Synchronous, BLDC and PMSM Drives

The torque equation loses a term. With the flux fixed by the magnets, only the quadrature current matters:

One current, one constant — simpler even than the induction machine's vector control, because \(i_d\) is set to zero rather than to a magnetising value. Below base speed a PMSM drive commands only \(i_q\).

The electrical frequency at base speed:

Two hundred hertz — four times the mains frequency, from a machine turning at 3000 rpm. High pole counts give high torque density and demand a fast inverter, which is why PMSM drives switch at 10–20 kHz where induction drives manage at 4.

The back EMF:

And this exists whenever the rotor turns, whether the drive is running or not — a safety consideration with no counterpart in an induction machine. A PMSM spun by its load with the inverter disabled produces voltage at its terminals and, if the diodes conduct, charges the DC link.

The terminal voltage adds the reactance drop, in quadrature:

Hence the DC link, using the 0.707 utilisation of Set 26:

Note how directly the machine's design determines the converter's. Choosing a higher \(\lambda_m\) for more torque per amp raises the back EMF and therefore the link voltage — the two are inseparable, and machine and inverter must be specified together.

Compare with the induction machine of Set 35:

The seventh row removes Set 35's hardest problem outright. In a PMSM the flux is locked to the rotor, so an encoder reading the rotor angle gives the flux angle directly — no slip calculation, no \(R_r\), no detuning.

The problem at double speed:

Without intervention no current can be forced into the machine at all — the machine is generating more voltage than the inverter can produce. Torque is not merely reduced; it is zero.

The remedy: negative \(i_d\). Current along the negative d-axis produces flux opposing the magnets:

Nearly seven amps of demagnetising current, producing no torque whatsoever — pure overhead, dissipating \(i_d^2R_s\) in the stator for the sole purpose of cancelling flux the magnets insist on providing.

The contrast is stark:

The first two machines weaken their field by supplying less; the PMSM must supply more. That inversion is the fundamental penalty of permanent excitation, and it grows with speed — at four times base speed the demagnetising current would dominate the machine's rating entirely.

The current budget at 6000 rpm. With a 25 A limit:

Only 4% of the torque capability lost at double speed, because \(i_d\) is still small compared with the limit. The penalty is modest here and becomes severe only when \(i_d\) approaches the current limit — which defines the machine's maximum usable speed.

The hazard: uncontrolled generation. If the inverter shuts down — a fault, a protection trip, a loss of gate supply — while the machine is spinning fast:

The line-to-line EMF exceeds the link voltage, so the inverter's freewheeling diodes form an uncontrolled rectifier and pump power into the link. With no load to absorb it the capacitor overvolts, and there is no way to stop it short of stopping the rotor. Every magnet drive above base speed lives with this.

Mitigations, none of them free:

The first row is what production drives do: shorting the three phases together forces the machine into a controlled short circuit, where the current is limited by the synchronous reactance and the torque is a modest braking torque. It requires that the gate drives keep working during the fault, which is why they have their own energy reserve.

The BLDC principle. The machine is wound so that its back EMF is trapezoidal — flat over 120° of each half cycle rather than sinusoidal:

Two phases conduct at a time and the third floats, exactly the 120° conduction mode of Set 26 — which was dismissed there for losing 13.4% of the output. Here it is the whole point, because the machine has been designed to match it.

What the controller needs. Only six commutation instants per electrical cycle:

The cost difference is real. A BLDC controller for a fan or a pump can be built around a few pounds of silicon; a PMSM drive needs a resolver or a good encoder plus the processing of Set 35.

Where the torque ripple comes from. Two mechanisms, both at the commutation instants:

During commutation the outgoing phase's current decays while the incoming phase's rises, and the two do not match — so the total torque dips or peaks. The result is a ripple at six times the electrical frequency, typically 10–15% of the mean.

The comparison:

The third row is worth explaining. A trapezoidal machine puts all its conductors under full flux for 120° instead of distributing them sinusoidally, so it extracts more torque from the same copper and iron — about 15% more. That advantage is why BLDC persists despite its ripple.

Where each belongs:

The distinction has blurred: the same magnet machine can be driven either way, and many controllers run BLDC commutation at low speed and switch to sinusoidal control above it — taking the simplicity where it helps and the smoothness where it matters.

Sensorless BLDC is unusually easy, which reinforces the cost argument:

One phase is always open, so its back EMF can be measured directly — no observer, no model, no parameters. Detect the zero crossing, wait 30 electrical degrees, commutate. It fails at standstill, where there is no EMF, so the machine is started open-loop and handed over once it is moving — which is why some fans hesitate briefly on start-up.

