MC-C3 · Induction equivalent circuit

The Induction Equivalent Circuit and R2/s

The trick that turns a spinning rotor into a single static circuit lies in slip. When the rotor resistance becomes R2/s, see what that one term holds by varying the slip and watching the split between loss and output.

What does the single term R2/s hold?

Referring the rotor makes its resistance R2/s. This bar consumes the whole air-gap power Pag that crossed the gap. Move the slip slider and watch the bar split in two. Which side is the copper loss thrown away as heat, and which is the mechanical output sent to the shaft?

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 real rotor copper loss, the part thrown away as heat. The mechanical output appears separately as a fictitious resistance with slip woven in.

Which part of R2/s holds the mechanical output?

Misaligned

Turn the spinning rotor into a static circuit

The flux the rotor bars see has the slip frequency sf. So the rotor EMF is sE2 and the rotor reactance is sX2, both carrying slip. The rotor current is I2 = sE2 / (R2 + jsX2). Dividing numerator and denominator by s gives I2 = E2 / (R2/s + jX2). Now the frequency is gone and the rotor looks stationary, at the price of its resistance changing from R2 to R2/s. The speed information has moved into a single resistance term.

ObserveR_rotor = R2s

Referring makes the rotor resistance 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)

The air-gap power Pag the stator sends across the gap is all consumed in R2/s. The share in R2 is the rotor copper loss Pcu2 = sPag, and the share in R2(1-s)/s is the mechanical output Pmech = (1-s)Pag. So Pag : Pcu2 : Pmech = 1 : s : (1-s), and the rotor-side efficiency is (1-s). Larger slip means larger copper loss. At the instant of starting, s = 1, all the air-gap power becomes heat and the output is zero; in normal running the slip must be small for high efficiency.

Fill inPcu2 = s ?

The rotor copper loss is 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 slip. Cancelling the slip frequency the rotor sees makes the spinning rotor look like a static circuit, at the cost of its resistance swelling to R2/s, with the speed information held in that one term. In R2/s = R2 + R2(1-s)/s the first is the real copper loss thrown away as heat, the second the fictitious resistance standing in for the mechanical output to the shaft. The air-gap power splits 1 : s : (1-s) into loss and output, so slip is the very measure of loss. At starting (s = 1) it is all heat, and the slip must be small for high efficiency. This one circuit becomes the tool that draws the next unit’s torque-speed curve.

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).