Induction Motor Drives — V/f Control

Reduce to a Thevenin source. Everything to the left of the rotor branch collapses into one source and one impedance:

The magnetising reactance is large enough that \(V_{th}\) is 96% of the phase voltage and \(R_{th}\) is 93% of \(R_1\) — so the reduction changes little numerically but makes the torque expression a single formula rather than a network solution.

Torque at rated slip:

Note that \(R_2'/s = 20\ \Omega\) dominates everything else in the denominator — the machine at rated slip is essentially a 20 Ω resistor across 231 V. That is why the torque–slip curve is so nearly linear near the origin.

Breakdown. Maximum power transfers to \(R_2'/s\) when it matches the magnitude of everything else:

The machine can produce nearly three times its rated torque — briefly, since it would draw far more than rated current doing so. That margin is the reason induction motors tolerate shock loads that would stall a carefully sized servo.

Starting torque, at \(s = 1\):

Half again its rated torque at standstill — enough to start most loads, which is why direct-on-line starting works at all. It comes with the 6× current of Set 30, because at \(s = 1\) the rotor branch is only 0.8 Ω.

The essential observation for everything that follows:

The machine normally lives on the segment between \(s = 0\) and \(s = 0.04\) — a tiny fraction of the curve, where torque is nearly proportional to slip and the operating point is stable. Everything a drive does is an attempt to keep it there while moving the synchronous speed.

Why constant V/f. The airgap flux is set by the voltage and frequency together:

Holding that ratio holds the flux, and holding the flux preserves the machine's torque capability at every frequency — in principle. The reactances all scale with frequency, so the whole equivalent circuit scales together. Except for one element.

At 25 Hz everything halves except \(R_1\) and \(R_2'\):

Notice \(R_{th}\) is essentially unchanged at 0.927 while \(X_{th}\) has halved. The resistance is now a much larger fraction of the total impedance, and that is the whole problem.

The breakdown torque falls:

Constant V/f was supposed to preserve the torque capability and has lost a quarter of it at half frequency. The cause is entirely the stator resistance: it does not scale, so at low frequency it consumes a growing share of an ever-smaller applied voltage.

The trend is severe:

At 5 Hz the machine retains barely a seventh of its torque capability — less than its rated torque. A conveyor that needs full torque at crawl speed simply will not start.

The fix: voltage boost. Add back what the stator resistance takes:

A third of the applied voltage is lost in the stator winding before any flux is produced. Adding it back restores the airgap voltage and with it the flux — and the same 13.6 V at 50 Hz would be only 3% of 415, which is why boost matters only at the bottom of the range.

How boost is implemented, and its risk:

The first row is a real failure mode. A fixed boost sized for full load at 5 Hz over-excites the machine when it runs unloaded at the same frequency — the flux rises, the core saturates, the magnetising current grows sharply, and the motor overheats while doing no work at all. That is why cheap drives with a manual boost setting have a maximum recommended value.

The argument. Write the torque in terms of the rotor branch and substitute \(s = f_{sl}/f\):

Every \(f\) has cancelled. At constant flux the torque is a function of the slip frequency alone — the difference between the stator field's speed and the rotor's — and it does not matter at all what the absolute frequency is.

The rated slip frequency:

Two hertz of slip frequency produces rated torque, at any stator frequency the drive chooses. That single number characterises the machine's torque production far better than the 4% slip does, because 4% means different things at different speeds.

What that implies at low speed:

Forty per cent slip at 5 Hz. Set 30 established that efficiency cannot exceed \(1-s\), so this looks like the same disaster — and it is not.

Why 40% slip is acceptable here and catastrophic there. The rotor loss is a fraction of the airgap power, and the airgap power has fallen with the speed:

Four per cent — entirely manageable. The machine is turning at a tenth of its speed producing full torque, so it is handling a tenth of the power, and 40% of a tenth is small.

The contrast with Set 30 is the whole point:

Reducing the voltage forces high slip against a synchronous speed that never moves; a drive moves the synchronous speed down to meet the rotor, so the slip stays small in absolute terms. That distinction — changing the field's speed rather than weakening it — is the entire justification for variable-frequency drives.

And it makes slip frequency the natural control variable:

A closed-loop V/f drive with a speed sensor does exactly this: measure the rotor speed, compute the slip frequency needed for the demanded torque, add them, and apply the result. That is slip-frequency control — and it is one small step from the indirect field orientation of Set 35, which computes the same quantity and uses it to locate the flux rather than merely to set the frequency.

The voltage ceiling. Set 26 established it precisely: a 415 V drive on a 587 V link reaches exactly 415 V with space vector modulation and no more:

So above base speed the machine is deliberately under-excited — field weakening, exactly as the DC machine of Set 33 achieved by reducing its field current. The mechanism differs; the consequence is identical.

