MOTOR-11 — PMSM, SPMSM, and IPMSM

FACT — In the global electric vehicle traction market (Tesla Model 3/Y rear drive, Porsche Taycan, Hyundai E-GMP, BYD Blade, VW MEB platform), over 85% of primary traction drives utilize Permanent Magnet Synchronous Machines (PMSM), specifically the Interior Permanent Magnet (IPMSM) topology.

In this lesson, we break down the physics of sinusoidal PM machines, contrast surface vs. interior magnet rotors, and analyze the reluctance torque mechanism that makes IPMSM the undisputed champion of EV powertrains:

1. Sinusoidal PMSM Fundamentals and Synchronous Operation
2. SPMSM (Surface Magnet): Architecture, Benefits, and High-Speed Limits
3. IPMSM (Interior Magnet) Geometries (V-Shape, Double-V / Delta Cavities)
4. Magnetic Saliency and the $L_q > L_d$ Asymmetry
5. Dual Torque Components: Magnetic vs. Reluctance Torque
6. Maximum Torque Per Ampere (MTPA) Current Angle Optimization ($\beta$)
7. Why Automotive e-Axles Standardized on IPMSM

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1. Sinusoidal PMSM Operating Principle

A PMSM features distributed or hairpin-wound three-phase stator windings supplied by balanced sinusoidal AC currents, paired with a permanent magnet rotor.

The balanced stator currents generate a rotating magnetic flux density wave ($\vec{B}_s$) of constant amplitude rotating at synchronous electrical speed ($\omega_e = 2\pi f = p \cdot \omega_m$). The rotor magnets establish a fixed magnetic field ($\vec{B}_r$).

Electromagnetic mutual attraction locks the rotor into lock-step synchronism with the stator field. Unlike induction motors, there is zero rotor slip ($s = 0$):

$n_s = \frac{60 \cdot f_e}{p} \quad [\text{rpm}]$

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2. SPMSM (Surface Permanent Magnet Synchronous Motor)

In an SPMSM, permanent magnets are bonded directly to the outer cylindrical circumference of the rotor lamination stack using structural adhesives.

SPMSM Cross-Section:
           [Stator Teeth]
      --------------------------- (Air Gap ~1 mm)
      [N] [S] [N] [S] [N] [S]      (Surface-Mounted Magnets)
      =========================== (Rotor Steel Core)
                 (Shaft)

SPMSM Characteristics and Constraints:

1. Uniform Magnetic Air Gap: NdFeB magnets have a relative magnetic permeability nearly identical to air ($\mu_{r,mag} \approx 1.05 \approx \mu_0$). Magnetically, the direct axis ($d$-axis) and quadrature axis ($q$-axis) present the same effective magnetic air gap thickness.
2. Non-Salient Machine: Direct and quadrature inductances are virtually equal: $L_d \approx L_q$
3. Zero Reluctance Torque: Torque production is purely driven by the magnetic Lorentz interaction.
4. Mechanical Speed Limitation: At high rotational speeds ($>10,000\text{ rpm}$), massive centrifugal force exceeds adhesive tensile shear limits. Retaining the magnets requires wrapping the rotor in a carbon-fiber sleeve or non-magnetic stainless steel sleeve (e.g. Tesla Model S Plaid carbon-wrapped rotor).
5. Demagnetization Vulnerability: Direct exposure to the stator air gap leaves magnets vulnerable to counter-armature reaction fields during faults and direct air-gap heat flux.

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3. IPMSM (Interior Permanent Magnet Synchronous Motor)

In an IPMSM, magnets are buried inside cavities punched within the rotor electrical steel laminations.

Modern EV traction units employ flat-bar, single-V, or double-V / delta cavity configurations:

IPMSM Rotor Geometries:
  a) I-Bar (Flat):       b) V-Shape:               c) Double-V / Delta:
      ---[ Magnet ]---           \  Magnet  /             \  /  \  /
                                  \   [V]   /               \/    \/
                                    \     /

Structural Advantages of IPMSM:

• Outstanding Mechanical Integrity: Magnets are structurally contained by rotor steel bridges, easily withstanding $18,000 - 22,000\text{ rpm}$ centrifugal stress without external sleeves.
• Thermal and Demagnetization Shielding: Rotor iron bridges shunt high-frequency stator tooth harmonics away from the magnet core.
• Direct Rotor Oil Cooling: Transmission fluid (ATF) pumped through hollow shafts can circulate directly through rotor cooling passages adjacent to magnet pockets.

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4. Magnetic Saliency and Inductance Disparity

What makes IPMSM unique is the magnetic asymmetry of its rotor (magnetic saliency):

• Direct Axis ($d$-axis): The axis aligned with the magnet polarization vector. Because magnets act like magnetic air ($\mu_r \approx 1.0$), the magnetic flux path faces high reluctance. Consequently, direct-axis inductance is low ($L_d$).
• Quadrature Axis ($q$-axis): The axis passing through the solid rotor iron between magnet pockets. High-permeability electrical steel ($\mu_{Fe} > 2000 \cdot \mu_0$) presents a low-reluctance path for magnetic flux. Consequently, quadrature-axis inductance is substantially higher ($L_q$).

