Synchronous, BLDC, and PMSM Drives

Because a synchronous machine has no self-starting mechanism at all. Its torque depends on the angle between the rotor and the stator field, and if the drive does not know that angle it cannot choose which way to push.

Contrast the two families squarely:

• Induction machine. Energise the stator at any frequency and the rotor is dragged along by induction. The slip provides the torque automatically; no position information is needed to start.
• Synchronous machine. Apply a rotating field to a stationary rotor and the torque alternates in direction as the field sweeps past. The average is zero, and the rotor merely vibrates.

So the drive must know where the rotor is before it applies anything. The methods, in order of cost:

1. Fit an absolute encoder or resolver. Position is known at power-up, and this is the only fully reliable answer for a machine that must produce full torque from standstill under load.
2. Hall sensors give the sector, which is enough to start in the right direction and to run six-step commutation. This is why BLDC machines nearly always have them.
3. Alignment. Apply a DC current along an arbitrary axis and wait; the rotor pulls into alignment with it, and the drive then knows the angle. Simple, but it moves the shaft — unacceptable for a hoist or a positioning axis — and it needs the load to be light enough to allow the movement.
4. Signal injection. Inject a high-frequency voltage and measure the resulting current. In a salient machine the current response depends on the angle between the injection and the d axis, so the position can be extracted at zero speed without moving the shaft. This works only because \( L_d \ne L_q \), so it is available on an IPM and not on a surface-magnet machine.
5. Back-EMF observers give excellent position above a few percent of rated speed and nothing at all at standstill, because the back-EMF they observe is proportional to speed. Hence the standard architecture: start with one of the methods above, then hand over to the observer.

The general statement: a PM machine and its drive are a single system. The machine cannot run without the drive, and the drive must be matched to that specific machine's parameters. An induction machine and its drive are far more loosely coupled, which is a large part of why induction drives dominate general industry.

Because the magnet layer is present in every direction, so the flux path looks the same whichever way the stator field points.

Work through the geometry. Magnets have \( \mu_r \approx 1.05 \), so magnetically they behave almost exactly like air.

• Surface magnets form a continuous shell of magnet material over the rotor. Flux entering along the d axis crosses the magnet thickness; flux entering along the q axis crosses the same magnet thickness. \( L_d \approx L_q \), and the machine is magnetically round.
• Interior magnets are buried in slots inside the iron. Along the d axis the flux must cross the magnets — a large effective air gap, so \( L_d \) is small. Along the q axis it can travel entirely through iron between the magnet pockets — a small effective gap, so \( L_q \) is large.

The counter-intuitive part is the direction of the inequality. In a wound-field synchronous machine \( L_d > L_q \), because the d axis lies along the pole where the iron is. In an interior-magnet machine \( L_q > L_d \), because the magnets sit on the d axis and block it. The saliency is inverted relative to the classical machine, and the sign of \( (L_d - L_q) \) in the torque equation follows.

The consequence for control: reluctance torque requires \( i_di_q(L_d-L_q) > 0 \). With \( L_q > L_d \) this needs \( i_d \) negative — that is, the current vector advanced ahead of the q axis. Conveniently, negative \( i_d \) is also what field weakening needs, so in an IPM the MTPA trajectory and the field-weakening trajectory lie in the same quadrant and blend into one another smoothly. In a classical wound-field machine they would oppose.

And a practical note: surface-magnet machines are not inferior — they are simpler to build, have lower cogging and are entirely adequate where the constant-power range is narrow. They are the right choice for fans, pumps and low-speed direct drives. The interior magnet earns its complexity only where field weakening or peak torque density is genuinely needed.

From the fact that the current vector can only occupy six positions, so it is held still for 60° while the rotor keeps turning.

The geometry. With two phases conducting, the stator current vector points in one of six directions, 60° apart. Ideally it should lead the rotor flux by exactly 90° at all times; in practice it is placed at 90° in the middle of each sector and drifts \( \pm30° \) either side.

The extremes:

• Maximum \( T_{pk} \) at mid-sector, where the alignment is ideal.
• Minimum \( T_{pk}\cos30^\circ = 0.866\,T_{pk} \) at each commutation instant.

The average:

So the peak-to-peak ripple as a fraction of the mean is \( (1-0.866)/0.955 = 14.0\% \), and it repeats six times per electrical cycle.

Three things follow that are worth remembering:

• It is inherent. It comes from the quantisation of the current vector, not from any imperfection. A faster current loop, a better sensor or a higher switching frequency change nothing.
• Real machines are worse. The figure above assumes ideal trapezoidal EMF and instantaneous commutation. Finite phase inductance means the outgoing current cannot fall instantly, producing a commutation notch on top of this — 20% total ripple is common in practice.
• The cure is to stop using six steps. Sinusoidal commutation, with the position interpolated between Hall edges, reduces the ripple to a few percent at the cost of a small drop in torque per ampere and a more capable controller. This is now standard in appliance drives.

What is bought is an unbounded field-weakening range and inherent fault safety. What is given up is magnet strength, and therefore torque per ampere at low speed.

