MC-C3 · Induction equivalent circuit

The Induction Equivalent Circuit and R2/s

The key to turning a spinning rotor into a single stationary circuit is the slip. See what this one term holds once the rotor resistance becomes R2/s, and watch for yourself how loss and output divide as the slip changes.

What does the single term R2/s hold?

Referring the rotor to the stator side turns the rotor resistance into R2/s, and the whole bar is this resistance, which consumes the entire air-gap power Pag that crosses the gap. Move the slip slider and the bar splits into a red part and a gold part; which is the true copper loss thrown away as heat and which is the mechanical output going out through the shaft is the question of this lesson. Below, you can first pick the term that carries the mechanical output.

Slip ss = 0.50

Slide for continuous slip. Red is rotor copper loss, gold is mechanical output.

Split of the air-gap power

Pag : Pcu2 : Pmech = 1 : 0.50 : 0.50

R2 alone is just the true rotor copper loss, the loss thrown away as heat. The mechanical output shows up separately, as a fictitious resistance tangled up with the slip.

Which part of R2/s holds the mechanical output?

Misaligned

Turn the spinning rotor into a static circuit

The frequency of the flux seen by the rotor conductors is the slip frequency sf. So the rotor emf is sE2 and the rotor reactance is sX2, both carrying the slip. The rotor current is I2 = sE2 / (R2 + jsX2), and dividing top and bottom by s gives I2 = E2 / (R2/s + jX2). The frequency has now vanished and the rotor looks as if it were standing still, at the price of the resistance changing from R2 to R2/s. The speed information has moved into a single resistance term.

ObserveR_rotor = R2s

Referred to the stator, the rotor resistance becomes R2/s.

Splitting R2/s into loss and output

Write R2/s as R2 + (R2/s - R2), and the second term tidies to R2(1-s)/s. The first term R2 is the real copper loss vanishing as heat in the actual rotor winding. The second term R2(1-s)/s consumes power like a resistor on the diagram, but that power is really the mechanical output leaving on the rotor shaft. A single fictitious resistance in a static circuit stands in for the work the spinning machine does.

ChooseR2s = R2 + ?

Subtract the real loss R2 from R2/s and the output term remains.

The power flow is 1 : s : (1-s)

All the air-gap power Pag that the stator sends across the gap is consumed in R2/s. The share taken by R2 is the rotor copper loss Pcu2 = sPag, and the share taken by R2(1-s)/s is the mechanical output Pmech = (1-s)Pag. So Pag : Pcu2 : Pmech = 1 : s : (1-s), and the efficiency in the rotor is (1-s). The larger the slip, the larger the copper loss, so at the moment of starting, s = 1, all the air-gap power becomes heat and the output is 0, and efficiency is high only when the slip is small in normal running. Friction and windage losses still have to be subtracted from Pmech to get the actual shaft output, and speed control that inserts resistance in a wound rotor to raise the slip throws away, as heat in that resistance, the air-gap power matching the added slip.

Fill inPcu2 = s ?

The rotor copper loss is the slip times the air-gap power.

On your ownPmech = (1-s) ?

The mechanical output is (1-s) times the air-gap power.

Back to the first screen

What split the R2/s bar in two was the slip. Cancel out the slip frequency the rotor sees, and the spinning rotor looks like a stationary circuit, but in return its resistance swells to R2/s and the speed information is carried in that one term. In R2/s = R2 + R2(1-s)/s, the first part is the true copper loss thrown away as heat, and the second is a fictitious resistance standing in for the mechanical output going out through the shaft. Since the air-gap power divides 1 : s : (1-s) between copper loss and output, the slip is the measure of the loss: at starting (s=1) it is all heat, and efficiency is high only when the slip is small. This one circuit becomes the tool for drawing the torque-speed curve in the next lesson.

The induction equivalent circuit: referring the rotor cancels the slip frequency, the rotor resistance becomes R2/s, and it folds into one static circuit. This is R2/s = R2 + R2(1-s)/s, the first term the real rotor copper loss, the second a fictitious resistance standing in for the mechanical output. In power, Pag : Pcu2 : Pmech = 1 : s : (1-s), with rotor efficiency (1-s).

Once you hold this circuit

The equivalent circuit is the machine that turns torque into a function of slip. Carrying the mechanical output into torque gives the form T ∝ sE2²R2 / (R2² + (sX2)²): at small slip the torque is proportional to slip, at large slip inversely so, and somewhere between the torque is maximum. The next unit reads the torque-speed curve, the maximum torque and the starting torque off this expression (MC-C4).