is s = R2/X2 means that raising the rotor resistance makes the maximum torque occur at a larger slip. The whole curve shifts proportionally along the slip axis while the height of the hill stays put, and this is called proportional shifting. Inserting external resistance in series with a wound rotor to raise R2 lets the maximum torque be set to occur at standstill slip (s = 1), giving a large starting torque and a small starting current together, and once starting is over the resistance is removed to make the running slip small and recover efficiency. A squirrel-cage rotor has fixed resistance and cannot be adjusted this way, but deep bars or a double cage are designed so that the effective resistance is high only at starting, when the rotor frequency is high, for a similar effect.

On your ownTmax ∝ ?

In the maximum-torque value the rotor resistance cancels and disappears.

Back to the first screen

Raising the rotor resistance changed the position of the hill, not its height. The slip at maximum torque, s = R2/X2, moves in proportion to the resistance, but put that slip back into the torque formula and R2 cancels, so the maximum torque Tmax ∝ E2²/(2X2) does not depend on the resistance. The resistance sets only the position of the hill, not its height, and thanks to this proportional shifting, with the whole curve sliding proportionally along the slip axis, a wound-rotor motor uses external resistance at starting to put the maximum torque at standstill slip for a large starting torque, then removes the resistance in running to keep efficiency with a small slip. Where the torque hill sits is set by the rotor resistance.

The torque-speed curve: T ∝ sE2²R2 / (R2² + (sX2)²) is hill-shaped, peaking at the maximum-torque slip s = R2/X2. The peak value Tmax ∝ E2²/(2X2) is independent of R2. The rotor resistance sets only the position of the hill, not its height, so raising R2 shifts the curve proportionally along the slip axis (the proportional shift). A wound rotor uses external resistance to raise the starting torque up to the maximum torque.

Once you hold this curve

The torque-speed curve is the map of how to work an induction machine. Its left end (standstill) fixes the starting torque and current, and its right end (near synchronous) fixes the operating point. But applying full voltage straight from standstill makes the starting current surge to five or six times rated. The next unit covers how to hold down that starting current (star-delta, reactor, autotransformer) and how to move the whole curve to change speed (voltage, frequency and pole control) (MC-C5).