MOTOR-19 — MTPA and Field Weakening

FACT — An electric vehicle requires enormous launch torque to accelerate off the line, but cruising at $180\text{ km/h}$ pushes motor Back-EMF against the battery DC voltage ceiling. The algorithms that allow a single traction motor to deliver both massive low-end torque and high-speed highway cruising are Maximum Torque Per Ampere (MTPA) and Field Weakening.

In this lesson, we analyze the geometry of the $d-q$ current plane, shrinking voltage ellipses, and optimal operating trajectories for EV traction:

1. The Physical Boundaries: Current Circle vs. Voltage Ellipse
2. Region I: MTPA Control Below Base Speed (Ohmic Loss Minimization)
3. Base Speed Transition and Voltage Saturation
4. Region II: Negative $I_d$ Field Weakening (Constant Power Speed Ratio)
5. Region III: Deep Weakening & MTPV (Maximum Torque Per Voltage)
6. High-Speed Functional Safety: Uncontrolled Generator (UCG) Mode & Active Short Circuit (ASC)

---

1. Physical Boundaries in the $d-q$ Current Plane

Two hardware constraints dictate the permissible current vector operating envelope:

d-q Current Plane Operating Trajectories:
                 I_q (Torque Current)
                  ^
                  |        Current Limit Circle (I_max)
                  |          . - - - .
                  |        '     |     '
                  |      /       |       \
                  |     /   MTPA |        \
                  |    |      .  |         |
                  |----+----*----+---------+----> I_d (Flux Current)
                 /     |   /     |         |  0
  Voltage       /      |  /FW    |         \
  Ellipse      |       | .       |          |
  Center       |       |/ MTPV   |          |
 (-psi/Ld, 0) -*-------+---------+----------'
               |       |

A. Current Limit Circle (Thermal Ceiling)

The thermal limit of the inverter silicon switches (IGBT/SiC) and motor windings imposes a circular current boundary:

$I_d^2 + I_q^2 \le I_{max}^2$

B. Voltage Limit Ellipse (Battery DC Bus Ceiling)

The maximum output voltage is constrained by battery voltage ($V_{dc}$) and SVPWM linear limits:

$V_{max} = \frac{V_{dc}}{\sqrt{3}}$

Neglecting small stator resistance drops at high speeds ($R_s \approx 0$):

$V_d \approx -\omega_e L_q I_q$ $V_q \approx \omega_e (L_d I_d + \psi_{pm})$

Enforcing $V_d^2 + V_q^2 \le V_{max}^2$ yields the Voltage Limit Ellipse:

$(\omega_e L_d I_d + \omega_e \psi_{pm})^2 + (\omega_e L_q I_q)^2 \le V_{max}^2$

$\frac{(I_d + \frac{\psi_{pm}}{L_d})^2}{\left(\frac{V_{max}}{\omega_e L_d}\right)^2} + \frac{I_q^2}{\left(\frac{V_{max}}{\omega_e L_q}\right)^2} \le 1$

CRITICAL — Three fundamental characteristics of the Voltage Limit Ellipse:

1. Fixed Center: The ellipse center is permanently anchored at $(- \frac{\psi_{pm}}{L_d}, 0)$ along the direct axis.
2. Horizontal Compression: In an IPMSM ($L_q > L_d$), the vertical semi-axis along $q$ is shorter than the horizontal semi-axis along $d$.
3. Speed Inversion: As electrical rotor speed ($\omega_e$) increases, the semi-axes shrink inversely ($1/\omega_e$), compressing the ellipse toward its center!

---

2. Region I: MTPA Operation Below Base Speed

At low speeds, Back-EMF is modest and the voltage ellipse encompasses the entire current circle. Voltage is unconstrained.

The control objective is producing required shaft torque with the absolute minimum stator current magnitude, minimizing $I^2 R$ copper losses. This trajectory is termed Maximum Torque Per Ampere (MTPA).

