Permanent-Magnet Synchronous Motors

i Learning Objectives

By the end of this chapter you should be able to:

Describe surface-mounted and interior PMSM rotor constructions and their saliency.
Write the d–q voltage and torque equations of a PMSM.
Separate magnet torque and reluctance torque, and compute each.
Explain field-oriented control, maximum-torque-per-ampere operation and flux weakening.

Section 90-1

Construction

A PMSM has a conventional three-phase distributed stator winding, like an induction motor or alternator, and a rotor carrying permanent magnets in place of a field winding. The magnets are shaped to produce a sinusoidal back-EMF.

Surface PMSM (SPMSM)

Magnets glued on the rotor surface. Magnets have permeability close to air, so the effective air gap is the same on both axes: \(L_d = L_q\) — non-salient. Simple, low ripple; magnets need a retaining sleeve at high speed.

Interior PMSM (IPMSM)

Magnets buried in slots inside the rotor iron. The d-axis passes through the magnets (high reluctance), the q-axis through iron: \(L_q > L_d\) — inverse saliency. Robust at high speed and produces extra reluctance torque.

Note the reversal. In the wound-field salient-pole machine of Chapter 76, \(X_d > X_q\). In an IPMSM the magnet sits on the d-axis and behaves like air, so \(L_d < L_q\).

Section 90-2

The d–q Model

Transforming the three-phase quantities to a frame rotating with the rotor (Park's transformation), with the d-axis aligned to the magnet flux \(\lambda_f\) and \(\omega_e = p\,\omega_m\) (\(p\) = pole pairs):

\[v_d = R_si_d + L_d\frac{di_d}{dt} - \omega_eL_qi_q\]

\[v_q = R_si_q + L_q\frac{di_q}{dt} + \omega_e(L_di_d + \lambda_f)\]

Electromagnetic torque

Magnet torque + reluctance torque

\[T_e = \frac{3}{2}\,p\left[\lambda_fi_q + (L_d - L_q)\,i_di_q\right]\]

For an SPMSM \(L_d = L_q\) and torque is simply proportional to \(i_q\). For an IPMSM, \(L_d - L_q < 0\), so a negative \(i_d\) adds reluctance torque.

Amplitude-invariant transformation is assumed (hence the factor 3/2); \(i_d\), \(i_q\) are peak phase-current components and \(\lambda_f\) is the peak flux linkage per phase.

1 Worked Example 90.1 — Magnet and Reluctance Torque

Problem. An 8-pole PMSM (\(p = 4\)) has \(\lambda_f = 0.1\) Wb. Find the torque for \(i_q = 10\) A when (a) it is an SPMSM with \(i_d = 0\); (b) it is an IPMSM with \(L_d = 2\) mH, \(L_q = 5\) mH and \(i_d = -5\) A. (c) Find the peak phase back-EMF at 3000 rev/min.

\[\text{(a)}\quad T_e = \tfrac32(4)(0.1)(10) = 6.0~\text{N·m}\]

\[\text{(b)}\quad T_\text{rel} = \tfrac32(4)(0.002 - 0.005)(-5)(10) = 0.9~\text{N·m}, \qquad T_e = 6.0 + 0.9 = 6.9~\text{N·m}\]

\[\text{(c)}\quad \omega_e = 4 \times \frac{2\pi(3000)}{60} = 1256.6~\text{rad/s}, \qquad E_\text{pk} = \omega_e\lambda_f = 125.7~\text{V}\]

The IPMSM produces 15 % more torque for only a modest increase in current magnitude (\(\sqrt{10^2 + 5^2} = 11.2\) A).

Section 90-3

Field-Oriented Control

Because \(i_d\) and \(i_q\) control flux and torque independently, a PMSM can be controlled exactly like a separately excited DC motor. An encoder or resolver measures rotor angle; Park's transformation converts measured phase currents to \(i_d\), \(i_q\); two PI regulators set \(v_d\), \(v_q\); the inverse transformation and space-vector PWM generate the inverter switching.

\(i_d = 0\) control: optimal for an SPMSM — all current produces torque.
Maximum torque per ampere (MTPA): for an IPMSM, choose the negative \(i_d\) that maximises torque for a given current magnitude.
Flux weakening: above base speed the back-EMF \(\omega_e\lambda_f\) approaches the inverter's voltage limit. Injecting negative \(i_d\) opposes the magnet flux, reducing the stator flux linkage \(L_di_d + \lambda_f\) and extending the speed range at constant power.

Section 90-4

Merits and Applications

Merits

Highest efficiency and power density of common industrial motors (no rotor copper loss, no magnetising current); smooth torque; operation at unity power factor possible; excellent dynamic response.

Limitations

Cost and supply risk of rare-earth magnets; demagnetisation risk at high temperature; cannot be started directly on the grid without a cage (line-start PMSM); requires an inverter and position sensing for variable speed.

Applications: electric-vehicle traction (mostly IPMSM), servo drives and robotics, CNC spindles, washing-machine direct drives, compressors, direct-drive wind generators.

Section 90-5

Summary and Common Mistakes

Key Points

SPMSM: \(L_d = L_q\); IPMSM: \(L_q > L_d\).
\(T_e = \tfrac32p[\lambda_fi_q + (L_d - L_q)i_di_q]\).
FOC decouples flux (\(i_d\)) and torque (\(i_q\)); negative \(i_d\) gives MTPA and flux weakening in an IPMSM.

Common Mistakes

Using the number of poles instead of pole pairs for \(p\).
Assuming \(L_d > L_q\) for an IPMSM, by analogy with wound-field machines.
Applying positive \(i_d\) in an IPMSM — it reduces torque and strengthens the flux.

Section 90-6

Chapter Review

Practice Problems

P90.1 A 6-pole SPMSM has \(\lambda_f = 0.08\) Wb. What \(i_q\) gives 9 N·m?

Show answer
\(p = 3\); \(i_q = 9/(1.5 \times 3 \times 0.08) = 25\) A.
P90.2 For the IPMSM of Example 90.1, find the total torque with \(i_q = 10\) A and \(i_d = -10\) A.

Show answer
\(T_\text{rel} = 1.5(4)(-0.003)(-10)(10) = 1.8\) N·m; \(T_e = 7.8\) N·m.

Multiple-Choice Questions

MCQ 1. In a surface-mounted PMSM:
(a) \(L_d > L_q\) (b) \(L_d < L_q\) (c) \(L_d = L_q\) (d) \(L_d = 0\)

Show answer
(c)
MCQ 2. Flux weakening in a PMSM is achieved with:
(a) positive \(i_d\) (b) negative \(i_d\) (c) zero \(i_q\) (d) higher frequency only

Show answer
(b)

Conceptual Questions

Why is the PMSM more efficient than an induction motor of the same rating?
Explain why buried magnets give inverse saliency.
Why is flux weakening necessary above base speed?

Looking Ahead

Remove the magnets altogether and rely only on reluctance torque, and you arrive at the simplest and most rugged rotor of all — the switched reluctance motor of Chapter 91.