This gives rise to the defining characteristic of an IPMSM:

$L_q > L_d$

Saliency Ratio ($\xi$):

$\xi = \frac{L_q}{L_d} \approx 1.5 - 2.5$

FACT — In a typical passenger EV traction motor (such as the Nissan Leaf or BMW i3), $L_d \approx 1.2\text{ mH}$ while $L_q \approx 2.8\text{ mH}$, yielding a saliency ratio $\xi \approx 2.33$.

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5. Torque Components and Reluctance Torque

In the synchronous $d-q$ reference frame, total electromagnetic torque is expressed as:

$T_e = \frac{3}{2} p \left[ \underbrace{\psi_{pm} I_q}_{\text{Magnetic Torque}} + \underbrace{(L_d - L_q) I_d I_q}_{\text{Reluctance Torque}} \right]$

Where:

• $p$: Pole-pair count
• $\psi_{pm}$: Permanent magnet stator flux linkage ($Wb$)
• $I_d, I_q$: Direct and quadrature stator currents ($A$)

Dual Physical Torque Mechanisms:

1. Magnetic Torque ($T_{mag} = \frac{3}{2} p \psi_{pm} I_q$): Mutual alignment torque between rotor magnet flux and quadrature stator current. Scales linearly with $I_q$.
2. Reluctance Torque ($T_{rel} = \frac{3}{2} p (L_d - L_q) I_d I_q$): Physical tendency of the rotor iron to align its path of minimum reluctance ($q$-axis) with the stator field. Because $L_q > L_d$, the term $(L_d - L_q)$ is negative. Producing positive reluctance torque requires injecting a negative direct current ($I_d < 0$)!

$(- \times - = +) \implies T_{rel} > 0$

Torque Components vs. Current Advance Angle Beta:
  Torque [Nm]
    ^
    |          Total Torque Te (Peak at ~45°-55°)
    |             .---.
    |            /     \
    |     Magnetic Torque \     Reluctance Torque Trel
    |       .--------.    \         .----.
    |      /          \    \       /      \
    |     /            \    \     /        \
    +----+--------------+----+---+----------+----> Beta (Current Advance Angle)
   0°   15°            30°  45°  60°        90°
(Pure Iq)                     (Optimum MTPA)

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6. Current Advance Angle ($\beta$) and MTPA

Let total stator current magnitude be $I_s = \sqrt{I_d^2 + I_q^2}$. Defining $\beta$ as the angle by which current leads the $q$-axis:

$I_d = -I_s \sin\beta$ $I_q = I_s \cos\beta$

Substituting into the torque equation yields:

$T_e = \frac{3}{2} p \left[ \psi_{pm} I_s \cos\beta + \frac{1}{2}(L_q - L_d) I_s^2 \sin(2\beta) \right]$

• In SPMSM ($L_d = L_q$): Reluctance torque is zero. Peak torque occurs at $\cos\beta = 1$, i.e. $\beta = 0^\circ$ ($I_d = 0$).
• In IPMSM ($L_q > L_d$): Reluctance torque peaks at $\beta = 45^\circ$. Combining magnetic and reluctance torque shifts the optimal torque angle to $\beta = 45^\circ - 55^\circ$!

This optimal trajectory is known as Maximum Torque Per Ampere (MTPA) control. It minimizes required stator current for any torque demand, drastically lowering $I^2 R$ copper losses across drive cycles.

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7. Comparative Summary: SPMSM vs. IPMSM

Feature	SPMSM	IPMSM
Magnet Placement	Bonded to rotor surface	Embedded inside rotor steel
Inductance Ratio	$L_d \approx L_q$ ($\xi \approx 1.0$)	$L_q > L_d$ ($\xi \approx 1.5 - 2.5$)
Reluctance Torque	Zero ($0\%$)	$25\% - 45\%$ of total torque
Rotor High-Speed Limit	Moderate (Demands composite sleeve)	Exceptional ($>20,000\text{ rpm}$)
Field Weakening Range	Narrow (High risk of demagnetization)	Wide Constant Power Speed Ratio (CPSR)
Rare-Earth Magnet Usage	High (100% of torque from PMs)	Optimized (Reluctance torque reduces magnet mass)
Manufacturing	Simple lamination, complex sleeving	Precision stamping, magnet insertion
Benchmark EV Applications	Rimac Nevera, Plaid carbon-sleeved	Tesla Model 3/Y rear, Porsche Taycan, Hyundai E-GMP

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8. Summary

• PMSM machines deliver zero rotor slip, high efficiency, and class-leading power density.
• SPMSM features non-salient rotors ($L_d \approx L_q$) and relies entirely on magnetic torque, making high-speed retention challenging.
• IPMSM buries magnets within the rotor steel, creating strong magnetic saliency ($L_q > L_d$).
• IPMSM exploits both magnetic torque and reluctance torque via negative $I_d$ current injection, maximizing torque per ampere (MTPA).
• Superior high-speed mechanical robustness, wide field-weakening capability, and reduced magnet usage cement IPMSM as the gold standard of electric vehicle propulsion.

Next Lesson

• MOTOR-12 — Asynchronous (Induction) Motors: Squirrel-cage rotor construction, slip dynamics, and the freewheeling disconnect advantage on secondary EV axles.