Take the field-weakening argument first. The d-axis flux is \( \psi_m + L_di_d \). Setting \( i_d = -i_{ch} = -\psi_m/L_d \) makes it exactly zero. With the flux cancelled, the back-EMF no longer rises with speed, and the voltage limit is never reached — so in theory the machine can run at any speed, and the constant-power region has no upper bound. In practice mechanical and loss limits intervene, but ratios of 4:1 or more become achievable.

Now the fault argument, which is the one that actually gets written into the specification. If the inverter trips at high speed, the freewheel diodes turn the machine into an uncontrolled generator. The steady-state short-circuit current of a PM machine converges to \( i_{ch} \). So:

• If \( i_{ch} \le I_{max} \), the fault current is within what the machine and the devices can survive, and the braking torque is bounded.
• This makes the active short circuit a legitimate protection strategy: on detecting a fault, turn all three low-side transistors on. The machine is shorted, the current settles at \( i_{ch} \), and the DC link is completely isolated from the back-EMF.
• If instead \( i_{ch} \gg I_{max} \), neither option is safe — the short circuit would produce a destructive current, and leaving the diodes to rectify would destroy the link.

The price. \( i_{ch} = \psi_m/L_d \) is made small by weakening the magnets or by increasing \( L_d \). Weaker magnets mean less magnet torque for the same current, so the design must recover it elsewhere:

1. More saliency, so the reluctance term contributes a larger share. This pushes the design towards a strongly salient IPM or a PM-assisted reluctance machine.
2. More current, which means more copper loss and more cooling.
3. More poles or a longer stack, which is a size and cost decision.

A further honest cost: holding \( i_d = -i_{ch} \) at high speed dissipates copper loss continuously while producing no torque at all. Efficiency at high speed and light load — motorway cruising, in a vehicle — is measurably worse than it would be with a machine that did not need so much weakening current. Real designs therefore compromise: \( i_{ch} \) close to \( I_{max} \) but not below it, tuned against the drive cycle the vehicle is actually expected to see.

Because the closed-form expression assumes \( \psi_m \), \( L_d \) and \( L_q \) are constants, and in a real machine none of them is.

What actually varies, and by how much:

• \( L_q \) saturates heavily. The q-axis flux path is all iron, so at high \( i_q \) that iron approaches saturation and \( L_q \) can fall by 30–40% from its small-signal value. The saliency \( \Delta L \) therefore shrinks exactly where the reluctance torque is largest.
• \( L_d \) varies much less, because its path already crosses the magnets — a large effective air gap that dominates the reluctance and keeps the path linear.
• Cross-saturation. \( L_d \) depends on \( i_q \) and \( L_q \) depends on \( i_d \), because both fluxes share the same iron. There is no way to represent this with two independent inductances.
• \( \psi_m \) falls with temperature — about 0.12% per kelvin for NdFeB — so a hot machine has measurably less magnet flux and a slightly different optimum angle.

What that does to the formula. Substituting small-signal inductances typically gives an MTPA angle several degrees away from the true optimum at high current. The torque error is second-order near a maximum — a 5° error costs well under 1% — so it is not catastrophic, but it is measurable, and in a traction application 1% of peak torque is worth having.

How real drives handle it:

1. A two-dimensional table of \( i_d \) and \( i_q \) against torque demand and speed, populated from finite-element analysis and then corrected on a dynamometer.
2. Temperature correction of \( \psi_m \) from a rotor thermal model or a magnet temperature estimate.
3. Online search in some high-end drives — perturb the angle slightly and observe whether the current needed for the commanded torque rises or falls, then walk towards the minimum. Slow, but it tracks everything automatically.

The closed-form expression is still worth knowing, because it tells you the trend: the angle starts at zero, grows with current, and approaches 45°. That is what lets you sanity-check a table and recognise when one is wrong.

Torque is always flux times current at right angles, and every drive is an argument about how to arrange that.

How each family achieves the orthogonality:

• DC machine (Chapter 25). Mechanically, by the commutator. The brushes physically switch the armature current so its MMF stays 90° from the field, whatever the rotor does. Simple, effective, and it wears out.
• Induction machine (Chapter 26). Not at all, in scalar control — which is precisely why V/f is slow and cannot command torque. Field-oriented control supplies it computationally, by finding the rotor flux angle and resolving the current about it.
• PM machine (Chapter 27). The flux is fixed to the rotor by the magnets, so the drive must place the current relative to a measured rotor position. And because the flux is no longer adjustable, the current angle acquires a second job: trading magnet torque for reluctance torque, and later for field weakening.

The second connecting idea is the two-region picture, which appeared in all three chapters in the same shape:

• Constant torque below base speed, where flux is held at its maximum and torque is limited by current — that is, by heating.
• Constant power above base speed, where voltage is at its ceiling and flux must be reduced.
• And in every case a limit on how far the second region can extend: commutation in the DC machine, pull-out torque falling as \( 1/f^2 \) in the induction machine, and the characteristic current in the PM machine.

The third is the cascade. Every drive in this part is built the same way: an inner current loop, tuned fast, that protects the semiconductors and makes the machine look like a torque source; a speed loop outside it; and often a position loop outside that. The machine changes, the physics changes, the transformations change — the control architecture does not.

And the practical thread: in all three chapters the decisive question for choosing a drive was rarely efficiency. It was what happens at zero speed, what happens when the load drives the machine, and what happens when the electronics stop.