MOTOR-17 — Field-Oriented Control (FOC)

FACT — In virtually every modern road-going electric vehicle (Tesla, Porsche, Hyundai, BYD), the Motor Control Unit (MCU) runs Field-Oriented Control (FOC) at its core. By decoupling complex AC dynamics into orthogonal magnetic flux and electromagnetic torque axes, FOC translates driver pedal demands into instantaneous, ripple-free wheel torque within fractions of a millisecond.

In this lesson, we examine the complete closed-loop architecture of FOC, the dynamics of cross-coupling decoupling, and analytical PI loop tuning for high-performance automotive traction:

1. FOC Closed-Loop Architecture & Signal Flow
2. Synchronous $d-q$ Voltage State Equations
3. Cross-Coupling Dynamics & Feedforward Decoupling Networks
4. Current Loop Synthesis & Analytical PI Tuning ($K_p, K_i$)
5. Managing 10–20 kHz Discrete Sampling & Execution Delays
6. Inverter Integration & ISO 26262 ASIL-D Plausibility Monitoring

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1. FOC Architecture and Closed-Loop Signal Flow

The core principle of FOC is to orient the stator current vector relative to the rotor magnetic field.

The signal chain executes cyclically:

1. Current Sensing: Stator phase currents ($i_a, i_b, i_c$) are sampled simultaneously by isolated Hall-effect or shunt sensors.
2. Rotor Angle Synchronization: A Resolver-to-Digital Converter (RDC) provides rotor electrical angle ($\theta_e$).
3. Clarke & Park: Currents are mapped into the synchronous rotating frame, yielding DC quantities $I_d$ and $I_q$.
4. Current Regulators: Dedicated PI controllers compare feedback currents ($I_d, I_q$) against torque-derived setpoints ($I_d^*, I_q^*$).
5. Decoupling Network: Speed-dependent coupling voltages are cancelled via feedforward injection.
6. Inverse Park & SVPWM: Demanded voltages ($V_d^*, V_q^*$) are rotated into $\alpha-\beta$ coordinates and modulated onto the physical inverter switches via Space Vector PWM.

FOC Signal Flow:
   Torque Request ---> [ MTPA Table ] ---> I_d*, I_q*
                                              |
      +---------------------------------------+
      |
      v
  I_d* --(+)--> [ PI_d ] --(+)--> V_d* -----\
          ^ -               ^ +              \
          |                 |                 \
         I_d       -w_e*L_q*I_q (Decoupling)   \
          |                                     \
          |                                      +--> [ Inv. Park ] ---> [ SVPWM ] ---> [ Inverter ] ---> Motor
          |                                     /          ^
         I_q       +w_e*(L_d*I_d + psi_m)      /           |
          |                 |                 /         theta_e
          v -               v +              /             |
  I_q* --(+)--> [ PI_q ] --(+)--> V_q* -----/              |
                                                           |
   [ ia, ib ] ----> [ Clarke ] ----> [ Park ] <------------+ [ Resolver: theta_e ]

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2. PMSM Voltage Equations in the $d-q$ Frame

In the synchronously rotating $d-q$ reference frame, the dynamic voltage equations of a PMSM are expressed as:

$V_d = R_s I_d + L_d \frac{dI_d}{dt} - \underbrace{\omega_e L_q I_q}_{\text{Cross-Coupling}}$

$V_q = R_s I_q + L_q \frac{dI_q}{dt} + \underbrace{\omega_e L_d I_d}_{\text{Cross-Coupling}} + \underbrace{\omega_e \psi_{pm}}_{\text{Back-EMF}}$

Where:

• $R_s$: Stator phase resistance ($\Omega$)
• $L_d, L_q$: Direct and quadrature inductances ($H$)
• $\omega_e$: Stator electrical angular velocity ($\omega_e = p \cdot \omega_m$)
• $\psi_{pm}$: Rotor permanent magnet flux linkage ($Wb$)

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3. Cross-Coupling Dynamics and Feedforward Decoupling

Notice the speed-dependent cross-coupling terms:

• The $V_d$ equation contains $I_q$ ($-\omega_e L_q I_q$).
• The $V_q$ equation contains $I_d$ ($+\omega_e L_d I_d$).

At high rotor speeds ($\omega_e$), these mutual coupling terms dominate. If left uncompensated:

• Step commands in $I_q$ (acceleration) induce severe voltage disturbances across the $d$-axis, corrupting the flux state.
• The two axes cross-interfere, leading to regulator fighting, severe current overshoot, and potential inverter overcurrent tripping.

