PMSM Electromagnetic Torque
Compute electromagnetic torque from d/q-axis currents, flux linkage and inductances.
Usually negative (flux-weakening / MTPA convention for Lq > Ld).
Assumption
Amplitude-invariant dq transform convention (peak Id, Iq, λm) — the same convention used throughout this toolkit. A power-invariant transform would remove the 1.5 factor.
1.5 · P · λm · Iq
1.5 · P · (Ld − Lq) · Id · Iq
Formula
PMSM electromagnetic torque (IPMSM, general form)
T = 1.5 · P · [λm · Iq + (Ld − Lq) · Id · Iq]
- T
- Electromagnetic torque [N·m]
- P
- Pole pairs [—]
- λm
- Permanent magnet flux linkage, peak [Wb]
- Id, Iq
- Peak d/q-axis currents [A]
- Ld, Lq
- d/q-axis inductances [H]
Amplitude-invariant (peak-preserving) dq transform convention — a power-invariant transform removes the 1.5 factor. This module does not support that convention.
SPMSM simplified form
T = 1.5 · P · λm · Iq
- T
- Electromagnetic torque [N·m]
- P
- Pole pairs [—]
- λm
- Permanent magnet flux linkage, peak [Wb]
- Iq
- Peak q-axis current [A]
Valid when Ld = Lq, the defining approximation for a surface-mounted PMSM — the reluctance term vanishes identically, so Id has no effect on torque.
SPMSM vs. IPMSM
In a surface-mounted PMSM (SPMSM), magnets sit on the rotor surface, so the magnetic path seen by the d- and q-axes is nearly identical: Ld ≈ Lq. There is effectively no saliency, so the reluctance torque term is negligible and torque is produced almost entirely by Iq — Id contributes nothing and is normally kept at zero.
In an interior PMSM (IPMSM), magnets are buried inside the rotor iron, which deliberately makes the q-axis magnetic path lower reluctance (higher inductance) than the d-axis: Lq > Ld. This saliency creates a second, reluctance torque contribution that depends on both Id and Iq — which is why IPMSM drives commonly run Id negative (flux-weakening / MTPA) to add reluctance torque on top of magnet torque, rather than leaving it at zero.
Worked example
An IPMSM with 4 pole pairs, λm = 0.012 Wb, Ld = 0.15 mH, Lq = 0.25 mH, driven at Iq = 15 A, Id = −5 A:
- Magnet torque: 1.5 × 4 × 0.012 × 15 = 1.08 N·m
- Reluctance torque: 1.5 × 4 × (0.00015 − 0.00025) × (−5) × 15 = 0.045 N·m
- Total electromagnetic torque: 1.08 + 0.045 = 1.125 N·m
Related calculators
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Convert between the back-EMF constant (Ke) and torque constant (Kt) of a PMSM or BLDC motor.
Motor Constant Converter
Convert Ke, Kt and flux linkage across RMS/peak, phase/line-line and speed-basis datasheet conventions, with every assumption stated explicitly.
Electrical Speed Calculator
Convert mechanical speed to electrical speed and frequency using the motor's pole pair count.