Why the interior machine is different. Burying the magnets inside the rotor makes the magnetic path direction-dependent:

The second term is reluctance torque — the rotor's tendency to align its low-reluctance axis with the stator field. Since \(L_d-L_q\) is negative, it contributes positively when \(i_d\) is negative.

Which is a happy coincidence. Negative \(i_d\) is exactly what field weakening required in Problem 2:

The surface-mount machine's demagnetising current was pure loss; here it does useful work. That is why traction drives use interior magnets almost universally — the machine that most needs field weakening is the one that profits from it.

The MTPA condition. Minimising \(\left|\vec i_s\right|\) for a given torque gives:

The torque produced:

Eleven per cent of the torque from saliency alone, with no magnet material involved. Machines designed for higher saliency — more flux barriers in the rotor — push this to 30–50%, reducing the magnet content and with it the cost and the dependence on rare-earth supply.

The current saving:

Against producing the same torque with \(i_d = 0\):

Seven per cent less current and thirteen per cent less copper loss, from choosing the current's direction rather than merely its magnitude. That is free efficiency, requiring only a lookup table in the controller.

The complete operating strategy:

All three regions are usually precomputed into a two-dimensional lookup table indexed by torque demand and speed, because solving the constrained optimisation online is expensive. Calibrating that table is a substantial part of commissioning a traction drive.

The topology. Two six-pulse thyristor bridges back to back with a DC link inductor — a current source inverter, as in Set 29:

Twelve thyristors instead of the cycloconverter's thirty-six, with a DC link inductor holding the current constant. The supply-side bridge controls the current; the machine-side bridge steers it into the right phases.

How commutation happens without any commutation circuit. Set 6 established that a thyristor needs reverse voltage for its turn-off time:

An over-excited synchronous machine behaves capacitively at its terminals — it supplies reactive power rather than consuming it. That leading power factor provides exactly the reverse voltage the outgoing thyristor needs, so the machine's own EMF performs the commutation. No forced commutation circuit, no self-commutating devices, no snubber energy.

Which is why LCIs reach ratings nothing else approaches:

The fourth row echoes Set 29: the DC link inductor makes a short circuit harmless. For a 40 MW compressor drive that robustness is worth far more than waveform quality.

The problem at low speed. The commutating voltage is the machine's EMF, which is proportional to speed:

At standstill there is no EMF at all, so a machine that must be started from rest cannot commutate its own inverter. Something else must do it.

The solution: DC link current pulsing. Force the link current to zero at each commutation instant:

The supply-side bridge is driven into inversion for a few milliseconds, collapsing the link current so the machine-side thyristors turn off naturally. Torque is then pulsating and the acceleration is rough — but it only has to work up to about 10% speed, after which the machine's own EMF takes over. It is why an LCI-driven compressor sounds distinctly unhappy for the first few seconds of a start.

Against the cycloconverter:

The fourth row divides the applications cleanly. A gas compressor spends its life near rated speed and starts unloaded, so pulsed starting is acceptable; a gearless mill must be inched into position and produce full torque at 0.2 Hz, which only the cycloconverter does well. Neither displaces the other.

What traction demands is an unusually broad set of requirements:

The second row is what makes traction distinctive. A single-speed transmission means the machine must cover the whole speed range electrically — up to five times base speed — where a conveyor or a pump needs one.

The four candidates:

The fifth row deserves emphasis. An induction machine that loses its inverter simply stops exciting:

Safe by construction. A permanent-magnet machine at 12,000 rpm keeps generating and keeps braking, so a fault produces both an overvoltage and an uncommanded retarding torque on a driven wheel — which is why an active short-circuit strategy, and gate drives that survive the fault, are safety items in every EV.

Why the SRM keeps not winning. On paper it should:

The simplest, cheapest and most robust rotor imaginable — a lump of laminated steel — tolerant of high temperature and safe under any fault. And it is defeated by rows seven, eight and nine together: the doubly salient geometry produces large torque ripple and radial forces that make the stator ring audibly, and it needs a non-standard converter that no one makes in volume. Decades of development have narrowed but not closed those gaps.

What actually decides it:

The third row is a genuinely elegant engineering answer. Using a magnet machine on one axle for efficiency at cruise and an induction machine on the other for cost and for its zero standby drag — it can be de-energised completely, whereas a PMSM always drags its magnets past the stator — captures the strengths of both.

And the requirement that dominates everything. Traction efficiency is judged over a drive cycle, not at one point:

So the map of efficiency across the whole torque–speed plane matters far more than the peak figure, and the machine that wins is the one whose high-efficiency island sits where the drive cycle actually lives. That is where permanent magnets are strongest — they have no excitation loss at light load, exactly where an induction machine is still paying for its magnetising current.