Feedforward Decoupling Synthesis:

To present the linear PI regulators with purely uncoupled, first-order $R-L$ plants, the cross-coupling terms are cancelled feedforward:

$V_d^* = \underbrace{u_d}_{\text{PI Output}} - \omega_e L_q I_q$

$V_q^* = \underbrace{u_q}_{\text{PI Output}} + \omega_e (L_d I_d + \psi_{pm})$

This decoupling cancels the nonlinearities, yielding two perfectly uncoupled SISO linear plants:

$\frac{I_d(s)}{u_d(s)} = \frac{1}{L_d s + R_s}$

$\frac{I_q(s)}{u_q(s)} = \frac{1}{L_q s + R_s}$

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4. Current Loop Synthesis & Analytical PI Tuning

With the plant decoupled into a standard first-order lag, controller synthesis proceeds via pole-zero cancellation:

The PI regulator transfer function:

$C(s) = K_p + \frac{K_i}{s} = \frac{K_p s + K_i}{s}$

Placing the controller zero directly over the motor electrical pole ($s = -R_s/L$):

$\frac{K_i}{K_p} = \frac{R_s}{L} \implies K_i = K_p \frac{R_s}{L}$

The open-loop transfer function reduces to a pure integrator:

$G_{ol}(s) = C(s) \cdot \frac{1}{L s + R_s} = \frac{K_p}{L \cdot s}$

The resulting closed-loop transfer function exhibits clean first-order low-pass filter dynamics:

$G_{cl}(s) = \frac{G_{ol}(s)}{1 + G_{ol}(s)} = \frac{\frac{K_p}{L}}{s + \frac{K_p}{L}} = \frac{\omega_{bw}}{s + \omega_{bw}}$

Where $\omega_{bw}$ is the targeted angular closed-loop current bandwidth ($\text{rad/s}$).

The PI gains are calculated analytically:

$K_p = \omega_{bw} \cdot L$

$K_i = \omega_{bw} \cdot R_s$

EXAMPLE — For a traction e-motor with $R_s = 15\text{ m}\Omega$ and $L_q = 300\ \mu\text{H}$, selecting a closed-loop current bandwidth $f_{bw} = 1000\text{ Hz}$ ($\omega_{bw} = 2\pi \times 1000 \approx 6283\text{ rad/s}$) yields:

$K_{p,q} = 6283 \times 300 \times 10^{-6} \approx 1.885\text{ V/A}$

$K_{i,q} = 6283 \times 0.015 \approx 94.25\text{ V/(A}\cdot\text{s)}$

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5. Discrete Sampling and Timing Delays

Automotive traction inverters typically switch at $f_{sw} = 10\text{ kHz}$ ($16 - 20\text{ kHz}$ for Silicon Carbide).

In digital embedded execution, discrete time delays inevitably affect phase margin:

1. Sampling & Hold Delay ($T_s / 2$): Currents are sampled at the PWM carrier midpoint to avoid switching noise.
2. Computational Processing Delay ($T_s$): Transform calculations and PI loops execute during the cycle and update the PWM registers at the next carrier boundary.

The total effective delay is approximately $1.5 \cdot T_s$ ($150\ \mu s$ at $10\text{ kHz}$). To maintain robust stability margins ($>60^\circ$ phase margin), the current loop bandwidth is rule-of-thumb constrained to:

$f_{bw} \le \frac{f_{sw}}{10}$

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6. ISO 26262 ASIL-D Safety Architecture

Under automotive functional safety requirements, the MCU executes continuous plausibility monitoring:

• Torque Plausibility Check: Estimated instantaneous shaft torque ($T_e$) is compared against driver demand. Discrepancies exceeding safety limits trigger an automatic shutdown into Active Short Circuit (ASC) within $50\text{ ms}$.
• Voltage Circle Limiter: When demanded vector voltage exceeds physical DC rail capability ($V_d^{*2} + V_q^{*2} > V_{max}^2$), the voltage vector is scaled down radially to maintain stability without windup.

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7. Summary

• FOC provides decoupled, high-dynamic control of magnetic flux ($I_d$) and electromagnetic torque ($I_q$).
• Feedforward decoupling cancels speed-induced cross-coupling terms ($\omega_e L_q I_q$ and $\omega_e L_d I_d$), converting coupled nonlinear dynamics into linear SISO plants.
• Analytical pole-placement matches PI gains directly to motor electrical constants ($K_p = \omega_{bw} L$, $K_i = \omega_{bw} R_s$).
• Executing at $10 - 20\text{ kHz}$, the current loop delivers millisecond-grade torque responsiveness, maximum drive-cycle efficiency, and smooth traction control.

Next Lesson

• MOTOR-18 — Space Vector PWM (SVPWM): Modulating $d-q$ voltage commands into physical inverter switching times, DC bus utilization benefits, and dead-time distortion compensation.