From the IPMSM torque equation:

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

Setting $dT_e / d\beta = 0$ yields the optimal direct-axis current command:

$I_d = \frac{\psi_{pm}}{2(L_q - L_d)} - \sqrt{\frac{\psi_{pm}^2}{4(L_q - L_d)^2} + I_q^2}$

• For SPMSM ($L_d = L_q$): $I_d = 0$, operating purely along the $q$-axis ($\beta = 0^\circ$).
• For IPMSM ($L_q > L_d$): Current advances into the second quadrant ($\beta \approx 45^\circ - 55^\circ$), actively recruiting reluctance torque.

---

3. Base Speed and Voltage Saturation

As vehicle speed ramps up, the voltage ellipse contracts toward the origin.

At Base Speed ($n_{base}$), the boundary of the shrinking voltage ellipse touches the peak MTPA operating point on the current limit circle. The inverter has exhausted its available DC bus voltage ($V_{max}$).

To accelerate beyond base speed, the operating point must abandon the MTPA curve.

---

4. Region II: Field Weakening via Negative $I_d$

Above base speed, preventing Back-EMF from exceeding $V_{max}$ requires injecting large negative direct current ($I_d \ll 0$).

This negative $I_d$ current sets up a demagnetizing stator MMF directly opposing the permanent magnet flux:

$\psi_{net,d} = \psi_{pm} + L_d I_d \quad (\text{Since } I_d < 0, \text{ net flux is suppressed!})$

Consequently, net Back-EMF is clamped:

$E = \omega_e \cdot \psi_{net} \le V_{max}$

• The operating point migrates along the intersection of the Current Circle and the contracting Voltage Ellipse, shifting leftward into deep negative $I_d$.
• Because total current is capped at $I_{max}$, increasing $|I_d|$ forces torque current to contract ($I_q = \sqrt{I_{max}^2 - I_d^2}$).
• Torque rolls off inversely with speed ($T \propto 1/n$) while shaft power remains constant (Constant Power Speed Ratio - CPSR).

---

5. Region III: Maximum Torque Per Voltage (MTPV)

At extreme rotational speeds ($>15,000\text{ rpm}$), the voltage ellipse shrinks entirely inside the current limit circle.

The machine transitions from being current-limited to strictly voltage-limited. The inverter can no longer draw full current $I_{max}$ without violating voltage ceilings.

The optimal trajectory shifts to the Maximum Torque Per Voltage (MTPV) curve, tracking the points where the shrinking voltage ellipse tangentially touches constant-torque hyperbolas heading toward $(- \psi_{pm}/L_d, 0)$:

$T_e \propto \frac{1}{\omega_e^2}$

Power rolls off as the machine approaches its ultimate physical speed limit.

---

6. High-Speed Hazard: Uncontrolled Generator (UCG) Mode

If an inverter hardware trip forces all semiconductor gates off while cruising at $180\text{ km/h}$ in deep field weakening:

• Negative $I_d$ vanishes instantaneously.
• The unshielded rotor magnet flux generates raw open-circuit Back-EMF far exceeding battery voltage ($E_{bemf} \gg V_{dc}$).
• The freewheeling body diodes become forward-biased, dumping massive uncontrolled regen current into the battery pack (UCG Mode).
• This creates severe overvoltage risks for battery cells and induces sudden, violent driveline drag torque.

FACT (ASIL-D Mitigation) — Automotive traction inverters integrate hardware-level Active Short Circuit (ASC) protection. Upon critical faults at high speed, the three high-side or three low-side switches are clamped closed simultaneously, trapping Back-EMF currents harmlessly within stator copper resistance instead of overcharging the battery.

---

7. Summary

• MTPA optimizes the current angle ($\beta$) below base speed to minimize stator $I^2 R$ copper losses.
• Base speed marks the transition where Back-EMF consumes full battery DC bus voltage.
• Field Weakening injects negative $I_d$ to oppose magnet flux, enabling constant power operation across broad speed ranges.
• Deep high-speed operation tracks the MTPV locus as the voltage ellipse shrinks inside the current circle.
• High-speed loss of field weakening triggers hazardous Uncontrolled Generator (UCG) mode, mandating ASIL-D Active Short Circuit (ASC) failsafes.

Next Lesson

• MOTOR-20 — Cascaded Torque, Speed, and Current Loops: Multi-loop control hierarchies, anti-windup clamping, and active driveline anti-jerk damping filters.