MC-C4 · Torque-speed curve

The Torque-Speed Curve and Proportional Shift

An induction machine’s torque draws a hill against slip. Vary the rotor resistance, watch how the hill moves, and separate for yourself which of the maximum torque’s height and position is tied to resistance.

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

The torque curve is a hill: it rises in proportion to torque at small slip and falls at large slip. Move the rotor-resistance slider and watch where the hill goes. Of the hill’s height (the maximum torque) and its position (the slip of maximum torque), which one changes?

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

The maximum torque does not grow when you raise the rotor resistance. The hill keeps its height and only shifts sideways. In the value of the maximum torque, R2 cancels out and disappears.

Raise R2 — the maximum torque is?

Misaligned

Torque is a hill of slip

Carrying the equivalent circuit’s mechanical output into torque gives the form T ∝ sE2²R2 / (R2² + (sX2)²). At small slip the R2² in the denominator dominates and the torque rises almost in proportion to slip (the running region). At large slip the (sX2)² dominates and the torque falls inversely with slip (just after starting). So the torque curve is a hill that peaks somewhere between.

ObserveT ∝ sE2²R2(R2² + (sX2)²)

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

The peak forgets R2

To find the slip of maximum torque, differentiate the expression with respect to slip and set it to zero; it cleanly gives R2 = sX2, so the maximum-torque slip is s = R2/X2. Putting this slip back into the torque expression cancels the R2 in numerator and denominator, and the maximum torque becomes Tmax ∝ E2²/(2X2), with R2 gone. The maximum torque is independent of rotor resistance and is set only by voltage and leakage reactance.

Choosemax T: R2 = ?

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

Fill insmax = R2 / ?

The maximum-torque slip is resistance over reactance.

Buy starting torque with the proportional shift

That the maximum-torque slip is s = R2/X2 means raising the rotor resistance makes the maximum torque occur at a larger slip. The whole curve shifts proportionally along the slip axis while the hill keeps its height. This is the proportional shift. Adding external resistance in series with a wound rotor to raise R2 lets you place the maximum torque at standstill slip (s = 1), giving large starting torque and small starting current at once. After starting, the resistance is removed to make the running slip small and recover efficiency. A squirrel cage has fixed resistance and cannot do this directly, but a deep bar or double cage is designed so the effective resistance is large only at starting, achieving 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 hill’s position, not its height. The maximum-torque slip is s = R2/X2 and moves in proportion to resistance, but putting that slip back into the torque expression cancels R2, so the maximum torque is Tmax ∝ E2²/(2X2) and forgets the resistance. So resistance sets not the height but only the position of the hill. Thanks to this proportional shift of the whole curve along the slip axis, a wound rotor uses external resistance to place the maximum torque at standstill slip for large starting torque, and removes it in running for small slip and kept efficiency. Where to put the hill of torque is what the rotor resistance decides.

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).