Two different quantities fall at two different rates:

The capability falls as \(1/f^2\) while the requirement falls only as \(1/f\). They start far apart — capability is 2.77 times requirement at base speed — and they converge.

Where they meet:

At 138 Hz the breakdown torque has fallen to exactly the constant-power torque, so the machine is operating at its peak with no margin whatsoever — any disturbance stalls it. Practical drives stop well short.

The three regions:

Constant power is usually limited to about twice base speed — 100 Hz here — where the breakdown torque has fallen to 31.5 N·m against a required 22.8, leaving a margin of 1.4. Beyond that the drive must follow the breakdown curve down, so torque falls as \(1/f^2\) and power as \(1/f\).

What limits the top of the range, beyond the electrical argument:

A standard machine should not be run far above its nameplate speed without checking the manufacturer's limit — the cage endrings are the usual constraint, and their failure at speed is destructive. Inverter-duty machines are qualified for a stated maximum frequency.

Comparing with the DC machine:

The third row is a genuine advantage of the induction drive: field weakening requires no extra hardware at all, because it happens automatically once the voltage saturates. The fourth row is the cost — a DC machine's independent field gives it a wider constant-power range.

The fundamental limitation. An open-loop V/f drive commands the stator frequency, and the rotor runs slower by whatever slip the load demands:

So the speed error is the slip, and it depends on the load. The drive does not know the rotor speed and cannot correct it — it is a frequency source, not a speed controller.

The resulting accuracy:

The absolute error is constant — 60 rpm, the rated slip — so it becomes an ever-larger fraction as the speed falls. A drive that holds 4% at rated speed holds only 40% at a tenth of it, which is why open-loop V/f is unsuitable for anything needing speed accuracy at low speed.

Slip compensation improves this without a sensor:

The drive estimates the slip from the current it is supplying and adds it to the frequency command. That typically takes the error from 4% to under 1% at rated speed — but it is an estimate based on \(R_2'\), which drifts by 50% between cold and hot, so it degrades exactly when the machine is working hardest.

The complete list of limitations:

The fifth row disqualifies V/f from hoisting outright. At zero commanded frequency the machine produces zero torque, so a suspended load simply falls — which is why the elevator of Set 33 needed either a DC drive or the vector control of Set 35.

The low-frequency instability. Between roughly 5 and 20 Hz, a lightly loaded V/f drive can break into a sustained oscillation of a few hertz:

The mechanism is a lightly damped interaction between the machine's electrical dynamics (the rotor time constant) and its mechanical inertia, with the inverter's constant-voltage output providing no damping at all. At light load there is nothing to absorb the energy, so the oscillation grows until it is limited by nonlinearity — audible as a rhythmic growl and visible as a fluctuating link current.

How it is damped. Feed the disturbance back with the opposite sign:

The DC link current carries a clear signature of the oscillation, so subtracting a high-pass-filtered version of it from the frequency command adds damping without affecting the steady state. Every commercial V/f drive includes this, usually under a name like "stabilisation" or "load oscillation damping", and it is one of the few settings that genuinely needs adjusting on site.

The affinity laws. For a centrifugal machine, flow follows speed, head follows speed squared, and power follows speed cubed:

The cube is what makes variable-speed pumping so profitable. It also explains why Set 30 tolerated stator voltage control for fan loads — the load power collapses so fast that even a poor control method survives.

With a drive, at 80% flow:

A 20% flow reduction halves the power. That is the whole argument, and it is why pump and fan applications dominate the installed base of variable-speed drives.

With a throttling valve, the story is completely different. The pump still runs at full speed; the valve simply adds resistance:

Reducing the flow by 20% reduces the power by only 10%, because the pump is working against a higher head. All the energy difference is dissipated across the valve as turbulence and heat — the hydraulic equivalent of controlling a motor with a series resistor.

The saving:

The sensitivity to operating point:

The saving grows dramatically as the throttling deepens, so the drive's value depends entirely on the duty profile. A pump that runs at full flow all year saves nothing; one that spends its time half throttled saves more than half its energy. Assessing the profile is the first step of any proposal.

Two caveats that reduce the calculated saving:

The cube law assumes the pump works against friction alone. A system lifting water to a fixed height has a static head that does not fall with flow, so the speed cannot be reduced proportionally and the saving is smaller. A system that is mostly static head may save very little.

The drive itself is 97–98% efficient and the motor's efficiency falls somewhat at reduced load — costing perhaps 4–5% of the saving, which the arithmetic above ignores.

And what the drive brings besides energy:

The last row is the honest entry. A drive replaces a motor's clean sinusoidal current with the distorted current of a capacitor-input rectifier, and a site with many drives needs the mitigation of Set 32 — a DC link choke at minimum. That cost belongs in the assessment.