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Permanent-magnet synchronous motors (PMSM)

Sinusoidal back-EMF, SPM vs IPM saliency, dq-axis torque equation, field-oriented control with MTPA and field weakening. The dominant EV traction motor (Prius, Tesla Model 3) and industrial servo.

Junior ~12 min

Step 1 — PMSM: sinusoidal back-EMF, synchronous PM motor

Animation speed 0.55×

i_d — i_q — ω —

Reference notes

A permanent-magnet synchronous motor (PMSM) is a synchronous machine with rotor permanent magnets and sinusoidal back-EMF. Use Next → to compare surface-PM and interior-PM constructions, the dq-axis torque equation, field-oriented control with MTPA and field weakening, and where PMSM sits in modern EVs and industrial servos.

PMSM overview

Stator: 3-phase distributed windings (vs BLDC's concentrated). Slotted laminated iron.
Rotor: high-energy NdFeB permanent magnets. Distributed winding + shaped magnets give sinusoidal back-EMF e(θ).
Sinusoidal back-EMF requires sinusoidal stator currents for smooth torque → continuous PWM with field-oriented control (FOC), high-resolution encoder for rotor θ_r.
Result: very smooth, very quiet, very efficient — at the cost of more complex drive electronics than BLDC's 6-step inverter.

SPM vs IPM construction

Type	Magnet placement	Saliency	Speed limit
SPM	Surface-bonded to rotor outer face	Non-salient (L_d = L_q)	~10 000 RPM (centripetal stress limits)
IPM	Buried inside rotor iron (V-shape, radial, or spoke)	Salient (L_d < L_q)	~30 000 RPM (magnets protected)

IPMs are the dominant choice for premium EV traction (Toyota Prius generation 1 in 1997, Tesla Model 3, Nissan Leaf, BMW i3) and high-speed industrial servos. The buried-magnet construction protects magnets from demagnetization AND exploits magnetic asymmetry to produce additional reluctance torque.

dq-axis model (Park transform)

The Park transformation rotates the 3-phase stationary frame into a 2-axis frame fixed to the rotor:

d-axis — aligned with the rotor magnet flux λ_PM.
q-axis — 90 electrical degrees ahead of d.

In this rotating frame, 3-φ AC quantities become DC. Stator current decomposes into i_d (flux-producing) and i_q (torque-producing). Torque equation:

T = (3/2)(P/2) [λ_PM · i_q + (L_d − L_q) · i_d · i_q]

Magnet torque — λ_PM · i_q: q-axis current acting on rotor magnet flux. Present in all PMSMs.
Reluctance torque — (L_d − L_q) · i_d · i_q: only for salient (IPM) machines, since L_d = L_q for SPM kills this term. Positive when i_d < 0 (because L_d − L_q < 0 for IPM).

Field-Oriented Control (FOC)

Measure i_a, i_b, i_c and rotor position θ_r via encoder/resolver.
Clarke transform: i_abc → i_αβ.
Park transform: i_αβ → i_dq using θ_r.
Outer speed-loop PI → i_q^* reference.
i_d^* set by control strategy: i_d = 0 for SPM, MTPA (small −i_d) for IPM.
Inner PI loops drive i_d, i_q → outputs v_d^*, v_q^*.
Inverse Park → inverse Clarke → space-vector PWM into 3-φ inverter.

Typical sample rates: current loops 10–20 kHz, speed loop 1–2 kHz.

MTPA — Maximum Torque Per Ampere

For an IPM, MTPA finds the i_d, i_q split that maximizes T for a given |i_s| (∂T/∂i_d = 0 at constant |i_s|). The optimum applies a small NEGATIVE i_d to recruit the reluctance torque term — more shaft torque per amp of stator current means less I²R loss for the same load.

Field weakening (above base speed)

At base speed, back-EMF approaches V_DC of the inverter — no voltage headroom remains for more current. Above base speed, inject NEGATIVE i_d to create a d-axis stator flux that opposes the magnet flux λ_PM:

Net air-gap flux drops → back-EMF drops → inverter can deliver current at higher ω.
i_q must shrink (inverter current limit) → torque falls with speed.
Constant-POWER region extends to 2–3× base speed. Limited above that by mechanical strength and magnet demagnetization risk.
IPMs handle field weakening better than SPMs because buried magnets resist demagnetization more robustly.

PMSM in modern applications

EV traction — Toyota Prius (1997+), Tesla Model 3 and onwards, Nissan Leaf, BMW i3. IPM dominant for combined-cycle efficiency over induction.
Industrial servos — CNC machine tool axes, robot joint actuators (FANUC, KUKA, ABB), precision packaging.
Premium HVAC — variable-speed compressors in air conditioners, heat pumps. 5–15 % efficiency gain over induction.
Direct-drive wind turbines — multi-pole PMSM eliminates gearbox.

PMSM vs BLDC vs induction

vs BLDC: sinusoidal back-EMF + FOC → smoother torque, lower acoustic noise. Cost: complex drive, encoder. BLDC dominates cost-sensitive high-volume (drones, HDDs, power tools).
vs induction: 5–10 % higher η, higher power density. Induction depends on rotor slip to induce rotor currents, producing rotor copper loss proportional to slip × P_input. PMSM rotor runs at synchronous speed (zero slip) with no rotor copper loss at all. Cost: PMSM needs expensive rare-earth magnets with supply-chain risk; induction uses only iron + copper and wins on raw cost and material robustness for large frames.

Take-away. PMSM = synchronous PM machine with sinusoidal back-EMF, driven by field-oriented control with continuous PWM. Surface-PM (SPM) is non-salient; interior-PM (IPM) is salient (L_d < L_q) and produces both magnet torque and reluctance torque. MTPA picks the optimal i_d/i_q split; field weakening injects −i_d to extend speed range above base. IPM-PMSM is the dominant EV traction motor — Prius, Tesla Model 3 — and the standard industrial servo motor. More complex drive than BLDC, more expensive (magnets) than induction, but the best smoothness + efficiency combination.

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The rotor diagram alternates between SPM and IPM constructions; the dq panel shows the live current vector decomposition.